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(3.52)
where a, b, c, and d are constants. 3.6.1 Mixers The voltage V applied to a mixer diode is the sum of the voltages of two or more input RF signals to be mixed. The first term of (3.52) is the dc offset. This term contains no RF signals. The second term, because it is linear, contains only signals at the same frequencies as those of the input signal components of the applied voltage V. Higher order terms in (3.52) produce mixer products. The squared term is of interest for many mixing applications. Typically, a fixed-frequency local oscillator (LO) signal is mixed with a relatively lower level received signal to produce a difference-frequency output called the IF signal. A receiver that uses a mixer in this manner to translate a frequency band of signals down in frequency to a convenient IF band is called a superheterodyne (superhet) receiver. Today's technology makes it possible to translate wideband signals at all microwave and millimeter-wave bands of interest for high-resolution applications. Up-translation is also common for translating low-level reference waveforms up to the transmitted carrier frequency. Mixer performance is often analyzed by assuming that the diode is biased to operate primarily in a current-versus-voltage region represented by the third term of (3.52). (Bias is not actually needed to obtain the desired performance for most applications.) The diode is then said to be operating in its square-law region. Higher order terms produce mixer products containing generally unwanted signals that are filtered out. The first two terms, if present, are not of interest for mixing because they do not produce mixer products. In the square-law region, for two input signal voltages V, and V , the diode current represented by the third term is 2
/ = c(V, +
2
V,) = c(V] + 2V, V
1
2
+ V )
(3.53)
Only the product term of (3.53) is normally of interest in mixer applications. The other two terms contain second-harmonic frequencies of the two input signals, respectively, which are filtered out. Consider two input signals expressed as V, = B cos(27r/,r + ^ , )
(3.54)
Vj = tT cos(27r/ f + ipi)
(3.55)
and 2
81
where ift, and i// are the relative phases of V, and V , respectively. The product term of the square-law current of (3.53) produces a voltage across the impedance Z expressed as 2
2
0
sM = IZ = 2cV,V Z 0
2
(3.56)
0
with V, and V from (3.54) and (3.55), the product term becomes 2
s,(t) = 2cBB'Za[cos(27r/,r + ifi )\ x [COS(2TT/ ( + {
2
fc)]
(3.57)
By using the trigonometry identity for the product of two cosine functions, and after dropping the constants 2c and Z , 0
s,(t) = BB'
COS[2TT(/, -
f )t + 2
fa-
fa \ 2
(3.58)
+ BB' cos|27r(/, +f )t + fa + fa ] 2
2
Two input signals produce a mixer product, which is seen to contain frequencies equal to the sum and difference of the two input signal frequencies. The mixer output illustrated in Figure 3.12(a) for down-conversion to IF is filtered as shown in Figure 3.12(b) such that only the difference frequency signal appears at the output of the filter. Signals produced by other product terms of (3.52) are also filtered out. In the standard superheterodyne configuration, one of the mixer inputs is the LO signal and the other is the received signal. The output IF signal s (i) can be seen in (3.58) to be proportional to the amplitudes B and B' of the two input signals, respectively. Therefore, for a constant-amplitude LO signal, the IF signal is linearly related to the input RF echo signal amplitude. r
3.6.2 Quadrature Detection Quadrature detection is used in various types of coherent radar systems to recover received signal phase relative to the transmitted carrier. For high-resolution applications, such as for a chirp-pulse or short-pulse radar, amplitude and phase are required as a function of range delay, along a selected range-delay extent of the received response, relative to the fixed-frequency carrier of the transmitted pulse. In other applications, received amplitude and phase are required relative to the transmitted phase for each of a set of narrowband transmitted pulses spread over a wide band of discrete frequencies. Quadrature detection can be thought of as a mixing operation that translates the received signal to baseband to recover amplitude and phase in the form of quadrature components. A quadrature detector is illustrated in Figure 3.12(c). For quadrature mixing, both the reference LO signal and signal carrier are at the same frequency, except for
82
B'COS (2rtf,t+ y ) 2
BCOS(2nf,t+v,)
•{
•S(t) = BB' COS[2n(f,-f )t + ( v , - V a ) ] 2
+ BB' COS[2n(f,+f )t + ( v , + 2
(a) IF SIGNAL AT f, - i WITH PHASE V , 2
V
2
LO SIGNAL @ 'a.Va BANDPASS FILTER
RECEIVED SIGNAL
(b) n/2 REF.
-MLPF
- Q = BB' SIN ( , V
v ) 2
B' COS (2nft + y,) o LPF
• I = BB' COS ( v , - V ) 2
SIG. BCOS(2trft +
y )02
(c)
INPUT RF PULSE
DETECTED VIDEO DETECTOR DIODE
M
»•
LPF
Figure 3.12 Mixing and detection: (a) mixer square-law products of two sinusoidal input signals; (b) superhet mixer; (c) quadrature detector, (d) video detector.
83
Doppler shift. The output of the lower mixer in Figure 3.12(c), following low-pass filtering, is then represented by the first term of (3.58) with /, = / ; in effect, 2
sfr) = BB' cosM -
fr)
(3.59)
This signal is called the inphase (I) output of the mixer. A second mixer with the reference signal delayed by irtl rad of phase produces a quadrature (Q) output. The / and Q output video pair is called the baseband signal. A Doppler-shifted echo signal will produce a baseband signal at the Doppler frequency. The transmitted signal, for pulsed-Doppler radar, is amplitude modulated into discrete pulses at some pulse repetition interval (PR1). The Doppler shift then appears as a pulse-to-pulse phase shift. Figure 3.13(a,b) illustrates quadrature detection. Practical systems are likely to operate as shown in Figure 3.13(c), so that filtering and amplification can be done more conveniently at lower frequencies.
3.6.3 Quadrature Detector Errors Figure 3.14 illustrates a quadrature detector with an input signal at +/from the reference signal frequency/. Signal amplitude is A and signal phase is 2irfrelative to the reference phase. An ideal quadrature detector produces / and Q outputs x = A cos lirft and y = A sin lirft, respectively. An actual system will exhibit gain and phase imbalance between the two channels and bias in each channel. Output for gain imbalance d, phase imbalance S, /-channel bias p. , and Q-channel bias py, can be expressed as x
x = A cos lirft + p,
(3.60)
y = A(1 +d) sin[27r/Y+ S] + /x,
(3.61)
x
and
It has been shown [4] that the effect of gain and phase imbalance in a quadrature detector is to generate "images" in the spectral domain of the complex output signal. An input signal at +/has a main response at +/and an "image" response at -/. Consider, for example, the Fourier transform processing of the quadrature-detected signal in Figure 3.13(c) produced by a target signal that is Doppler-shifted from the carrier by f . The Fourier transform will produce a main response at +/ and an image response at - / . SAR and ISAR processing commonly involves a discrete Fourier transformation of baseband sample data sets. The result, when amplitude and/or phase imbalance exists, is the appearance of "ghost targets" in the presence of real targets in the high-resolution display, SAR map, or ISAR image display. D
0
0
84
REFERENCE LO@f
lo @
Q
4cos2rc(/+/)Nonideal system: x = Acos 2ntt + fi y = A(1 +d)sm
t
(2nft
+ &)+M
y
Effect:
Relative image power from gain unbalance is —
Relative image power from phase unbalance is —
2
Relative DC power from bias is (jij A)
Figure 3.14 Quadrature detector errors.
where fi = yjfi]
86
relatively simple because imbalances d and S and biases fi and /t, can be treated as constants. A suggested measurement approach to be summarized below is to analyze a large set of statistically independent samples x and y of / and Q data obtained by sampling baseband outputs produced, for example, by an input test signal slightly offset in frequency by / from that of the reference to the quadrature detector. Let true / = cos 27r/t and true Q = sin 2irft with amplitude A = 1. Then, by trigonometric identity, (3.60) and (3.61) can be written as s
2
x = I + ti,
(3.62)
y = G[Q cos S + I sin S] + n,
(3.63)
and
where G = 1 + d is the ratio of Q- to /-channel gain. /- and Q-channel biases are simply calculated as fi, = x and /x, = y, where a bar over the symbol indicates average value. To obtain expressions for amplitude and phase imbalances, we rewrite (3.62) and (3.63) as x =I +x
(3.64)
y = CQ + DI + y
(3.65)
and
for C = G cos S and D = G sin S. With C and D defined in this way, we can show from trigonometry that gain and phase imbalance can be expressed, respectively, as d = G - 1 = VC
2
2
+D - 1
(3.66)
and fi=tan-'^
(3.67)
where expected values of C and D are experimentally obtained from random samples of x and y. The quantities C and D are first expressed in terms of the statistical values x-x and y - y by writing (3.64) and (3.65), respectively, as / =x - x 2. The analysis below is based on unpublished notes by Barry Hunt of San Diego.
(3.68)
87
and Z = y-y
(3.69)
= CQ + Dl
Next we solve (3.68) and (3.69) for D in terms of Z, C, Q, and /. Then, by recognizing that the expected value TQ of the product of true quadrature outputs / and Q, respectively, for many random samples of x and y is zero, we can obtain the expected value of D as -_IZ_(x-X)(y-y) 2
I
(3.70) 2
I
By solving (3.68) and (3.69) for C in terms of Z, D, Q, and / and noting that the expected values ~P and Q are equal and expected values 7 and Q are zero, we obtain 2
(3.71) In summary, the set of statistically independent x, y values are processed as follows. 1. Calculate /-channel bias as the expected value fi = x and £)-channel bias as /x, = x
y. 2. Calculate the set of expected values P = (x - X) , Z (x - X)(y -J). 3. Solve for D from (3.70) and C from (3.71). 4. Determine imbalances d and 8 from (3.66) and (3.67). 2
2
2
- (y - y") , and IZ =
Once the constants d, 8, fi„ and //, are determined from analysis of the test signal data set, they can be incorporated in a preprocessor to convert baseband data represented by (3.62) and (3.63) into approximations to true / and Q values x = A cos 2 77/1 and y = A sin lirft, respectively. Collection of the independent data set can often be conveniently obtained in a number of ways as a radar system test or receiver system test without requiring direct access to the quadrature detector. 3.6.4 Square-Law and Linear Detection Detectors using microwave diodes operating in their square-law region are used in highresolution radars, as well as in radars generally, for envelope detection of processed responses at RF or IF. Square-law characteristics are approximated for low signal levels. Detected video output current in the square-law region is proportional to input RF power. Relatively flat response over octaves of bandwidth is possible with square-law detectors. Linear detectors operate in the linear-current-versus-voltage region of the diode by using high signal levels biased so that only positive swings conduct. The output envelope of
88
an ideal linear detector, following low-pass filtering, is represented by the second term of (3.52). Detector video output current in the linear region is proportional to RF voltage. Operation of a video detector is illustrated in Figure 3.12(d). 3.7 SELECTION OF LOCAL-OSCILLATOR FREQUENCY The LO frequency of a superheterodyne receiver for a wideband radar must be carefully selected so as to avoid responses to signals in the preselector bandwidth that are not related to the echo signal. These responses, called spurious responses, become more of a problem as percentage bandwidth increases. Selection of an LO frequency can be made on the basis of calculations for forbidden zones of the LO frequency that result in spurious responses in the IF passband. These spurious responses occur at frequencies equal to the differences in frequency between harmonics m and n of the unwanted signal and LO frequencies, respectively. Once the forbidden zones are located for the radar center frequency and bandwidth, it is possible to select an optimum LO frequency that at least avoids the low-order spurious responses. This in turn determines the center of the IF passband. Assume that the LO frequency f is chosen so that an echo signal at frequency / appears at an intermediate frequency / = \f-fw\- Spurious responses then occur for the following two cases [5]. w
;
Case
= n/Lo - mf
I:/-/lo
Case I I : / - /
(3.72)
= mf - nf^
L 0
where m and n are harmonic numbers starting with zero, and/* is the frequency of an unwanted signal within the preselector passband that results in a spurious output within the receiver's IF bandwidth. Forbidden LO frequencies for case / occur at
J
'"> TTT
•
(3 73)
-
n+1 The minimum LO frequency for the harmonic set (m, n) that will result in a spurious output can be seen to occur w h e n / a n d / ' are minimum (i.e., both are at the low end'of the preselector passband). The maximum LO frequency for spurious response will occur w h e n / a n d / " are maximum (i.e., both are at a high end of the passband). Therefore, for a receiver with a preselector band covering a frequency range of / , to /, + [}, forbidden LO frequencies produced by the case / conditions will lie between the values m
, , . , ft + f< /u)
(min) = —
f
m+ 1 =— / ,
(3.74)
89
and , . . f. + P + « ( / . + P) m+\, Ao(max) = — =
, m+ 1 „ +— 3 f
/
(3.75)
for the harmonic set {m, n) of the signal and LO frequencies, respectively. Forbidden LO frequencies for case II occur at mf'-f n-\
(3.76)
The minimum LO frequency that will result in a spurious output will occur for f at the minimum preselector frequency / , and the desired signal frequency / at / + B. The maximum LO frequency for spurious response will occur for/* at the maximum preselector frequency f, + /3 and the desired signal frequency / at /,. Thus, forbidden LO frequencies produced by the case II condition lie between the values mf
( /
+
{ m
l)f
/ „x = - ^> — ~ ' j A - - —~ — ' ~ — ? y / (m.n) f
L0
mi
n (3.77) 77^
and , ,
,
Ao(max) =
n>a+B)-f, n
X
(m - l)f, , mB =— j -
+—
(3.78)
No significant harmonics of undesired low-level signals exist above m = 1, but particular care must be taken to avoid (m, n) harmonics of (0, 1), (0, 2), (0, 3), and (0, 4). The z t T O t h harmonic (a dc component) of signal frequency may be present, regardless of signal level. Figure 3.15 indicates forbidden LO frequency choices for an experimental HRR radar with 600-MHz bandwidth and a center frequency of 3.2 GHz. For this radar, it was decided to use an LO frequency of /uo = 4.55 GHz. This resulted in an IF band of 1.05 to 1.65 GHz. The only possible spurious signals from this choice of LO frequency up to the fourth harmonics are seen to result from (m, n) values of (2, 1), (3, 2), and (4, 3). Spurious response, according to Lepoff [5], drops off at 10 dB per harmonic order. Unavoidable spurious responses can be suppressed by using balanced mixers. 3.8 DATA SAMPLING 3.8.1 Time-Domain Sampling Wideband radar signals are often sampled, then converted to digital quantities, before data processing for target detection or imaging. The type of sampling required depends
90
3
Qco-
o:0M\ and |/«(/)| are both real. Figure 3.22(b) illustrates the two-sided density S / / J and Figure 3.22(c) illustrates the one-sided density ££(/.). Figure 3.23 is a phasor-diagram illustration of the relationship between i£(/„) and S+(f ). The two counter-rotating phasors represent the respective noise voltages relative to the carrier in a 1-Hz bandwidth at modulation frequency f each side of the carrier. Their phasor sum, for small modulation index and no amplitude modulation, is the phasor with a sinusoidally varying magnitude at the quadrature phase to the carrier that generates phase noise T, is expressed as 4
3T+T
T is averaging time T is sampling interval
Figure 3.26 Allan variance.
6. Center frequency denoted b y / i s not to be confused here with average frequency deviation/((, r) from center frequency.
1)2
a'\n, 7, r) = ^ { l l / O T . r ) )
2
(3.113)
If n — 2, (3.113) becomes
2
a\2, 7. r) = (1/2)|7(0. r) -f(T. T)}
(3.114)
This definition represents one estimate of frequency variance during time interval r, based on two samples / ( 0 , r) and 7(7, r) of the frequency deviation, separated by the sampling spacing 7. The measurements indicated in (3.114) are conveniently made by connecting the output of the two identical sources to be measured to a mixer followed by a low-pass fdter. A precise digital counter is used to measure the time r required for some set number of periods of the low-pass filter's output voltage. Average phase drift 1.5 X 1.5 x 5 x 5 x 5 x 1x 5 x
10"' sec 10* sec 10"' sec 1 sec 10 sec 100 sec 24 hr
10-'° 10-" 10" I0" 10" 10-" 10IJ
10
3.9.7 Cumulative Phase Noise From Allan Variance From the fractional Allan variance we obtain the average rms frequency noise cr = aj about transmitted carrier frequency / . The corresponding cumulative phase noise of the transmitted frequency for noise-free frequency translation is a = va= lirraj
(3.116)
c
where *>is the phase constant 27TTat delay r. Consider the following application. A 94-GHz spaceborne radar is to provide SAR mapping of a planet's surface at ranges of up to 10*m. The frequency stability of the radar is expected to be established primarily by a stable oscillator for which the fractional frequency deviation is specified as in Table 3.3. It is determined that the rms phase noise needs to be held to less than 15 deg (0.262 rad) to achieve the desired image quality in terms of contrast, which is defined as the ratio of peak signal to phase noise floor. The problem is to determine if the synthesizer of Table 3.3 is adequately stable to meet the contrast requirements. To determine needed fractional rms frequency deviation, we refer to (3.116) to obtain cr^-^-z
(3.117)
6
Range delay at 10 m is 6.67 ms. For this delay and the phase noise criteria of cr = 0.262 rad, we obtain the maximum tolerable fractional frequency deviation as c
0
2
6
2
3
. = 6.7X10-
2TT(6.67 x 10' )(94 x 10')
which appears to be roughly the limit of the synthesizer performance of Table 3.3.
114
3.10 FREQUENCY SYNTHESIZERS Modern frequency synthesizers make it possible for the radar designer to design radars that tune pulse to pulse over a wide band of discrete frequencies and operate with precisely defined but adaptable pulse or continuous waveforms of all forms.
3.10.1 Direct and Indirect Synthesizers Two basic classes of frequency synthesizers are the direct and indirect types. For wideband radar applications, the direct synthesizer has been of greater interest because it is able to switch from frequency to frequency in a short time as compared with typical radar PRIs. Indirect synthesizer outputs are generated from tunable oscillators that are phase locked in frequency increments related to a stable reference. The phase-locking process typically requires from 0.5 to 5 ms to establish a lock for each new frequency. The typical direct synthesizer, by comparison, can change frequencies in less than 1 LIS. Output frequencies produced with the direct method can be synthesized from a basic reference signal by selecting combinations of frequency addition, subtracting, multiplication, and division using mixers, multipliers, and dividers. Actually, the direct synthesizer was developed earlier. R. R. Stone of the U.S. Naval Research Laboratory (NRL) developed it in 1949 [9]. Hewlett-Packard (HP) built the 5100 series of direct synthesizers as part of their line of general-purpose test equipment in the early 1960s [101. The first ISAR images of ships and aircraft were generated in 1975 from radar data collected at the Naval Ocean Systems Center (NOSC) in San Diego using a pulse-to-pulse stepped-frequency radar waveform obtained using a 5100 series frequency synthesizer. It is worthy of note that the indirect synthesizer was developed after the direct synthesizer in order to reduce spurious response and phase noise, which were problems with the early direct synthesizers. The indirect synthesizer developed later was also less costly than the early add-and-divide type of synthesizer developed by Stone of NRL. Later, R. I. Papaieck developed a binary-coded-decimal (BCD) type of direct synthesizer [11], which combines most of the advantages of direct and indirect synthesizers. Both the add-and-divide and BCD direct synthesizer designs require a set of comb frequencies. Output frequencies for the add-and-divide design are synthesized from selections from 10 comb frequencies using divide-by-ten decade units. Output frequencies for the BCD design are synthesized from selections of two comb frequencies. Basic operation of the Stone and Papaieck types of direct synthesizers is illustrated in Figures 3.27 and 3.28, respectively. These are greatly simplified design examples, both producing only 1,000 frequency combinations. Each example provides synthesized output frequencies of 20.000 to 20.999 MHz in 1,000-Hz steps.
115
COMB FREQUENCIES I, — — 18.0 18.1
18.9 MHz
MATRIX SWITCH FOR COMB FREQUENCY SELECTION TO DECADES
l„,=2 MHz OUTPUT FREQUENCY (20.000 TO 20.999 MHz)
TO OBTAIN LOWEST TO OBTAIN HIGHEST DECADE OUTPUT FREQUENCY OUTPUT FREQUENCY COMB FREQUENCY SELECTIONS
»1 »2 13
18.0 MHz 18.0 MHz 18.0 MHz
DECADE #1 OUT DECADE »2 OUT DECADE # 3 OUT
(18t2)+10 = 2 (l»*2) + 1 0 » 2 (18*2) 120
OUTPUT FREQUENCY
20.000 MHz
TO OBTAIN 20.583 MHz
18.9 MHz 18.9 MHz 18.9 MHz
18.3 MHz 18.8 MHz 18.5 MHz
(I8.9.2)+10 = 2.08 (18.9.2.09)tI0 = 2.099 (18.9.2.099) =20.999
(I8.3.2) 10 i 2.03 (18 8 . 2 03) + 10= 2.083 (18.5.2.083) =20.583
20.999 MHz
20.583 MHz
T
Figure 3.27 Add-and-divide direct synthesizer example.
3.10.2 Add-and-Divide Design (Stone) Examples of comb frequency settings for three output frequencies are shown in Figure 3.27 for the add-and-divide type. A set of 10 comb frequencies for this example are spaced by the frequency increment A / = 0.1 MHz. This is the increment of the lowest order decade (decade #1). A value for the input frequency/„ was selected to result in the desired output base frequency of 20 MHz. Assume that by suitable filtering, only sum frequencies are allowed out of the mixers. The lowest frequency, 20.000 MHz, will then occur for/„ = 2 MHz when the block of n = 10 comb frequencies are selected according to the expression
116
^^CpUB
FREQUENCIES^^
(, = 37 MHz
f, = 38 MHz
t,„ = 20 MHz
x4
•A< 38
°A' 37
37
38
38 37
37 38
- BINARY DIGIT
(a) SIGNAL DECADE. 37
= 20 MHz
OUTPUT FREQUENCY (20.000 FOR SWITCHES AS SHOWN)
38
BCD DECADE #1 110 0
37
20.3 MHz
12 4 8
38
BCD DECADE #2 0 001
37
20.83 MHz
38
BCD DECADE #3 10 1 0
12 4 8
12 4 8
OUTPUT FREQUENCY (20.583 FOR BINARY CODES AS SHOWN)
(b) THREE CASCADED DECADES.
Figure 3.28 BCD direct synthesizer example: (a) signal decade; (b) three cascaded decades.
/ = / o + /A/. / = 0 , 1 , 2 = 18.0, 18.1
9
(3.118)
18.9 MHz
These selections are consistent with Stone's formula [9]: fo +/.
, — /in
(3.119)
Other comb and input frequency combinations can be found to produce the same output frequencies. Actual designs are based on practical considerations such as the ability to
117
filter spurious responses. Filters, which pass only the sum-frequency outputs of the mixers, are not shown in Figure 3.27. Additional decimal places for increments of output frequency selection are possible by adding more decades in the series. 3.10.3 Binary-Coded-Decimal Design (Papaieck) The BCD synthesizer of Figure 3 . 2 8 uses three BCD decades to synthesize the same 1,000 frequencies between 2 0 . 0 0 0 and 2 0 . 9 9 9 MHz, as shown for the add-and-divide example in Figure 3.27. This is done with the BCD design by using only two comb frequencies, selected according to the formulas /,=
1.8/„ + 1 0 A /
(3.120)
and JWD+10A/
(3.121)
where/„ is both the input frequency and the lowest output frequency, and A/is the singledecade frequency-step size. The two comb frequencies for our simplified design example are/o = 3 7 MHz and/, = 3 8 MHz with A / = 0.1 MHz. The BCD settings that result in the lowest output frequency of 2 0 . 0 0 0 MHz for a single decade are indicated in Figure 3.28(a). The settings correspond to the binary number 0 0 0 0 . This and other frequencies can be checked by following through the single decade with the switches set according to four-digit binary numbers associated with the output frequency to be selected. Filters following each of the mixers allow only difference frequency outputs. The filters themselves are not shown. Binary settings for an output frequency of 2 0 . 5 MHz for the single BCD decade, for example, are 1 0 1 0 in the 1, 2 , 4 , and 8 binary digits, respectively (left to right. Figure 3.28(a)). This corresponds to the binary number 0 1 0 1 , which is 5 in decimal form (i.e., 5 of the 0.1-MHz increments above the lowest output frequency, 20.000 MHz). The highest output frequency for decimal stepping of a single decade is 20.999 MHz for binary settings 1001, the binary number equivalent to the decimal number 9. (A single BCD decade could actually produce 1 6 frequency increments, 2 0 to 2 1 . 5 MHz, if it were not to be used in the decimal system.) Other frequency settings are possible by setting in the corresponding binary numbers. Multiple decades in series add further decimal places of output frequency selection increments. Suppose an output frequency of 2 0 . 5 8 3 MHz were desired, as was the case for the add-and-divide synthesizer in the first example. The cascaded BCD decade binary settings in Figure 3.28(b) can be seen from left to right to be 1100, 0 0 0 1 , and 1010, respectively, corresponding to binary numbers 0 0 1 1 , 1000, and 0 1 0 1 , respectively. In decimal form, these are the numbers, 3 , 8, and 5 , which result in the 0 . 5 8 3 MHz added to 2 0 . 0 0 0 MHz.
118
3.10.4 Direct Digital Synthesizer At the time of writing, the direct digital synthesizer (DDS) is rapidly becoming the preferred method of frequency and waveform synthesis for many applications, including high-resolution radar. The DDS generates fully synthesized, digitally controlled output signals with precise control of frequency, phase, and amplitude. Frequency can be changed and returned to a previous frequency without losing coherence. There is no frequency addition or subtraction by mixing or frequency multiplication and division except as needed by the user to multiply or up-convert the synthesized frequency for a specific application. The DDS, unlike the add-and-divide and BCD synthesizers, can generate pulse waveforms, such as chirp pulses, as well as discrete frequencies. Figure 3.29 illustrates a basic DDS that includes a frequency control word register, a phase accumulator, read-only memory (ROM) or random access memory (RAM), a digital-to-analog (D/A) converter, and required auxiliary circuit elements. The low-pass filter and clock are selected by users for their specific applications. The synthesized output frequency is set by the frequency control word. Frequency resolution is the clock frequency divided by the size of the accumulator. For example, a 32-bit accumulator clocked at 40 MHz can set up frequencies as finely spaced as (40 x lO )/! = 0.0093 Hz. Output frequency is frequency resolution times the frequency control word. A frequency control word of 2 for the above example produces a frequency of [(40 x lO )/^ ] x 2 = 10 MHz. From Nyquist's theory, the maximum frequency is one-half the clock frequency. Practical systems synthesize frequencies up to about 45% of the clock rate. Contents of the frequency control word register are accumulated in the phase accumulator. When the control frequency remains constant, the output of the accumulator is a contiguous series of phase ramps, each ramp scaled from 0 to 2 ir radians. Ramp length is the accumulator size divided by the frequency control word size. For the above example, the number of phase words per phase ramp is 2 /2 = 4 phase words. Each ramp is one 6
32
30
6
32
10
32
M
Clock | Phase accumulator
Sine map in ROM or RAM
Frequency control
Digital to analog converter
Output
1111111
Figure 3.29 Direct digital synthesizer. (Courtesy of Sciteq Electronics, Inc., from "Frequency Synthesize! Strategies for Wireless," Microwave J., June 1993, p. 26. Reprinted with permission.)
119
cycle of the synthesized output signal. Thus, the above four phase words composing one phase ramp occur at the above synthesized frequency of (40 x ltfJAt = 10 MHz. The ROM or RAM maps the contiguous series of phase ramps into a contiguous series of single-cycle sinusoids at the control frequency. The D/A converter converts the digital amplitude words into a continuous analog signal at the control frequency. Finally, the low-pass filter removes the clock frequency and other aliases. Chirp waveforms can be generated by adding another accumulator called a frequency accumulator before the phase accumulator. The output of the frequency accumulator represents the needed instantaneous frequency to produce phase control words representing the quadratic phase associated with the chirp waveform. System phase and amplitude equalization is possible. Phase compensation can be performed by an adder between the phase accumulator and the memory. Amplitude compensation can be provided by a digital multiplier function between the memory and the D/A converter. Theory and design of direct digital synthesizers is covered in detail by Goldberg [12]. 3.10.5 Summary Each of the above synthesizer approaches can be extended to cover any desired range of output frequencies spaced by any desired frequency increment. For example, the output of a frequency synthesizer that can step in 0.1-MHz increments over a frequency band of 50.0 to 59.9 MHz can be translated to a frequency band of 3.0500 to 3.0599 GHz that steps in 0.1-MHz increments by mixing with a stable 3-GHz source and passing the sum signal. This signal can in turn be doubled in a frequency doubler to 6.1000 to 6.1 198 stepped in 0.2-MHz increments. Furthermore, the entire frequency selection process can be digitally controlled or programmed to produce any desired sequence of frequencies. 3.11 TRANSMISSION LINES FOR WIDEBAND RADAR Radar systems usually use either coaxial transmission line or waveguides to conduct transmitter power to the antenna and signals from the antenna to the receiver. Coaxial line size is selected for transverse electric and magnetic (TEM) propagation, which is nondispersive unless there are multiple mismatches along the line, as discussed above. However, for longer lines, loss increase with frequency, unless equalized, can become a source of signal distortion in wideband systems. Low-loss coaxial runs of up to above 100 feet, well matched at each end, allow radar bandwidths of 25% or more at frequencies up to the top end of the S-band (3.70 GHz). High-power radars operating above S-band nearly always require waveguides for transmission over more than a few feet. This is because of the reduced peak-power handling capability of the smaller diameter coaxial lines required at higher microwave frequencies to avoid losses associated with propagation at unwanted higher order modes.
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Waveguides also have the advantage over coaxial lines in having lower loss and better impedance matching over wide bandwidths. Unfortunately, for wideband radar applications, waveguides are dispersive because they propagate in the dispersive transverseelectric (TE) and transverse-magnetic (TM) modes instead of the nondispersive TEM mode. Of most interest to radar designers is the rectangular waveguide propagating in its lowest order mode, TE, - The propagation constant in radians per meter of rectangular waveguides operating below cutoff frequency in this mode is expressed as 0
(3.122) assuming a lossless waveguide, where/is the frequency of propagation,/ is the waveguide cutoff frequency, and v is the propagation velocity of the medium inside the guide. Distortion produced by transmission through waveguides can be evaluated in terms of waveguide insertion phase characteristics. As an example, insertion phase versus frequency was calculated from (3.122) for the WR-284 S-band waveguide. The cutoff frequency of this guide, when propagating in the TE, mode, is 2.078 GHz. Phase linearity will be examined over a 400-MHz bandwidth from 3.05 to 3.45 GHz. Table 3.4 lists values, for several frequencies in this band, of (1) the propagation constant from (3.122), (2) linear phase (with slope equal to the difference in WR-284 phase constants at the two band edges divided by the bandwidth), and (3) deviation from linear phase. The bandedge phase deviation can be seen to be 6.8 deg/m when the phase reference is zero at the band center. Phase deviation from a linear best fit is one-half this amount, ±3.4 deg/m at the band edges. For a 30m WR-284 waveguide run, the effective phase deviation will be about 100 deg. The paired-echo theory discussed above for this deviation would predict extreme distortion of a short RF pulse containing frequencies from 3.05 to 3.45 GHz. Only 2m of waveguide would retain ±6.8-deg band-edge deviation. From Figure 3.3, this amount of phase deviation would result in time-sidelobe levels of -25 dB if no other sources of distortion existed in the radar. Distortion predicted by paired-echo analysis is 0
Table 3.4 Frequency Dispersion in WR-284 Waveguide
Frequency (GHz)
Propagation Constant (deg/m)
Linear Phase (deg/m)
Phase Deviation from Linear (deg/m)
3.05 3.15 3.25 3.35 3.45
2,679.1 2.840.8 2,998.7 3,153.2 3.304.8
2,685.9 2,842.3 2,998.7 3.155.1 3.311.6
-6.8 -1.5 0.0 -1.9 -6.8
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pessimistic, however, because phase deviation in waveguide transmission lines from the linear phase mostly consists of what is called quadratic-phase error, which will be discussed later in relation to pulse-compression systems. As much as ir/4 rad of quadraticphase error (measured at the band edges) can be tolerated by unweighted signals before significant distortion occurs. Weighted signals are even more tolerant because weighting reduces the effect of phase deviation at the band edges where the deviation is highest. Some methods for reduction of waveguide-produced distortion are the selection of a smaller waveguide size to operate farther from cutoff frequency, the use of waveguide equalization filters, and FM slope adjustment (on chirp-pulse-compression radars).
3.12 WIDEBAND MICROWAVE POWER TUBES The design of most moderate to high-power wideband radar systems, like their narrowband counterparts, is built around the venerable transmitter tube, which has its origins in the 1940s. Fortunately, for wideband radar designers, power tubes are available with wide bandwidths at all microwave frequency bands. Radars tend to be divided into two categories based on transmitter type: coherent radars and noncoherent radars. Coherent radars transmit signals by power amplification of an input RF drive signal. In other words, phase coherence is maintained through the transmitter. Coded waveforms are generated at low power, and LO signals for the receiver can be coherently related to the transmitted signal. Noncoherent radars transmit signals from power oscillators (usually magnetrons). The frequency depends on the power oscillator characteristics as well as the applied voltage and current. Waveforms for noncoherent radars are normally limited to either monotone pulse or monotone CW. Klystron power amplifiers are inherently narrowband devices, but stagger-tuned, linear-beam klystrons may have up to 10% bandwidth. The klystron was invented in 1939 by W. W. Hanson, R. H. Varian, and S. F. Varian. The traveling-wave tube (TWT) is a linear-beam tube, some types of which can operate over very wide bandwidths. The TWT was invented in 1940 by R. Kompfner [13]. This amplifier is characterized by the continuous interaction of an electron beam with a helix for low- to moderate-power applications, and coupled-cavity or other slow-wave structures for high-power applications. Octave bandwidths are possible with helix TWTs, but at relatively low power. About 10% to 30% bandwidth is possible at higher power with coupled-cavity TWTs. The magnetron power oscillator was the device that made microwave airborne radar possible during World War II, and was at one time so closely associated with radar that common microwave ovens, because they use magnetron output power to heat food, were sometimes called radar ovens. The magnetron was invented in 1921 by A. W. Hull, but the device was not used for radar until 1939 when J. T. Randal and H. A. H. Boot of the United Kingdom invented the resonant-cavity traveling-wave magnetron, which operated at about 0.1m. Raytheon Company in the United States developed the means for highvolume production of magnetrons at low cost. This made it possible for the United States
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and the United Kingdom to field thousands of microwave radars for surface and airborne platforms during World War II. Pulse magnetrons can generate pulses as short as SO ns, equivalent to a range resolution of about 25 feet. Frequency-agile coaxial magnetrons can produce pulse-to-pulse frequency-agile bandwidths of up to 400 MHz at frequencies within X-band (8.5 to 10.68 GHz). Figure 3.30(a) shows a cut-away view of a tunable coaxial magnetron. Slow tuning is done by adjustment of the tuning piston. A frequency-agile magnetron operates by MAGNETIC FIELD LINES r COUPLING SLOT ' r ANODE RESONATOR VANE /-CAVITY MODE ATTENUATOR ! "N^TEon MODE ELECTRIC FIELD LINES
I
IJ^Li^V V
V/ ^ NV * /4P^'. ' !\ ' t i l
i.&jH^L y N,[
T E
0 1 1 STABILIZING CAVITY
[
OUTPUT WAVEGUIDE
—
V ^ OUTPUT VACUUM \ - L - .VACUUM BELLOWS - INNER CIRCUIT MODE ATTENUATOR 1
y
l
TUNING PISTON
CATHODE
(a)
CAM
MOTOR
RESOLVER
SLOW TUNE \L< (BROADBAND) (i x
POTENTIOMETERS -WELDED BELLOWS V
WAVEGUIDE OUTPUT
, , L ANODE RESONATOR VANES TUNING J I ^CATHODE PLUNGER T E o MODE CAVITY L
l n
<W Figure 3.30 Tunable and frequency-agile magnetrons: (a) tunable magnetron; (b) frequency-agile magnetron. (From J. R. Martin, Varian Associates, The Frequency Agile Magnetron Story, pp. 9, 10.).
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driving the piston with a motor through a bellows, as shown in Figure 3.30(b). Typical tuning rates are 70 Hz for a 60-MHz tuning range. (One cycle of tuning takes the frequency from one end of the tuning range to the other and back.) Maximum tuning rates are lower at wider tuning ranges. In some designs, a servo motor provides control of the magnetron's frequency. Other frequency-agile techniques exist. While they do not possess signals of particularly wide instantaneous bandwidths, frequency-agile magnetrons can provide a radar with the advantage of improved detection and electronic counter-countermeasures performance. The potential for target imaging with frequency-agile magnetrons will be discussed in Chapter 9. The crossed-field amplifier (CFA) can produce microwave power levels that are high enough for a long-range air search. It is similar in some ways to a magnetron, but the CFA is an amplifier, not an oscillator. Characteristically, CFAs are relatively lowgain devices, but they operate at high efficiency and 10% or more bandwidth is possible. Amplitron amplifiers, also called backward-wave CFAs, were developed first and have the capability of a slightly wider bandwidth than the forward-wave CFAs. Amplitude and phase distortion occur in wideband signals transmitted through highpower transmitter tubes. Because power amplifier tubes are typically driven into saturation, the amplitude tends to be flat across the pulse width. It is, therefore, the phase ripple during the pulse or from pulse to pulse that is of most concern. Phase ripple produced by the ripple of the current or voltage applied to the tube is listed for various tubes in Table 3.1, obtained from Cook and Bernfeld [14]. The reader is referred to Chapter 4 of Skolnik's Radar Handbook [15] for more details on radar transmitters.
3.13 WIDEBAND SOLID-STATE MICROWAVE TRANSMITTERS When solid-state components emerged in the 1960s, they not only provided low-noise characteristics, small size, and low power requirements, but their flat response and low phase ripple simplified wideband radar-receiving system design. Higher power solid-state devices for use as imbedded active phased-array elements have since been developed for radar surveillance in the lower microwave bands [16]. Consumer applications of millimeterwave high-resolution radars have been tested experimentally with phase-locked gallium arsenide and indium phosphide devices as transmitters at up to 1W of pulse power. The radar's transmitter, whether in the form of a high-power microwave tube amplifier or a phased array of embedded lower power solid-state amplifiers, is today a major cost driver and prime power user. One alternative that may greatly reduce size, weight, cost, and power requirements of future radar transmitters is the electron-beamactivated diamond switch. This device can be configured in the form of a solid-state transmitter that possesses characteristics of conventional pulsed microwave tube transmitters in terms of gain, bandwidth, duty cycle, and power, but may be much more attractive in terms of size, weight, cost, and efficiency. The device, at the time of this writing, is
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in the early stages of experimental development and testing by the ThermoTrex Corporation of San Diego. 3.14 WIDEBAND ANTENNAS Simple microwave antennas, such as waveguide horns, slots, and dipoles, provide adequate instantaneous bandwidth with sufficiently low phase and amplitude ripple for most highresolution radar applications. When fed by these antennas, parabolic reflectors or lenses and various fixed-array antennas also can be designed to possess adequate instantaneous bandwidth. The problem comes with phase-steered arrays. A planar phase-steered array is steered by generating a phase slope across the aperture electronically to scan the beam off normal to broadside. Unfortunately, the insertion phase for any propagation path length / of the aperture and feed system is -lirllX. The phase slope for a given steering command is therefore affected by the signal frequency. Spectral components of a wideband transmitted waveform are radiated over a range of beam positions about the steered beam position at center frequency. The effect in both transmit and receive modes is reduced antenna efficiency and degraded beamwidth as the scan angle toward broadside increases. With equal-line-length feeds, the percentage bandwidth limitation at a scan angle of 60 deg, based on frequency scanning less than one-fourth the local beamwidth, is given by Cheston and Frank [17] in terms of beamwidth as % bandwidth = 2 x (beamwidth in deg)
(3.123)
for modulo 2 7rphase-shift scanning. The corresponding limit of range resolution as defined in (1.1) in terms of absolute bandwidth can be obtained from (3.123) from known aperture beamwidth and center frequency. From a time-domain viewpoint with equal-line-length feeds, the radar range-delay resolution is limited to the aperture fill time defined [17] as (3.124) for aperture dimension d and scan angle toward broadside. The limit of range resolution based on (3.124) is obtained as Ar = cA(/2. Problem 37 shows that the resolution limit based on (3.123) and (3.124) from the frequency- and time-domain viewpoints, respectively, are approximately equivalent. These equations account only for aperture effects on array bandwidth limitation. The effect of nonequal-line-length feed networks on array bandwidth limitation is also discussed by Cheston and Frank. Well-known methods for obtaining broadband phased arrays include (1) time-delay rather than phase steering, (2) time-delay steering of phase-steered subarrays, and (3) c
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focal plane arrays, where fixed beams are selected by activating the needed elements in the focal plane instead of steering a single beam by adjusting delay or phase of all elements in the aperture array. Another solution is to operate the radar with pulse-to-pulse frequencystepped waveforms to allow time between pulses to adjust phase shifters to compensate for frequency sensitivity. PROBLEMS Problem 3.1 What is the instantaneous frequency within the envelope a(t) of a chirp pulse defined as s(t) = a(t) exp(j27r(// + KtVl)}? Problem 3.2 What is the insertion phase and the group delay at 3 GHz through a 20m, air-filled, TEMmode transmission line? Problem 3.3 Derive the expressions for the inband phase delay and group delay through a pulsecompression filter that has a transfer function given by
Problem 3.4 a
A 300-m/s target observed with a 10-GHz Doppler radar begins accelerating at 5 m/s toward the radar at time t = 0. (a) What is the instantaneous frequency of the echo signal at / = 0 and t = 10 sec? (b) What is the instantaneous Doppler frequency shift in each case? Problem 3.5
The total transfer function of a 1 -GHz short-pulse radar receiver measured from receiving antenna terminals to display input has the following steady-state amplitude and phase characteristics over its pulse spectrum:
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A(a>) = 1 + 0.02 cos(8 x 10-"w) 4>(u) = -10-" u - 0.02 sin(8 x 10""w) What are the paired-echo amplitudes and delay positions relative to the main response to a 10-ns echo pulse from a point target? Problem 3.6 What is the delay position of the main response seen at the short-pulse radar display of Problem 3.5 relative to the antenna terminals, based on (3.12) (expressed in angular frequency at) and on (3.21)? Problem 3.7 Show that the value c in (3.17) with respect to the number of cycles of ripple across bandwidth B of the transfer function A(co)e' is given by c = (number of cycles of ripple)//3. M
Problem 3.8 A short-pulse radar receiver with a 1-GHz center frequency has linear-phase response, but 20 cycles of sinusoidal amplitude ripple appear in the receiver transfer function over its 10% bandwidth. The amplitude of the ripple is 20% of the average amplitude response. What are the amplitudes and positions of the resulting paired echoes relative to the main response to a short pulse at a 1-GHz center frequency with 5% bandwidth? Problem 3.9 Compute the maximum allowable amplitude ripple in a network that has zero phase ripple and the maximum allowable phase ripple in a network that has zero amplitude ripple if the sidelobes of the output response for each network are to be at least 46 dB below the peak. Check the results with Figure 3.3. Problem 3.10 A short-pulse radar receiver is to use the transversal equalization filter shown in Figure 3.4 to reduce time sidelobes introduced by phase and amplitude ripple in the radar system, (a) How many divider outputs are required to cancel five prominent time sidelobes? (b)
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If the input peak-to-sidelobe level for the highest sidelobe is 15 dB, what is the output peak-to-sidelobe level, assuming that the filter itself is distortion-free? Problem 3.11 A radar transmitter is connected to an antenna by a long transmission line. The input VSWR to the line varies from 1.0 to 1.4, and the output VSWR varies from 1.0 to 1.6 over the radar frequency band. What is the maximum possible phase deviation from linear? Assume zero loss. Problem 3.12 What is the maximum possible phase deviation from linear phase versus frequency produced by a long transmission line? Given are the following VSWR conditions: (a) Input VSWR = 1, output VSWR = 2; (b) Input VSWR = 2, output VSWR = 1; (c) Input and output VSWR = 2. Assume zero loss. Problem 3.13 Show that the expression for the video pulse that has a spectrum given by S(f) = rect(// B) (where rect(///3) = 1 for |(///3)| 1/2 and zero elsewhere) is 5(f) = (sin
ir/3t/(7rt))
Problem 3.14 Show that the expression for the RF pulse at carrier frequency / that has the spectrum S(f) equal to rect[(/-/)//J] is given by ., > sin irBt s(t) = e' f — 2
rrt
Assume that rect[(/ - f)ip\ = 1 for | ( / - ~f)ip\ < 1/2 and zero elsewhere. Problem 3.15 Show that the expression H(f) = recl( f//3) (rect(/7/7), defined as in Prob. 3.13) is the transfer function of a filter matched to a video pulse expressed as
128
sin irBt
Problem 3.16 A rectangular pulse at a carrier frequency / is expressed as 2
s(t) = rectCOe' "?' where rect(/) = I for |/| < 1/2 and zero elsewhere. Show that the normalized response of a filter matched to this pulse is given by h(t) = s(t). Problem 3.17 (a) Write the integral expression ^ r , 0) for the ambiguity surface of a rectangular pulse for f = 0. Let the rectangular pulse be represented by rect(/) = 1 for |/| < 1/2 and zero elsewhere, (b) Plot the graph of X(T, 0). D
Problem 3.18 A square-law detector is to be designed for envelope detection of microwave pulses of 10-ns duration at the half-power points. What are the approximate band-edge frequencies of the output Hlter that is matched to the detected video pulse? Problem 3.19 Two sinusoidal signals of voltages x, and x at frequencies f\ and / , respectively, are applied to a mixer operating in the square-law region. By using (3.53), show that the output spectrum contains the sum and difference frequencies of x and jt , and their second harmonics. Assume that x, = A cos lirft and x = B cos 27r/ /. 7
2
t
2
2
2
Problem 3.20 Show that the cubic and fourth-power terms of the current response to applied voltage cos lirft to a mixer produce the first and third harmonics, and the zeroth (dc), second, and fourth harmonics, respectively, of the input signal. Problem 3.21 J
A radar illuminates a lm target. The IF response is at 0.3V rms. The first target is replaced by a second target at the same range and the IF response goes up to 1.2V rms. Assume an ideal mixer and a linear receiver. What is the RCS of the second target?
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Problem 3.22 An HRR radar uses a wideband square-iaw detector to detect target-range profiles at long range. A detected profile, when amplified and displayed linearly on a wideband oscilloscope, shows two major peaks at 2.2V and 1.1 V, respectively. What is the ratio of RCS at the corresponding two resolved target locations? Assume that the receiver is operating in its linear range, except for square-law detection. Problem 3.23 The radar system of Figure 3.13(a) illuminates a fixed target at range R. (a) In terms of R, what is the phase, relative to the reference signal, of the received signal represented by the / and Q outputs of the synchronous detector? (b) Express the individual / and Q outputs for output signal magnitude A. Assume that the normalized reference signal is exp(j27r//) and ignore all delay, except for two-way propagation delay 2R/c, Problem 3.24 Assume that an ideal (square-law) mixer is to translate to an IF those signals appearing within a radar's receiving system RF passband of 500 MHz, centered at 3.20 GHz. The LO frequency is at 4.55 GHz. What are the band edges of an ideal (rectangular response) bandpass filter at the down-converted IF output that provides nonspurious signal translation for spurious signal and LO signal harmonics below the second harmonic? Problem 3.25 Calibration of a coherent narrowband radar receiver is performed by collecting thermal noise l/Q data with the transmitter disabled and receiver input blocked. Noise is adjusted by receiver gain control to have peak values below the 5 V maximum levels for digitization. Averaged inphase and quadrature-phase video is found to be +0.1 V and -0.05V, respectively. During the operation, a 100-m target at some range R is seen at a video power level of / + Q = 0.13W. What is the apparent zero-velocity target size in meters squared produced by the / and Q bias at the same range with no target present? 2
2
2
Problem 3.26 Calibration data collected from the radar of Problem 25 is further analyzed to determine relative image power. The averaged value of the square of the bias-corrected inphase and quadrature-phase video voltage is found to be 1.51 and 1.43, respectively. The average value of the product of bias-corrected inphase and quadrature-phase video voltage is found
130
to be 0.23. What is the equivalent target size at range R of the images produced by gain and phase imbalance, respectively? Problem 3.27 A short-pulse radar is to obtain 4m range resolution, (a) What is the required sampling rate, in terms of complex sample pairs per second, to sample the baseband range profiles? (b) What is the required sampling rate, in terms of real samples per second, if the echo signals are square-law detected before sampling? Problem 3.28 Range-profile video data are sampled and digitized with a 4-bit A/D quantization. What is the maximum dynamic range in decibels that can be sampled in terms of relative signal along the range profile? Problem 3.29 A ship model 5m in length sits on a turntable in an anechoic chamber. Frequency-domain reflectivity measurements are to be made at small increments in angle as the model is rotated through 360 deg relative to the radar. What is the maximum frequency-step size and minimum number of steps required to be able to obtain 0.1m range resolution unambiguously at all rotation positions? Problem 3.30 A steady target at a fixed range exhibits a 15-deg sector of rms phase noise seen on a polar display of the l/Q output. Radar PRF is 5,000 pulses per second in a pulsed-Doppler mode, (a) What is the expected reduction in signal-to-thermal-noise ratio, due to phase noise, of the response to the target obtained by FFT processing of data obtained during each beam dwell of 10 ms? (b) What is the phase noise floor of processed data relative to the single-target response? Problem 3.31 The source of phase noise in a 94-GHz coherent radar system is the transmitter, which at an offset frequency of 1 kHz from the carrier has a one-sided power spectral density ^ ( / • ) of ~60 dBm. What is the signal-to-thermal-noise loss and signal-to-phase-noise floor produced by cumulative phase noise on echoes from targets at 25 nmi, assuming white noise for FFT processing of 128 pulses?
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Problem 3.32 What is the standard deviation of frequency during an averaging time of 1 sec of an ideal 5.4-GHz transmitter driven from a 100-MHz frequency source multiplied up to the radar frequency? Assume that Table 3.2 is applicable for the frequency source. Problem 3.33 (a) What are the 10 comb frequency settings that must be made available to each decade of an add-and-divide synthesizer design that generates outputs of 10.0000 to 10.9999 MHz from a 1-MHz input? (b) How many decades are required? Problem 3.34 What are the binary digit settings in the order 8421 that will produce an output signal of 20.8 MHz from the synthesizer of Figure 3.28(a)? Problem 3.35 What is the control word in binary form for an 8-bit DDS clocked at 50 MHz that produces the nearest frequency to 10.5 MHz? What is the deviation from 10.5 MHz? Problem 3.36 Use the definition of group delay (3.12) and the expression (3.122) to show that the group delay per unit length of a rectangular waveguide operating below cutoff in the TE mode is given by l0
where f is the cutoff frequency and v is the velocity of propagation in the medium inside the guide. c
Problem 3.37 Fractional peak ripple voltage on the pulse modulator of a 10-GHz TWT amplifier is 1%. Assume that the TWT electrical length is about 15A and that several cycles of voltage
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ripple appear during the transmitted chirp-pulse interval. What is the paired-echo sidelobe level? Problem 3.38 What is the delay dispersion through 30m of the waveguide of Table 3.3 over the 400MHz frequency range? Problem 3.39 A 30m x 30m planar array scans 60 deg off broadside in one dimension. Compare the maximum resolution possible based on (3.123) and (3.124). Assume that broadside beamwidth is Aid rad for aperture dimension d, REFERENCES [II MacColl, L. A., unpublished manuscript, cited by C. R. Burrows in "Discussion of Paired-Echo Distortion Analysis," Proc. IRE (Correspondence), Vol. 27, June 1939. p. 384. [2] Wheeler, H. A., "The Interpretation of Amplitude and Phase Distortion in Terms of Paired Echoes," Proc. IRE, Vol. 27. June 1939. pp. 359-384. [3] Reed, J., "Long-Line Effect in Pulse-Compression Radar," Microwave J.. Sept. 1961, pp. 99, 100. [4) Churchill, F. E., G. W. Ogar, and B. J. Thompson, "The Correction of I and Q Errors in a Coherent Processor," IEEE Transactions on Aerospace and Electronic Systems, Vol. AES-17, No. 1, January 1981. (5) Lepoff, J. H , "Spurious Responses in Superheterodyne Receivers," Microwave J., June 1962, pp. 95-98. |6) Allan, D. W„ "Statistics of Atomic Frequency Standards," Proc. IEEE, Vol. 54, No. 2, Feb. 1966, pp. 221-230. [7J Papoulis, A., Probability, Random Variables, and Stochastic Processes, New York: McGraw-Hill, 1965. p. 144. [8] Papoulis, A., Probability, Random Variables, and Stochastic Processes, New York: McGraw-Hill, 1965, p. 246. 19) Stone, R. R., Jr., and H. F. Hastings, "A Novel Approach to Frequency Synthesis." Frequency, Sept. 1963, pp. 24-27. [10] " H P Direct-Type Frequency Synthesizers, Theory, Performance and Use," Frequency Synthesizers, Hewlett-Packard Application Note 96. Jan. 1969. [II] Papaieck, R. J., and R. P. Coe, "New Technique Yields Superior Frequency Synthesis at Lower Cost," Electronic Design News, 20 Oct. 1975, pp. 73-79. [12] Goldberg. B. G., Digital Frequency Synthesizers, Englewood Cliffs, NJ: Prentice Hall, 1993. [13] Gilmour, A. S., Jr., Microwave Tubes, Dedham, MA: Artech House, 1986. (14] Cook, C. E. and M. Bernfeld. Radar Signals, New York: Academic Press, 1967, p. 395 (republished by Artech House in 1993). [15] Weil, T. A., "Transmitters," Ch. 4 in Radar Handbook, 2nd edition, M. I. Skolnik, ed., New York: McGraw-Hill, 1990. [16] Borkowski, M. T„ "Solid-State Transmitters." Ch. 5 in Radar Handbook, 2nd edition, M. I. Skolnik, ed.. New York: McGraw-Hill, 1990. [17] Cheston, T. C , and J. Frank, "Phased Array Radar Antennas," Ch. 7 in Radar Handbook, 2nd edition, M. 1. Skolnik. ed.. New York: McGraw-Hill, 1990, p. 7.51.
Chapter 4 High-Range-Resolution Waveforms and Processing 4.1 INTRODUCTION The genesis of wideband radar came about at the end of World War II when the peakpower limitations of microwave-transmitting tubes were beginning to manifest themselves. There appeared to be a growing gap between the requirements of long-range detection and high resolution. In order to achieve the high resolution, shorter pulses were employed with the result that less energy was being transmitted per pulse. The need for unambiguous range measurement prevented raising the PRF, so that increasing the peak power seemed to be the only available option. The dilemma began to be resolved when it was realized that range resolution need not be limited by pulse length. If the frequency of ihe carrier, which usually had been constant, were instead varied over some frequency bandwidth, this bandwidth would determine range resolution, according to (1.1) of Chapter 1, written as (4.1) where fi is the frequency bandwidth and c is the propagation velocity. In principle, the range resolution can be made arbitrarily small by transmitting a signal of large enough bandwidth. The pulse length can then be stretched as much as necessary to radiate the energy required to detect distant and small targets without losing resolution. Consequently, microwave power tubes can be operated at the relatively high duty factors at which they tend to be most efficient. This is even truer of the solid-state power sources that are now beginning to supplant thermionic tubes. Also, high operating voltages, which had previously been a source of unreliability and even danger in the operation of tubes, could now be kept within manageable bounds. One of Ihe pioneers of the new type of radar, 133
134
apparently imagining himself to be able to hear both a typical short pulse as well as the new FM one, wrote a Bell Laboratories memorandum entitled: "Not With a Bang, But With a Chirp!" (B. M. Oliver, Bell Laboratories, 1951). This was the first use of the term chirp to describe linear FM of pulses for pulse compression [1]. To this day, chirp radars remain an important class of high-resolution radar. However, more recently, wideband processing to achieve HRR is carried out by using a variety of waveforms in addition to linear FM within each transmitted pulse, as is done in chirp radar. Waveform selection for any radar design is closely tied to transmitter type. The simplest type of radar transmitter is probably the magnetron oscillator. Magnetron radars are called noncoherent radars because the transmitted signal is determined only by the oscillation characteristics of the magnetron. By contrast, coherent radar systems using power amplifiers, such as a TWT or klystron amplifiers, generate the transmitted signal by power amplification of an input RF reference waveform. We shall see in this and subsequent chapters that methods exist to collect wideband reflectivity data from targets using a number of categories of waveforms. Given below are some common types of radar categorized according to transmitter type and listed with likely waveforms to achieve HRR: • Fixed-frequency magnetron—short pulse; • Dithered magnetron—coherent-on-receive magnetron imaging (described in Chapter 9); • Wideband, CW power amplifier—discrete frequency coding and digital phase coding; • Low PRF, wide-instantaneous-bandwidth power amplifier—chirp pulse, phase-coded pulse, and stretch; • High PRF, wideband power amplifier—discrete pulse-to-pulse frequency coding. Table 4.1 lists six waveforms for providing HRR capability. The first four are briefly discussed further in this chapter. Then, chirp-pulse waveforms and associated pulse compression processing will be discussed in more detail. Pulse-to-pulse, stepped-frequency waveforms are discussed in Chapter 5. Not included in Table 4.1 is a class of waveforms referred to as impulse or ultrawideband waveforms. Their common characteristic is large fractional bandwidth. Pulses contain one-half to several cycles of RF, and bandwidth is usually greater than 1 GHz. Renewed interest has come about in recent years as transmitting devices have improved to the point where sufficient pulse energy appears possible at high enough average power for some military applications. Potential applications are target classification, clutter discrimination, and improved performance against stealthy targets. Research is still in the early stage. 4.2 SHORT-PULSE WAVEFORMS HRR using short transmitted pulses is possible with both coherent and noncoherent radars. In coherent systems, very short RF pulses have been generated by using even shorter
135
ft E u
E _4> «j
•3 1
. 3 .•§
5.
M
3
"
S ( Z l )
^
m
(
4
-
4
l
>
It can be shown that is a function only of the factor D and ( / - / ) / A . The calculated spectra for three values of D are shown in Figures 4.13, 4.14, and 4.IS. Rectangular bandpass characteristics are shown comparatively for each value of D. We can easily show from the definition of a matched filter that, in terms of magnitude, a matched-filter transfer function and the normalized input spectrum to which it is matched are identical. Figures 4.13, 4.14, and 4.13 show that a rectangular passband is approached for large values of D. For smaller values of D, the mismatch to an idealized chirp pulse will result in reduced SNR. Figure 4.16, obtained from Klauder, illustrates this. We can see, however, that the degradation of SNR is small, even for relatively low dispersion. Thus, the approximation of a rectangular transfer function (sharp cutoff filter of width A) is normally attempted in the design of pulse-compression filters for systems where the chirp signal is typically a linearly swept, constant-level pulse. Complicated transfer functions with amplitude characteristics, such as those shown in Figures 4.13, 4.14, and 4.1S, are not normally needed. Amplitude weighting of the frequency response, to be discussed below, is often superimposed on the approximately rectangular response, usually by means of a separate weighting filter to reduce time sidelobes. The mismatch between an ideal rectangular chirp signal and a phase-equalization . network that is band-limited by a rectangular filter transfer function can also be viewed intuitively. The spectrum of the ideal chirp signal defined by (4.15) contains both the unlimited spectrum of frequency components produced by the assumed rect(f/7|) envelope
0
0.2
0.4 (f-i)/A
0.6
0.8
1.0
Figure 4.13 Spectral amplitude of a rectangular chirp signal and magnitude of the transfer function of its matched filler for D = 10.125 compared to a rectangular bandpass characteristic; shape symmetric about the point ( / - f V A = 0 (From [7], p. 756 (rectangular passband added and symbols modified). Reprinted with permission.)
159
1.5
I
IN O
RE CTANGUL AR SSBAND
ID
II Q
1.0
OC ui o 3 K -J
0.5
is,(>)-
OL
= |H(f)|
2
< 0.2
0.4
0.6
0.8
1.0
(f-f)/A
Figure 4.14 Spectral amplitude of a rectangular chirp signal and magnitude of the transfer function of its matched filter for D = 60.5 compared to a rectangular bandpass characteristic. (From [7), p. 757 (rectangular passband added and symbols modified). Reprinted with permission.) 1.5 RECTA NGULAR PASSB AND ll Q
111
a
1.0
1A A A A
0.5
3 1-
\
IS
=|H(f)| ,/"A
Zj a. 2
-^DupT|«-^
/
Point Tgt. Compressed Response @ IF / P o i n t Tgt. / Signal @ IF
A/WWlAr Compression Saw
LO
IF Rel. Alternative Galed IF Approach
PRF Trig.
LO
IF Ref.
t
I
t
RMO
IF Rel.
Fast Switch Waveforms
AAAAAAAAAAAr — I
-MyGated Pulse se / 3 IF
Figure 4.27 Coherent chirp-pulse generation.
Spectrum
170
with a gated pulse containing three cycles of the above 250-MHz chirp pulse at a 750MHz IF equals the 250-MHz spectral width needed to excite all components of the chirp pulse. However, achieving the required gate duration of fi/fa = 4.0 ns, though possible, is difficult at this time. 4.6.6 DDS Chirp Generation Basic limitations in the technology for analog passive and active chirp generation have in the past made it difficult to produce chirp pulses with the desired frequency linearity and pulse flatness, particularly for the large time-bandwidth products needed for highresolution radar imaging systems. The generation of adequately linear, large time-bandwidth-product chirp pulses using analog methods often requires frequency linearization, temperature control, and calibration procedures. The result is that these systems are complex and exhibit reliability problems. Application of DDS technology now makes it possible to generate chirp pulses that are nearly perfectly linear and repeatable. Furthermore, while the signal is in the digital domain, phase, frequency, amplitude, and on/off timing can be controlled with accuracy determined by a stable clock frequency. This allows adaptable compensation for phase and amplitude ripple of the radar system over the chirp bandwidth. Synthesized waveforms can be changed pulse to pulse and pulse-to-pulse jitter is minimal because on/off timing is determined by clock stability instead of by switch stability, as for the analog systems of Figure 4.25(a,b). Figure 4.28 is a block diagram of a DDS chirp synthesizer. The dual accumulator provides the function of dual integration of phase. A control word at the input to the frequency accumulator can be thought of as a fixed phase that is integrated to produce a linear-phase ramp, which, when integrated in the phase accumulator, produces the quadratic-phase ramp associated with linear FM. The input to the phase accumulator a frequency accumulator in the sense that it provides linearly ramped frequency control words to the phase accumulator instead of the selected frequency control word in the DDS of Figure 3.29. Accumulation of the selected frequency control words of Figure 3.29 produce the linear-phase ramps for the selected frequency. The accumulated frequency control words of the Figure 4.28 DDS produce the quadratic-phase ramps associated with linear FM. Device 1 -
Frequency accumulator
Phase accumulator
Phase adder
- Device 2 -
- Device 3 •
Memory (ROM)
DAC
Figure 4.28 DDS chirp generator. (Courtesy of Scitec Electronics, Inc., San Diego, California.)
171
QUADRATIC PHASE
ANGULAR FREQUENCY
Figure 4.29 Quadratic phase (shown as phase deviation from center frequency to the high end of the band over which pulse compression is to be carried out).
The adder following the phase accumulator provides for correction for radar system phase ripple and a multiplier (not shown) provides amplitude correction or control. Figure 4.28 is a simplified block diagram of a DDS chirp generator developed by Scitec Electronics, Inc., for high-resolution synthetic aperture radar. The clock rate is 500 MHz. Chirp bandwidth is 230 MHz. Sandia Laboratories of Albuquerque, New Mexico, tested a similar unit [14] for synthetic aperture radar application. 4.6.7 Quadratic-Phase Distortion The phase and amplitude ripple seen in the transfer function of a pulse-compression radar will produce distortion of the compressed signal in the form of paired echoes, as discussed in Chapter 3. Another type of distortion unique to chirp waveforms is called quadraticphase distortion, which is produced by any deviation from the quadratic-phase-versusdelay function for the matched condition. The distortion can be produced by either deviation from linear FM of the generated chirp signal or by dispersive components in the radar system (other than the phase equalizer), including RF transmission lines in the radar
172
system. Figure 4.29 illustrates deviation from linear phase versus frequency produced by unwanted dispersion in the radar system. We will now derive expressions that relate quadratic-phase error and delay mismatch for chirp-pulse systems. Tolerance to mismatch in terms of pulse widening and sidelobes will also be discussed. The ideal transfer function, given in (4.20) for phase equalization of a chirp waveform, is based on matching the delay-versus-frequency slope P to the inverse of the frequency-versus-delay slope K = A/7V From the definition (4.19) of dispersion factor, the ideal transfer function (4.20) can be rewritten in terms of chirp-pulse length T and dispersion factor D as x
«(/) = e x p j - ^ ( / - / )
2
(4.42)
This transfer function has the desired quadratic-phase characteristic, wherein phase, expressed in terms of angular frequency (radians per second), becomes #a>) = ^ ( w - 7 5 )
2
(4.43)
where a> is the instantaneous angular frequency and CJ is the center angular frequency. The desired instantaneous chirp delay, written in terms of angular frequency, becomes dS(oj)
T\
which is another form of (4.21). Similar expressions for undesired quadra'^-phase and equivalent delay error are ,) = I T — -
(4.47)
where = U(a> -a>)\
A = 2M-f)\
(4.48)
l
The constant To of (4.45) written in terms of band-edge phase error then becomes
7T A 2
2
The magnitude of the resulting delay error at either band edge ±co„ from (4.46) and (4.48), is \TA*>,)\
= Tl\(oj - 1S)\
(4.50)
t
= r 7rA 0
By substituting for T\ from (4.49) into (4.50), we have 2d>(u> ) r
M w
'
) {
=
( 4 5 I )
riT
Total chirp delay error over the entire band A (in hertz) is 4.)A|
Figure 4.30 Pulse widening and amplitude loss of mismatched chirp pulse with weighting. (From [IS]. Reprinted with permission.)
weighted pulses, values of ((o,) = ± 7 7 can be tolerated with less than about 1.5-dB loss in peak response, 40% pulse broadening, and sidelobe levels less than 38 dB [16]. For the 30m length of the WR-284 waveguide referred to in Chapter 3, the phase deviation is (6.8 deg/m) x (30m) = 200 deg. If a pulse-compression filter is originally matched to a 400-MHz chirp pulse, the chirp delay error introduced by this length of waveguide from (4.52) is 4x 2 | r
(
' "'
) l =
200x^
, r x 4 0 0 x 10*
(
4
5
3
)
= 0.01 1 /AS Quadratic distortion produced by the 30m waveguide length could be equalized by increasing the chirp-pulse duration by 2|TJ(«,)| = 0.011 /us with the same chirp bandwidth, as indicated in Figure 4.31. The reader will recall that, because the instantaneous delay of the waveguide approximates a linear delay-versus-frequency function, the waveguide can be used for passive generation of HRR chirp waveforms, but impractical waveguide lengths are needed to obtain significant energy transmitted per pulse. 4.7 DIGITAL PULSE COMPRESSION The chirp pulse is an analog signal and pulse compression, described in Section 4.6.4 to convert target return signals into HRR profiles, was performed with analog hardware. At
175
DELAY VS. FREQUENCY OF PULSE-COMPRESSION SYSTEM (WITH WAVEGUIDE)
| 2 T > , ) | = 0.011 us \ SYSTEM (NO WAVEGUIDE) ORIGINAL CHIRP , 3.05
3.25
3.45
FREQUENCY (GHz)
Figure 4.31 Waveguide quadratic-distortion correction by FM slope adjustment (for slope error produced by 30m of WR-284 waveguide).
this point, the high-resolution target range profiles are sampled and digitized for further processing to perform target recognition, target detection, or target imaging. The pulsecompression process can also be performed digitally on echo data sampled at baseband. The advantages are reduced quantization noise at the output for a given number of bits quantized from the A/D converter and the potential for adaptive control of the matchedfilter transfer function, including weighting, to improve resolution and sidelobe performance. The reduction in output quantization noise occurs because of the increased signalto-quantization noise provided by the signal-processing gain associated with convolution. Pulse compression was described above in terms of the mathematical process of convolution. Likewise, digital pulse compression is also a process of convolution. A digitized version of the echo pulse at baseband can be convolved with a digitized version of the matched filter's impulse response to produce a digitized HRR response. Digital convolution can employ a DFT process equivalent to convolution. The process is sometimes called/aif convolution. Although more complex, it is faster than direct discrete convolution for large data sets because of efficiencies obtained by using the FFT algorithm. The DFT equivalent to convolution can be described in terms of the convolution theorem, which relates the convolution expression and its Fourier transform. This is a very important relationship for many areas of engineering and scientific analysis. It states that the Fourier transform of the convolution of one function with another is the product of the Fourier transform of the first function multiplied by the Fourier transform of the second function. The convolution theorem, expressed in terms of an input echo signal sfj) and the impulse response h(t) of the matched filter, is FTfaM * MO] = £ ( / ) » ( / )
(4.54)
176
Thus, convolution in the time domain can be carried out by multiplication in the frequency domain. The quantity S,(f) is the spectrum of the echo signal from one transmitted pulse. The transfer function / / ( / ) is the Fourier transform of the impulse response of the matched filter. Following each transmitted pulse, the return signal in each quadrature channel out of a quadrature mixer is sampled at or above the Nyquist rate, which is A complex samples per second for chirp bandwidth A. Discrete samples, called range data, are collected over some desired range window corresponding to the target range extent to be processed and then converted into digital quantities by an A/D converter. The result is a digital version of the input signal for one transmitted pulse. The digital version of the matched filter's transfer function (also called reference function) can be stored directly as a series of digitized complex pairs. This transfer function will remain constant for a particular chirp waveform but can be controlled to correct for radar system phase and amplitude ripple. Weighting can also be included. Pulse compression, regardless of the method, refers to convolution of the received echo signal, after appropriate down-conversion, with the impulse response of the matched filter to the transmitted chirp pulse. For analog pulse compression, the convolution process is accomplished by simply passing the echo signal through a physical matched filter and an appropriate weighting filter. For digital pulse compression, the convolution process could be carried out by convolving the digitized input range data for each transmitted pulse with a digitized discrete version of the matched-filter impulse response to the transmitted pulse. If fast convolution is to be used, the digitized range data are convolved as shown in Figure 4.32 by first transforming to the frequency domain, then vector multiplying with the digital version of the transfer function / / ( / ) , and, finally, transforming back to the time domain, which is then the compressed range data. Not shown are lowpass filters at the / and Q outputs of the quadrature detector, which pass the chirp bandwidth at video but reject the sum signal and harmonics. LOCAL OSCILLATOH
ECHO SIGNAL
QUADRATURE! MIXER
RANGELINE BUFFER SAMPLING AND A/D CONVERSION
FFT
•
MULTIPLIER
DIGITAL REFERENCE GENERATOR (TRANSFER FCT OF MATCHED FILTER TO TRANSMITTED PULSE)
Figure 4.32 Digital pulse compression.
FFT-
*_! Ar
(4.56)
179
where R and R, are the edges of the range window to be processed. Zeros are added to the signal-data samples and to the T,/Af samples of the impulse response function, as shown in Figure 4.34(c), to produce the common period of length n. At this point, the two resulting data sets of Figure 4.34(c) could be convolved to produce the compressed response. However, use of the convolution theorem carried out digitally by the FFT algorithm for the DFT, although not shown, is implied. The DFTs of J,(iAr) and h(iAt) can be defined, respectively, as follows. 2
2
S,(iAf) = £ s,(lAt) expl^- j ^ ' / j . 0 < i < n - 1
(4.57)
WAf)
(4.58)
and = £ WAf) e x p / - j—17 j , 0 < i < n - 1
where A/ = l/(nAr). The FFT algorithm calculates (4.57) and (4.58) for values of n =V
(4.59)
where y is an integer. Equation (4.59) imposes a second requirement on the selection of n when the convolution theorem is to be applied with the FFT algorithm to generate the compressed-range data. The first requirement, (4.56), applies whether or not the FFT type of DFT is used. A third requirement in the selection of n is that the sampling rate /, equal or exceed the Nyquist sampling rate, which is related to the chirp bandwidth A as /, > 2A
(4.60)
The last criterion can be met by taking complex samples at baseband, spaced by 1/A. Application of (4.57) and (4.58) by using the FFT algorithm produces the discrete versions of the signal spectrum 5,(iA/) and matched-filter transfer function H(iAf), illustrated in Figure 4.34(d). Next, these quantities are vector multiplied to form the frequency spectrum of the range-compressed output. The final step is to perform the inverse (frequency-to-time) FFT of the output frequency spectrum to obtain the output range-delay response. Of interest is the response made up of the first n discrete values £ = 0 through n - 1. This result, illustrated in Figure 4.34(e), replicates that of Figure 4.33(c) for analog pulse compression when the criteria expressed in (4.56), (4.59), and (4.60) are met. Table 4.2 lists minimum acceptable values of period length n versus both sampling interval At and the sum of the sampled signal extent plus chirp-pulse length in seconds. Convolution and correlation by using the FFT are described in more detail by E.O. Bringham [17].
180
Table 4.2 Minimum Acceptable Period Lengths for Discrete Convolution (Assuming Complex Sampling) Chirp-Pulse Length Plus Sampled Range-Delay Extent T,
(
Minimum Acceptable Period Length n Versus Sampling Interval, Af
2(K, - R,)
2
1
10 ns
16
20
10
5 4
50
Al (ns)->
2
20 ns
32
16
4
2
50 ns
64
32
16
4
100 ns
128
64
32
16
4
2
200 ns
256
128
64
32
16
4
500 ns
512
256
128
64
32
16
1 /is
1.024
512
256
128
64
32
2
4.8 DISTORTION PRODUCED BY TARGET RADIAL MOTION Up to this point, our analysis of methods for obtaining HRR performance from radar systems has assumed a stationary target. We now consider the effect of Doppler shift produced by target radial velocity, which reduces peak response and degrades resolution. The nature of this distortion is probably best studied from the viewpoint of the ambiguity function. Two ideal waveforms will be considered: the short monotone pulse and the linear FM (chirp) pulse. Expressions for the rectangular envelopes of the two waveforms
sM = \fc
rect( - ]
(4.61)
for the monotone pulse, and
s,0) = A / F
RCCT
( ¥)
J
exp(j2,7r/Y/ /2)
(4.62)
for the chirp pulse. The term rect(//T|) is defined by (4.16). The waveforms are normalized according to the expression 2
jjs,(f)| df = 1
(4.63)
This normalization results in an ambiguity surface of unit height at the origin. The ambiguity surfaces for the rectangular monotone and rectangular chirp pulses are determined using (3.44), together with (4.61) and (4.62). Results are expressed as follows.
181
l-nvr,)] 7r/ r,(i - |r|/r,) . M < r,
sin[ff/i,r,(i -
Uv"i
= 0, for the monotone pulse, and \xir,fo)\=
|r|\sin[7r(/:r+/ )(r,-M)]] T,j TT(KT + f )(Ti - M)
(4.65)
0
1
D
M>r,
= 0,
for the chirp pulse, where T is the delay relative to the origin and f is the Doppler shift produced by the moving target. Critical features of the ambiguity functions, (4.64) and (4.65), can be discussed with reference to Figures 4.35 and 4.36. In each case, the ambiguity surface extends from -7", to +7, in range delay and -°° to + in Doppler. Doppler frequency response at zero delay points has (sin x)lx profiles for both the FM and monotone pulses. Also, the responses for both monotone and FM pulses are maximum at matched delay and Doppler shift points T = 0 and f = 0, respectively. Range-delay resolution is optimum at f = 0 and the response broadens as [f \ increases. A distinctive feature of the chirp-pulse ambiguity function is its range-Doppler coupling characteristic. A Doppler shift produces a range-delay shift in the response. Profiles normal to the Doppler axis for FM pulses maximize above the line f = - K T through the origin of the/ , rcoordinates. Profiles for the monotone pulses, by comparison, are maximized above t h e / = 0 axis. It is clear from Figures 4.35 and 4.36 that for either monotone or chirp pulses the pulse duration T, determines tolerance to Doppler shift. The response to a target observed with a monotone pulse degrades with target radial velocity. Resolution is reduced and sidelobes increase. The peak of the zero-Doppler response occurring at a given range delay is seen to go to zero at f = 1/7*|, and at that Doppler frequency the range-delay response bears no resemblance to the matched response at zero Doppler. By contrast, the chirp waveform is said to be Doppler-invariant or Doppler-tolerant. Location of the peak shifts with Doppler frequency, but the response remains relatively unaffected well beyond D
D
D
D
D
0
D
D
f =m. 0
A3 DISPLAY, RECORDING, AND PREPROCESSING OF HRR TARGET RESPONSES For simple viewing of a target's HRR profile generated by analog pulse compression, the output of the matched filter, such as a SAW device, can be envelope-detected and then displayed on a wideband oscilloscope activated by a range-delay trigger pulse. The detector and oscilloscope's phase and amplitude characteristics then become part of the total system
182
BASEBAND PULSE (FOR LARGE CARRIER FREQUENCY)
RECT
(f)
N
\
IX(T.« )I 0
| SIN n T , f | 0
|*| = D
-Hurl
A
Figure 4.35 Ambiguity surface for rectangular monotone pulse.
transfer function. Distortion, in terms of decreased resolution and time sidelobes, occurs in the manner discussed for RF components in Chapter 3. However, wideband video detectors and oscilloscopes are available today with sufficiently flat amplitude response and low phase ripple to view target range profiles obtained with greater than 1-GHz bandwidth. Display can be achieved by connecting the wideband video output to the y-axis of a wideband oscilloscope. The horizontal sweep is set to move across the x-axis during the time interval associated with the range window to be observed. The result is an Ascope display of the target's range profile. A range-delay trigger pulse starts the range window. The horizontal sweep time sets the extent of the range window delay. Rangedelay jitter must be about an order of magnitude better than the range resolution; otherwise blur will appear on the A-scope display.
JS3
lz(o,f )l = D
Figure 4.36 Ambiguity surface for rectangular chirp pulse.
Jitter-free range-delay trigger pulses to track moving targets can be generated by the circuit shown in Figure 4.37. A stable oscillator, followed by shaping and divider circuits, generates the radar's PRF. A VCO, in the form of a second stable oscillator, is adjusted in frequency slightly above and below that of the first oscillator to generate a variable delay trigger. The delay is continuously adjusted to track the target as it moves in range. Manual range tracking is carried out by setting the VCO voltage drive so that the delay trigger starts the oscilloscope sweep just ahead of the arrival of the target's compressed response. An earlier version of a range tracker used a motor-driven phase shifter, as shown in Figure 4.38, to generate the delay trigger from a single fixed oscillator. HRR target range profiles, as viewed on an oscilloscope, have had some limited value. Early work in the late 1960s and early 1970s in San Diego at the Naval Electronics Laboratory (NEL) and the Naval Electronics Laboratory Center (NELC) demonstrated that air and ship targets were largely made up of individual backscatter sources. Targets were found to be easily tracked through severe land clutter by manually tracking the target's range profile as it "moved through" a clutter background that produced much higher return than the target. It was also apparent that the range-profile signatures were unique to target type within a limited range of target aspect angles. Sea clutter showed up as individual scatterers (called spikes), which appeared and disappeared with lifetimes on the order of three to five seconds. Recording or HRR target signatures and clutter was originally done by photographing the A-scope display. It was soon found necessary to develop a digital recording capability
184
JUULT STABLE OSCILLATOR e.g., 10 MHz
PULSE SHAPER
_n_
LRTUU 1 VOLTAGECONTROLLED OSCILLATOR e.g., 10 MHz ± «
MANUAL VCO CONTROL
AUTOMATIC VCO CONTROL
MAIN TRIGGER
*N
_n_
PULSE SHAPER
DELAY TRIGGER
AUTOMATIC TRACKER
Figure 4.37 Range tracker for HRR radar.
in order to obtain suitable data for analysis to determine target recognition potential. Later, clutter analysis was also carried out by using digitized data. The digitizing of short-pulse or pulse-compression data requires samples of the detected envelope of the range profile at range-delay intervals separated by an amount equal to or less than the duration of the compressed response. For a 500-MHz pulsecompression radar, for example, the compressed pulse, duration will be about 2 ns. This corresponds to sampling the detected video at a rate of 500 x 10 per second. Sampling and A/D conversion at these rates has recently become possible, but the degree of amplitude quantization is limited, as indicated in Figure 3.17. An early method used at NEL to circumvent the requirement for a high-speed A/ D converter employed a serial sampling system closely related to the design of wideband sampling oscilloscopes. The concept is to sample the target signature at the radar's PRF while advancing the sample position for each pulse. In this way, the entire signature is sampled during n radar pulses, where n then becomes the number of samples that make up the range window. The technique allows data to be collected with a high degree of amplitude quantization for those target-signature features that do not vary significantly during n radar pulses. Range tracking was carried out as described above. 6
18S
PULSE SHAPER
MAIN TRIGGER
STABLE OSCILLATOR e.g., 10 MHz
MOTORDRIVEN PHASE SHIFTER
PULSE SHAPER
-i-N
DELAY TRIGGER
MANUAL MOTOR CONTROL
Figure 438 Range tracker for HRR radar using a motor-driven phase shifter.
This serial sampling method was used to collect aircraft and ship signature data from a ground site at NEL. The technique was used to collect the first dynamic HRR signature measurements of ships and aircraft targets in motion. A block diagram of the sampling system is shown in Figure 4.39. Also shown in the figure is a second sampling mode that is able to collect samples from a selected modulating portion of the rangeprofile video signature. In both modes, only a small segment of the signature is sampled for each pulse. The serial sampling technique, therefore, "throws away" signal energy, which, if sampled and processed, could provide a higher output SNR. The problems of sampling, digitizing, and processing HRR signatures obtained in the time domain remain formidable for resolution less than about one-third of a meter. For this reason, frequency-domain sampling techniques have been developed (e.g., for stretch and synthetic range-profile generation) which provide increased resolution over that possible with present technology for direct sampling of the compressed pulses. Examples of HRR signatures are shown in Figures 4.40, 4.41, and 4.42. Figures 4.40 and 4.41 were obtained by photographing range profiles appearing on a wideband
186
TARGET ANGULARPOSITION — AND AGC DATA FROM RADAR
DETECTED HRR SIGNATURE FROM RADAR
SAMPLING UNIT IN INCREMENTALDELAY ADVANCE MODE
INTERFACE EQUIPMENT AND A/D CONVERTER
DIGITAL MAGNETICTAPE RCDR
RANGEPROFILESIGNATURE RECORDING MODE
SAMPLING UNIT IN FIXED t DELAY MODE
INTERFACE EQUIPMENT AND AID CONVERTER
DIGITAL MAGNETICTAPE RCDR
MODULATION RECORDING MODE
DELAY TRIGGER FROM RANGE TRACKER
Figure 4.39 Target signature and modulation recording (serial sampling).
CRT. Figure 4.42 was generated from serial samples obtained using the range-profile recording system of Figure 4.39.
7' 8' 10'
6'
Figure4.40 HRR signature of T-28 at S-band (I-ft resolution, nose aspect).
Figure 4.41 HRR signature of C-45 aircraft at S-band (1-ft resolution, tail aspect).
189
190
PROBLEMS Problem 4.1 /2
Show that H(f) = e'* is the correct expression for the transfer function of the matched filter to a Gaussian-shaped video waveform expressed as
Problem 4.2 A filter that is driven by an ideal impulse has a rectangular bandpass filter characteristic with bandwidth fi and center frequency /. Use the Fourier shift theorem to show that the complex expression for the normalized output signal is given by sin ir/3l exp flirft irt Assume
f>P-
Problem 4.3 What is the highest sidelobe level in decibels of the envelope of the output pulse of Problem 4.2? Problem 4.4 Determine the half-power temporal resolution of the envelope of the monotone Gaussian pulse expressed by
i(/) = e-°'e'*' 2 7
Assume resolution
1//.
Problem 4.5 A Gaussian-shaped waveform is represented by
191
(a) What is the duration of the pulse envelope in terms of cr at the half-peak points? (b) What is the range resolution associated with this RF pulse at the half-peak points for a - 2 ns? Assume resolution < l/f. Problem 4.6 (a) What is the achievable compression ratio of a 5-fis, 32-bit binary phase-coded pulse waveform? (b) What is the range resolution? (c) What is the waveform bandwidth? Problem 4.7 (a) Write the complex expression for the baseband form of the waveform illustrated in Figure 4.4. (b) Write the expression for its matched filter. Problem 4.8 Using a block diagram like Figure 4.5, show that the binary phase-coded Barker code ( + + + - + ) has a peak response of +5 and peak-to-sidelobe ratio of +14 dB. Problem 4.9 Show that as the number of frequency steps n in a contiguous, discrete frequency-coded waveform approaches infinity, the envelope of the matched-filter response near the peak approaches that of a compressed chirp pulse of the same bandwidth. Assume both waveforms are matched-filtered but unweighted and that the frequency-segment length T\ is equal to the reciprocal of the frequency-step size. Problem 4.10 Show that the pulse-compression ratio of an n-element discrete frequency-coded pulse following matched-filter processing is approximately n for large n when the frequencysegment length T| is equal to the reciprocal of the frequency-step size. 1
Problem 4.11 A radar is to be designed for 5-ft (1.524m) range resolution. What are the required clock rates to generate the discrete delay segments of (a) a phase-coded waveform, and (b) a 32-element, contiguous, stepped-frequency-coded waveform, where segment duration
192
equals the reciprocal of frequency-step size? (Either coded pulses or periodic CW waveforms may be assumed.) Problem 4.12 We want to use a periodic stepped-frequency-coded CW waveform for unambiguous resolution of isolated targets of up to 300m in length with 10m resolution. Assume uniformly stepped frequencies in each period with step size set equal to the reciprocal of frequency-step duration, (a) What is the total bandwidth required? (b) What is the frequency-step size if frequency-step duration is matched to target length? (c) What is the waveform period in number of steps? (d) What is the waveform period in seconds? Problem 4.13 A radar transmits 100-/is pulses, each with a linear FM of 250 MHz over the pulse duration. Compression is to be accomplished using stretch processing by first mixing the return signal with a delayed reference having an identical FM slope. What is the timebandwidth product of the signal before and after mixing? Assume a point target. Problem 4.14 A stretch waveform is used to obtain signatures of space objects from earth-based radar stations. The waveform consists of 100-/AS linear FM pulses with 500-MHz bandwidth. Return signals are processed as in Problem 4.13 by mixing with a delayed reference that is a replica of the transmitted waveform: What is the total bandwidth seen at IF when a 30m target is to be observed? Problem 4.15 A radar transmits monotone pulses of 5-/AS duration, (a) What is its approximate slantrange resolution following matched-filter processing? (b) If the radar were redesigned so that the same pulse envelope is frequency modulated with linear FM over 100 MHz, what is the possible new range resolution? (c) What is the time-bandwidth product in each case? Problem 4.16 Dispersion D for a chirp-pulse radar's waveform is 100 and the point-target compressed response width is 2 ns. What is the approximate FM bandwidth across the response width?
193
Problem 4.17 A pulse-compression radar transmits rectangular chirp pulses of 500-MHz bandwidth. What is the approximate slant-range resolution after Dolph-Chebyshev frequency weighting that results in 30-dB sidelobes? Problem 4.18 With reference to the MacColl paired-echo analysis, compute the allowable amplitude deviation in a pulse-compression radar system if the sidelobes of the output response are to be at least 46 dB below the peak. Assume no phase ripple. Amplitude deviation is defined here as (1 + aja ), expressed in decibels. Calculate from the equations, then compare with Figure 3.3(b) of Chapter 3. a
Problem 4.19 A pulse-compression filter for a radar has a time-bandwidth product of 80. Two methods of chirp generation are being considered: (1) active generation with a VCO that produces a rectangular-envelope chirp, and (2) passive generation using a dispersive filter of the same time-bandwidth product. Assuming equal losses and no weighting in each method, use Figure 4.16 to compare the optimum SNR performance. Problem 4.20 A 2-jjs chirp pulse with chirp slope K = 5 x 10" Hz/s undergoes pulse compression in a phase equalizer exhibiting a delay-versus-frequency slope of P = 0.2 x 10 s/Hz, followed by a Gaussian weighting filter of 100-MHz bandwidth at the -8-dB points. No other band-limiting is involved, (a) What is the chirp-pulse FM bandwidth? (b) What is the degradation in SNR from that of an ideal matched filter to the chirp pulse? (c) What is the compressed pulse duration at the half-power points? (d) What are the peak-tosidelobe levels? Use Figures 4.18 and 4.19. 13
Problem 4.21 The pulse-compression receiver of a radar is matched to its transmitted 10-/JS chirp pulse of 200-MHz bandwidth centered at 3.25 GHz. The only source of distortion is 60m of WR-284 waveguide, (a) What is the approximate band-edge phase deviation from the best linear fit, based on Table 3.4 of Chapter 3? (b) What is the equivalent chirp-delay error? (c) What is the fractional pulse widening and amplitude loss based on Figure 4.30?
194
(d) What new pulse length of the same bandwidth is required to equalize the quadratic error produced by the waveguide?
Problem 4.22 Show that if 4>() j
- 2 ^ , ^ ( 2 7 , + m,T ) j
- - ^ O T . + m,r,) j
- 2 7 r / ^ 0 T ! + m.T,^'
- ^ ( 6 3 7 , + m,r,) j
- 2 7 r / ^ ( 6 3 7 - , + m,^) j
u
- 2 * / ^
63
f»
- Z ^ ^ T l ) ]
-2*17
i2
+ m,T,) j
t
a
Note that the sample time for surveillance in (5.33) is represented as_/T, + mT> with delay mTt, in contrast to iT + r + 2R/c with delay T, + 2R/c in (5.8) for sampling the response from an isolated target at range R. Inverse DFT processing, after frequency-reordering the velocity-corrected hoppedfrequency sample data of (5.33), produces synthetic profiles expressed (with j = i) in the same way as for stepped-frequency data in (5.14), shown to be the high-resolution response of (5.19). Consider the processed response to targets 1 and 2 of Figures 5.15 and 5.16 and target 1 of Table 5.1. The peak response for target 1 will occur at range and velocity position R,, v, in Figure 5.15. Other targets at or near range R may also contribute to the baseband response sampled at range sample position m,. Responses to these targets, however, will be suppressed unless their velocity is at v,. The peak response for target 2 will peak at range and velocity position /? , v if velocity correction is made for velocity v. 2
r
t
2
2
2
5.5 RANGE-EXTENDED TARGETS The narrowband echo signal resulting from the summation of the individual responses from scatterers of an extended target is illustrated in Figure 2.7 of Chapter 2. The angulai
221
extent of summation of scatterers from extended targets is limited by the radar beamwidth. The range extent of the summation of scatterers from extended targets is limited by the duration of the point target response seen at the baseband output of the low-pass filters (LPFs) in Figure 5.2(b). It is convenient to assume that the receiving system bandwidth is matched to the transmitted-pulse duration T,. Then the range extent over which individual scatterers contribute to the amplitude and phase of the response at baseband is cT|, which is c/2 times the duration 27, of the triangular matched-filter response to a point target illuminated by a rectangular transmitted pulse. For receiver bandwidth > 1/7",, the range extent over which scatterers contribute to the baseband response reduces toward cT,/2, which is the range extent of the transmitted-pulse width. Design considerations for sampling and processing two types of extended targets will now be discussed: (1) a single extended but isolated target, such as an aircraft or a ship, and (2) extended surveillance target areas in the presence of clutter.
5.5.1 Isolated Targets To obtain the undistorted range profile of an isolated target, the complex sample of reflectivity collected at each frequency in the burst sequence must approximate that obtained from a steady-state signal with uniform target illumination. Stated differently, echo signals arriving from each of the multiple reflection points of the target must be summed in the receiving system, before sampling, with nearly uniform weighting across the target's range and azimuth extent. This condition is met for a receiver bandwidth that is much less than the reciprocal of the target's range-delay extent, assuming that the crossrange extent of the target is uniformly illuminated by the radar's antenna beamwidth. When the receiving system bandwidth is matched to transmitted-pulse duration 7",, distortion-free range profiles are approximated for a target range-delay extent that is much less than the duration 27 , of the triangular baseband response. For example, to image a 300m ship target without distortion requires a matched transmitted-pulse duration exceeding 300/c = 1 /is. The matched pulse duration should be several times longer than the range-delay extent of expected targets to approach uniform weighting. Target range extent also sets an upper limit on frequency step size. From the first expression of (3.81), the maximum frequency step size for a 300m ship target was shown to be c/(2 x 300) = 0.5 MHz. The azimuth extent of target illumination must also exceed 300m. 1
5.5.2 Surveillance Applications We now consider applications such as ground mapping and other surveillance functions, wherein targets of interest are not isolated in range, but extend continuously over relatively large ranges compared to, say, the range extent of a ship or an aircraft. Requirements for
222
frequency step size and matched pulse duration based on surveillance range extent for these applications tend to be impractical. Practical stepped-frequency and hopped-frequency sequences can, however, be used for surveillance of range-extended target areas by creating a set of fixed range-sample positions extending over the selected surveillance range as illustrated in Figure 5.17. T/his allows unambiguous sampling at each sampling position, with pulse duration and frequency step size determined by a small segment of the extended range. Overlapped unambiguous segments of processed data from the multiple range-sample positions are summed to form the extended synthetic range profile. The top of Figure 5.17 illustrates one channel of the baseband signal produced by one pulse of a sequence of stepped-frequency or hopped-frequency echo pulses. Individual components of the baseband signal appear as triangular / and Q responses, each corresponding to the convolution of the rectangular baseband echo signal from an individual scatterer with the rectangular matched-filter impulse response of an idealized low-pass filter. The baseband signal itself, which is the sum of responses for individual scatterers, is shown as the dash-line signal. Triangular components, though all shown for convergence as I or Q for target A alone at / I or Q (or target B alone at // I or Q (or target C alone at / ;
Baseband I or Q output Transmitted pulse I
I
(
1-
-^f^K
-+r?H 1
2
3
4
5
1—*- Range
6
Unambiguous range window A
A
B
AIL
> >—•
A AB
- Range
• Range C
A
B
• Range
Hi(4)
B
F,
• Range C
• Range
Figure 5.17 Unambiguous range sampling for extended surveillance regions.
223
positive, would actually be bipolar. The unambiguous range windows shown in Figure 5.17 are generated by selecting a frequency step size equal to 1/(2T|), where 2T, is the duration of the triangular / and Q signal, before sampling, for an ideal matched filter matched to the transmitted pulse of duration T,. Processed responses from stepped-frequency data collected at f to /„_, from range gates 2. 3, 4, 5, and 6 are illustrated as H{2), Hi~i), H/iA), H{5), and 77,(6), respectively. Synthetic high-resolution responses to individual scatterers can be seen to occur at range positions corresponding to their true ranges within the unambiguous range windows centered for illustration at their respective range-gate sample positions. Actual positions of scatterers seen in the synthetic range profile domain will depend on the IDFT process, but the relative range alignment of the range profiles HJi2) through H16) remain constant. Dotted responses represent multiple ambiguous IDFT outputs that fall outside the ambiguity windows associated with the five sample positions, and they will not contribute to the sum of the overlapped segments H&) through H/(6). Consider first the H{2) profile, which was obtained by IDFT processing of frequencydomain data collected at sample position 2. Sample position 2 can be seen to sample the contributions to the baseband response from scatterers A and B only. The amplitude and phase of the sampled data are the complex sum of triangular components produced by echoes from scatterers A and B, respectively. Contributions from both of these components of the baseband response can be seen in range-sample position 2 to be sampled below their peak response. Resolved scatterers A and B are can therefore be seen in HIT) at a reduced level. Scatterer A can be seen at its maximum value in H^i) processed from range-sample position 3 data, where scatterer B also appears larger because range-sample position 3 is closer to the range position of scatterer B and thus closer to the peak of its contribution to the baseband response. Range-sample position 4 samples the baseband response where / and Q components are determined by scatterers A, B, and C. None of the contributions to the baseband response for these three scatterers are near the peak of range-sample position 4. As a result all are seen to appear reduced in amplitude. Profiles H,(5) and Hffi) can be understood in the same manner. Individual profiles Hi\) through Hffi) can be seen to possess the following characteristics: (1) resolved scatterers associated with sampling at the five range-sample positions appear in range alignment; (2) although a given scatterer appears at different amplitudes in different profiles, the complex sum of the profile amplitudes //XI) through 7/,(6) will show the three resolved scatterers at approximately their correct relative amplitudes; (3) no foldover appears from responses outside the unambiguous range windows. It is concluded that contiguous synthetic high-resolution profiles of extended-range surveillance regions can be obtained by complex summation of overlapped unambiguous segments of processed data from multiple range-sample positions. Ambiguity in the form of foldover of responses is eliminated by selecting frequency step size A/ so that the unambiguous range-delay window 1/A/is at least the duration 27", of the baseband response to a point target following ideal matched-filter low-pass filtering. 0
224
Two distortion effects are noted. First, some discrete scatterer positions could be found where separate portions of the response, though not ambiguous, would appear at both edges of one of the ambiguity windows. The summed responses that form the contiguous extended range profile would then include two responses to the single scatterer. This effect could be eliminated by complex summing of processed responses jfrom H, segments in Figure 5.17 which are reduced slightly from c7",. Secondly, some distortion is produced by the nonlinear insertion phase that exists in practical low-pass filters. Sampled baseband data, when phase nonlinearity exists, will shift in phase with the sample position, which introduces phase ripple. However, since the summed contiguous profiles contain scatterer responses from five sample positions, samples taken near the peak, being of larger magnitude, will dominate the sum, thus reducing the distorting effect of phase nonlinearity. In an actual system, baseband responses will deviate from the triangular shape illustrated in Figure 5.17. Sample spacing T} could vary from as small as desired to as large as the pulse duration T . The penalty for small sample spacing is increased complexity. The penalty for larger spacing is reduced fidelity with severe degradation appearing when 7/3 approaches 2T,. {
5.5.3 Surveillance Example Hopped-frequency parameter selection will now be illustrated for a hypothetical allweather landing radar that displays to the pilot the runway, surrounding fixed structures, moving ground vehicles, and moving aircraft on the runway during final approach from 2 km to 75m from touchdown. The requirement for maximum two-dimensional surface resolution cell is set as a function of distance from touchdown in three stages: (1) 20m by 20m for distances between 2 km and 1 km, (2) 10m by 10m between 1 km and 500m, and (3) 5m by 5m between 500m and 75m. Moving ground targets are to be displayed unambiguously over at least ±40 m/s with velocity resolution cells smaller than 3 m/s. The maximum antenna dimension to be accommodated in the aircraft nose is lm. The display over a forward-looking sector of 7r/4 rad is to be updated at the rate of one update per second. (The above application and requirements are selected for illustration only and' are not based on an analysis of any actual landing system needs.) Pulse duration T, is,selected to be 0.5 /AS, which allows surveillance at a range as small as c7",/2 = 75m without attempting to receive while transmitting. Time interval 7j between sample positions following each pulse is 0.25 /AS, which corresponds to about four samples per matched-filtered baseband (coarse range) resolution cell. Unambiguous range profiles are obtained for step size A/ = 1/(27*,) = 1 MHz. As the runway is approached, bandwidth is increased by increasing the number of pulses per burst in three stages to approximately match processed slant-range resolution, with cross-range resolution provided by the antenna beamwidth. From (5.21), the number of pulses, before rounding to the nearest V where 7 is an integer, is
225
(5.36)
2A/Ar,
The wavelength is chosen to be 8.57 mm, which corresponds to a frequency of 35 GHz. The azimuth beamwidth, for the allowable lm aperture at this wavelength with the approximation lc->"'
(5.49)
Let -2nA/R
+I
(5.50)
Then with (5.50) and taking terms not including i outside the summation, we write (5.49) as ^ ^ e ^ ^ X e i ^ K -
(5.51)
The expected value [3] of fifa) is ElMxd)
= J l j l
•• -
£/7Xx,Kro. x
^-OAtodjt, . . . dx..,
(5.52)
where p(x , x *„_,) is the joint probability density of random frequency error XQ, X,. x,-,. Substitution of H^xi) from (5.51) into (5.52) results in the expression a
EiHix,)]=r
r ••r
j
^
A
^
A
^
"
^
p^,
x
^.od^d*,... d*.., (5.53)
232
For p(x,) independent of p(Xj) for all i and j except i = integration, we can write (5.53) as W
]
=
and exchanging summation apd
Iff., . j V ^ e ^ Y -
(5.54)
x P(x )p(x,) ... p(x„-,)dxadx, . . . dx„_, 0
where pfo) is the probability density function of frequency error x,. By carrying out the summation in (5.54) for i = 0, 1, 2 . . . . , / » - 1. we have 3
,
1
EMU)] = j V 4 ^ e ~ ^ e + » p ( * ) x J^pfx.jdx, x j^pfxjjdxi... J^p(jc,.,)dx,., jd^fl fyr*-k'—'^ >x )
+
P
t
x ^£p(xo)dx x £_p(x )dx,... £p(x„.,)dx„_, jdx. 0
2
(5.55)
e"
p(x..,)
x j^p(x )dxo x £p(x,)dx, . . . £p(x„_,)dx„-i jdx.,, 0
For all i, we can write
v
£pfx,)dx, = I
(5.56)
£lH,(x,)l = " f e ^ ^ e ^ T c-^pMdx,
(5.57)
Then (5.55) with (5.56) becomes
3. The index ( = 0 is indicated* but not multiplied out. in the hope of maintaining clarity.
23}
A normal distribution of frequency error with standard deviation a and zero mean will be assumed for p(x,) for all /. The standard deviation is expressed as [4] !
p(x) = — ^ = e - * * " ' er-^2 JT
(5.58)
Equation (5.57) with p'x) from (5.58) becomes
£[///(*.)] = Ye"'' *"' IA "' \
!/
— =e->"e' "°''dx ~tryJ2ir 7
J
i«o
i
(5.59)
Terms not including x can be brought outside the summation. Also omitting the subscript i inside the integral, we obtain (5.59) as E\HAx,)) = t^
'*' Y e'^"'' f — ^ - ' " e - ' ^ ' d x o-\J2ir
(5.60)
Equation (5.60) can be simplified by using the notion of a characteristic function. The] characteristic function of the random variable x takes on the form of the inverse Fourier transform of its probability density function p(x) [5]. The integral term of (5.60) can be viewed as containing the characteristic function of the zero-mean normal probability density, which for standard deviation a is C,(t) = FT-'IpWl = f — ^ e ^ e - ' '
4 2
' ' ^ = e"
V / 3
(5.61)
where the symbol t is used here because the inverse Fourier transform commonly transforms from frequency/to time t. Thus, in (5.61) for / = -**, the integral term in (5.60) becomes C,(-p) = f - 4 = e - * " e ~~o~y2ir i
w
' d r = e^"'"
(5.62)
Equation (5.60) with (5.62) becomes E\HHx,)\ = e ' V " * ' / £ e ' " V** /,_,2*\
The application of the identity (5.16) to (5.63) yields
»-l
(5.63) Iwi
234 -
fij^-e^e-^e^-^fr
(5.64)
which is the expected value of the synthetic range profile H, expressed as a function of the variance a of the frequency error. The peak response of the expected value of the range profile occurs at y = 0, ± n, ± 2/i, ± 3n which are the same positions as for the ideal response (5. 19). The expected value given by (5.64) of the peak at y = 0 becomes 1
hT
Peak £[ff/jt,)] = ne^' \-'''''
(5.65)
In terms of absolute value with C from (5.62), f
Magnitude of Peak E[H,(Xi)] = nC,
(5.66)
1
From (5.66), if the frequency variance a were zero, the magnitude of the pea^response of the expected value of the range profile would be n. With random frequency error present, the peak value is reduced to nC - n exp(-p = xIR. Under this assumption, (6.46) becomes
(6.47)
If we let u = 2x/(AR), we can write (6.47) as
(6.48)
Rewriting (6.48) in the form of the Fourier transform of the product of two (identical) functions and employing the inverse of the convolution theorem expressed in the form of (4.54), we obtain
(6.49) 2
2
where rect(2y/) = 1 from -1/4 to +1/4 and zero elsewhere is the inverse Fourier transform of each of the (sin x)lx forms and the symbol * refers to convolution. Figure 6.13 illustrates
260
Figure 6.13 Normalized response of focused side-looking SAR for integration over the entire beam produced by a uniformly illuminated line antenna.
the triangular response Z(y)/Z(0) produced by the convolution of the two identical rectangular functions after normalization by Z(0) = 2A/V(v,i ). Resolution from Figure 6.13 is J
Ar, = ^ (measured at the half-amplitude points)
\6.50)
Ar = 0.29/ (measured at the half-power points)
(6.51)
and r
6.4.3 Equivalent Rectangular Beamwidth It is convenient to estimate SAR resolution based on SAR integration length determined by an equivalent rectangular real beamwidth defined in terms of the antenna gain response
|Z(0)P as
*' = J ^ D Z ( 4 W
^6.52)
The equivalent rectangular beamwidth of the one-way power gain response (6.5) for the unweighted line antenna using the definition (6.52), in radians, is
(6.53)
261
which from (6.11) is also the beamwidth defined at the 21 n points. SAR resolution can be expressed directly in terms of the illuminating antenna length / by substituting ££ = Rifi, = RXIl in the expression (6.20) for SAR Rayleigh resolution with uniform illumination. The result is 1/2, which happens to be identical to the half-amplitude resolution given by (6.50) for the nonuniform illumination produced by the unweighted line antenna for which All is the effective beamwidth. We conclude that 112 is a goodfirstapproximation to the SAR cross-range resolution. A more accurate estimate is the width of the response function (6.45) with |Z(y,/)| obtained from the actual power gain pattern of the illuminating antenna. Figure 6.14 compares the response to a point target for (a) a uniformly illuminated synthetic aperture, and (b) a synthetic aperture formed by the nonuniform beam pattern of an unweighted antenna of length /. For a side-looking focused SAR, we can see that the resolution is limited by the . real-aperture size. A small real aperture along the cross-range dimension results in better 2
Illumination function
Illumination
Response to a point target
(a)
se
i. 2se
nix
y
Figure (.14 Response for uniform and nonuniform illumination functions over which coherent integration is performed: (a) uniform gain segment or small real beam: (b) unweighted narrow-beam line antenna of length I.
262
SAR resolution, in contrast to real-aperture mapping, where a large azimuth aperture dimension produces better resolution. In summary, it is possible to increase the cross-range resolution of surface-mapping radars over that of real-aperture mapping radars by coherently integrating target echo signals as the radar platform passes by the area to be mapped. Maximum possible resolution occurs for focused SAR when quadratic-phase and phase errors caused by deviation from straight-line motion are corrected before integration. The SAR technique is essential for spaceborne radar mapping of the earth's surface, where useful resolution is not likely to be achieved with practical real apertures. The following equations were derived above for resolution associated with three types of apertures in increasing order of resolution. 1. Real aperture (6.14): &r » 0.64 « y r
(6.54)
2. Optimum unfocused SAR (6.35): Ar = 0.5 yfRA f
(6.55)
3. Focused SAR (6.50): Ar f
(6.56)
Resolution as a function of range for the above three types of apertures is plotted in Figure 6.15 for a 3m real aperture at A = 3m. 6.S SAR THEORY FROM DOPPLER POINT OF VIEW The focused aperture SAR concept can also be explained from the point of view of differential Doppler signals produced by scatterers separated in azimuth. j Consider the airborne side-looking SAR of Figure 6.16 at the instant that the aircraft is directly beamed on boresight to the center of two point targets, both at range R, which are located in the cross range at -y and +y, respectively, from boresight. Focusing can be thought of as correcting for the range deviation SR of the straight-line flight path from the constant radius dashed curve. The instantaneous velocity of the radar past the two targets will produce an echo signal containing a pair of instantaneous Doppler offset frequencies -2<wy/A and +2<wy/A for wavelength A, where to is the instantaneous angular velocity of the aircraft relative to the centroid of the two targets. The Doppler frequency separation is
263
1
10
100
1000
RANGE R (nmi) Figure 6.1S Resolution versus range for three generic types of mapping radars at 0.03m wavelength.
Sf = ja,y
(6.57)
D
For a Doppler frequency resolution A/ , the cross-range resolution becomes D
Ar = 2|>i = ~M r
D
(6.58)
264
Figure 6.16 Source of SAR cross-range Doppler.
6.5.1 Uniform Illumination Doppler frequency resolution A/ is also the Doppler frequency bandwidth of the spectrum of the echo signal from a point target. Figure 6.17(a) illustrates the envelop of a point target's echo signal and signal spectrum for uniform illumination over a selected small integration angle if/and for the corresponding integration time T. The two-way echo signal can be represented as the impulse response D
Z(r) = rect'^j = r c c t ^ T
(6.59) T
= 1 for r = - - < r < - and zero elsewhere The Doppler frequency spectrum of the point target response is FT[rect(//7")]. From (2.52), the Doppler frequency resolution A/„ associated with coherent integration of an
265
Illumination
Point target echo signal vs. angle
Normalized spectrum
SAR Impulse response
Doppler h(f ) Q
- 1 As/- _ J L _
A,
c~
2
0)
W
* D "
2
ror
(a)
i V i
.j
sin' A
(
^c = H
8
/
o =
T
M
(b)
2, L Figure 6.17 SAR resolution determined from Doppler frequency spectrum: (a) uniform gain segment of small real beam; (b) beam pattern of unweighted narrow-beam antenna of length I.
echo signal having a rectangular envelope is the bandwidth L\f = 1/7 at the 21 ir points of the (sin x)lx form of the Doppler frequency spectrum of the rectangular-shaped response. Therefore, from (6.58), the cross-range resolution obtained by coherent integration over a small uniform illumination segment of a real beam is D
A
1
A
(6.60)
For coherent integration over an integration angle if/, regardless of w, we have A
1
A
(6.61)
45.2 Nonuniform Illumination Figure 6.17(b) illustrates the envelope of the echo signal and its spectrum for the nonuniform illumination over the entire two-way response (6.4) of an unweighted line antenna
266
of length /. As for the above analysis carried out from the aperture viewpoint, we will assume that the SAR impulse response is determined primarily by illumination within a small angular deviation from boresight, so that in (6.4) sin tf> =» tf>. The two-way echo signal for = tut can then be represented as the normalized impulse response
(6.62)
which can be shown to have the triangular spectrum illustrated in Figure 6.17(b). The, Doppler frequency resolution at the half-amplitude bandwidth is lu>/A. The cross-range resolution from (6.S8) at the half-amplitude points is . A r
Ito
A X
-2^ T
i =
( 6 6 3 )
2
which checks with (6.50) determined from the aperture viewpoint. The expressions given above for cross-range SAR resolution were based on Ihe response produced by coherent integration during the real-beam dwell time of the reflected signals from point targets. From the Doppler viewpoint, coherent integration in the forjn of the Fourier transform of the reflected signal produced fine Doppler resolution, which was shown to be related directly to cross-range resolution. In practice, the coherent integration process may take several forms. A common approach (discussed later in this chapter) is to correlate the azimuthal signal data collected along known range-versusazimuth trajectories with a suitable azimuthal reference to achieve azimuth compression. 6.6 CHIRP-PULSE SAR 6.6.1 Resolution
\
Fine range resolution produced by conventional side-looking SAR, spotlight SAR, or Doppler beam sharpening is often obtained via some type of pulse-compression method, chirp-pulse compression being the most common. Later, we will discuss a stepped-frequency SAR concept, wherein range resolution is achieved synthetically as described in Chapter 5. The resolution Ar, in range for pulse-compression SAR is c/2/3, where j3 is the frequency-weighted bandwidth, which equals A for uniform weighting over the chirppulse frequency excursion A. Regardless of how slant-range resolution is achieved, the cross-range resolution for side-looking SAR produced by integration ever small real beams is approximately
267
A r , - - J
[
(664)
where iff, is the equivalent rectangular beamwidth of (6.53). The cross-range resolution possible with spotlight SAR for integration over a small angle tfi, in radians, is Ar = i^
(6.65)
r
6.6.2 Data Collection Up to this point it was assumed that echo signal data samples were so closely spaced that continuous integration could be assumed in the calculation of resolution in the slant range and the cross range. Requirements for data collection of discrete samples will now be defined for SAR using chirp-pulse-compression radar waveforms. The term data collection, as before, does not preclude real-time processing, but is intended to clarify separate requirements for signal sampling and signal processing. Figure 6.18(a) illustrates the process of collecting SAR data obtained with a chirppulse-compression radar. As the platform containing the radar travels above (at a small down-look angle) and alongside of the area to be mapped, chirp pulses are transmitted at some PRF, assumed here to be/onstant. The time between pulses is made sufficiently long to prevent ambiguous range responses, at least over the effective illuminated range extent wherein echo signals may appear above the noise. A slightly different patch of the earth's surface is illuminated by each transmitted pulse. Each time a pulse is transmitted, the echo signal is sampled at some range-sample spacing or continuously recorded over some portion of the illuminated range extent, called range swath. The collected data comprise a set of reflectivity measurements in two dimensions. The dimensions of the data format can be referred to in several ways: slant range versus cross range, range delay versus time history, fast time versus slow time, and range versus azimuth. A data record (Fig. 6.18(b)) will extend in the slant range over the range swath and continuously in the cross range along the flight path over which data are collected. Data sampled in azimuth at a given range sample position is called an azimuth data line, and data sampled from the echo signal for a single pulse is called a range data line. Typically, a data record will consist of unresolved dispersed responses from a continuum of scatterers. The data set before processing does not resemble a map of the terrain.. Rather, echoes from individual point targets are dispersed in both range and azimuth, as illustrated by the data collection element in Figure 6.18(b). Range and azimuth compression, to be described later, produce the desired maps. It will be convenient to refer to the approximatelyrectangulardata collection element illustrated in Figure 6.18(c) for side-looking SAR. This element is sampled by approximately TJ, x i t samples. This is the area bounded in the cross-range extent by two slantN
r
268
III CO -I o.
zer. I-a
DCO llllllllllllllllllllll llllllllllllllllllllll iiiiiiiliii>M!!:;tiii m«s Ullllllll Miirmii iiiiikiii llllllllllllllllllllll
| c
I 3 111
-I 3
z
(9
oz cT,/2. A total of % complex echo samples are collected in each data collection element during range integration time T, for each of N transmitted pulses occurring within the azimuthal integration angle if/. A total of •n = N samples is collected along the length Rip of each resulting range cell. t
c
270
Data collected from the slant-range and cross-range space indicated in Figure 6.18 are processed to achieve range and azimuth compression. Compression in range for each point-target response is from cT,/2 to Ar„ where Ar, is the slant-range resolution. The compression in azimuth is from Rift to Ar , where Ar is the cross-range resolution. Echo signals produced from each linear FM pulse of chirp bandwidth A to meet the Nyquist sampling criteria must be sampled by at least A complex samples per second. This corresponds to a complex sample spacing of 1/A in range delay and to a range resolution of Ar, = c/(2A). In other words, the dispersed range-delay signal produced by each chirp pulse is required to be sampled at slant-range spacing equal to or less than the slant-range resolution Ar, associated with the transmitted chirp bandwidth. Sampling requirements in the cross range are similar. At the nearest approach of a side-looking SAR in straight-line motion past a surface point target, the range rate and therefore the Doppler shift will vary approximately linearly with time (history), passing through zero frequency at boresight. During the target dwell time for small real beamwidth, the Doppler shift therefore approximates linear FM. Azimuth compression, then, can be thought of as compression of the FM Doppler signal produced during the integration length Rip. Therefore, the azimuth echo signal in each range cell isrequiredto be sampled at a cross-range spacing equal to or less than the cross-range resolution Ar as jciated with the Doppler FM seen across the real beamwidth during its dwell time at R? Unambiguous data sampling of the two-dimensional dispersed response occurs when r
r
r
.
vAr, * Y
(6.66)
r)Ar > Rip
(6.67)
and r
with one complex sample per resolution cell. 6.6J Slant-Range Sampling Criteria e minimum number ofrequiredsamples following each transmitted pulse for unambiguous slant-range sampling is obtained from (6.66) as
(»,.)... =
^
=
r
,
A
(6.68)
2. Low-level Doppler signals from sidelobes of the illuminating antenna produce Doppler frequencies outside the FM bandwidth seen across the main beam. Samples collected at cross-range spacing equal to the crossrange resolution A r produced by the Doppler FM across lite effective real beamwidth, therefore, do not strictly meet the Nyquist criteria. r
271
\
where A is the chirp-pulse bandwidth. In practice, the design of pulse-compression radars that use data sampling techniques is often limited in resolution by the maximum available A/D conversion rates. For example, if the A/D conversion rate is 100 megasamples per sec, this translates to about l.Sm slantrange resolution for unambiguous sampling, according to (6.66); that is.
Ar,=
2V, I
(6.69)
2 (y,fT,)
3 x 10*
I
100 x 10
6
= 1.5m
High-resolution SAR systems, to avoid sampling at very high rates, have tended in the past to rely on analog means for recording echo data on film. This is followed by optical processing. Stepped-frequency SAR, to be described later in this chapter, is a concept that avoids the requirement for high A/D converter rates to achieve high resolution with sampled data. High-speed A/D conversion can also be avoided by means of stretch waveforms, mentioned in Chapter 4. The rate at which analog data can be sampled, converted into digital quantities, and stored is increasing rapidly at this writing. Trends in high-speed sampling and A/D conversion are indicated in Figure 3.17. 6.6.4 Cross-Range (Azimuth) Sampling Criteria The minimum number of samples required for unambiguous azimuth sampling of the azimuthal integration length Rip at each range position is the integration length divided by the cross-range resolution associated with the integration length of the entire real beam. From (6.67) and (6.64), we obtain Rip
Rip
(»7c)». = £7 = j - J samples
(6.70)
Integration for side-looking SAR may be carried out over the entire real beam. An estimate of the minimum number of samples can then be made under the assumption that the effective integration beamwidth is given by the equivalent rectangular beamwidth tp, of (6.53). For ib = rad from boresight, is (4viA)v Ti sin for a PRI of r . When the phase difference is 2w, the unwanted signal from the off-boresight scatterer is coherently integrated, producing a spurious response referred to by antenna f
2
designers as a grating
lobe.
Grating lobes occur for (6.76) where n is an integer. The first grating-lobe angle at n = 1 of (6.76) is
ECHO RADIATION V FROM ON-BORESK3H1 SCATTERER
FOR RADIATION FROM + | CR - d tin + • v , T tin | t
Figure 6.20 SAR grating-lobe geometry.
275
tf J • A
(TJ,)*.
2v,», 2v, A " / c>_ c A
J_ T, 0
"7
=
ifc..
0.5"V"M
IR'frh A 2
which is the range associated with the time between pulses, as for chirp-pulse SAR. 6.7.3 Cross-Range (Azimuth) Sampling Criteria and PRF The required number of samples in each resolved range cell per azimuthal integration length Rip is given by (6.70). Azimuth sampling occurs at the burst rate U(nTi). The minimum number of bursts required for unambiguous sampling during integration angle ip in radians at range R from (6.70) with y} -N bursts becomes c
2RM
NZ.—^
bursts
(6.87)
where 1
A
at
c
A
"7
«
A
A
O.S-JRA
_ 2*vJcos*| 2v,|cos* 2n • - : • n ' . X t c$ c7VJcos ft A XX A
is
n V\ /—
283
n is the number of pulses per burst. Compare (6.90) and (6.75). Pulse-tb-pulse frequency separation, however, may allow operation in the otherwise range-ambiguous region of PRF. Equations in Table 6.2, as for Table 6.1 for chirp-pulse SAR, apply for focused and unfocused SAR, small integration angle, and Rayleigh resolution. Waveforms for unambiguous and ambiguous range are illustrated in Figures 6.23 and 6.24, respectively. In the side-looking mode of stepped-frequency SAR, each burst produces k sets of n complex echo samples, spread throughout the desired swath-delay interval. Figure 6.23 illustrates sampling when the PRF corresponds to the unambiguous range so that PRF < c/(2A/?,), where A/?, is the illuminated range extent over which significant echo power is received. Figure 6.24 illustrates sampling when PRF > i7(2A/? ). In this figure, echo foldover is avoided by frequency separation between pulses. In this way, the PRF might be made sufficiently high to avoid synthetic aperture grating lobes while also avoiding range ambiguity. Receiver blanking would likely be required during each transmitted pulse. Further study is needed to characterize degradation of system performance by relative motion of the target and radar platform during the burst time, and to develop appropriate motion-compensation algorithms. In addition, further study is needed in the areas of memory and computation speed requirements for mapping operations. Figure 6.25 is a generic block diagram of a stepped-frequency system. Similarity to the block diagram in Figure 6.21 for pulse-compression SAR is apparent. The key difference is the means for achieving the fine resolution in the slant range. A controlledfrequency synthesizer is used in stepped-frequency SAR to generate the waveforms for synthetic range-profile processing, instead of a chirp generator and the pulse-compression scheme as for pulse-compression processing. It may be possible to avoid platform motion compensation by using a variation of the technique to be described in Chapter 7 for ISAR data motion compensation with ;
SWATH DELAY INTERVAL
ECHO
COARSE RANGE-DELAY CELL EXTENT
. £
TIME
ONE BURST OF n PULSES-
Figure 6.23 Stepped-frequency SAR sampling (unambiguous range).
284
ECHO
SWATH INTERVAL
1
I
Miiliiiii^ t t . 0
t
DELAY
tk-1
-COARSE RANGE-DELAY CELL EXTENT
~32^JIIIIIII
tk-1
2T,t„t,...
-3T,
-I
tk-1
ilWlftnnmr t,t,...
START OF NEW BURST
I
T,
Figure 6.24 Stepped-frequency SAR sampling (ambiguous range).
stepped-frequency waveforms. This is suggested by Che dotted lines associated with motion compensation in Figure 6.2S. The hopped-frequency alternative to stepped-frequency waveforms, discussed in Chapter 5, may be able to provide advantages, such as improved electronic countercountermeasures (ECCM) performance and ability to unambiguously sample Doppler
QUAORATU MIXER
oc
SAMPLER
285
UjCC_J
? OC < H I T J D-UJOT y/v,,
Figure 6.35 Two-dimensional quadratic-phase response to a point target observed with a side-looking SAR assuming small range curvature.
309
target will extend over the uncompressed pulse duration T, and will be centered in range delay at the target's range-delay position r. The signal at the target's range delay will extend in azimuth over integration time 7", corresponding to the synthetic aperture size, and will be centered at the target's azimuth delay. It is assumed that the delay r is essentially constant during each echo pulse, but varies according to (6.122) during the target's dwell time T. Symbols /, and h in Figure 6.35, sometimes called fast time and slow time, refer to range delay and time history, respectively. The third exponential term of (6.123) is a phase term dependent on the closest approach in range of the radar platform to the point target, a term ideally made a constant by flying the platform in a straight line. The response of a single point target in terms of data collection space will extend over the integration lengths in range and azimuth. The area cT,/2 x v,7"is the data collection element (for small curvature) that contains the dispersed response to a point target.
6.11.2 Optical Processing Conventional optical SAR processing is carried out on film rolls that contain the twodimensional phase history of the response of target scatterers, which were produced as the SAR platform traveled above and alongside the range swath to be mapped. Both range and azimuth compression can be performed optically. Film rolls are exposed on an optical film scanner, illustrated in Figure 6.36. The input to the scanner is the coherent signal heterodyned down to bipolar video. Light intensity from a CRT in the scanner is modulated by the bipolar video signal. This corresponds to the baseband signal produced by reflection from multiple scatterers on the illuminated earth surface. The film roll is exposed as it moves past the CRT in a direction perpendicular to the range sweep of the intensitymodulated light spot. A bias voltage may be used to produce the desired film exposure. After recording the SAR phase history, the input film roll is brought to an optical bench, where it is focused to form the output film roll, which is the SAR strip map. Data recording of the response from a single point target is illustrated in Figure 6.37. An actual film record would contain the phase histories of the numerous scatterers on the surface to be mapped. Phase histories of individual scatterers are likely to overlap one another, but will ideaily focus to individual points. Recorded phase history on film is similar to the Fresnel zone plates used in optics. The quadratic nature of the phase response makes "it possible to diffract collimated coherent light passing through the film to produce focused images. Phase history recorded on a SAR film roll focuses incident coherent light at different focal lengths in azimuth and elevation. The situation can be thought of as astigmatism, which can be optically corrected by the use of cylindrical lenses. Separate focal lengths occur because recorded phase is the sum of separate phase components in each dimension. From (6.124), the slant-range component of a recorded phase of the echo from a point target is 4irK(t - r)V4. This dimension of phase is recorded in the range dimension at a
310 T
Figure 636 SAR optical film scanner.
x
sweep velocity of v,. Also from (6.124), the azimuth component of a phase of the same point target is -4ir\v,t - y)V(2XR). This component of phase is recorded in the azimuth dimension at the film transport velocity of v.. The resulting two-dimensional phase history in range and azimuth focuses collimated light passing through the film at different focal lengths in range and azimuth. The focal length associated with SAR phase history recorded on film can be compared to other, more familiar optical focusing mechanisms. Figure 6.38 illustrates three equivalent focusing mechanisms (each shown for one dimension). Light in each mechanism propagates in a manner so as to encounter quadratically distributed delay in the cross-axis dimension labeled x. The quadratic-phase function in'Figure 6.38(a,b) results from qua- . dratic variation of delay along the x-dimension. Figure 6.38(c) illustrates one dimension of quadratically distributed phase history recorded on SAR film. F^rr each case, the oneway phase function for light at wavelength X, is )=
^ -xA
(6,25)
where 9 is the optical focal length. This result for the reflector in Figure 6.38(b) is directly analogous to that found from Figure 6.10 and the accompanying discussion regarding the quadratic-phase response produced by the SAR platform moving past a point target on
311
RANGE DELAY, ti
AZIMUTH TIME HISTORY, t,
Figure 6.37 Optical film record of the phase history of a single point target (side-looking SAR with insignificant range curvature).
the earth's surface. With the proper optics, focusing results when collimated coherent light is passed through the zone plate formed by the film record of the quadratically distributed responses to individual scatterers. For sweep velocity v, and film transport velocity v„ the ^-dimension of the recorded signal is x, = vft, - T) in range and jr = v,(r - ylv ) in azimuth. Therefore, uV(jr) of (6.125), written in terms of I, and /j, becomes 2
2
r
In v|((, - r)
A, in the range dimension, and
29,
a
(6.126)
312
LENS
(»)
REFLECTOR
(b)
(c)
Figure 6.38 Equivalent focusing mechanisms.
in the azimuth dimension, where /, and fj refer to time associated with range delay and time history, respectively. Optical focal lengths 9 , and JF, can be expressed in terms of radar parameters by setting the magnitude of the two optical phase ct ^ponents given in (6.126) and (6.127) equal to their corresponding RF phase components from (6.124) as follows.
2 7 r v
in range, and
'(>.-r)»
„(», -
T)
1
(6.128)
313
^""AT
2*.
~T~2R
"T—*—
( 6 1 2 9 )
in azimuth. By solving for the two focal lengths and recalling from Chapter 4 that chirp slope K = A/7",, we obtain
" A,K ~ A,A
(6.130)
for the optical focal length in the range dimension, and
9
(6.131)
for the optical focal length in the azimuth dimension. The two focal lengths are illustrated in Figure 6.39. Azimuth focal length varies linearly with range across the width of the film because of increased radius of range curvature of input data at increasing range. SAR optical processing corrects for the astigmatism by using cylindrical lenses. In addition, conical or tilted cylindrical lenses correct for the linear variation of azimuth focal length with range. Figure 6.40 illustrates a simplified configuration. The data film on the left has a vertical range focal plane followed by the tilted azimuth focal plane. A cylindrical lens is oriented so that its input focal plane coincides with the tilted azimuth focal plane of the data film to coliimate rays in the azimuth dimension. A second cylindrical lens, further to the right, is placed so that its input focal plane coincides with the vertical range focal plane of the data film to coliimate rays in the range dimension. With both dimensions collimated, targets will be focused at infinity to the right. A spherical lens focuses targets on the SAR image plane. Actual optical processors are far more complicated in practice. Usually, the SAR image is made continuously. Both the SAR data film and output SAR image film are driven, and a slit in the range dimension produces continuous exposure of the SAR image.
6.113 Digital Processing The generation of SAR images is a two-dimensional process, regardless of the processing technique employed. Optical SAR processors process the range-azimuth analog data simultaneously in time. Digital SAR processors often resort to a series of two one-
314
Figure 639 Range and azimuth focal lines of point-target phase history. (Modification of Fig. 23. p. 1191 • (10). Reprinted with permission.).
dimensional processes to produce the two-dimensional result from digitized input data. The advantages of increased accuracy and flexibility in digital processing are obtained at the expense of considerable complexity. It is beyond the scope of this section to cover the field of SAR digital processing. Rather, a two-dimensional | prrelation method of processing that is applicable to chirp-pulse-compression SAR wnl be discussed in a t attempt to report some of the important issues. Two-dimensional correlation achieves pulse compression
in the slant range (range compression) and azimuth compression m
the cross range (azimuth compression). The idealized response to a single point target viewed with a chirp radar wat expressed in (6.124). This equation contains similar quadratic terms in both range-delay and time-history dimensions. Lenses are able to perform the two-dimensional compressioa in optical SAR processors. The lenses were shown to possess quadratic-phase functions, which collimated the light through the data film so that individual target responses could be focused into points on the image film. This process has also been described as two-
315
RANGE
A
Z
|
M
u
T
H
AZIMUTH FOCAL PLANE
RANGE COLLIMATOR
SARIMAGE RECONSTRUCTION PLANE o
p
T
|
c
A
L
/AXIS
COLLIMATED COHERENT LIGHT RANGE FOCAL PLANE
INFINITE CONJUGATE IMAGING LENS
TILTED AZIMUTH COLLIMATOR
Figure 6.40 Simple optical SAR processor. (From (101, Fig. 24. p. 1191. Reprinted with permission.)
dimensional optical convolution [3]. Digital processors for pulse-compression SAR, in an analogous process, may convolve the digitized two-dimensional data with a digitized two-dimensional, matched-filter impulse response function instead of lenses. The response function, in general, is made up of the impulse response h(t ) of the chirp signal for range compression and a similar function /t('i) for azimuth compression. As before, t, refers to range delay (fast time) and tj refers to time history (slow time). SAR processing, however, is often described in terms of correlation rather than convolution. Instead of referring to the impulse response of the matched Filter in range delay or in azimuth time history, the concept of range and azimuth reference functions B used. The equivalent reference functions in range and azimuth are the time inverses of the complex conjugates of the respective matched-Filter impulse responses. Correlation of the range-delayed signal with a range reference is the equivalent of convolution of the same signal with the impulse response of the matched filter to the transmitted waveform. A similar equivalence holds in the azimuth dimension. The reference function for range correlation is the point-target response in range. The reference function for azimuth correlation is the point-target response in azimuth. A two-dimensional reference function is the dispersed response in range and azimuth. Azimuth and range compression of two-dimensional signal data will now be described for Ihe ideal case in which the two dimensions of the reference function can be defined independently. This idealization is valid for the processing of a data block for which range and azimuth extent is sufficiently small that range curvature and range walk can be neglected. Then a single azimuth reference produces azimuth focusing at all ranges in ne block. Figure 6.41 illustrates a block of digitized two-dimensional data that includes the idealized response from a single point target at delay rand azimuth position y . Each t
316
Hrl RANGE-DELAY EXTENT OF INPUT DATA BLOCK
POINTTARGET UtRESPONSEl
TIME HISTORY -EXTENT OF INPUTDATA BLOCK
RANGE DELAY, t, (FROM TRANSMIT)
AZIMUTH TIME HISTORY, t, (FROM BORESIGHT)
Figure 6.41 SAR dan Mock for chirp waveform showing response lo a point target centered at /, = r, fi« -y/v (small range curvature). f
resolved element contains a complex data sample. T h e response tn two separate point targets is illustrated in Figure 6.42(a). Two-dimensional correlation with the two-dimensional reference produces an image block containing the two targets in focus as indicated in Figure 6.42(c). Columns of range data lines are first correlated against the range reference. The correlated result for each range data line is a set of range-compressed data lines. Range-correlated results are shows in Figure 6.42(b). Rows of azimuth data lines are then correlated against the azimuth reference to obtain two-dimensional correlated results, shown in Figure 6.38(c). The two one-dimensional processes produce the required two-dimensional image of Figure 6.42(c) without distortion because the same range reference was assumed valid for all range columns and the same azimuth reference was assumed valid for all azimuth rows.
317
RANGE DELAY t,
AZIMUTH TIME HISTORY t ,
DISPERSED RESPONSE F R O M TARGET 1
DISPERSED RESPONSE FROM TARGET 2
n-1
(a)
RANGE REFERENCE 1
(b) 02
N-1
1 3 AZIMUTH REFERENCE
(e)
Figure 6.42 Processing of SAR input data containing two point targets (small range curvature): (a) input data • block; (b) range-correlated data; (c) image frame.
318
6.11.4 Nonindependent References Independent range and azimuth references were employed in Figures 6.41 and 6.42. This was possible because of the stated assumption of sufficiently small range curvature, range swath, azimuth integration angle, and range walk. Consider the case in which range compression produces such closely spaced azimuth lines that azimuth responses are not contained along individual lines. This occurs when range migration M' of (6.98) exceeds unity. Azimuth compression for each image pixel must then be carried out along curved paths in range to achieve full resolution capability. The azimuth reference is also range-dependent. Figure 6.43 reprr ^nts the phase history of a chirp-pulse response to two point targets at the same azimuth position, but separated in range at opposite edges of a SAR range swath. The phase history for both targets remains quadratic in both range and azimuth (as viewed along their curved range responses), but we can see that the azimuth reference needed for azimutfi focusing at near range differs from that at far range. The curve is longer but less pronounced for the response to the target at far range. Therefore, an azimuth reference for range R matches a larger FM Doppler slope than that for /? . The range reference, because it is determined only by radar waveform, is independent of azimuth position. Finally, range walk caused by cross-track earth motion beneath a satellite SAR, unless corrected, produces responses that walk through range cells. Range curvature and range walk result in responses from individual scatterers that travel through range cells and require a range-dependent reference. In principle, image formation is still possible by using two-dimensional processing from known geometry. For example, after range compression, the reflectivity for a selected two-dimensional resolution cell could be established by processing range-compressed data obtained along the range-azimuth path on which a scatterer would travel to produce a response in the selected resolution cell. The process would be repeated for each cell. This approach is avoided in practical processors because of its complexity. An example of a shortcut method for carrying out two-dimensional processing is that for the SEASAT digital SAR processor, described below. Another method is polar reformatting, which is described for ISAR in Chapter 7. t
2
6.11.5 Fast Correlation We discussed the convolution of sampled and digitized target signal data produced by a chirp radar using FFT processing in Chapter 4. The method was called fast convolution. Fast convolution of digitized data was accomplished with a digital version of the matchedfilter impulse response to the chirp-pulse waveform. The same process could have been described in terms of fast correlation with a reference function equal to the time inverse of the conjugate of the digitized point-target response in range. SAR data sets can be processed by using a two-dimensional fast-correlation method. Such a method, because
Figure 6.43 Response tt> two point Urgets at the same aiimuth position but separated in range (chirp-pulsecompression SAR).
320
of the use of the FFT algorithm, is usually faster than direct correlation, just as fast convolution is faster than direct convolution. Fast convolution is based on the discrete form of the convolution theorem, which for input signal s,(t) convolved with impulse response h(t), was expressed in ( 4 . 5 4 ) as FTta(f) * /.(f)] = SAf) x / / ( / )
(6.132)
where / / ( / ) is the Fourier transform of h(t) and SAf) is the Fourier transform of 5,(r). An equivalent expression can be written in terms of correlation. From the definitions of convolution and correlation, the following equivalence can be written: FTUAD * HO] = F T U W ® h'(-t)]
(6133)
where denotes cross correlation. The matched-filter transfer function H(f) for the transmitted waveform s\(i) is S",{f), and the time inverse of the complex conjugate /i'(-f) of the matched-filter impulse response h(t) is 5,(0, where s,(t) is the point-target response, which becomes the reference function. Equation (6.132) for the convolution theorem, therefore, can be rewritten as the correlation theorem, expressed as FT[5,
(854 x 10 )(2T»/360)(4)(0.3) 6,600
(6.136)
= 2.71 sec 4. Input data of the actual processor were sampled at an offset frequency from baseband. Equivalent complei data will be assumed for this example.
325
a
-SECT. 1
JTL
J
r—SECT. 2
H-SECT. 3—1 -SECT. 4 -
M
204S EACH SECTION, 768 OVERLAP
0>)
768
.—
1280
- DISCARD
(e)
768
NOTE: 768,1280 ft 2048 REFER TO NUMBER OF COMPLEX SAMPLES OR COMPUTED VALUES
1260 ^DISCARD ?
768
1280 DISCARD
768
1280
UNRECOVEREO PORTION (END EFFECT) 6 3
(0
-»«( 1280.
12801280,
1280
ETC
,1
RANGE SUBIMAQE LINES
Figure 6.46 SEASAT range correlation of data line containing dispersed response to eight (hypothetical) point targets: (a) range data line; (b) range reference; (c) section I correlation; (d) section 2 correlation; (e) section 3 correlation; (0 composite correlation (one image line).
326
As with the range reference function, the azimuth reference function is represented by complex digital values of the same time spacing as that of the data. Data spacing in azimuth is the reciprocal of the 1,500 complex azimuth samples per second (one sample per PRI) produced in each range cell, which is 4,065 samples per azimuth line, generated during the 2.71 sec of beam dwell time at beam center. Actually, 4,096 complex values are used to represent four coherent-integration looks. At the 4.4m sample spacing in azimuth, this covers an azimuth extent equal to the four-look cross-range integration length of about 18 km. Image quality depends on the accuracy with which the azimuth reference function represents the phase history of point targets in the real beam. Azimuth phase history can be represented by a quadratic function analogous to the quadratic function that represents the range reference. The azimuth reference function can be defined if the Doppler frequency at the azimuth center of illumination (Doppler centroid) and Doppler frequency slope (hertz per second) are known. This corresponds to the requirement that the instantaneous frequency at the center of the FM chirp pulse and FM slope (hertz per second) be known. Uncertainties in SAR platform attitude and Doppler echo spectrum produced by the earth's rotation below the satellite can require special preprocessing programs to generate the azimuth reference function. However, no such variations occur in the range reference, because the FM chirp generator in the radar determines the reference independently of platform attitude and orbit considerations. Clutter lock and autofocusing are methods used to estimate the Doppler centroid and Doppler frequency slope, respectively, based on the SAR data [6,10]. The clutterlock method sets the Doppler centroid of the reference to that of the received spectral response from the illuminated surface area. Autofocusing sets the Doppler frequency slope to produce minimum azimuth blur in the processed image, as determined by spatial frequency analysis, or by adjusting for minimum azimuth registration error between looks. Range correlation is performed first. The sectioned range data are illustrated in Figure 6.46(a). The reference function in range is shown in Figure 6.46(b). The range reference in the speclral domain can be separately generated in a preprocessing program. Fast correlation is performed on each of the 2,048-element sections of the input data, with the results as indicated in Figure 6.46(c-e). The composite result for one range data line is shown in Figure 6.46(f). The overlap-save process results in 2,048 - 768 = 1,280 complex values saved from each section. Range walk in the IDP is corrected to the first order by sliding the range lines as needed to align their starting samples. Finer correction is carried out by selecting from one of several range reference functions that vary by a fraction of a range cell in delay. This provides range-walk interpolation to within a fraction of a range cell. Correlated range data is stored in a corner-turn memory, then read out in the azimuth dimension. The data read out are transformed line by line (or column by column, as in Figure 6.44) into azimuth spectral data. Range curvature in the IDP processor is compensated for in the azimuth spectral domain by using a process that is efficient in terms of processor time, covering the known range curvature [13]. This process is illustrated in
327
Figure 6.47. The range curvature of a particular point target is shown plotted in the azimuth spectral data domain as range delay versus Doppler frequency. Because quadraticphase history is assumed, the range delay of a point target versus Doppler frequency, and its range delay versus azimuth time history are represented by the same curve, except for a constant factor. To correlate the near-range azimuth spectral line of data in Figure 6.47, the spectral form of the azimuthal reference requires vector multiplication by the spectral data that appear along the curve path. The product comprises the composite spectral line in the lower part of the figure. Another composite spectral line is obtained from spectral data that appear along the same curve when it is shifted outward in range to the next azimuth spectral line. The process of shifting to the next azimuth data line is repeated until all the curved data in the spectral domain have been converted into composite lines free of curvature. The piecewise-linear approximation of the curved delay provides advantages in terms of memory storage requirements and flexibility in reference updating [13]. RANGE
AZIMUTH (SPECTRAL DOMAIN)
CURVED RANGE DELAY OF
COMPOSITE AZIMUTH SPECTRAL LINE
Figure 6.47 Range curvature compensation in the azimuth spectral domain. (Based on Fig. 3 from C. Wu et al.. " S E A S A T Synthetic-Aperture Radar Data Reduction Using Parallel Programmable Array Processors," IEEE Trans. Geoscience and Remote Sensing, Vol.GE-20, No. 3. July 1982. Reprinted with permission.)
328
At this point, azimuth correlation can be thought of as being performed on curvaturefree azimuth time-history data. The fast correlation process proceeds for each composite line by inverse-transforming spectral regions for each of four looks. Figure 6.48 illustrates correlation for one look. Figure 6.48(a) illustrates a single azimuth line of curvature-free data from which an image line is to be generated. Individual looks at a given point target occur at separate portions of the total Doppler spectral response to the target. The first look at the leading edge of the beam contains only positive Doppler shift because range decreases during the first look. The last look contains only negative Doppler shift because range increases. The spectrum of the reference functions for each look likewise occupies a separate portion of the spectrum of a hypothetical reference for the total beam response, as is illustrated in Figure 6.48(b). Fast correlation for each look uses the overlap-save process to correlate azimuth data sections of2,048 elements, each with its 1,024-element, single-look reference function. The result is 2,048 - 1,024 = 1,024 azimuth values saved per subimage data line. Only 2S6 of the 1,024 values ultimately must be saved, however. This can be understood by recalling that for the SEASAT velocity and PRF, the total synthetic aperture length of 18 km is sampled with 4,096 complex samples spaced 4.4m apart. Pixels produced by one look will represent the equivalent of about 22m in resolution. In other words, azimuth data is oversampled by about a factor of four for integration during each look. Azimuth data are originally sampled at or above the Nyquist rate for the total aperture because the actual phase history is that of the Doppler spectrum produced by the total aperture. Therefore, 2,048 complex values per section are retained up to the point where the inverse FFT is performed in the azimuth compression process. The inverse FFT then must be performed on only 512 of the 2,048 spectral values for unambiguous representation of the reduced single-look resolution. Of the 512 resulting time-domain values, only 256 are saved, as indicated in Figure 6.48 (c-e), which are detected (converted to magnitude only) to form image pixels. Figure 6.48(f) indicates that saved azimuth subimage lines register side by side to form a contiguous azimuth image line, as in the range domain. A contiguous set of subimages of 1,280-by-256 azimuth-elevation pixels are generated. The corresponding subimages from four looks are overlapped and noncoherently summed. Subimages are assembled to form an image of 5,800-by-5,144 azimuth/elevation pixels. This is called a SEASAT-A SAR frame and covers an area of 100 by 100 km with about 25m by 25m resolution. A four-look SEASAT image of the San Diego, California, area, obtained from data collected during revolution 107, is shown in Figure 6.49.
6.12 DOPPLER BEAM SHARPENING The third type of SAR mentioned at the beginning of this chapter was called Doppler beam sharpening (DBS). It is discussed separately here from side-looking and spotlight SAR because the theory takes a somewhat different form. To date, DBS radar has been
329
(I) AZIMUTH OATA LINE
(b) AZIMUTH REFERENCE
EEL
^ 1 " II I L -SECT. —SECT. 2—J H—SECT. -—SECT. -"—SECT. 4 — —
1st LOOK
T I
2nd I 3rd I 4th , LOOK | LOOK | LOOK I 1
Ul
1
-4V/ 2048 EACH SECTION. \ 1024 OVERLAP
1st LOOK
_L
1024 SPECTRUM OF AZIMUTH REFERENCE \ 256 *
(c) SECTION 1 LOOK1
i
li .
.
25S (d) SECTION 2 LOOK1 CORRELATION
NOTE: 256. 512,1024, & 2048 REFER TO NUMBER OF COMPLEX SAMPLES OR COMPUTED VALUES
'CARD; 256
(I) SECTION 3 LOOK 1 CORRELATION
-£ARD'
/////, UNRECOVERED PORTION (END EFFECT)
256
(I) COMPOSITE CORRELATION (LOOK1)
AZIMUTH SUBIMAGE LINES
Figure 6.4U SEASAT azimuth correlation of data line containing dispersed response to seven (hypothetical) point targets: (a) azimuth data line; (b) azimuth reference; (c) section I look I correlation: (d) section 2 look I correlation: (e) section 3 look I correlation: ( 0 composite correlation (look I).
no
\ •
ft
7-
..•*
•••
•
»;* .... .'•if*'/'
.>•..;•;.•;
^
v
: . i ? • Sr, we obtain
346
-\»/2. . Scatter 2 ^"-•^^^^
Center of rotatk rotation
Radar-
Scatter 1 Target 91 - 0
Figure 7.4 Unfocused ISAR.
v*=f— j
(7.8)
Assume, as we did for SAR, a maximum allowable two-way phase deviation of iri% rad as the criteria for focus. The corresponding allowable range deviation Sr is A/32. The maximum integration angle before defocusing occurs, from (7.8), then becomes
The maximum target size in terms of radius r, from (7.9), before defocusing begins to occur is expressed in terms of focused resolution Ar from (7.6), as r
(7.10)
Processing to correct for range curvature is required to obtain focused imagery for data collected over larger integration angles than indicated by (7.9) or from targets with larger radii than indicated by (7.10). 7.2.2 Optimum Unfocused ISAR Integration Angle The optimum unfocused integration angle and associated cross-range resolution for ISAR are analogous to the respective SAR parameters. Figure 7.4 for ISAR is analogous to
347
Figure 6.10 and the associated analysis for SAR, where scatterer I in Figure 7-4 corresponds to the SAR scatterer on boresight in Figure 6.10. The phase advance with time t for the echo signal from scatterer 1 located at rotation angle d» = 0 when / = 0 is
,
W
.
q
.
-
£
»
.
_
£
= tot with constant angular rotation rate to. The two-way phase advance for scatterer 2, which is located at r, -d> when I = 0 is
W
»»,(/,-T= ^correspond to (7.5) and (7.6) for a nonfocused, small-integration-angle ISAR. The uniform illumination assumption for SAR used to obtain resolution (6.39) and (6.40), while useful for analysis, does not accurately represent illumination by practical physical SAR apertures, but the uniform illumination assumption does apply to most practical ISAR situations where the target azimuth extent is small compared to the illumination beamwidth. Thus, (7.18) and (7.19) more closely represent observed resolution for ISAR than (6.39) and (6.40) do for SAR. A fundamental difference should be noted between procedures required for focusing side-looking SAR data and those for ISAR data. Range focusing in both cases is performed based on the range-independent point-target response determined by the radar waveform. Azimuth focusing for side-looking SAR can be performed by correlation to the rangedependent point-target reference response determined by the SAR geometry. However, the azimuth point-target response for ISAR systems used to image ships and aircraft in operational environments is determined by the angular rotation part of the target geometry, which is generally not known a priori with sufficient accuracy for useful target imaging. Fortunately, azimuth focusing is not required for many smaller targets that meet the criteria of (7.10). We will see later how rotational motion occurring during data collection from larger targets can be determined from the data by seeking the rotational motion solution that results in the sharpest focus. In addition, some ISAR waveforms such as steppedfrequency waveforms require that collected data be corrected for target translational motion, which is also not generally known a priori with sufficient accuracy to focus in range. Translational motion solutions for these ISAR waveforms can also be generated from the data.
349
13 RANGE-DOPPLER IMAGING
Resolution and sampling requirements for ISAR are probably most easily understood in terms of range-Doppler imaging. Assume the target model of Figure 7.5, consisting of a three-dimensional rigid set of scatterers from which wideband echo data is collected during target rotation about a fixed rotation axis in the far field. The target is assumed to be uniformly illuminated, and processing to obtain an image is assumed to be performed on data collected during target rotation through a small integration angle segmentfathat meets the criteria of (7.9). This is approximately equivalent to assuming that the integration angle is small enough that the slant-range and Doppler frequency of scatterers at the target extremes shift less than the corresponding processed slant-range and Doppler resolution. The processed image consists of estimates of the magnitude and position of scatterers in the slant range and cross range. The slant range is the radar LOS range dimension.
SCATTERER
10
•—RADAR
RANGE-PROFILE SAMPLE NUMBER l
0^VV"' V
h T 1» N
l f t
"
(I OR Q AMPLITUDE)
* «nT«1 RADIAN > INTEGRATION TIME * NUMBER OF RANGE SAMPLES = NUMBER OF PROFILES PER INTEGRATION TIME
Figure 7 J Range-Doppler sampling of a routing target.
350
The cross range is the dimension lying normal to the plane contained by the radar LOS and target rotation axis. The range-Doppler model will be used to develop expressions for slant-range and cross-range resolution, sampling requirements, and the target image plane. Basic principles of processing will be described. This will be followed in succeeding sections by analysis of the defocusing effects of target translational motion and processing over integration angles that exceed the criteria of (7.9). Methods for translational motion correction (TMC) and rotational motion correction (RMC) will be described for the practical situation in which target motion is not known a priori. After this, a generalized target model will be developed that includes both target translational and rotational motion. Automatic focusing methods using this model will be described for applications where the focusing criteria of (7.9) and (7.10) for maximum integration angle and target size are exceeded. Range-Doppler imaging is further discussed by Ausherman et al. [7], Chen and Andrews [2], and Walker [3].
7.3.1 Basic ISAR Theory for Small Integration Angle Figure 7.S suggests a series of range profiles produced by an HRR radar as it observes a rotating target. Range sample increments correspond to target dimensions in meters, and profile-to-profile increments correspond to time in seconds. The signal along one range profile is illustrated, and the response to a resolved scatterer at one range position is shown in time history. The response to the scatterer can be seen to produce a few cycles of Doppler shift during integration time T while the target rotates through if rad. Not shown in the figure are Doppler responses produced in other range cells corresponding to other scatterers on the target. Data for one image are sampled at baseband with rj, inphase and quadrature-phase (/ and Q) samples per range profile for each of N range profiles obtained during time T. Waveforms used to obtain the range profiles and sampling criteria will be discussed later. Doppler frequency shift produced by a given slant-range resolved scatterer for small if is proportional to the target angular rotation rate as well as to the cross-range distance between the scatterer and the center of the target rotation. One or more Doppler spectral lines can exist for each slant-range cell, one for each Doppler-resolved scatterer. The magnitude of a spectral line is proportional to the reflectivity of the resolved sotterer. The target's reflectivity, therefore, can be mapped in both the slant range and cross range with the cross-range scale factor dependent on the target angular rotation rate. Target track and other data can be used in some applications to estimate rotation rate and orientation of the rotation axis relative to the radar LOS. The orientation of the target's rotation relative to the radar establishes the orientation of the image plane. The image is bounded by slant-range and cross-range windows, the significance of which will be discussed later.
351
13.2 Cross-Range Resolution The basic relationship between target rotational motion, scatterer position, and the resulting Doppler frequency shift can be seen by referring to Figure 7.6. Neither the radar nor the target has any translational motion in this example. Radar LOS is in the plane of the paper. The target rotates at a constant angular rotation rate to in radians per second about a fixed axis perpendicular to the plane of the paper. A single scatterer at a cross-range distance r can be seen with instantaneous velocity tor, toward the radar. The instantaneous Doppler frequency shift is c
(7.20) where / is the carrier or center frequency of the radar, A is the wavelength, and c is the propagation velocity. We will initially assume that f is constant during the viewing-angle change that occurs during a small integration time T. Later, we will show that defocusing is produced by variation of the Doppler frequency, which increases as processing is performed over larger viewing angles. If two scatterers in the same slant-range cell are separated in the cross range by a distance Sr„ then the separation between the frequencies of the received signals, from (7.20), is D
-rtoSr,
Sf
D
(7.21)
so that (7.22)
Sr.
INSTANTANEOUS SCATTERER VELOCITY TOWARD THE TARGET
LOS RADAR
Figure 7.6 Radial velocity produced by a scatterer on a rotating target.
352
Then, for a radar that has a Doppler frequency resolution of Lfo, we have a crossrange resolution given by
The cross-range resolution Ar can be seen to be dependent on the resolvable difference in the Doppler frequencies from two scatterers in the same slant-range cell. The Doppler resolution in turn can be related to the available coherent integration time T of a constant-level signal. The relationship, from (2.S2) of Chapter 2, in terms of the Rayleigh resolution, is A/ = \IT. The coherent integration time will also be called the image frame time. The cross-range resolution obtained by coherent integration of the echo signal received during the viewing-angle change if/ that occurs during integration time T is thus obtained from (7.23) as c
D
Ar
--5^-53f-B
(724)
Where coherent angle ifr= a>T for uniform target rotation. Equation (7.24) is the same as (7.6) and (7.19) obtained from the aperture viewpoint of ISAR, and the same as (6.60) and (6.61) for SAR. Typically, a DFT process, in the form of an FFT, is used to convert the set of timehistory samples collected in each range cell during the time segment T into a discrete Doppler spectrum. Resolution from (7.24) assumes uniform weighting during integration. The precise relationship between Doppler frequency resolution and integration time depends on the type of transform and the window function used to weight the segment of time-history response. Figure 7.7(a) illustrates a series of echo samples available in the same range cell of N range profiles. The DFT of the data is illustrated in Figure 7.7(b). 7.3 J Slant-Range Resolution As for SAR, the slant-range resolution for ISAR is obtained by using wideband waveforms. Regardless of the type of waveform, the achievable range resolution is approximately cl (20), where 0 is the waveform bandwidth. In principle, any of the waveforms discussed in Chapters 4 and 5 would be suitable. Only two are discussed here: chirp pulse and stepped frequency; these are the same two waveforms discussed for SAR in Chapter 6. While chirp-pulse and stretch waveforms are the most common for SAR, stepped-frequency waveforms have been found to be useful for ISAR when the application requires extreme resolution. Rayleigh resolution for chirp waveforms is (7.25)
353
SAMPLES OF RESPONSE
TIME HISTORY
DISCRETE FOURIER TRANSFORM
I
Figure 7.7 Sampled lime history and associated Doppler spectrum in one range cell (illustrated for the case where one range profile is generated from the received response from each chirp pulse).
where A is the chirp bandwidth. Synthetic processing of stepped-frequency waveforms, as discussed in Chapter 5, requires the conversion of echo data, collected in the frequency domain, into synthetic range profiles. This is typically carried out by using a DFT process, as illustrated in Figure 7.8. The resolution for n steps of A/Hz each, from (5.21), is (7.26) Synthetic ISAR involves two dimensions of the Fourier transform: (1) frequencydomain reflectivity into range-delay reflectivity for each burst to resolve targets in range.
354
Figure 7.8 Echo spectrum and associated synthetic range (delay) profile for a single burst.
followed by (2) time-domain reflectivity in each range cell into Doppler frequency-domain reflectivity for each range cell to resolve targets in the cross range. The above twodimensional transformation process, in the most fundamental sense, transforms reflectivity data obtained in frequency and viewing-angle space into object-space reflectivity estimates.
7.3.4 Slant-Range Sampling Targets to be imaged using ISAR are usually isolated moving targets, in contrast to the large fixed surface areas to be mapped with SAR. For ISAR, therefore, we assume that some type of angle and range tracking is used to keep a selected target immersed in the radar antenna beam during data collection. Samples from each of N range profiles, regardless of waveform, will be assumed as the input data shown in Figure 7.5 for one image. The n, samples of the range profile produced by real processing of the received response from a transmitted chirp pulse are collected directly in the time domain. When stepped-frequency bursts are transmitted, sampling can be said to occur in the frequency domain. The synthetic range profile obtained from each burst by the DFT process is effectively~sampled by n pulses of each burst so that r\, = n. A slant-range window will now be defined for each type of waveform.
355
A sampled target range profile received from the transmission of a single chirp pulse is illustrated in Figure 7.9. The unambiguously sampled slant-range extent, called the slant-range
window, is given by cL\t
c
w. = V.-J- = l . ^
(727)
for T}, complex samples spaced by Ar sec in range delay. Samples are obtained, as described in Chapter 4, by using some form of range tracking that starts the first sample just before the target echo arrives from each pulse. Additional samples are collected during a total delay interval corresponding to the slant-range window given by (7.27). To meet Nyquist's criterion, the complex / and Q sampling rate during this interval must equal or exceed A complex samples per second. At least one complex sample of the baseband response for each pulse of a pulseto-pulse stepped-frequency burst is required for unambiguous sampling of targets of rangedelay extent less than the duration of the baseband response. In other words, we require at least one complex sample of the target signal produced at each frequency. As discussed in Chapter 3, a target's reflectivity for unambiguous sampling in the frequency domain requires complex sample pairs spaced by A/ £ l/(5r), where St is the range-delay extent over which the target reflects incident waves. As stated in terms of an unambiguous range window, also called range ambiguity window, we write
=
T
=
2A?
(
7
2
8
)
The synthetic range window is effectively sampled by the t), = n samples per burst collected over bandwidth 0 = nA/. Targets that exceed the slant-range window defined by (7.27) for sampling of real profiles produced by chirp-pulse radars will be imaged over only that portion of the rangedelay extent of the target where samples were taken. Targets that exceed the range ambiguity window defined by (7.28) for stepped-frequency pulse sequences will produce images that are folded over within the range window.
Figure 7.9 Sampled range (delay) profile.
356
The slant-range integration length for chirp-pulse waveforms of pulse duration T, is cT,f2, as for SAR. The sampling window is unaffected by the integration length. When stepped-frequency sequences are transmitted, the attempt is typically to sample at the delay position corresponding to the center of the narrowband / and Q video response. The slant-range integration length, when the receiver bandwidth before sampling is perfectly matched to the transmitted pulse duration T,, is approximately cT,, which is the approximate effective length associated with the duration 2T, of the triangular matched-niter output pulse. This integration length, for one sample per frequency step, needs to equal or exceed the target length in order to image the entire target. To approach uniform weighting over the target's range extent requires that integration length exceed target length by a factor of two or more. The integration length for a given transmitted pulse duration can be increased before sampling by reducing receiving system bandwidth to less than that for the matched-filter case. The integration length approaches cT, when samples of extended targets taken at multiple coarse-range sample positions are superimposed as described in Chapter 5. 7.3.5 Cross-Range Sampling Cross-range sampling refers to sampling along time history in each resolved range cell. Samples are separated in time by the radar PRI 7j for chirp-pulse waveforms and by nT for stepped-frequency waveforms. Analogously to SAR, a cross-range ambiguity window for ISAR can be defined as the largest cross-range target extent that can be unambiguously sampled for a given PRF, viewing-angle rotation rate, and wavelength. For ISAR, however, the target is usually immersed in the illuminating antenna beamwidth so that the ambiguity window refers to the target size in the cross range, rather than to the instantaneous illuminated cross-range extent on the earth's surface, as is the case for SAR. From (7.20), which expresses the Doppler frequency produced by a single scatterer on a uniformly rotating target, we can show that the Doppler frequency bandwidth produced by scatterers extending over a cross-range window w, is liowJK. The PRF required for unambiguous sampling of reflectivity data produced from a chirp-pulse radar when viewing a target of cross-range extent w„ therefore, is t
1
2a>w,
(7.29)
assuming complex samples are collected, one sample in each range cell for each transmitted pulse. Synthetic processing of stepped-frequency bursts of n pulses per burst requires a PRF of 1
2n cow.
(7.30)
357
Later in this chapter, we will estimate PRF requirements for ISAR images of ships and aircraft. Regardless of waveform, the number of range profiles needed for unambiguous sampling of a target of cross-range extent w„ based on the requirement that NITtfo, is (7.31) The unambiguous cross-range length w = A/Ar is the maximum cross-range extent of a target that can be examined unambiguously with N stepped-frequency bursts of n pulses for synthetic processing or with N = n pulses for real processing, one profile for each transmitted chirp pulse. For a small integration angle o>T = ifr, the unambiguous crossrange window A/Ar, with Ar from (7.24) or from (7.31) is expressed as f
r
c
Wavelength A, when referring to chirp-pulse or stepped-frequency waveforms, is the wavelength at the center frequency. Narrow fractional bandwidth is assumed in both cases. 73.6 Square Resolution Square resolution with ISAR, as for SAR, is defined as equal resolution in the slant range and cross range. Required bandwidth caiTIk to obtain square resolution is obtained by solving for bandwidth A or nA/ for chirp-pulse or stepped-frequency waveforms, respectively, for which the cross-range resolution given by (7.24) equals the slant-range resolution given by (7.25) and (7.26). A summary of basic ISAR equations written in terms of chirp and stepped-frequency waveforms is given in Table 7.1. 7.4 SOURCES OF TARGET ASPECT ROTATION So far, we have considered only the viewing-angle rotation produced by target rotational motion. Target aspect change is also produced by the tangential translation of the target relative to the radar. Radial translation (motion along the radar LOS) produces no viewingangle change, but tangential translation (motion normal to the LOS), like target rotation, produces a viewing-angle rotation that results in a Doppler gradient associated with target scatterers distributed in the cross range. Figure 7.10 illustrates how a differential Doppler shift is produced between two scatterers of a radar target that has a tangential velocity component relative to the radar. The scale in this drawing has been exaggerated to clarify the relationship between radial velocities v«, and v»».
358
Table 7.1 Summary of Equations for ISAR Waveform Symbol
Chirp-Pulse
Stepped-Frequency
Slant-range resolution* Slant-range window
Ar, w.
Slam-range integration length Cross-range resolution*' Cross-range ambiguity window*
c/(2nt\f) c 2A/ cT,/2
Ar
c/(2A) c "•2A cT,/2 A/(2t» A/A A IhiT loiTi uT r,A
AM _ ,» 2o)7" 2n
