Introductory Topics in Electronics and Telecommunications: Noise v. 6 (Introductory topics in electronics & telecommunication)

•

I

r

r

I

F. R. Connor 1982

1979 Second edition 1982

may be retI"ieval . . "... or transmitted in any means, ..........'''' ......."" . . . . '..... 1I1ecJt1al1l1CalJ, p,n01:0CIDp~(lnJ~, nth.;coT''UI,nc:l'.;co without the l.PubHs:hers) Ltd. L .......... _

British

t.:altaU)2UID2 in Publication Data

F. R. Noise.-2nd ed. 1. Electronic circuits-Noise I. Title 621.3815'3 TK7867.5 ISBN 0-7131-3459-3

A"''''V& , .....,'"',

Scotland

Preface

In this new edition, certain parts of the text have been extensively revised. A new section on random variables is introduced in Chapter 2 and some basic ideas concerning matched filtering, decision theory, and estimation theory are presented in Chapter 3. A further treatment of circuit noise is made in Chapter 4 and a new section on low-noise amplifiers is included in Chapter 5. In Chapter 6, a comparative study of the signal-to-noise performance of various systems has been extended to cover digital systems and satellite systems. As an alternative approach, the energy-to-noise density ratio and its effect on the bit error rate is also included. A further feature of the book is the extended use of appendices to cover such topics as narrowband noise, decision theory, estimation theory, and the probability of error. It is intended for the reader seeking a deeper understanding of the text and is supplemented by a large number of useful references for further reading. The book also includes several worked examples and a set of typical problems with answers. The aim of the book is the same as in the first edition, with the difference that Higher National Certificates and Higher National Diplomas are being superseded by Higher Certificates and Higher Diplomas of the Technician Education Council. In conclusion, the author wishes to express his gratitude to those of his readers who so kindly sent in various corrections for the earlier edition. 1982

FRC

Preface to the first edition This is an introductory book on the important topic of Noise. Electrical noise is of considerable importance in communication systems and the book presents basic ideas in a coherent manner. Moreover, to assist in the understanding of these basic ideas, worked examples from past examination papers are provided to illustrate clearly the application of fundamental theory. The book begins with a survey of the various types of electrical noise found in communication systems and this is followed by a description of some mathematical ideas concerning random variables; Circuit noise, noise factor, and noise temperature are considered in the following chapters, and the book ends with a comparative study of some important communication systems.

IV

1

1

1 2 4

1.1 1 1.3

and

2

6 6 7 9 10 12 17

3

4 4.1

C·onlenlS

VI

4.4

5

tranMstor noise FET nOIse measurement

5.1 63

6 6.1

B C D E F H I J

L

m

B C

]

F

Vlll

Symbols

Pc PD PF PT R Req

R(r) R.Jt) Rxy{-r;) S

SIN StiNt SolN 0 S(w)

T

average carrier power detection probability false alarm probability transmitted power bit rate resistance equivalent thermal noise resistance autocorrelation function autocorrelation function of variable x cross-correlation function of variables x and y average signal power signal-to-noise ratio input signal-to-noise ratio output signal~to-noise ratio power spectral density absolute temperature periodic time antenna noise temperature effective noise temperature receiver noise temperature system temperature energy highest modulating frequency noise power of standard source , noise power of antenna

y

Y-factor

rx

transistor forward current gain transistor d.c. forward current gain Dirac delta function wavelength frequency of events correlation coefficient standard deviation time interval angular frequency

cX o

£5 (t)

A v p (J

L OJ

= ------------

I

III

I

2

1.1

3

o~O'-1----------~~-----------1~O------------1~O~O f(GHz)---

1.2

4

5

il

statistics. 6

if rnB is

occurrences

=

n- co

n

Bwe

statistics

p

n events

+

p

B mAB n

-------n

p

\..'IIJUJl.iIIJl.IJlk

the

two

P

v\..I1I.IUI.I.\..II.&..J

P

IA)P

or are

p A

Three coins are tossed all heads or all tails?

'nr1jQ ...... jQni'"1jQj"'lt

=1'

=p B is

random. What is the

Drclbaloili1tv of all tails P sum of P

events

Pr 0 and they are therefore statistically independent. From Fig. 3.7(a) it will be observed that the average power, which is given by

Pay =

+- f+ "" .... 1f

- oc

S(w)dw

Correlation techniques

31

R(.)

I

No/2 S (Ul) .....

____

L _ _ _ __

0

............... -Lv

+Ul(b)

(a)

~vl

R(.)

1 ("I o

~B

B

f-

T-

(d)

(c)

Fig. 3.7

becomes infinite and cannot be physically realised in any practical circuits. As most communication circuits are band-limited, it is more practical to consider the results of passing white noise through a filter with some defined bandwidth. The output noise is then called band-limited white noise or coloured noise. 3.7

Band-limited white noise

Ifwhite noise is passed through a low-pass ideal filter with a bandwidth ± B Hz, the output noise can be obtained by means of the transfer function H(w) of the filter. Hence, we have SI)(W) Sj(w)

= IH(w)12

where S, (w) and So(w) are the input and output power spectral densities and IH(w)1 = 1, with 1 r+ cc No No f+B 1 "+cc. Pav = 2 So(W) dw =? I -2 dw = df = NoB watts n -00 _n,; _ -oo 2_B

J

and is illustrated in Fig. 3.7(c). The autocorrelation function R (r) of the filtered white noise is R(r)

1 =;;-.t..1t

1

=-

2n

f+

00

-co

So(w)eiW!dw

= -1

f+

00

2n:",_oo

Sj(w) IH(w)1 2 e JWt dw

J"+ ~eJW!dw=~! N. N i+B N [eJ21r

•

L,To

l"s

•

Tp,G

I

I

I

Tr

I

Fig. 5.14

Example 5.4 An antenna with a noise temperature of 57 K is connected by means of a cable to a preamplifier and receiver. The cable loss is 1 dB and the preamplifier has a 20 dB power gain and a noise temperature of90 K. If the effective temperature of the receiver is 290 K, what is the system noise temperature? Assume an ambient temperature of 290 K. Solution We have

Ts

=

Ta + (L -l)To + LTp + LTr/G

with

L = 1 dB = 1·26

and

G = 20dB

=

100

Hence

Ts = 57 + (1'26 -1'0)290 + (1'26 x 290) + (1'26 x 290)/100 or

Ts

=

57 + 75·4+ 113-4+ 3·65

=

249·5K

Noise factor L. nOIse

the

is t43rnn,o.ro:Jtl1'·43

of the receiver

28

>

1

> d> c w

C)

>

C)

d> c w

- - - ' ' - - - - - - ' - - - - W, Th ree-Ievel maser

maser

5.15

71

.5.16

72

I

t _______ Output

5.17

measurement Input

Input Idler circuit

5.18

74

Noise

performance and is fabricated by the planar process. The thin conducting GaAs epitaxial layer is deposited on a semi-insulating GaAs substrate and the Schottky barrier gate is formed by the deposition of a metal on the epitaxial layer. The construction of a typical device is shown in Fig. 5.19. Schottky barrier

s

\

\

G

D

\ s

D

Semi-insulating GaAs

-1~ Gate length

Fig. 5.19

At present, the most important application of GaAs FETs is in low-noise amplifiers. GaAs FETs designed for low-noise applications have Schottky barrier gates with typical gate length of 0·5 ,urn. Noise figures range from 1 dB at 4GHz to about 2dB at 12GHz with a power gain of about 10-12 dB. A microwave low-noise amplifier may employ one or two GaAs FET stages followed by a bipolar amplifier. Alternatively, the first stage can be an ultra lownoise parametric amplifier followed by a low-noise FET stage. An overall noise figure of 3 dB can' be achieved over a 1 IvlHz band in the freqi,lency range 7·25-7·75 GIlz. FET amplifiers have other advantages, such as long shelf and working lives, small size, and low power dissipation. They are finding increasing use in the front end of various kinds of microwave receivers for both radar and satellite communications. In earth station applications, they are a very reliable and costeffective way of implementing a low-cost satellite ground station with an excellent figure of merit GIT.

Example 5.5 Sketch and discuss the variation of the sky noise temperature as a function of frequency. A receiver has a system noise factor of 10 dB and it is proposed to improve its sensitivity by adding a preamplifier of 3 dB noise factor and 10 dB power gain.

Noise measurement

75

Solution

minimum when horizon. behaves

Problem of aerial without

290

2900K ----=841K

1rn.r"\rr'l.''I7o.r'\'''!ol'>'nt

factor

2900

3-45

2668 K with pre:amlplrDer

+----

7.,

lrn'l"'Iorr'l.'l.l01r'\"'!ot:>lnt

factor

K 6·4dB

the latter

dB when the

76

Noise what

Assume that the bandwidth of 290 K. defined with reference

linear

IS

the main same and noise

K

1

5.20

Lis

with

When referred

Hence

we have

the mput

290

=1

280

or Also or

40dB

the 9

F=

and or

or

Noise measurement

5.8

77

Noise temperature measurement 31

The effective noise temperature Ta of an antenna is usually measured using the 'Y-factor method'. The principle of the method is to compare the noise power received by the antenna to the noise power generated by a standard noise source and, from the ratio cif these noise powers, Ta can be determined. The circuit arrangement used is shown in Fig. 5.21.

Low-noise preamplifier

rn~~wer

I

! meter , ! i

Matched load

Fig. 5.21

With the standard liquid-cooled load connected to the input, the output noise power Ns is noted on the meter. The antenna is now connected to the input and the output noise power Na is noted on the meter. If the ratio NjNa is denoted by Yand the receiver noise temperature is TR we have k(T, + T R)B Y - -Ns - -'---- Na - k(Ta + TR)B

where B is the relevant system bandwidth. Hence

Y= Ts+TR Ta+TR or

YTa + YTR = 1~ + T R

T = ~+TR(l- Y)

and

a

Y

To measure TR , the output noise power N h of a 'hot load', i.e. a load at room temperature, is compared with the noise power output N s of the standard cooled load. If the ratio of these noise powers is Y' then we have Y' with or

N = k(Th+ T)B R Ns k(Ts + TR)B

=_h

Y'Ts + Y'T R = Th + TR Y'T R - TR = Th - 'f'T,

78

Noise

Th - Y'Ts

T ------R - (Y'-1)

Hence

which can be determined since Y', T h , and Ts are known by direct measurement. The value ofTa can then be determined from the previous expression using the known values of 7'., TR , and Y. 5.9

Excess noise ratio (ENR)32

Microwave noise generators are usually of the solid-state or gas discharge types which use room temperature as the standard reference by simply de-energising the noise source and assuming room temperature as the standard value To. In this case, the effective receiver temperature T R is related to the noise factor F by T = R

with or

T - Y'T h

(Y' _ 1) F _

° = (F -1)1'°

_ (Th/1'o) - Y' 1- (Y'-l)

F

=

_(T-"h-,---IT--,o,-,-)-,---_1 (Y' -1)

where Th is the hot-load temperature of the energised source. The quantity in the nUlneratQr is a measure of the power output of the :Q.Qise source and is called th~_~xc,=-~s noise rqUo.(ENR) which is giveI1l:>i" -and

ENR = 10 log (ThiTo -1)dB F = ENR - 10 10g(Y' - 1) dB

Solid-state noise sources are semiconductor p--n diodes operating in the avalanche region. The randomness of the avalanche multiplication process produces fluctuations in the avalanche current which generate random noise over a wide frequency range. Typically, the diodes operate at voltages of around 20 to 30 volts and are driven from a constant current source from between 5 to 20 rnA. The choice between solid-state and gas discharge noise sources is based on frequency coverage. For laboratory work below 18 GHz requiring operation at many frequencies, solid-state units otTer economic advantages, in addition to small size, low mass, and low power consumption. Usually, solid-state noise sources are calibrated at several frequencies and, typically, may have an ENR of around 15 dB in the frequency range 1-12·4 GHz or, in certain cases, an ENR as high as 40 dB. They can be used to determine the noise figures of amplifiers, mixers, or receivers and also to check the performance of radar and communication systems.

34

80

is

~

'0

components 1------

>

o 6.1

f--

Systems

81

If S (f) is the noise power spectral density, then the power in a bandwidth (~f associated with the single 'impulse' is S (f) (jj = No (5j where No is the power spectral density in watts/Hz. If the corresponding single noise component is vn(r) we have V,,(t) = Vn sin 2nUIF+ jn)t where };F is the IF frequency and fn is the frequency of the particular noise frequency component with a peak value Vn and No (5} = V~/2. When of --+ 0, all the noise components cover the IF bandwidth 2B continuously and the total noise power N is N, =

S

and

J::

I

No 6} = 2NoB

m2 V 2

1/\';, =

m 2 V2

J

~j

2N oB =

m2 P

8No~ = 4No~

where Pc is the average carrier power. To obtain the output noise power oflhe detector, each noise component v,,(t) within a bandwidth ~rwill beat with the carrier and the resultant voltage is given approximately by Vn(t) = Vc [1 +(Vn/Vc)coswntJsinwct

if Vc no(t)

}>

Vn . When this is applied to the detector, the output noise voltage

= Vncoswnt. Hence, the average noise power (jNo in a bandwidth of is 6N o

=

V;/2

= No~f

(as obtained earlier)

and for the whole IF bandwidth this becomes

f

+B

No

=

No

df =

2N () B

-B

Hence

S m2 V 2 I m l V2 m2 P -;'- = _ _c / 2N 0 B = - - " = - - " No 2 4N oB 2NoB

I

which is equal to twice S,/N, or (So/NolAM = 2(SjNJAM

and amounts to a 3 dB improvement at the detector. The 3 dB improvement is due to arithmetic addition of power in the sidebands and quadratic addition of the independent noise in each sideband.

DSBSC system The only difference between an AM system and a DSBSC system is the carrier power which is present in the former. Hence, for equal average power in the

X2

N01.\e

ratlO5J

the

the two

must

same,

and

amounts to a 6

on . . . . . . . . . . .,. . . . . '. . . """ ...., it can

opf~ratlon

an

of the

en~/eU)ne

amlplltu(le-lmodu.latc~d

detectors for use in the wave under conditions of very low

C'uY"U''''hrn1nnllC'

83

or

v(t)

carner

The

and with the

and the

blocked

84

Noise

[ ...... "' ..... .,. ....~ •• ,...,.. """I .. '

and

from

to

due to

If since

is small. Hence cos does not contain the modulation ,...,,,,"'I"1""'A'11C" from 0 to

The

'\1 however, PCM appears to offer the best advantage. As a comparison, the ideal system shown is still about 8 to 10 dB better than the FM or PCM system.

LpCM AM

-----;~-i----:;7~1 O.dlglt code)

30dB

..Y.

per"l (5·dlglt code)

o

(S;iN,) dB

Fig. 6.5

The most familiar pulse systems used are pulse amplitude modulatJOfl (PAM). pulse position modulation (PPM), and pulse code modulation (peM). " Sec F R Connor. ModliiarlOli. Edward Arnold (19R2)

88

Noise

PAM system Pulse amplitude modulation is normally employed in the early stages of PPM or PCM systems since it is easy to multiplex PAM pulses. It is generally not used as the final system, however, since the SIN ratio obtainable is not as good as those of the other pulse systems. It can be shown that PAM gives results very similar to those obtained previously for AM. Since noise directly affects the amplitude of the pulses, it appears as direct amplitude modulation in the system, as with the modulating signal. Hence, we have

PPM system In pulse position modulation, the modulating voltage causes a time displacement of the pulse. To evaluate the SIN ratio, assume that the maximum time displacement due to the modulation is to and so the peak signal volts is equal to K Co where K is a constant of proportionality. Hence, the average signal power at the output of the detector is

So = K2t5/2 The effect of noise in the signal is to alter the time displacement which leads to an error e, as shown in Fig. 6.6. The rms noise volts causing the error e produces an rms time displacement lit such that e Vc

=

I1t tr

where Vc is the peak pulse volts and which carries the modulation.

tr

is the rise-time of the leading pulse edge

v

..../ "

/With noise

t-

Fig. 6.6

The output rms noise volts is thus K I1t and the average output noise power at the detector is

Hence .....o r•• ' .. 't'"orl

IS

or

Quantlsed Analogue signal

E~============~

Sampling

Quantised signal

6.7

to

mean

90

Noise

The error c: can take on all possible values between -/1 V/2 and + /1 V/2 and may be considered as due to added noise in the signal. Hence, the mean-square value of the error gives the mean-square value of 'quantisation noise'. To calculate it, assume that over a long period of time all levels have an equal probability of occurrence and so the occurrence of any level is the same. Hence, we obtain

N

or

L1 V 2

o

= -12-

watts

(for a 1 Q load)

To calculate the signal power for q levels spaced L1 V volts apart we have

v=

(q -1).6. V

volts

Assuming further that bJpolar pulses are used (since less power is consumed), the pulse heIghts are ± 6 r /2. ± 311 Ii /2 ..... ± (q - 1)11 V/2. For equal probabIllty of occurrence of all levels 111 a long message. we obtain the average Signal power as

or

51 -"

61

2

0

= - 2q -- [! - +

_,

j -

+

_0

+. .. +

y

(I} -

1) 2 J

This may be written as

NOlA

from whlc]) we obtaIn

+ 2- + ... + (If -

0 '

1-

_,

.

1~+2~+.,.+

,q(q-l)(2(/-1) 1)- = ...._ 6

(l -q _2')" ..... . \ 2 /

Hence

L1 j So = -'1")

2

,61

'2

q(ij - l)(q - 2)

4x6

[1/((/ - 1)(2ij - 1) - (j(q - l)(Cf - 2)J

Lit

=

._-

12

(lj-l)I'{2(j- IJ-Uf-2)]

'..

,

Systems

91

AV 2

=U(q-l)(q+l) AV 2 =_(q2_1)

12 or

AV2 So ~ 12 q2

(for q

~

1)

Hence, the signal-to-noise ratio due to quantisation noise becomes (S o

AV2

/N

)PCM ~ _ _ (q2 0

12

-

/AV 2

-1) - ! 12

~(q2_1)

(for q

or

~

1)

The result depends on the square of the number oflevels used and so a large number of levels is required for a large SIN ratio. As an example, if q = 128, (So/No) ~ (128)2 ~ 42dB. This requires theuseofa 7-digit code since 27 = 128 levels.

According to the Hartley-Shannon law of information, the communication capacity C of a system with a bandwidth B and a signal-to-noise ratio SIN is given by C = B log] (1 + SIN) bits/s This rate of information transmission may be regarded as the ideal if it is assumed that the error rate is less than 1 in 10 5 bits/so Comparing this with a binary PCM system, we observe that, for a sampling frequency of 2 HI where HI is the highest modulating frequency and q quantised levels are used, the amount of information H transmitted is given by H

=

log2 q bits

For a sampling frequency of 2 W, at least one pulse is sent in each sampling period and so the total number of pulses sent per second is n = 2 W. Hence

C = H' = nH = 2Wlog 2 qbits/s or

C

=

B 10g2 q2 bits/s

since B = 2 HI is the system bandwidth. Previously it was shown that

AV2 So = 12 (q2 -1)

C

B

C=B

ratio

was obtained """..."'........."......

ratio of where q is the number of . . . ",....... f-'C"oA For

nB or

n

group or

1.. ,

Ideal

5 PCM

40

PCM

N

-

:::c 23

:3

8 peM

Error

o 6.8

10·

A

rot

=0 ~

two

p

';:).lJ:.".l.lU.l';:)

and

one

well

the

bit ratio

1,

is the same.

we

-E/No

Noise Table 6.1

1

'2

SER 10- 3

6.9

Systems

97

budget, the critical parameter to be considered is the effective isotropic radiated power (EIRP) at the satellite which is important in achieving maximum power output from its transmitter. It is given by the expression EIRP

PTG T

=----------------

LFLS or where

EIRP = PT+GT-LF-L s dB P T = transmitter power LF = feeder and diplexer loss Ls = free space loss G T = transmitting antenna gain

Given below are typical values of the relevant parameters at an operating frequency of 6 GHz for a geostationary satellite orbiting at a distance of about 36000 km above the earth. The losses include the large space loss (201 dB) and the miscellaneous losses due to transmitter ageing (1 dB), antenna pointing error (2 dB), and rain attenuation margin (2 dB). Transmitter power Transmitting antenna gain EIRP Free space loss (4nd 21A2) Satellite antenna gain Miscellaneous losses Received carrier power Satellite noise power Carrier-to-noise power density ratio Carrier-to-noise power ratio

23dBW 60dB 83dBW -201 dB 28dB -5dB -95dBW -126dBW 97dBHz 31 dB

The critical downlink parameter is the figure of merit G IT since it directly determines the carrier-to-noise power density ratio (C / No), which is the ultimate criterion at the receiver's demodulator for analogue signals, or the bit energy-to-noise power density (E I No) for digitaJ signals. Hence, it is easily shown that C No or

C -N - 0

and

EIRP X GR kTsLsLm M

= EIRP+GR-kTs-Ls-Lm-M dBHz

98

Noise

C = carrier power at the demodulator No = noise power spectral density N = noise power EIRP = effective isotropic radiated power G R = receiving antenna gain k = Boltzmann's constant T, = system noise temperature Ls = free space loss Lm = miscellaneous losses 114 = margin for multiple carriers J-V = relevant bandwidth

where

Typical figures for a ground station receiving a signal at 4 GHz from a geostationary satellite are given below. It is convenient here to consider the parameter (C I No) for analogue signals since it is independent of the system bandwidth used. For digital signals, the ratio (EI No) is easily determined from C

Cr

1

ER

-=-x-=No No r No

E C -=--dB No NoR

or where

T

is the bit duration, E is the energy per bit, and R is the bit rate.

Satellite EIRP 18 dBW Free space loss ·····197 dB Receiving antenna gain 60dB System noise temperature 70 K GIT ratio 41-5dB/K Miscellaneous losses - 5 dB -124dBW Received carrier power Noise power density -210dBW/Hz Carrier-to-noise power density ratio 86 dB Hz 20dB Carrier-to-noise power ratio Bit rate (digital systems)* 8·5 x 10 6 bits/s Energy-to-noise power density ratio 17 dB BER 1 x 10"4 Here, the system noise temperature includes the antenna noise temperature and that of the subsequent receiver chain, the losses include the large space loss (197 dB) and the miscellaneous losses due to atmospheric attenuation (2 dB), antenna aperture efficiency (1 dB), and mispointing and polarisation loss (2 dB). ,. 132 channels

communication

Qlagra.ms the arr'an,gerneIlt of a microwave satellite an active earth in geo-

.........'...., .. I.... ...., ......... ,"'...

par'amete:rs and

s

6.1 chain and . . a,....,.,.l',.",,,,, modulators drive The ",.n,n~rol'Ufj"'r arnpl1lher .. """" ......... &r, C''lIf-,o.lh1~a the trallsrrUSSlon back earth. received with a bandwidth up to MHz band at 4 GHz and so low-noise wideband are ""' ...... ,o.n1l· ....

station which are isolated IF of 70 MHz

eqlUl]:lm 1, (d) the effect on (a), (b), and (c) of adding a preamplifier with a noise temperature of 100 K and gain of 20 dB. The receiver system is at a (eE.l.) temperature of 290 K. 12 A signal s(t) is a triangular pulse of the form s(t) = Kt s(t)

13

=

0

O~t~T

at all other values of time

where K is a constant. Determine the output of a filter matched to this signal. If Gaussian white noise of zero mean value and noise spectral density No (positive frequencies only) is added to the signal, what is the maximum signal-to-noise ratio at the output of the matched filter? In a pulse radar system, the observed signal is received in the presence of Gaussian noise of zero mean and unit variance. Assuming the received signal is of 2 volt amplitude, determine for a Neyman-Pearson receiver

Problems

103

the probability of detection when the false alarm probability is set at 0·2. A 30 channel PCM system with uniform quantisation and a 7-bit binary code has an output bit rate of 1·5 Mbits!s. Determine (a) the maximum information band\vidth over which satisfactory operation is possible, (b) the output signal-to-quantising noise ratio for an input sinusoidal signal at a frequency of 3 kHz and maximum design amplitude. 15 Calculate the signal-to-noise ratio for a sinusoidal signal quantised into M levels given that the total mean-square quantising noise voltage (T2 = 0·083(I\V)2 where 11v is the step size. What assumption has been made about the quantisation? Hence, estimate the number of digits per character required in a PCM system carrying the above signal if the quality has to be satisfactory for the telephone system. Discuss whether speech processed in a similar manner has the same quality. Explain how the signal-to-noise ratio for speech can be improved by analogue or digital signal processing. (C.E.I.) 16 A coherent binary data system uses on-otT pulses varying in amplitude from 0 to V volts. The probability of a 0 or 1 in the presence of Gaussian noise is the same. For a peak signal power to average noise power ratio of 13 dB, calculate the probability of error. Also, for a probability of error of 10 - 5, determine the signal-to-noise amplitude threshold required. 17 An FSK communication channel transmits binary information at a bit rate of 100 kbits!s in the presence of Gaussian noise with a spectral density of 10- 19 W 1Hz. If the signal is transmitted with a peak voltage level of 1 volt, determine for a probability of error of 10 - 4 the path loss of the channel for bit by bit detection with (a) incoherent FSI(, (b) coherent FSK. 18 Explain why communications satellites fulfil an important role in worldwide communications. Mention, particularly, aspects of the role that cannot be readily fulfilled by alternative systems. A satellite in a geostationary orbit at 35800 km has a 4 GHz downlink transmitter which feeds 25 watts into an antenna with 20 dB gain. The ground station receiving system has a total noise figure of 1·5 dB. Calculate the antenna gain necessary at the ground station to maintain a 30 dB input signal-to-noise ratio over a 12 MHz signal band. (eE.L) 19 The parameters of a satellite-to-ship link are as follows 14

Satellite RF transmitter power per channel Satellite aerial gain relative to isotropic Free space path loss at 1·5 GHz Propagation margin at 5° elevation for 99 % of the time Ship aerial gain relative to isotropic RF input power to ship receiver

+PsdBW + 17 dB -189 dB +5dB +GrdB

-152dBW

20

4 5

8 9 11

McGraw-Hill Communication

STEINBERG, J

2

and Communication

Inter-

4 5

7 8

Modulation

9

1 12

JOHNSON,

cation

13 14

r"h't:ln1n.t:lIl

Generalised harmonic Korrelationstheorie 1934.

WIENER. N

KHINTCHINE, A

15 16

and RAJASEKARAN, John Thermal n"".1'" .... 1'" .............

flP1JllOQlUJns.

,,1" • B

... \J.J" ....." ...

18

.J

...nf"t:lf" • ...,'n

t:lIlt:lIt"t1"'1r"1i"u

in C01001uct:or:s, ...,"',,,,,.ro,. . ,

of electric

and

20 21

Electronic Advances

t'r4ryCE~ealn~7S

24 25

Electrical

26

Ultra low-noise

28

lortrnnlr

and

29 30

33 34

35 36

arnlpllhelrs in communication 1971. The

Sa(~eUl1[e

Communication t'rllcedeOljrzas Institute

ROWE, H E

of GaAs FET low-noise

POSNER, R

arrlpllne:rs~

31

1I"'\f"lI1.. ,nr""'rrl

A.1

s

(a)

s

(b)

A.2

s

not

110

Noise

theorem.

was

Error . . ,," ............ '1 ..........

To

Power

x we 1 T IS

x

Appendices Table A.1 x

erf x

x

erf x

0 005 010 0'15 020 025 030 035 040 045 050 055 060 065 070 075 080 085 090 095 1 00 1 05

0 00563 01124 01679 02227 02763 () 3286 03793 04283 04754 05205 05663 06038 o 642() 06778 07111 074;;.1 On06 07969 08208 08427 08624 08802 08961 09103 09229 09340 09437 () 9522 09596

1 50 1 55 1 60 1 65 1 70 1 75 1 80 1 85 1 90 1 95 200 205 210 2 15 220 22E, 2 30 2. 35 240 245 250 255 260 265 270 275 280 285 290 295 300

09661 09716 09763 09803 09837 09866 09890 09911 09927 09941 09953 09962 09970 09976 09981 () 9985 09988 o 9991 09993 09994 09995 09996 09997 09998 099986 099989 099992 099994 099995 099996 099997

110

1 15 1 20 1 25 1 30 1 35 1 40 1 45

xiI)

X{;)

,!

.V'\

-f~O

3T

-"2

T

2"

t~

fb)

(a)

Fig, A,3

, Hence, rearranging the order of integration then yields

r

+ T I L2. 1 +W [ X (t)dt = -7F(w)dw 7J2 ~rrT • - '"

f

-I

TIL-

-TIL

X (t)ej

oc;

The quantity in the centre is the average power of the periodic signal x(t) and so we obtain

P

.

dV

= Jlln -

J'

1

7-+0C;2n

+

08

1

F(w)

T

-00

12-dw = -1

:

+

00

I

S(w)dw

2nj_00

from the defmition of S(w). Hence, by inspection, we obtain S(w) = lim

1F(w) 12

l-+OC;

V oltage spectral density The voltage spectral density

Svtfl

7

is defined by

f::SJf)dj=V~ where v ~ is the mean-square noise voltage. In the case of thermal noise, the noise spectrum is constant over a finite bandwidth B. Hence

f

+B

-B

or

SvCf)

Sy(j)df=4k7BR

f

+B

-B

dI =

4kTBR

1

Hence or

we

or

e

14 ~"I'·I\'U/h~nil1

noise

f

--IJIoo-

f--..... (b)

A.4

+

115

n or

R

Hence

/

A.5

116

Noise

lDDlendlx

F: a

+

s

or

and

IH

N

Appendices

117

Furthermore, if E is the signal energy we have E =

f

+OO -

1

f+oo

2n

-

S2(t) dt = -

00

IS(wW dw

00

and substituting this expression into (S/N)max yields

a result which depends on the signal energy but is independent of its waveform. If the impulse response of the matched filter is h(t), the Fourier transform of an impulse YI' hypothesis H2 is chosen and, if y(t) < Y1' hypothesis HI is chosen. To show the relationship with Bayes' criterion, we obtain the derivatives of

d

or

x

(z

I

(k

or 1

we

or

In

2

where

=0

a

=0

1

1

we

{

=1

1

as is

as a minimum

o

........----- I (a)

-------t

(b)

A.9

or

m

or a

dW

or

N=

watts

dW=--e

t

126

Noise

where f is the frequency of oscillation and h is Planck's constant. For frequencies up to about 1013 Hz, L1 W :::::: kT which is the classical value. Appendix J: Shot noise The shot noise rms current Is in a diode is due to the random emission of electrons from the cathode. Each electron arriving at the anode carries a discrete electronic charge e which gives rise to a current pulse i(t) in the anode during the transit time T, as shown in Fig. A.l0(a). The actual shape of the current pulse is immaterial if the time-average interval chosen is such that T ~ T. Each pulse can be regarded as a Dirac delta function (5(t) and approximated by a short rectangular pulse, as illustrated in Fig. A.I0(b). Hence, we have

l+:

b{t) dt

=

e

l.e. the area of the rectangular pulse is such that e/T x

~ 0

~ r-

T

-1

T

= e.

t~

la)

Fig. A.10

If F (w)

IS

the Fourier transform of b(t) then

F (OJ) =

f+ -

and

'L

.

b (t) e - Jrot d t = e

00

sin OJ, /2 . OJT/2

IF (wW = e2 [~-i-~;;¥~T

where IF (OJW is the energy spectral density and is shown in Fig. A.10(c). From Fig. A.lO(c) we observe that if the transit time T is very small (about 10 - 9 s) then liT:::::: 10 9 Hz and the spectral density over a bandwidth L1f = B is fairly constant, especially at lower frequencies. Hence, the total energy W in a bandwidth B is given by

W= or

00 f+oo 12 d f = 2 J1'+_""IF(w) IF(wWdf 0

Appendices

127

If n electrons arrive at the anode in time T where T is sufficiently large, the average shot noise power in a 1 Q load becomes I; = n ~VIT = n2e 2 BIT

and substituting for the average anode current 1a = ne IT yields

I; = 2el"B ~

Is =y'2el a B

or Appendix K: Noise factors

Grounded-cathode circuit

Fig. A.11

Grounded-cathode amplifier The equivalent circuit is shown in Fig. A.lI where Rs is the source resistance, Rg is the grid-leak resistance, and Req is the equivalent shot noise resistance of the valve. If their rms noise voltages are 1.:" vg, and Veq respectively, the total noise voltage of the amplifier is Vo and for an ideal amplifier it is V1 where VI is due to the source resistance only. Hence

v;

mean-square noise voltage of amplifier F=·································· ........................······........ _ ...................................................... = mean-square noise voltage due to source vf

4kTB[Req + RsRg/{Rs + Rg)] [vsR~/(Rs + Rg)] 2 Since

v; = 4kTBR

s,

we obtain

F =

or

[Req (Rs + Rg) + RsRgJ (R, + Rg) RsR;

--'---=---"----"-'----::---"-=----'-~

+i

F= 1+RjRg

q [1+RjR g]2

s

128

Noise

Grounded-grid amplifier The eq uivalent circuit is shown in Fig. A.Il. Since the effective amplification factor of the grounded-grid amplifier is (/1 + 1), the rms noise voltages Vs and Vg appear as (/1 + 1)u, and (/1 + 1)v~ in the equivalent circuit. The grid noise voltage veq , however, is common to both grid and anode circuits and so its value is (p + l)vg - ug = pUg in the equivalent circuit. Hence, we obtain mean-square noise voltage of amplifier mean-square noise voltage due to source

F=--------------------------------

/124kTBReQ + (p + If 4kTBR sR g/(R, + Rg) (/1 + 1)24k7BR, [Rg/(R, + Rg) J2 [/1/{fl + 1) J2 RCq + R,Rg/(R, + Rg) ················i(tRJO~·:·+··R~")ji---

or

F = 1 + Rj Rg + (_/1_)2 Req [1 \/1 + 1 R,

+ R,,/R gJ2

Common-base transistor

Common base circuit

Common·emltter circuit

Fig. A.12

The equivalent T-circuit is shown in Fig. A.12 where R, is the source resistance and rc , rb, and rc are the emitter, base, and collector resistances respectively. The rms t101Se voltages v, and Vb are due to thermal noise in the source and base resistances respectively, whIle ve and 1\ are due to shot noise and partition noise Il1 the emitter and collector regions respectively. Hence, we have for a bandwidth B and absolute temperature T

v: = 4kTBR, v~ = 4kTBrb

vc2 =

[2,.2 ,e

= 2kTBr c

since f: = 2efoB and re = kT/elt if Ie is the d.c. emitter current Also, the

is

(Ial =

+

F = -------------source

or

+ source

or

F

+

Noise varIOUS

(1 F =

1+

]

re

+-+

Shot

(kHz)--

A.13

if

Vg

A.14

current

F=

----+ F

1

t A.15

For ance, gi

t

to zero, we

error IS ..........

2E

,L ........................ " " .....

]

two

the

was

If/=

or

-J it

] /2

I

VUJ.JlUVA"-'

noise power,

55

Band-limited

coloured communication conditional nrl"\t'VJIt'\111T'U correlation

91

113, 126

91

14 UU ....,LIV.U.

9, 17

108

Low-noise . . . . . . 'I"\h1ho,...

lUCl.l\.lU.lUlJU-llA'-'lUI\...'VU

74

'-'.:>lJIIIClllVJLI.

41

Random

7

Ul.:>IL.llU'ULlVll.

15

10

112

White Wiener-Khintchine 110

1-1->,..,.,..._,... .........

13