Electrical Engineering

First Edition, 2007 ISBN 978 81 89940 49 2 © All rights reserved. Published by: Global Media 1819, Bhagirath Palace,...

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First Edition, 2007

ISBN 978 81 89940 49 2

© All rights reserved.

Published by: Global Media 1819, Bhagirath Palace, Chandni Chowk, Delhi-110 006 Email: [email protected]

Table of Contents 1. General Theories 2. Resistance, Capacitance, Inductances 3. Time Constants 4. Telecommunication 5. Electrical Engineering 6. Power Engineering 7. Photodiode 8. Photomultiplier 9. Digital Circuit 10. Boolean Algebra 11. Logic Analyzer 12. Logic Gate 13. Programmable Logic Device 14. Reconfigurable Computing 15. Analogue Electronics 16. Artificial Intelligence 17. Control System 18. Control Theory 19. Control Engineering 20. Programmable Logic Controller 21. Building Automation 22. HVAC Control System 23. Signal Processing 24. LTI System Theory 25. Fourier Transform 26. Signal (Electrical Engineering)

Notation The library uses the symbol font for some of the notation and formulae. If the symbols for the letters ‘alpha beta delta’ do not appear here [α β δ] then the symbol font needs to be installed before all notation and formulae will be displayed correctly. E G I R P

voltage source conductance current resistance power

[volts, V] [siemens, S] [amps, A] [ohms, Ω] [watts]

V X Y Z

voltage drop reactance admittance impedance

[volts, V] [ohms, Ω] [siemens, S] [ohms, Ω]

Ohm’s Law When an applied voltage E causes a current I to flow through an impedance Z, the value of the impedance Z is equal to the voltage E divided by the current I. Impedance = Voltage / Current

Z=E/I

Similarly, when a voltage E is applied across an impedance Z, the resulting current I through the impedance is equal to the voltage E divided by the impedance Z. Current = Voltage / Impedance

I=E/Z

Similarly, when a current I is passed through an impedance Z, the resulting voltage drop V across the impedance is equal to the current I multiplied by the impedance Z. Voltage = Current * Impedance

V = IZ

Alternatively, using admittance Y which is the reciprocal of impedance Z: Voltage = Current / Admittance

V=I/Y

Kirchhoff’s Laws Kirchhoff’s Current Law At any instant the sum of all the currents flowing into any circuit node is equal to the sum of all the currents flowing out of that node: ΣIin = ΣIout Similarly, at any instant the algebraic sum of all the currents at any circuit node is zero: ΣI = 0 Kirchhoff’s Voltage Law At any instant the sum of all the voltage sources in any closed circuit is equal to the sum of all the voltage drops in that circuit: ΣE = ΣIZ Similarly, at any instant the algebraic sum of all the voltages around any closed circuit is zero: ΣE - ΣIZ = 0

Thévenin’s Theorem Any linear voltage network which may be viewed from two terminals can be replaced by a voltage-source equivalent circuit comprising a single voltage source E and a single series impedance Z. The voltage E is the open-circuit voltage between the two terminals and the impedance Z is the impedance of the network viewed from the terminals with all voltage sources replaced by their internal impedances.

Norton’s Theorem Any linear current network which may be viewed from two terminals can be replaced by a current-source equivalent circuit comprising a single current source I and a single shunt admittance Y. The current I is the short-circuit current between the two terminals and the admittance Y is the admittance of the network viewed from the terminals with all current sources replaced by their internal admittances.

Thévenin and Norton Equivalence The open circuit, short circuit and load conditions of the Thévenin model are: Voc = E Isc = E / Z Vload = E - IloadZ Iload = E / (Z + Zload) The open circuit, short circuit and load conditions of the Norton model are: Voc = I / Y Isc = I Vload = I / (Y + Yload) Iload = I - VloadY

Thévenin model from Norton model Voltage = Current / Admittance Impedance = 1 / Admittance

E=I/Y Z = Y -1

Norton model from Thévenin model Current = Voltage / Impedance Admittance = 1 / Impedance

I=E/Z Y = Z -1

When performing network reduction for a Thévenin or Norton model, note that: - nodes with zero voltage difference may be short-circuited with no effect on the network current distribution, - branches carrying zero current may be open-circuited with no effect on the network voltage distribution.

Superposition Theorem In a linear network with multiple voltage sources, the current in any branch is the sum of the currents which would flow in that branch due to each voltage source acting alone with all other voltage sources replaced by their internal impedances.

Reciprocity Theorem If a voltage source E acting in one branch of a network causes a current I to flow in another branch of the network, then the same voltage source E acting in the second branch would cause an identical current I to flow in the first branch.

Compensation Theorem If the impedance Z of a branch in a network in which a current I flows is changed by a finite amount δZ, then the change in the currents in all other branches of the network may be calculated by inserting a voltage source of -IδZ into that branch with all other voltage sources replaced by their internal impedances.

Millman’s Theorem (Parallel Generator Theorem) If any number of admittances Y1, Y2, Y3, ... meet at a common point P, and the voltages from another point N to the free ends of these admittances are E1, E2, E3, ... then the voltage between points P and N is: VPN = (E1Y1 + E2Y2 + E3Y3 + ...) / (Y1 + Y2 + Y3 + ...) VPN = ΣEY / ΣY The short-circuit currents available between points P and N due to each of the voltages E1, E2, E3, ... acting through the respective admitances Y1, Y2, Y3, ... are E1Y1, E2Y2, E3Y3, ... so the voltage between points P and N may be expressed as: VPN = ΣIsc / ΣY

Joule’s Law When a current I is passed through a resistance R, the resulting power P dissipated in the resistance is equal to the square of the current I multiplied by the resistance R: P = I2 R By substitution using Ohm’s Law for the corresponding voltage drop V (= IR) across the resistance: P = V2 / R = VI = I2R

Maximum Power Transfer Theorem When the impedance of a load connected to a power source is varied from open-circuit to short-circuit, the power absorbed by the load has a maximum value at a load impedance which is dependent on the impedance of the power source. Note that power is zero for an open-circuit (zero current) and for a short-circuit (zero voltage). Voltage Source When a load resistance RT is connected to a voltage source ES with series resistance RS, maximum power transfer to the load occurs when RT is equal to RS. Under maximum power transfer conditions, the load resistance RT, load voltage VT, load current IT and load power PT are: RT = RS VT = ES / 2 IT = VT / RT = ES / 2RS PT = VT2 / RT = ES2 / 4RS Current Source When a load conductance GT is connected to a current source IS with shunt conductance GS, maximum power transfer to the load occurs when GT is equal to GS. Under maximum power transfer conditions, the load conductance GT, load current IT, load voltage VT and load power PT are: GT = GS IT = IS / 2 VT = IT / GT = IS / 2GS PT = IT2 / GT = IS2 / 4GS Complex Impedances When a load impedance ZT (comprising variable resistance RT and variable reactance XT) is connected to an alternating voltage source ES with series impedance ZS (comprising resistance RS and reactance XS), maximum power transfer to the load occurs when ZT is equal to ZS* (the complex conjugate of ZS) such that RT and RS are equal and XT and XS are equal in magnitude but of opposite sign (one inductive and the other capacitive). When a load impedance ZT (comprising variable resistance RT and constant reactance XT) is connected to an alternating voltage source ES with series impedance ZS (comprising resistance RS and reactance XS), maximum power transfer to the load occurs when RT is equal to the magnitude of the impedance comprising ZS in series with XT: RT = |ZS + XT| = (RS2 + (XS + XT)2)½ Note that if XT is zero, maximum power transfer occurs when RT is equal to the magnitude of ZS: RT = |ZS| = (RS2 + XS2)½ When a load impedance ZT with variable magnitude and constant phase angle (constant power factor) is connected to an alternating voltage source ES with series impedance ZS, maximum power transfer to the load occurs when the magnitude of ZT is equal to the

magnitude of ZS: (RT2 + XT2)½ = |ZT| = |ZS| = (RS2 + XS2)½

Kennelly’s Star-Delta Transformation A star network of three impedances ZAN, ZBN and ZCN connected together at common node N can be transformed into a delta network of three impedances ZAB, ZBC and ZCA by the following equations: ZAB = ZAN + ZBN + (ZANZBN / ZCN) = (ZANZBN + ZBNZCN + ZCNZAN) / ZCN ZBC = ZBN + ZCN + (ZBNZCN / ZAN) = (ZANZBN + ZBNZCN + ZCNZAN) / ZAN ZCA = ZCN + ZAN + (ZCNZAN / ZBN) = (ZANZBN + ZBNZCN + ZCNZAN) / ZBN Similarly, using admittances: YAB = YANYBN / (YAN + YBN + YCN) YBC = YBNYCN / (YAN + YBN + YCN) YCA = YCNYAN / (YAN + YBN + YCN) In general terms: Zdelta = (sum of Zstar pair products) / (opposite Zstar) Ydelta = (adjacent Ystar pair product) / (sum of Ystar)

Kennelly’s Delta-Star Transformation A delta network of three impedances ZAB, ZBC and ZCA can be transformed into a star network of three impedances ZAN, ZBN and ZCN connected together at common node N by the following equations: ZAN = ZCAZAB / (ZAB + ZBC + ZCA) ZBN = ZABZBC / (ZAB + ZBC + ZCA) ZCN = ZBCZCA / (ZAB + ZBC + ZCA) Similarly, using admittances: YAN = YCA + YAB + (YCAYAB / YBC) = (YABYBC + YBCYCA + YCAYAB) / YBC YBN = YAB + YBC + (YABYBC / YCA) = (YABYBC + YBCYCA + YCAYAB) / YCA YCN = YBC + YCA + (YBCYCA / YAB) = (YABYBC + YBCYCA + YCAYAB) / YAB In general terms: Zstar = (adjacent Zdelta pair product) / (sum of Zdelta) Ystar = (sum of Ydelta pair products) / (opposite Ydelta) Electrical Circuit Formulae

Notation The library uses the symbol font for some of the notation and formulae. If the symbols for the letters ‘alpha beta delta’ do not appear here [α β δ] then the symbol font needs to be installed before all notation and formulae will be displayed correctly. C E

capacitance voltage source

[farads, F] [volts, V]

Q q

charge instantaneous Q

[coulombs, C] [coulombs, C]

e G I i k L M N P

instantaneous E conductance current instantaneous I coefficient inductance mutual inductance number of turns power

[volts, V] [siemens, S] [amps, A] [amps, A] [number] [henrys, H] [henrys, H] [number] [watts, W]

R T t V v W Φ Ψ ψ

resistance time constant instantaneous time voltage drop instantaneous V energy magnetic flux magnetic linkage instantaneous Ψ

[ohms, Ω] [seconds, s] [seconds, s] [volts, V] [volts, V] [joules, J] [webers, Wb] [webers, Wb] [webers, Wb]

Resistance The resistance R of a circuit is equal to the applied direct voltage E divided by the resulting steady current I: R=E/I

Resistances in Series When resistances R1, R2, R3, ... are connected in series, the total resistance RS is: RS = R1 + R2 + R3 + ...

Voltage Division by Series Resistances When a total voltage ES is applied across series connected resistances R1 and R2, the current IS which flows through the series circuit is: IS = ES / RS = ES / (R1 + R2) The voltages V1 and V2 which appear across the respective resistances R1 and R2 are: V1 = ISR1 = ESR1 / RS = ESR1 / (R1 + R2) V2 = ISR2 = ESR2 / RS = ESR2 / (R1 + R2) In general terms, for resistances R1, R2, R3, ... connected in series: IS = ES / RS = ES / (R1 + R2 + R3 + ...) Vn = ISRn = ESRn / RS = ESRn / (R1 + R2 + R3 + ...) Note that the highest voltage drop appears across the highest resistance.

Resistances in Parallel When resistances R1, R2, R3, ... are connected in parallel, the total resistance RP is: 1 / RP = 1 / R1 + 1 / R2 + 1 / R3 + ... Alternatively, when conductances G1, G2, G3, ... are connected in parallel, the total conductance GP is: GP = G1 + G2 + G3 + ... where Gn = 1 / Rn For two resistances R1 and R2 connected in parallel, the total resistance RP is: RP = R1R2 / (R1 + R2) RP = product / sum The resistance R2 to be connected in parallel with resistance R1 to give a total resistance RP is: R2 = R1RP / (R1 - RP) R2 = product / difference

Current Division by Parallel Resistances When a total current IP is passed through parallel connected resistances R1 and R2, the voltage VP which appears across the parallel circuit is: VP = IPRP = IPR1R2 / (R1 + R2) The currents I1 and I2 which pass through the respective resistances R1 and R2 are: I1 = VP / R1 = IPRP / R1 = IPR2 / (R1 + R2) I2 = VP / R2 = IPRP / R2 = IPR1 / (R1 + R2) In general terms, for resistances R1, R2, R3, ... (with conductances G1, G2, G3, ...) connected in parallel: VP = IPRP = IP / GP = IP / (G1 + G2 + G3 + ...) In = VP / Rn = VPGn = IPGn / GP = IPGn / (G1 + G2 + G3 + ...) where Gn = 1 / Rn Note that the highest current passes through the highest conductance (with the lowest resistance).

Capacitance When a voltage is applied to a circuit containing capacitance, current flows to accumulate charge in the capacitance: Q = ∫idt = CV Alternatively, by differentiation with respect to time: dq/dt = i = C dv/dt Note that the rate of change of voltage has a polarity which opposes the flow of current. The capacitance C of a circuit is equal to the charge divided by the voltage: C = Q / V = ∫idt / V Alternatively, the capacitance C of a circuit is equal to the charging current divided by the rate of change of voltage: C = i / dv/dt = dq/dt / dv/dt = dq/dv

Capacitances in Series When capacitances C1, C2, C3, ... are connected in series, the total capacitance CS is: 1 / CS = 1 / C1 + 1 / C2 + 1 / C3 + ... For two capacitances C1 and C2 connected in series, the total capacitance CS is: CS = C1C2 / (C1 + C2) CS = product / sum

Voltage Division by Series Capacitances When a total voltage ES is applied to series connected capacitances C1 and C2, the charge QS which accumulates in the series circuit is: QS = ∫iSdt = ESCS = ESC1C2 / (C1 + C2) The voltages V1 and V2 which appear across the respective capacitances C1 and C2 are: V1 = ∫iSdt / C1 = ESCS / C1 = ESC2 / (C1 + C2) V2 = ∫iSdt / C2 = ESCS / C2 = ESC1 / (C1 + C2) In general terms, for capacitances C1, C2, C3, ... connected in series: QS = ∫iSdt = ESCS = ES / (1 / CS) = ES / (1 / C1 + 1 / C2 + 1 / C3 + ...) Vn = ∫iSdt / Cn = ESCS / Cn = ES / Cn(1 / CS) = ES / Cn(1 / C1 + 1 / C2 + 1 / C3 + ...) Note that the highest voltage appears across the lowest capacitance.

Capacitances in Parallel When capacitances C1, C2, C3, ... are connected in parallel, the total capacitance CP is: CP = C1 + C2 + C3 + ...

Charge Division by Parallel Capacitances When a voltage EP is applied to parallel connected capacitances C1 and C2, the charge QP which accumulates in the parallel circuit is: QP = ∫iPdt = EPCP = EP(C1 + C2) The charges Q1 and Q2 which accumulate in the respective capacitances C1 and C2 are: Q1 = ∫i1dt = EPC1 = QPC1 / CP = QPC1 / (C1 + C2) Q2 = ∫i2dt = EPC2 = QPC2 / CP = QPC2 / (C1 + C2) In general terms, for capacitances C1, C2, C3, ... connected in parallel: QP = ∫iPdt = EPCP = EP(C1 + C2 + C3 + ...) Qn = ∫indt = EPCn = QPCn / CP = QPCn / (C1 + C2 + C3 + ...) Note that the highest charge accumulates in the highest capacitance.

Inductance When the current changes in a circuit containing inductance, the magnetic linkage changes and induces a voltage in the inductance: dψ/dt = e = L di/dt Note that the induced voltage has a polarity which opposes the rate of change of current.

Alternatively, by integration with respect to time: Ψ = ∫edt = LI The inductance L of a circuit is equal to the induced voltage divided by the rate of change of current: L = e / di/dt = dψ/dt / di/dt = dψ/di Alternatively, the inductance L of a circuit is equal to the magnetic linkage divided by the current: L=Ψ/I Note that the magnetic linkage Ψ is equal to the product of the number of turns N and the magnetic flux Φ: Ψ = NΦ = LI

Mutual Inductance The mutual inductance M of two coupled inductances L1 and L2 is equal to the mutually induced voltage in one inductance divided by the rate of change of current in the other inductance: M = E2m / (di1/dt) M = E1m / (di2/dt) If the self induced voltages of the inductances L1 and L2 are respectively E1s and E2s for the same rates of change of the current that produced the mutually induced voltages E1m and E2m, then: M = (E2m / E1s)L1 M = (E1m / E2s)L2 Combining these two equations: M = (E1mE2m / E1sE2s)½ (L1L2)½ = kM(L1L2)½ where kM is the mutual coupling coefficient of the two inductances L1 and L2. If the coupling between the two inductances L1 and L2 is perfect, then the mutual inductance M is: M = (L1L2)½

Inductances in Series When uncoupled inductances L1, L2, L3, ... are connected in series, the total inductance LS is: LS = L1 + L2 + L3 + ... When two coupled inductances L1 and L2 with mutual inductance M are connected in series, the total inductance LS is: LS = L1 + L2 ± 2M The plus or minus sign indicates that the coupling is either additive or subtractive, depending on the connection polarity.

Inductances in Parallel When uncoupled inductances L1, L2, L3, ... are connected in parallel, the total inductance LP is: 1 / LP = 1 / L1 + 1 / L2 + 1 / L3 + ...

Time Constants Capacitance and resistance The time constant of a capacitance C and a resistance R is equal to CR, and represents the time to change the voltage on the capacitance from zero to E at a constant charging current E / R (which produces a rate of change of voltage E / CR across the capacitance). Similarly, the time constant CR represents the time to change the charge on the capacitance from zero to CE at a constant charging current E / R (which produces a rate of change of voltage E / CR across the capacitance). If a voltage E is applied to a series circuit comprising a discharged capacitance C and a resistance R, then after time t the current i, the voltage vR across the resistance, the voltage vC across the capacitance and the charge qC on the capacitance are: i = (E / R)e - t / CR vR = iR = Ee - t / CR vC = E - vR = E(1 - e - t / CR) qC = CvC = CE(1 - e - t / CR) If a capacitance C charged to voltage V is discharged through a resistance R, then after time t the current i, the voltage vR across the resistance, the voltage vC across the capacitance and the charge qC on the capacitance are: i = (V / R)e - t / CR vR = iR = Ve - t / CR vC = vR = Ve - t / CR qC = CvC = CVe - t / CR Inductance and resistance The time constant of an inductance L and a resistance R is equal to L / R, and represents the time to change the current in the inductance from zero to E / R at a constant rate of change of current E / L (which produces an induced voltage E across the inductance). If a voltage E is applied to a series circuit comprising an inductance L and a resistance R, then after time t the current i, the voltage vR across the resistance, the voltage vL across the inductance and the magnetic linkage ψL in the inductance are: i = (E / R)(1 - e - tR / L) vR = iR = E(1 - e - tR / L) vL = E - vR = Ee - tR / L ψL = Li = (LE / R)(1 - e - tR / L) If an inductance L carrying a current I is discharged through a resistance R, then after time t the current i, the voltage vR across the resistance, the voltage vL across the inductance and the magnetic linkage ψL in the inductance are: i = Ie - tR / L vR = iR = IRe - tR / L vL = vR = IRe - tR / L ψL = Li = LIe - tR / L Rise Time and Fall Time The rise time (or fall time) of a change is defined as the transition time between the 10%

and 90% levels of the total change, so for an exponential rise (or fall) of time constant T, the rise time (or fall time) t10-90 is: t10-90 = (ln0.9 - ln0.1)T ≈ 2.2T The half time of a change is defined as the transition time between the initial and 50% levels of the total change, so for an exponential change of time constant T, the half time t50 is : t50 = (ln1.0 - ln0.5)T ≈ 0.69T Note that for an exponential change of time constant T: - over time interval T, a rise changes by a factor 1 - e -1 (≈ 0.63) of the remaining change, - over time interval T, a fall changes by a factor e -1 (≈ 0.37) of the remaining change, - after time interval 3T, less than 5% of the total change remains, - after time interval 5T, less than 1% of the total change remains.

Telecommunication

Copy of the original phone of Graham Bell at the Musée des Arts et Métiers in Paris Telecommunication is the transmission of signals over a distance for the purpose of communication. In modern times, this process almost always involves the sending of electromagnetic waves by electronic transmitters but in earlier years it may have involved the use of smoke signals, drums or semaphore. Today, telecommunication is widespread and devices that assist the process such as the television, radio and telephone are common in many parts of the world. There is also a vast array of networks that connect these devices, including computer networks, public telephone networks, radio networks and television networks. Computer communication across the Internet, such as e-mail and instant messaging, is just one of many examples of telecommunication. Telecommunication systems are generally designed by telecommunication engineers. Major contributors to the field of telecommunications include Alexander Bell who invented the telephone (as we know it), John Logie Baird who invented the mechanical television and Guglielmo Marconi who first demonstrated transatlantic radio communication. In recent times, optical fibre has radically improved the bandwidth available for intercontential communication helping to facilitate a faster and richer Internet experience and digital television has eliminated effects such as snowy pictures and ghosting. Telecommunication remains an important part of the world economy and the telecommunication industry’s revenue has been placed at just under 3% of the gross world product.

Key concepts The basic elements of a telecommunication system are: • • •

a transmitter that takes information and converts it to a signal for transmission a transmission medium over which the signal is transmitted a receiver that receives and converts the signal back into usable information

For example, consider a radio broadcast. In this case the broadcast tower is the transmitter, the radio is the receiver and the transmission medium is free space. Often

telecommunication systems are two-way and devices act as both a transmitter and receiver or transceiver. For example, a mobile phone is a transceiver. Telecommunication over a phone line is called point-to-point communication because it is between one transmitter and one receiver, telecommunication through radio broadcasts is called broadcast communication because it is between one powerful transmitter and numerous receivers. Signals can either be analogue or digital. In an analogue signal, the signal is varied continuously with respect to the information. In a digital signal, the information is encoded as a set of discrete values (e.g. 1’s and 0’s). A collection of transmitters, receivers or transceivers that communicate with each other is known as a network. Digital networks may consist of one or more routers that route data to the correct user. An analogue network may consist of one or more switches that establish a connection between two or more users. For both types of network, a repeater may be necessary to amplify or recreate the signal when it is being transmitted over long distances. This is to combat attenuation that can render the signal indistinguishable from noise. A channel is a division in a transmission medium so that it can be used to send multiple independent streams of data. For example, a radio station may broadcast at 96 MHz while another radio station may broadcast at 94.5 MHz. In this case the medium has been divided by frequency and each channel received a separate frequency to broadcast on. Alternatively one could allocate each channel a recurring segment of time over which to broadcast. The shaping of a signal to convey information is known as modulation. Modulation is a key concept in telecommunications and is frequently used to impose the information of one signal on another. Modulation is used to represent a digital message as an analogue waveform. This is known as keying and several keying techniques exist — these include phase-shift keying, amplitude-shift keying and minimum-shift keying. Bluetooth, for example, uses phase-shift keying for exchanges between devices (see note). However, more relevant to earlier discussion, modulation is also used to boost the frequency of analogue signals. This is because a raw signal is often not suitable for transmission over long distances of free space due to its low frequencies. Hence its information must be superimposed on a higher frequency signal (known as a carrier wave) before transmission. There are several different modulation schemes available to achieve this — some of the most basic being amplitude modulation and frequency modulation. An example of this process is a DJ’s voice being superimposed on a 96 MHz carrier wave using frequency modulation (the voice would then be received on a radio as the channel “96 FM”).

Society and telecommunication Telecommunication is an important part of many modern societies. In 2006, estimates place the telecommunication industry’s revenue at $1.2 trillion or just under 3% of the gross world product.Good telecommunication infrastructure is widely acknowledged as important for economic success in the modern world both on a micro and macroeconomic scale.And, for this reason, there is increasing worry about the digital divide.

This stems from the fact that access to telecommunication systems is not equally shared amongst the world’s population. A 2003 survey by the International Telecommunication Union (ITU) revealed that roughly one-third of countries have less than 1 mobile subscription for every 20 people and one-third of countries have less than 1 fixed line subscription for every 20 people. In terms of Internet access, roughly half of countries have less than 1 in 20 people with Internet access. From this information as well as educational data the ITU was able to compile a Digital Access Index that measures the overall ability of citizens to access and use information and communication technologies. Using this measure, countries such as Sweden, Denmark and Iceland receive the highest ranking while African countries such as Niger, Burkina Faso and Mali receive the lowest.Further discussion of the social impact of telecommunication is often considered part of communication theory.

History

A replica of one of Chappe’s semaphore towers.

Early telecommunications Early forms of telecommunication include smoke signals and drums. Drums were used by natives in Africa, New Guinea and South America whereas smoke signals were used

by natives in North America and China. Contrary to what one might think, these systems were often used to do more than merely announce the presence of a camp. In 1792, a French engineer, Claude Chappe built the first fixed visual telegraphy (or semaphore) system between Lille and Paris. However semaphore as a communication system suffered from the need for skilled operators and expensive towers often at intervals of only ten to thirty kilometres (six to nineteen miles). As a result, the last commercial line was abandoned in 1880.

Telegraph and telephone The first commercial electrical telegraph was constructed by Sir Charles Wheatstone and Sir William Fothergill Cooke and opened on 9 April 1839. Both Wheatstone and Cooke viewed their device as “an improvement to the [existing] electromagnetic telegraph” not as a new device. On the other side of the Atlantic Ocean, Samuel Morse independently developed a version of the electrical telegraph that he unsuccessfully demonstrated on 2 September 1837. Soon after he was joined by Alfred Vail who developed the register — a telegraph terminal that integrated a logging device for recording messages to paper tape. This was demonstrated successfully on 6 January 1838. The first transatlantic telegraph cable was successfully completed on 27 July 1866, allowing transatlantic telecommunication for the first time. The conventional telephone was invented by Alexander Bell in 1876. Although in 1849 Antonio Meucci invented a device that allowed the electrical transmission of voice over a line. Meucci’s device depended upon the electrophonic effect and was of little practical value because it required users to place the receiver in their mouth to “hear” what was being said. The first commercial telephone services were set-up in 1878 and 1879 on both sides of the Atlantic in the cities of New Haven and London.

Radio and television In 1832, James Lindsay gave a classroom demonstration of wireless telegraphy to his students. By 1854 he was able to demonstrate a transmission across the Firth of Tay from Dundee to Woodhaven, a distance of two miles, using water as the transmission medium. In December 1901, Guglielmo Marconi established wireless communication between Britain and the United States earning him the Nobel Prize in physics in 1909 (which he shared with Karl Braun).

On March 25, 1925, John Logie Baird was able to demonstrate the transmission of moving pictures at the London department store Selfridges. Baird’s device relied upon

the Nipkow disk and thus became known as the mechanical television. It formed the basis of experimental broadcasts done by the British Broadcasting Corporation beginning September 30, 1929. However for most of the twentieth century televisions depended upon the cathode ray tube invented by Karl Braun. The first version of such a television to show promise was produced by Philo Farnsworth and demonstrated to his family on September 7, 1927.

Computer networks and the Internet On September 11, 1940 George Stibitz was able to transmit problems using teletype to his Complex Number Calculator in New York and receive the computed results back at Dartmouth College in New Hampshire. This configuration of a centralized computer or mainframe with remote dumb terminals remained popular throughout the 1950s. However it was not until the 1960s that researchers started to investigate packet switching — a technology that would allow chunks of data to be sent to different computers without first passing through a centralized mainframe. A four-node network emerged on December 5, 1969; this network would become ARPANET, which by 1981 would consist of 213 nodes.

ARPANET’s development centred around the Request for Comment process and on April 7, 1969, RFC 1 was published. This process is important because ARPANET would eventually merge with other networks to form the Internet and many of the protocols the Internet relies upon today were specified through this process. In September 1981, RFC 791 introduced the Internet Protocol v4 (IPv4) and RFC 793 introduced the Transmission Control Protocol (TCP) — thus creating the TCP/IP protocol that much of the Internet relies upon today. However not all important developments were made through the Request for Comment process. Two popular link protocols for local area networks (LANs) also appeared in the 1970s. A patent for the Token Ring protocol was filed by Olof Soderblom on October 29, 1974. And a paper on the Ethernet protocol was published by Robert Metcalfe and David Boggs in the July 1976 issue of Communications of the ACM. These protocols are discussed in more detail in the next section.

Modern operation Telephone

Optic fibres are revolutionizing long-distance communication In a conventional telephone system, the caller is connected to the person they want to talk to by the switches at various exchanges. The switches form an electrical connection between the two users and the setting of these switches is determined electronically when the caller dials the number based upon either pulses or tones made by the caller’s telephone. Once the connection is made, the caller’s voice is transformed to an electrical signal using a small microphone in the telephone’s receiver. This electrical signal is then sent through various switches in the network to the user at the other end where it transformed back into sound waves by a speaker for that person to hear. This person also has a separate electrical connection between him and the caller which allows him to talk back.[28] Today, the fixed-line telephone systems in most residential homes are analogue — that is the speaker’s voice directly determines the amplitude of the signal’s voltage. However although short-distance calls may be handled from end-to-end as analogue signals, increasingly telephone service providers are transparently converting signals to digital before converting them back to analogue for reception.

Mobile phones have had a dramatic impact on telephone service providers. Mobile phone subscriptions now outnumber fixed line subscriptions in many markets. Sales of mobile phones in 2005 totalled 816.6 million with that figure being almost equally shared amongst the markets of Asia/Pacific (204 m), Western Europe (164 m), CEMEA (Central Europe, the Middle East and Africa) (153.5 m), North America (148 m) and Latin America (102 m). In terms of new subscriptions over the five years from 1999, Africa has outpaced other markets with 58.2% growth compared to the next largest market, Asia, which boasted 34.3% growth.[30] Increasingly these phones are being serviced by digital systems such as GSM or W-CDMA with many markets choosing to depreciate analogue

systems such as AMPS.[31] By digital it is meant the handsets themselves transmit digital not analogue signals. However there have been equally drastic changes in telephone communication behind the scenes. Starting with the operation of TAT-8 in 1988, the 1990s saw the widespread adoption of systems based upon optic fibres. The benefit of communicating with optic fibres is that they offer a drastic increase in data capacity. TAT-8 itself was able to carry 10 times as many telephone calls as the last copper cable laid at that time and today’s optic fibre cables are able to carry 25 times as many telephone calls as TAT-8.[32] This drastic increase in data capacity is due to several factors. First, optic fibres are physically much smaller than competing technologies. Second, they do not suffer from crosstalk which means several hundred of them can be easily bundled together in a single cable.[33] Lastly, improvements in multiplexing have lead to an exponential growth in the data capacity of a single fibre. This is due to technologies such as dense wavelength-division multiplexing, which at its most basic level is building multiple channels based upon frequency division as discussed in the Key concepts section.[34] However despite the advances of technologies such as dense wavelength-division multiplexing, technologies based around building multiple channels based upon time division such as synchronous optical networking and synchronous digital hierarchy remain dominant.[35] Assisting communication across these networks is a protocol known as Asynchronous Transfer Mode (ATM). As a technology, ATM arose in the 1980s and was envisioned to be part of the Broadband Integrated Services Digital Network. The network ultimately failed but the technology gave birth to the ATM Forum which in 1992 published its first standard.[36] Today, despite competitors such as Multiprotocol Label Switching, ATM remains the protocol of choice for most major long-distance optical networks. The importance of the ATM protocol was chiefly in its notion of establishing pathways for data through the network and associating a traffic contract with these pathways. The traffic contract was essentially an agreement between the client and the network about how the network was to handle the data, if the network could not meet the conditions of the traffic contract it would not accept the connection. This was important because telephone calls could negotiate a contract so as to guarantee themselves a constant bit rate, something that was essential to ensure a call could take place without the caller’s voice being delayed in parts or cut-off completely.[37]

Radio and television

Digital television standards and their adoption worldwide.

The broadcast media industry is at a critical turning point in its development, with many countries starting to move from analogue to digital broadcasts. The chief advantage of digital broadcasts is that they prevent a number of complaints with traditional analogue broadcasts. For television, this includes the elimination of problems such as snowy pictures, ghosting and other distortion. These occur because of the nature of analogue transmission, which means that perturbations due to noise will be evident in the final output. Digital transmission overcomes this problem because digital signals are reduced to binary data upon reception and hence small perturbations do not affect the final output. In a simplified example, if a binary message 1011 was transmitted with signal amplitudes [1.0 0.0 1.0 1.0] and received with signal amplitudes [0.9 0.2 1.1 0.9] it would still decode to the binary message 1011 — a perfect reproduction of what was sent. From this example, a problem with digital transmissions can also be seen in that if the noise is great enough it can significantly alter the decoded message. Using forward error correction a receiver can correct a handful of bit errors in the resulting message but too much noise will lead to incomprehensible output and hence a breakdown of the transmission.[38]

In digital television broadcasting, there are three competing standards that are likely to be adopted worldwide. These are the ATSC, DVB and ISDB standards and the adoption of these standards thus far is presented in the captioned map. All three standards use MPEG2 for video compression. ATSC uses Dolby Digital AC-3 for audio compression, ISDB uses Advanced Audio Coding (MPEG-2 Part 7) and DVB has no standard for audio compression but typically uses MPEG-1 Part 3 Layer 2.[39][40] The choice of modulation also varies between the schemes. Both DVB and ISDB use orthogonal frequency-division multiplexing (OFDM) for terrestrial broadcasts (as opposed to satellite or cable broadcasts) where as ATSC uses vestigial sideband modulation (VSB). OFDM should offer better resistance to multipath interference and the Doppler effect (which would impact reception using moving receivers).[41] However controversial tests conducted by the United States’ National Association of Broadcasters have shown that there is little difference between the two for stationary receivers.[42] In digital audio broadcasting, standards are much more unified with practically all countries (including Canada) choosing to adopt the Digital Audio Broadcasting standard (also known as the Eureka 147 standard). The exception being the United States which has chosen to adopt HD Radio. HD Radio, unlike Eureka 147, is based upon a transmission method known as in-band on-channel transmission — this allows digital information to “piggyback” on normal AM or FM analogue transmissions. Hence avoiding the bandwidth allocation issues of Eureka 147 and therefore being strongly advocated National Association of Broadcasters who felt there was a lack of new

spectrum to allocate for the Eureka 147 standard.[43] In the United States the Federal Communications Commission has chosen to leave licensing of the standard in the hands of a commercial corporation called iBiquity.[44] An open in-band on-channel standard exists in the form of Digital Radio Mondiale (DRM) however adoption of this standard is mostly limited to a handful of shortwave broadcasts. Despite the different names all standards rely upon OFDM for modulation. In terms of audio compression, DRM typically uses Advanced Audio Coding (MPEG-4 Part 3), DAB like DVB can use a variety of codecs but typically uses MPEG-1 Part 3 Layer 2 and HD Radio uses HighDefinition Coding. However, despite the pending switch to digital, analogue receivers still remain widespread. Analogue television is still transmitted in practically all countries. The United States had hoped to end analogue broadcasts by December 31, 2006 however this was recently pushed back to February 17, 2009.[45] For analogue, there are three standards in use (see a map on adoption here). These are known as PAL, NTSC and SECAM. The basics of PAL and NTSC are very similar; a quadrature amplitude modulated subcarrier carrying the chrominance information is added to the luminance video signal to form a composite video baseband signal (CVBS). On the other hand, the SECAM system uses a frequency modulation scheme on its colour subcarrier. The PAL system differs from NTSC in that the phase of the video signal’s colour components is reversed with each line helping to correct phase errors in the transmission. For analogue radio, the switch to digital is made more difficult by the fact that analogue receivers cost a fraction of the cost of digital receivers. For example while you can get a good analogue receiver for under $20 USD[46] a digital receiver will set you back at least $75 USD.[47] The choice of modulation for analogue radio is typically between amplitude modulation (AM) or frequency modulation (FM). To achieve stereo playback, an amplitude modulated subcarrier is used for stereo FM and quadrature amplitude modulation is used for stereo AM or C-QUAM (see each of the linked articles for more details).

The Internet

The OSI reference model Today an estimated 15.7% of the world population has access to the Internet with the highest concentration in North America (68.6%), Oceania/Australia (52.6%) and Europe (36.1%).[48] In terms of broadband access, countries such as Iceland (26.7%), South Korea (25.4%) and the Netherlands (25.3%) lead the world.[49]

The nature of computer network communication lends itself to a layered approach where individual protocols in the protocol stack run largely independently of other protocols. This allows lower-level protocols to be customized for the network situation while not changing the way higher-level protocols operate. A practical example of why this important is because it allows an Internet browser to run the same code regardless of whether the computer it is running on is connected to the Internet through an Ethernet or Wi-Fi connection. Protocols are often talked about in terms of their place in the OSI reference model — a model that emerged in 1983 as the first step in a doomed attempt to build a universally adopted networking protocol suite.[50] The model itself is outlined in the picture to the right. It is important to note that the Internet’s protocol suite, like many modern protocol suites, does not rigidly follow this model but can still be talked about in the context of this model. For the Internet, the physical medium and data link protocol can vary several times as packets travel between client nodes. Though it is likely that the majority of the distance travelled will be using the Asynchronous Transfer Mode (ATM) data link protocol across optical fibre this is in no way guaranteed. A connection may also encounter data link protocols such as Ethernet, Wi-Fi and the Point-to-Point Protocol (PPP) and physical media such as twisted-pair cables and free space. At the network layer things become standardized with the Internet Protocol (IP) being adopted for logical addressing. For the world wide web, these “IP addresses” are derived from the human readable form (e.g. 72.14.207.99 ) using the Domain Name System. At the moment the most widely used version of the Internet Protocol is version four but a move to version six is imminent. The main advantage of the new version is that it supports 3.40 × 1038 addresses compared to 4.29 × 109 addresses. The new version also adds support for enhanced security through IPSec as well as support for QoS identifiers.[51] At the transport layer most communication adopts either the Transmission Control Protocol (TCP) or the User Datagram Protocol (UDP). With TCP, packets are retransmitted if they are lost and placed in order before they are presented to higher layers (this ordering also allows duplicate packets to be eliminated). With UDP, packets are not ordered or retransmitted if lost. Both TCP and UDP packets carry port numbers with them to specify what application or process the packet should be handed to on the client’s computer.[52] Because certain application-level protocols use certain ports, network administrators can restrict Internet access by blocking or throttling traffic destined for a particular port.

Above the transport layer there are certain protocols that loosely fit in the session and presentation layers and are sometimes adopted, most notably the Secure Sockets Layer (SSL) and Transport Layer Security (TLS) protocols. These protocols ensure that the data transferred between two parties remains completely confidential and one or the other is in use when a padlock appears at the bottom of your web browser. Security is generally based upon the principle that eavesdroppers cannot factorize very large numbers that are the composite of two primes without knowing one of the primes. Another protocol that loosely fits in the session and presentation layers is the Real-time Transport Protocol (RTP) most notably used to stream QuickTime.[53] Finally at the application layer are many of the protocols Internet users would be familiar with such as HTTP (web browsing), POP3 (e-mail), FTP (file transfer) and IRC (Internet chat) but also less common protocols such as BitTorrent (file sharing) and ICQ (instant messaging).

sLocal area networks

A local area network. Despite the growth of the Internet, the characteristics of local area networks (computer networks that run over at most a few kilometres) remain distinct. In the mid-1980s, several protocol suites emerged to fill the gap between the data link and applications layer of the OSI reference model. These were Appletalk, IPX and NetBIOS with the dominant protocol suite during the early 90s being IPX due to its popularity with MS-DOS users. TCP/IP existed at this point but was typically only used by large government and research facilities.[54] However as the Internet grew in popularity and a larger percentage of local area network traffic became Internet-related, LANs gradually moved towards TCP/IP and today networks mostly dedicated to TCP/IP traffic are common. The move to TCP/IP was helped by technologies such as DHCP introduced in RFC 2131 that allowed TCP/IP clients to discover their own network address — a functionality that came standard with the AppleTalk/IPX/NetBIOS protocol suites. However it is at the data link layer that modern local area networks diverge from the Internet. Where as Asynchronous Transfer Mode (ATM) or Multiprotocol Label Switching (MPLS) are typical data link protocols for larger networks, Ethernet and

Token Ring are typical data link protocols for local area networks. The latter LAN protocols differ from the former protocols in that they are simpler (e.g. they omit features such as Quality of Service guarantees) and offer collision prevention. Both of these differences allow for more economic set-ups. For example, omitting Quality of Service guarantees simplifies routers and the guarantees are not really necessary for local area networks because they tend not to carry real time communication (such as voice communication). Including collision prevention allows multiple clients (as opposed to just two) to share the same cable again reducing costs. Though both Ethernet and Token Ring have different frame formats, it is in terms of collision prevention that the two present the greatest difference. With Token Ring a token circulates the network and clients only transmit when they have the token. The token must be managed to ensure it is not lost or duplicated. With Ethernet any client can transmit if it thinks the medium is idle, but clients listen for collisions and if one is detected suspend communication for a random amount of time.[55] Despite Token Ring’s modest popularity in the 80’s and 90’s, with the advent of the twenty-first century, the majority of local area networks have now settled on Ethernet. At the physical layer most Ethernet implementations use copper twisted-pair cables (including the common 10BASE-T networks). Some early implementations used coaxial cables. And some implementations (especially high speed ones) use optical fibres. Optical fibres are also likely to feature prominently in the forthcoming 10-gigabit Ethernet implementations.[56] Where optical fibre is used, the distinction must be made between multi-mode fibre and single-mode fibre. Multi-mode fibre can be thought of as thicker optical fibre that is cheaper to manufacture but that suffers from less usable bandwidth and greater attenuation.

Electrical engineering

Electrical Engineers design power systems…

… and complex electronic circuits. Electrical engineering (sometimes referred to as electrical and electronic engineering) is a professional engineering discipline that deals with the study and application of electricity, electronics and electromagnetism. The field first became an identifiable occupation in the late nineteenth century with the commercialization of the electric telegraph and electrical power supply. The field now covers a range of sub-disciplines including those that deal with power, optoelectronics, digital electronics, analog electronics, artificial intelligence, control systems, electronics, signal processing and telecommunications. The term electrical engineering may or may not encompass electronic engineering. Where a distinction is made, electrical engineering is considered to deal with the problems associated with large-scale electrical systems such as power transmission and motor control, whereas electronic engineering deals with the study of small-scale electronic systems including computers and integrated circuits.Another way of looking at

the distinction is that electrical engineers are usually concerned with using electricity to transmit energy, while electronics engineers are concerned with using electricity to transmit information.

History History of electrical engineering

Early developments Electricity has been a subject of scientific interest since at least the 17th century, but it was not until the 19th century that research into the subject started to intensify. Notable developments in this century include the work of Georg Ohm, who in 1827 quantified the relationship between the electric current and potential difference in a conductor, Michael Faraday, the discoverer of electromagnetic induction in 1831, and James Clerk Maxwell, who in 1873 published a unified theory of electricity and magnetism in his treatise on Electricity and Magnetism.

During these years, the study of electricity was largely considered to be a subfield of physics. It was not until the late 19th century that universities started to offer degrees in electrical engineering. The Darmstadt University of Technology founded the first chair and the first faculty of electrical engineering worldwide in 1882. In 1883 Darmstadt University of Technology and Cornell University introduced the world’s first courses of study in electrical engineering and in 1885 the University College London founded the first chair of electrical engineering in the United Kingdom.The University of Missouri subsequently established the first department of electrical engineering in the United States in 1886.

Thomas Edison built the world’s first large-scale electrical supply network During this period, the work concerning electrical engineering increased dramatically. In 1882, Edison switched on the world’s first large-scale electrical supply network that provided 110 volts direct current to fifty-nine customers in lower Manhattan. In 1887, Nikola Tesla filed a number of patents related to a competing form of power distribution known as alternating current. In the following years a bitter rivalry between Tesla and Edison, known as the “War of Currents”, took place over the preferred method of distribution. AC eventually replaced DC for generation and power distribution, enormously extending the range and improving the safety and efficiency of power distribution.

Nikola Tesla made long-distance electrical transmission networks possible. The efforts of the two did much to further electrical engineering—Tesla’s work on induction motors and polyphase systems influenced the field for years to come, while Edison’s work on telegraphy and his development of the stock ticker proved lucrative for his company, which ultimately became General Electric. However, by the end of the 19th century, other key figures in the progress of electrical engineering were beginning to emerge.

Modern developments Emergence of radio and electronics During the development of radio, many scientists and inventors contributed to radio technology and electronics. In his classic UHF experiments of 1888, Heinrich Hertz transmitted (via a spark-gap transmitter) and detected radio waves using electrical equipment. In 1895, Nikola Tesla was able to detect signals from the transmissions of his

New York lab at West Point (a distance of 80.4 km). In 1897, Karl Ferdinand Braun introduced the cathode ray tube as part of an oscilloscope, a crucial enabling technology for electronic television.John Fleming invented the first radio tube, the diode, in 1904. Two years later, Robert von Lieben and Lee De Forest independently developed the amplifier tube, called the triode. In 1920 Albert Hull developed the magnetron which would eventually lead to the development of the microwave oven in 1946 by Percy Spencer. In 1934 the British military began to make strides towards radar (which also uses the magnetron), under the direction of Dr Wimperis culminating in the operation of the first radar station at Bawdsey in August 1936. In 1941 Konrad Zuse presented the Z3, the world’s first fully functional and programmable computer. In 1946 the ENIAC (Electronic Numerical Integrator and Computer) of John Presper Eckert and John Mauchly followed, beginning the computing era. The arithmetic performance of these machines allowed engineers to develop completely new technologies and achieve new objectives, including the Apollo missions and the NASA moon landing. The invention of the transistor in 1947 by William B. Shockley, John Bardeen and Walter Brattain opened the door for more compact devices and led to the development of the integrated circuit in 1958 by Jack Kilby and independently in 1959 by Robert Noyce. In 1968 Marcian Hoff invented the first microprocessor at Intel and thus ignited the development of the personal computer. The first realization of the microprocessor was the Intel 4004, a 4-bit processor developed in 1971, but only in 1973 did the Intel 8080, an 8-bit processor, make the building of the first personal computer, the Altair 8800, possible.

Education Electrical engineers typically possess an academic degree with a major in electrical engineering. The length of study for such a degree is usually four or five years and the completed degree may be designated as a Bachelor of Engineering, Bachelor of Science, Bachelor of Technology or Bachelor of Applied Science depending upon the university. The degree generally includes units covering physics, mathematics, project management and specific topics in electrical engineering. Initially such topics cover most, if not all, of the sub-disciplines of electrical engineering. Students then choose to specialize in one or more sub-disciplines towards the end of the degree. Some electrical engineers also choose to pursue a postgraduate degree such as a Master of Engineering/Master of Science, a Master of Engineering Management, a Doctor of Philosophy in Engineering or an Engineer’s degree. The Master and Engineer’s degree may consist of either research, coursework or a mixture of the two. The Doctor of Philosophy consists of a significant research component and is often viewed as the entry point to academia. In the United Kingdom and various other European countries, the Master of Engineering is often considered an undergraduate degree of slightly longer duration than the Bachelor of Engineering.

Practicing engineers In most countries, a Bachelor’s degree in engineering represents the first step towards professional certification and the degree program itself is certified by a professional body. After completing a certified degree program the engineer must satisfy a range of requirements (including work experience requirements) before being certified. Once certified the engineer is designated the title of Professional Engineer (in the United States, Canada and South Africa ), Chartered Engineer (in the United Kingdom, Ireland, India and Zimbabwe), Chartered Professional Engineer (in Australia and New Zealand) or European Engineer (in much of the European Union). The advantages of certification vary depending upon location. For example, in the United States and Canada “only a licensed engineer may seal engineering work for public and private clients”. This requirement is enforced by state and provincial legislation such as Quebec’s Engineers Act. In other countries, such as Australia, no such legislation exists. Practically all certifying bodies maintain a code of ethics that they expect all members to abide by or risk expulsion. In this way these organizations play an important role in maintaining ethical standards for the profession. Even in jurisdictions where certification has little or no legal bearing on work, engineers are subject to contract law. In cases where an engineer’s work fails he or she may be subject to the tort of negligence and, in extreme cases, the charge of criminal negligence. An engineer’s work must also comply with numerous other rules and regulations such as building codes and legislation pertaining to environmental law. Professional bodies of note for electrical engineers include the Institute of Electrical and Electronics Engineers (IEEE) and the Institution of Electrical Engineers (IEE). The IEEE claims to produce 30 percent of the world’s literature in electrical engineering, has over 360,000 members worldwide and holds over 300 conferences annually. The IEE publishes 14 journals, has a worldwide membership of 120,000, and claims to be the largest professional engineering society in Europe. Obsolescence of technical skills is a serious concern for electrical engineers. Membership and participation in technical societies, regular reviews of periodicals in the field and a habit of continued learning are therefore essential to maintaining proficiency. In countries such as Australia, Canada and the United States electrical engineers make up around 0.25% of the labour force (see note). Outside of these countries, it is difficult to gauge the demographics of the profession due to less meticulous reporting on labour statistics. However, in terms of electrical engineering graduates per-capita, electrical engineering graduates would probably be most numerous in countries such as Taiwan, Japan and South Korea.

Tools and work From the Global Positioning System to electric power generation, electrical engineers are responsible for a wide range of technologies. They design, develop, test and supervise the

deployment of electrical systems and electronic devices. For example, they may work on the design of telecommunication systems, the operation of electric power stations, the lighting and wiring of buildings, the design of household appliances or the electrical control of industrial machinery.

Satellite Communications is one of many projects an electrical engineer might work on Fundamental to the discipline are the sciences of physics and mathematics as these help to obtain both a qualitative and quantitative description of how such systems will work. Today most engineering work involves the use of computers and it is commonplace to use computer-aided design programs when designing electrical systems. Nevertheless, the ability to sketch ideas is still invaluable for quickly communicating with others. Although most electrical engineers will understand basic circuit theory (that is the interactions of elements such as resistors, capacitors, diodes, transistors and inductors in a circuit), the theories employed by engineers generally depend upon the work they do. For example, quantum mechanics and solid state physics might be relevant to an engineer working on VLSI (the design of integrated circuits), but are largely irrelevant to engineers working with macroscopic electrical systems. Even circuit theory may not be relevant to a person designing telecommunication systems that use off-the-shelf components. Perhaps the most important technical skills for electrical engineers are reflected in university programs, which emphasize strong numerical skills, computer literacy and the ability to understand the technical language and concepts that relate to electrical engineering. For most engineers technical work accounts for only a fraction of the work they do. A lot of time is also spent on tasks such as discussing proposals with clients, preparing budgets and determining project schedules. Many senior engineers manage a team of technicians or other engineers and for this reason project management skills are important. Most engineering projects involve some form of documentation and strong written communication skills are therefore very important. The workplaces of electrical engineers are just as varied as the types of work they do. Electrical engineers may be found in the pristine lab environment of a fabrication plant, the offices of a consulting firm or on site at a mine. During their working life, electrical

engineers may find themselves supervising a wide range of individuals including scientists, electricians, computer programmers and other engineers.

Sub-disciplines Electrical engineering has many sub-disciplines, the most popular of which are listed below. Although there are electrical engineers who focus exclusively on one of these subdisciplines, many deal with a combination of them. Sometimes certain fields, such as electronic engineering and computer engineering, are considered separate disciplines in their own right.

Power Power engineering

Power engineering deals with the generation, transmission and distribution of electricity as well as the design of a range of related devices. These include transformers, electric generators, electric motors and power electronics. In many regions of the world, governments maintain an electrical network called a power grid that connects a variety of generators together with users of their energy. Users purchase electrical energy from the grid, avoiding the costly exercise of having to generate their own. Power engineers may work on the design and maintenance of the power grid as well as the power systems that connect to it. Such systems are called on-grid power systems and may supply the grid with additional power, draw power from the grid or do both. Power engineers may also work on systems that do not connect to the grid, called off-grid power systems, which in some cases are preferable to on-grid systems.

Control Control engineering

Control engineering focuses on the modelling of a diverse range of dynamic systems and the design of controllers that will cause these systems to behave in the desired manner. To implement such controllers electrical engineers may use electrical circuits, digital signal processors and microcontrollers. Control engineering has a wide range of applications from the flight and propulsion systems of commercial airliners to the cruise control present in many modern automobiles. It also plays an important role in industrial automation. Control engineers often utilize feedback when designing control systems. For example, in an automobile with cruise control the vehicle’s speed is continuously monitored and fed back to the system which adjusts the motor’s speed accordingly. Where there is regular feedback, control theory can be used to determine how the system responds to such feedback.

Electronics Electronic engineering

Electronic engineering involves the design and testing of electronic circuits that use the properties of components such as resistors, capacitors, inductors, diodes and transistors to achieve a particular functionality. The tuned circuit, which allows the user of a radio to filter out all but a single station, is just one example of such a circuit. Another example (of a pneumatic signal conditioner) is shown in the adjacent photograph. Prior to the second world war, the subject was commonly known as radio engineering and basically was restricted to aspects of communications and radar, commercial radio and early television. Later, in post war years, as consumer devices began to be developed, the field grew to include modern television, audio systems, computers and microprocessors. In the mid to late 1950s, the term radio engineering gradually gave way to the name electronic engineering. Before the invention of the integrated circuit in 1959, electronic circuits were constructed from discrete components that could be manipulated by humans. These discrete circuits consumed much space and power and were limited in speed, although they are still common in some applications. By contrast, integrated circuits packed a large number— often millions—of tiny electrical components, mainly transistors, into a small chip around the size of a coin. This allowed for the powerful computers and other electronic devices we see today.

Microelectronics Microelectronics

Microelectronics engineering deals with the design of very small electronic components for use in an integrated circuit or sometimes for use on their own as a general electronic component. The most common microelectronic components are semiconductor transistors, although all main electronic components (resistors, capacitors, inductors) can be created at a microscopic level. Microelectronic components are created by chemically fabricating wafers of semiconductors such as silicon (at higher frequencies, gallium arsenide and indium phosphide) to obtain the desired transport of electronic charge and control of current. The field of microelectronics involves a significant amount of chemistry and material science and requires the electronic engineer working in the field to have a very good working knowledge of the effects of quantum mechanics.

Signal processing Signal processing

Signal processing deals with the analysis and manipulation of signals. Signals can be either analog, in which case the signal varies continuously according to the information, or digital, in which case the signal varies according to a series of discrete values representing the information. For analog signals, signal processing may involve the amplification and filtering of audio signals for audio equipment or the modulation and demodulation of signals for telecommunications. For digital signals, signal processing may involve the compression, error detection and error correction of digitally sampled signals.

Telecommunications Telecommunications engineering

Telecommunications engineering focuses on the transmission of information across a channel such as a coax cable, optical fibre or free space. Transmissions across free space require information to be encoded in a carrier wave in order to shift the information to a carrier frequency suitable for transmission, this is known as modulation. Popular analog modulation techniques include amplitude modulation and frequency modulation. The choice of modulation affects the cost and performance of a system and these two factors must be balanced carefully by the engineer. Once the transmission characteristics of a system are determined, telecommunication engineers design the transmitters and receivers needed for such systems. These two are sometimes combined to form a two-way communication device known as a transceiver. A key consideration in the design of transmitters is their power consumption as this is closely related to their signal strength. If the signal strength of a transmitter is insufficient the signal’s information will be corrupted by noise.

Instrumentation engineering Instrumentation engineering

Instrumentation engineering deals with the design of devices to measure physical quantities such as pressure, flow and temperature. The design of such instrumentation requires a good understanding of physics that often extends beyond electromagnetic theory. For example, radar guns use the Doppler effect to measure the speed of oncoming vehicles. Similarly, thermocouples use the Peltier-Seebeck effect to measure the temperature difference between two points. Often instrumentation is not used by itself, but instead as the sensors of larger electrical systems. For example, a thermocouple might be used to help ensure a furnace’s temperature remains constant. For this reason, instrumentation engineering is often viewed as the counterpart of control engineering.

Computers Computer engineering

Computer engineering deals with the design of computers and computer systems. This may involve the design of new hardware, the design of PDAs or the use of computers to control an industrial plant. Computer engineers may also work on a system’s software. However, the design of complex software systems is often the domain of software engineering, which is usually considered a separate discipline. Desktop computers represent a tiny fraction of the devices a computer engineer might work on, as computerlike architectures are now found in a range of devices including video game consoles and DVD players.

Related disciplines Mechatronics is an engineering discipline which deals with the convergence of electrical and mechanical systems. Such combined systems are known as electromechanical systems and have widespread adoption. Examples include automated manufacturing systems, heating, ventilation and air-conditioning systems and various subsystems of aircraft and automobiles. The term mechatronics is typically used to refer to macroscopic systems but futurists have predicted the emergence of very small electromechanical devices. Already such small devices, known as micro electromechanical systems (MEMS), are used in automobiles to tell airbags when to deploy, in digital projectors to create sharper images and in inkjet printers to create nozzles for high-definition printing. In the future it is hoped the devices will help build tiny implantable medical devices and improve optical communication. Biomedical engineering is another related discipline, concerned with the design of medical equipment. This includes fixed equipment such as ventilators, MRI scanners and electrocardiograph monitors as well as mobile equipment such as cochlear implants, artificial pacemakers and artificial hearts.

Power engineering

Rural 3 phase distribution transformer Power engineering is the subfield of electrical engineering that deals with power systems, specifically electric power transmission and distribution, power conversion, and electromechanical devices. Out of necessity, power engineers also rely heavily on the theory of control systems. A power engineer supervises, operates, and maintains machinery and boilers that provide heat, power, refrigeration, and other utility services to heavy industry and large building complexes.

History Power engineering was one of the earliest fields to be exploited in electrical engineering. Early problems solved by engineers include efficient and safe distribution of electric power. Nikola Tesla was a notable pioneer in this field.

Power

Transmission lines transmit power across the grid. Power engineering deals with the generation, transmission and distribution of electricity as well as the design of a range of related devices. These include transformers, electric generators, electric motors and power electronics.

In many regions of the world, governments maintain an electrical network that connects a variety electric generators together with users of their power. This network is called a power grid. Users purchase electricity from the grid avoiding the costly exercise of having to generate their own. Power engineers may work on the design and maintenance of the power grid as well as the power systems that connect to it. Such systems are called on-grid power systems and may supply the grid with additional power, draw power from the grid or do both. Power engineers may also work on systems that do not connect to the grid. These systems are called off-grid power systems and may be used in preference to on-grid systems for a variety of reasons. For example, in remote locations it may be cheaper for a mine to generate its own power rather than pay for connection to the grid and in most mobile applications connection to the grid is simply not practical. Today, most grids adopt three-phase electric power with an alternating current. This choice can be partly attributed to the ease with which this type of power can be generated, transformed and used. Often (especially in the USA), the power is split before it reaches residential customers whose low-power appliances rely upon single-phase electric power. However, many larger industries and organizations still prefer to receive the three-phase power directly because it can be used to drive highly efficient electric motors such as three-phase induction motors. Transformers play an important role in power transmission because they allow power to be converted to and from higher voltages. This is important because higher voltages suffer less power loss during transmission. This is because higher voltages allow for lower current to deliver the same amount of power as power is the product of the two. Thus, as the voltage steps up, the current steps down. It is the current flowing through the components that result in both the losses and the subsequent heating. These losses, appearing in the form of heat, are equal to the current squared times the electrical resistance through which the current flows. For these reasons, electrical substations exist throughout power grids to convert power to higher voltages before transmission and to lower voltages suitable for appliances after transmission.

Components Power engineering is usually broken into three parts:

Generation Generation is converting other forms of power into electrical power. The sources of power include fossil fuels such as coal and natural gas, hydropower, nuclear power, solar power, wind power and other forms.

Transmission Transmission includes moving power over somewhat long distances, from a power station to near where it is used. Transmission involves high voltages, almost always higher than voltage at which the power is either generated or used. Transmission also includes connecting together power systems owned by various companies and perhaps in different states or countries. Transimission includes long meduim and short lines.

Distribution Distribution involves taking power from the transmission system to end users, converting it to voltages at which it is ultimately required.

Optoelectronics Optoelectronics is the study and application of electronic devices that interact with light, and thus is usually considered a sub-field of photonics. In this context, light often includes invisible forms of radiation such as gamma rays, X-rays, ultraviolet and infrared. Optoelectronic devices are electrical-to-optical or optical-to-electrical transducers, or instruments that use such devices in their operation. Electro-optics is often erroneously used as a synonym, but is in fact a wider branch of physics that deals with all interactions between light and electric fields, whether or not they form part of an electronic device. Optoelectronics is based on the quantum mechanical effects of light on semiconducting materials, sometimes in the presence of electric fields. •

•

• • •

Photoelectric or photovoltaic effect, used in: o photodiodes (including solar cells) o phototransistors o photomultipliers o integrated optical circuit (IOC) elements Photoconductivity, used in: o light-dependent resistors o photoconductive camera tubes o charge-coupled imaging devices Stimulated emission, used in: o lasers o injection laser diodes Lossev effect, or radiative recombination, used in: o light-emitting diodes or LED Photoemissivity, used in o photoemissive camera tube

Important applications of optoelectronics include:

• •

Optocoupler optical fiber communications

Photodiode

A photodiode

Photodiode closeup A photodiode is a semiconductor diode that functions as a photodetector. Photodiodes are packaged with either a window or optical fibre connection, in order to let in the light

to the sensitive part of the device. They may also be used without a window to detect vacuum UV or X-rays. A phototransistor is in essence nothing more than a bipolar transistor that is encased in a transparent case so that light can reach the base-collector junction. The phototransistor works like a photodiode, but with a much higher sensitivity for light, because the electrons that are generated by photons in the base-collector junction are injected into the base, and this current is then amplified by the transistor operation. However, a phototransistor has a slower response time than a photodiode.

Principle of operation A photodiode is a p-n junction or p-i-n structure. When light of sufficient photon energy strikes the diode, it excites an electron thereby creating a mobile electron and a positively charged electron hole. If the absorption occurs in the junction’s depletion region, these carriers are swept from the junction by the built-in field of the depletion region, producing a photocurrent. Photodiodes can be used under either zero bias (photovoltaic mode) or reverse bias (photoconductive mode). In zero bias, light falling on the diode causes a voltage to develop across the device, leading to a current in the forward bias direction. This is called the photovoltaic effect, and is the basis for solar cells — in fact, a solar cell is just a large number of big, cheap photodiodes. Diodes usually have extremely high resistance when reverse-biased. This resistance is reduced when light of an appropriate frequency shines on the junction. Hence, a reversebiased diode can be used as a detector by monitoring the current running through it. Circuits based on this effect are more sensitive to light than ones based on the photovoltaic effect. Avalanche photodiodes have a similar structure, but they are operated with much higher reverse bias. This allows each photo-generated carrier to be multiplied by avalanche breakdown, resulting in internal gain within the photodiode, which increases the effective responsivity of the device.

Materials The material used to make a photodiode is critical to defining its properties, because only photons with sufficient energy to excite an electron across the material’s bandgap will produce significant photocurrents. Materials commonly used to produce photodiodes:

Material

Wavelength range (nm)

Silicon

190–1100

Germanium

800–1700

Indium gallium arsenide 800–2600

Lead sulfide

0, then the first conditions result in what is sometimes referred to as undersampling, or using a sampling rate less than the Nyquist rate 2fH obtained from the upper bound of the spectrum. See aliasing for a simpler formulation of this Nyquist criterion that specifies the lower bound on sampling rate (but is incomplete because it does not specify the gaps above that bound, in which aliasing will occur). Alternatively,

for the case of a given sampling frequency, simpler formulae for the constraints on the signal’s spectral band are given below.

Spectrum of the FM radio band (88–108 MHz) and its baseband alias under 44 MHz (N– n = 4) sampling. An anti-alias filter quite tight to the FM radio band is required, and there’s not room for stations at nearby expansion channels such as 87.9 without aliasing.

Spectrum of the FM radio band (88–108 MHz) and its baseband alias under 56 MHz (N– n = 3) sampling, showing plenty of room for bandpass anti-aliasing filter transition bands. The baseband image is frequency-reversed in this case (odd N–n). Example: Consider FM radio to illustrate the idea of undersampling. In the US, FM radio operates on the frequency band from fL = 88 MHz to fH = 108 MHz. The bandwidth is given by

The sampling conditions are satisfied for

Therefore N=4, r=8 MHz and n = 0,1,2,3. The value n = 0 gives the lowest sampling frequencies interval and this is a scenario of undersampling. In this case, the signal spectrum fits between and 2 and 2.5 times the sampling rate (higher than 86.4–108 but lower than 88-110 MHz). A lower value of N will also lead to a useful sampling rate, equivalent to picking a nonzero n. For example, using N–n = 3, the FM band spectrum fits easily between 1.5 and 2.0 times the sampling rate, for a sampling rate near 56 MHz (multiples of the Nyquist frequency being 28, 56, 84, 112, etc.). See the illustrations at the right.

When undersampling a real-world signal, the sampling circuit must be fast enough to capture the highest signal frequency of interest. Theoretically, each sample should be taken during an infinitesimally short interval, but this is not practically feasible. Instead, the sampling of the signal should be made in a short enough interval that it can represent the instantaneous value of the signal with the highest frequency. This means that in the FM radio example above, the sampling circuit must be able to capture a signal with a frequency of 108 MHz, not 43.2 MHz. Thus, the sampling frequency may be only a little bit greater than 43.2 MHz, but the input bandwidth of the system must be at least 108 MHz. If the sampling theorem is interpreted as requiring twice the highest frequency, then the required sampling rate would be assumed to be greater than the Nyquist rate 216 MHz. While this does satisfy the last condition on the sampling rate, it is grossly oversampled. Note that if a band is sampled with a nonzero N, then a band-pass filter is required for the anti-aliasing filter, instead of a lowpass filter. As we have seen, the normal baseband condition for reversible sampling is that outside the open interval: And the reconstructive interpolation function, or lowpass filter impulse response, is . To accommodate undersampling, the generalized condition is that the union of open positive and negative frequency bands

outside

for some nonnegative integer . which includes the normal baseband condition as case N=0 (except that where the intervals come together at 0 frequency, they can be closed). And the corresponding interpolation function is the bandpass filter given by this difference of lowpass impulse responses:

. On the other hand, reconstruction is not usually the goal with sampled IF or RF signals. Rather, the sample sequence can be treated as ordinary samples of the signal frequencyshifted to near baseband, and digital demodulation can proceed on that basis.

Quantization (signal processing)

Quantized signal

Digital signal In digital signal processing, quantization is the process of approximating a continuous range of values (or a very large set of possible discrete values) by a relatively-small set of discrete symbols or integer values. More specifically, a signal can be multi-dimensional and quantization need not be applied to all dimensions. Discrete signals (a common mathematical model) need not be quantized, which can be a point of confusion. See ideal sampler. A common use of quantization is in the conversion of a discrete signal (a sampled continuous signal) into a digital signal by quantizing. Both of these steps (sampling and quantizing) are performed in analog-to-digital converters with the quantization level specified in bits. A specific example would be compact disc (CD) audio which is sampled at 44,100 Hz and quantized with 16 bits (2 bytes) which can be one of 65,536 (i.e. 216) possible values per sample.

Mathematical description The simplest and best-known form of quantization is referred to as scalar quantization, since it operates on scalar (as opposed to multi-dimensional vector) input data. In general, a scalar quantization operator can be represented as

where •

x is a real number to be quantized,

•

•

is the floor function, yielding an integer result referred to as the quantization index, f(x) and g(i) are arbitrary real-valued functions.

that is sometimes

The integer-valued quantization index i is the representation that is typically stored or transmitted, and then the final interpretation is constructed using g(i) when the data is later interpreted. In computer audio and most other applications, a method known as uniform quantization is the most common. There are two common variations of uniform quantization, called mid-rise and mid-tread uniform quantizers. If x is a real-valued number between -1 and 1, a mid-rise uniform quantization operator that uses M bits of precision to represent each quantization index can be expressed as

. In this case the f(x) and g(i) operators are just multiplying scale factors (one multiplier being the inverse of the other) along with an offset in g(i) function to place the representation value in the middle of the input region for each quantization index. The value 2 − (M − 1) is often referred to as the quantization step size. Using this quantization law and assuming that quantization noise is approximately uniformly distributed over the quantization step size (an assumption typically accurate for rapidly varying x or high M) and further assuming that the input signal x to be quantized is approximately uniformly distributed over the entire interval from -1 to 1, the signal to noise ratio (SNR) of the quantization can be computed as

. From this equation, it is often said that the SNR is approximately 6 dB per bit. For mid-tread uniform quantization, the offset of 0.5 would be added within the floor function instead of outside of it. Sometimes, mid-rise quantization is used without adding the offset of 0.5. This reduces the signal to noise ratio by approximately 6.02 dB, but may be acceptable for the sake of simplicity when the step size is small. In digital telephony, two popular quantization schemes are the ‘A-law’ (dominant in Europe) and ‘µ-law’ (dominant in North America and Japan). These schemes map discrete analog values to an 8-bit scale that is nearly linear for small values and then increases logarithmically as amplitude grows. Because the human ear’s perception of

loudness is roughly logarithmic, this provides a higher signal to noise ratio over the range of audible sound intensities for a given number of bits.

Quantization and data compression Quantization plays a major part in lossy data compression. In many cases, quantization can be viewed as the fundamental element that distinguishes lossy data compression from lossless data compression, and the use of quantization is nearly always motivated by the need to reduce the amount of data needed to represent a signal. In some compression schemes, like MP3 or Vorbis, compression is also achieved by selectively discarding some data, an action that can be analyzed as a quantization process (e.g., a vector quantization process) or can be considered a different kind of lossy process. One example of a lossy compression scheme that uses quantization is JPEG image compression. During JPEG encoding, the data representing an image (typically 8-bits for each of three color components per pixel) is processed using a discrete cosine transform and is then quantized and entropy coded. By reducing the precision of the transformed values using quantization, the number of bits needed to represent the image can be reduced substantially. For example, images can often be represented with acceptable quality using JPEG at less than 3 bits per pixel (as opposed to the typical 24 bits per pixel needed prior to JPEG compression). Even the original representation using 24 bits per pixel requires quantization for its PCM sampling structure. In modern compression technology, the entropy of the output of a quantizer matters more than the number of possible values of its output (the number of values being 2M in the above example). In order to determine how many bits are necessary to effect a given precision, logarithms are used. Suppose, for example, that it is necessary to record six significant digits, that is to say, millionths. The number of values that can be expressed by N bits is equal to two to the Nth power. To express six decimal digits, the required number of bits is determined by rounding (6 / log 2)—where log refers to the base ten, or common, logarithm—up to the nearest integer. Since the logarithm of 2, base ten, is approximately 0.30102, the required number of bits is then given by (6 / 0.30102), or 19.932, rounded up to the nearest integer, viz., 20 bits. This type of quantization—where a set of binary digits, e.g., an arithmetic register in a CPU, are used to represent a quantity—is called Vernier quantization. It is also possible, although rather less efficient, to rely upon equally spaced quantization levels. This is only practical when a small range of values is expected to be captured: for example, a set of eight possible values requires eight equally spaced quantization levels—which is not unreasonable, although obviously less efficient than a mere trio of binary digits (bits)— but a set of, say, sixty-four possible values, requiring sixty-four equally spaced quantization levels, can be expressed using only six bits, which is obviously far more efficient.

Relation to quantization in nature At the most fundamental level, all physical quantities are quantized. This is a result of quantum mechanics (see Quantization (physics)). Signals may be treated as continuous for mathematical simplicity by considering the small quantizations as negligible. In any practical application, this inherent quantization is irrelevant for two reasons. First, it is overshadowed by signal noise, the intrusion of extraneous phenomena present in the system upon the signal of interest. The second, which appears only in measurement applications, is the inaccuracy of instruments. Thus, although all physical signals are intrinsically quantized, the error introduced by modeling them as continuous is vanishingly small.

LTI system theory In electrical engineering, specifically in circuits, signal processing, and control theory, LTI system theory investigates the response of a linear, time-invariant system to an arbitrary input signal. Though the standard independent variable is time, it could just as easily be space (as in image processing and field theory) or some other coordinate. Thus a better, albeit less common, term is linear translation-invariant. The term linear shiftinvariant is the corresponding concept for a discrete-time (sampled) system.

Overview The defining properties of any linear time-invariant system are, of course, linearity and time invariance: •

Linearity means that the relationship between the input and the output of the system satisfies the scaling and superposition properties. Formally, a linear system is a system which exhibits the following property: if the input of the system is

then the output of the system will be for any constants A and B, where yi(t) is the output when the input is xi(t). •

Time invariance means that whether we apply an input to the system now or T seconds from now, the output will be identical, except for a time delay of the T seconds. More specifically, an input affected by a time delay should effect a corresponding time delay in the output, hence time-invariant.

The fundamental result in LTI system theory is that any LTI system can be characterized entirely by a single function called the system’s impulse response. The output of the system is simply the convolution of the input to the system with the system’s impulse response. This method of analysis is often called the time domain point-of-view. The same result is true of discrete-time linear shift-invariant systems, in which signals are discrete-time samples, and convolution is defined on sequences.

Relationship between the time domain and the frequency domain Equivalently, any LTI system can be characterized in the frequency domain by the system’s transfer function, which is the Laplace transform of the system’s impulse response (or Z transform in the case of discrete-time systems). As a result of the properties of these transforms, the output of the system in the frequency domain is the product of the transfer function and the transform of the input. In other words, convolution in the time domain is equivalent to multiplication in the frequency domain. For all LTI systems, the eigenfunctions, and the basis functions of the transforms, are complex exponentials. This is, if the input to a system is the complex waveform Aexp(st) for some complex amplitude A and complex frequency s, the output will be some complex constant times the input, say Bexp(st) for some new complex amplitude B. The ratio B / A is the transfer function at frequency s. Because sinusoids are a sum of complex exponentials with complex-conjugate frequencies, if the input to the system is a sinusoid, then the output of the system will also be a sinusoid, perhaps with a different amplitude and a different phase, but always with the same frequency. LTI system theory is good at describing many important systems. Most LTI systems are considered “easy” to analyze, at least compared to the time-varying and/or nonlinear case. Any system that can be modeled as a linear homogeneous differential equation with constant coefficients is an LTI system. Examples of such systems are electrical circuits made up of resistors, inductors, and capacitors (RLC circuits). Ideal spring–mass–damper systems are also LTI systems, and are mathematically equivalent to RLC circuits.

Most LTI system concepts are similar between the continuous-time and discrete-time (linear shift-invariant) cases. In image processing, the time variable is replaced with 2 space variables, and the notion of time invariance is replaced by two-dimensional shift invariance. When analyzing filter banks and MIMO systems, it is often useful to consider vectors of signals.

Continuous-time systems Time invariance and linear transformation Let us start with a time-varying system whose impulse response is a 2-dimensional function and see how the condition of time invariance helps us reduce it to one dimension. For example, suppose the input signal is x(t) where its index set is the real line, i.e., . The linear operator represents the system operating on the input signal. The appropriate operator for this index set is a 2-dimensional function

Since is a linear operator, the action of the system on the input signal x(t) is a linear transformation represented by the following superposition integral

If the linear operator

is also time-invariant, then

If we let

then it follows that

We usually drop the zero second argument to h(t1,t2) for brevity of notation so that the superposition integral now becomes the familiar convolution integral used in filtering

Thus, the convolution integral represents the effect of a linear, time-invariant system on any input function. For a finite-dimensional analog, see the article on a circulant matrix.

Impulse response If we input a Dirac delta function to this system, the result of the LTI transformation is known as the impulse response because the delta function is an ideal impulse. We illustrate this idea as follows:

(by the sifting property of the delta function). Note that

so that h(t) is the impulse response of the system. The impulse response can be used to find the response of any input in the following way. Again using the sifting property of the δ(t), we can write any input as a superposition of deltas:

Applying the system to the input,

(because

is linear and can pass inside the integral)

(because x(τ) is constant in t and

is linear)

(by definition of h(t)) All information about the system is contained in the impulse response h(t).

Exponentials as eigenfunctions An eigenfunction is a function for which the output of the operator is the same function, just scaled by some amount. In symbols,

, where f is the eigenfunction and λ is the eigenvalue, a constant. The exponential functions est, where , are eigenfunctions of a linear, timeinvariant operator. A simple proof illustrates this concept. Suppose the input is x(t) = est. The output of the system with impulse response h(t) is then

which is equivalent to the following by the commutative property of convolution

, where

is dependent only on the parameter s. So, est is an eigenfunction of an LTI system because the system response is the same as the input times the constant H(s).

Fourier and Laplace transforms The eigenfunction property of exponentials is very useful for both analysis and insight into LTI systems. The Laplace transform

is exactly the way to get the eigenvalues from the impulse response. Of particular interest are pure sinusoids, i.e. exponentials of the form exp(jωt) where and . These are generally called complex exponentials even though the argument is purely imaginary. The Fourier transform

gives the eigenvalues

for pure complex sinusoids. Both of H(s) and H(jω) are called the system function, system response, or transfer function. The Laplace transform is usually used in the context of one-sided signals, i.e. signals that are zero for all values of t less than some value. Usually, this “start time” is set to zero, for convenience and without loss of generality, with the transform integral being taken from zero to infinity (the transform shown with lower limit of integration of negative infinity is formally known as the bilateral Laplace transform). The Fourier transform is used for analyzing systems that process signals that are infinite in extent, such as modulated sinusoids, even though it can not be directly applied to input and output signals that are not square integrable. The Laplace transform actually works directly for these signals if they are zero before a start time, even if they are not square integrable, for stable systems. The Fourier transform is often applied to spectra of infinite signals via the Wiener–Khinchin theorem even when Fourier transforms of the signals do not exist. Due to the convolution property of both of these transforms, the convolution that gives the output of the system can be transformed to a multiplication in the transform domain, given signals for which the transforms exist

Not only is it often easier to do the transforms, multiplication, and inverse transform than the original convolution, but one can also gain insight into the behavior of the system from the system response. One can look at the modulus of the system function |H(s)| to see whether the input exp(st) is passed (let through) the system or rejected or attenuated by the system (not let through).

Examples A simple example of an LTI operator is the derivative:

When the Laplace transform of the derivative is taken, it transforms to a simple multiplication by the Laplace variable s.

That the derivative has such a simple Laplace transform partly explains the utility of the transform. Another simple LTI operator is an averaging operator

. It is linear because of the linearity of integration

. It is time invariant too

. Indeed,

can be written as a convolution with the box function Π(t).

, where the box function is

.

Important system properties Some of the most important properties of a system are causality and stability. It is more or less necessary for a system to be causal in order for it to be implemented in the real world. Non-stable systems can be built and can be useful in many circumstances. Even non-real systems can be built and are very useful in many contexts. Causality Causal system

A system is causal if the output depends only on present and past inputs. A necessary and sufficient condition for causality is

where h(t) is the impulse response. It is not possible in general to determine causality from the Laplace transform, because the inverse transform is not unique. When a region of convergence is specified, then causality can be determined. Stability BIBO stability

A system is bounded input, bounded output stable (BIBO stable) if, for every bounded input, the output is finite. Mathematically, if

and

(i.e., the maximum absolute values of x(t) and y(t) are finite), then the system is stable. A necessary and sufficient condition is that h(t), the impulse response, satisfies

In the frequency domain, the region of convergence must contain the imaginary axis s = jω.

Discrete-time systems Almost everything in continuous-time systems has a counterpart in discrete-time systems.

Discrete-time systems from continuous-time systems In many contexts, a discrete time (DT) system is really part of a larger continuous time (CT) system. For example, a digital recording system takes an analog sound, digitizes it, possibly processes the digital signals, and plays back an analog sound for people to listen to. Formally, the DT signals studied are almost always uniformly sampled versions of CT signals. If x(t) is a CT signal, then an analog to digital converter will transform it to the DT signal x[n], with x[n] = x(nT), where T is the sampling period. It is very important to limit the range of frequencies in the input signal for faithful representation in the DT signal. Due to the sampling theorem, a DT signal can only contain a frequency range of 1 / (2T). Other frequencies are aliased to the same range.

Time invariance and linear transformation Let us start with a time-varying system whose impulse response is a two dimensional function and see how the condition of time-invariance helps us reduce it to one dimension. For example, suppose the input signal is x[n] where its index set is the integers, i.e., . The linear operator represents the system operating on the input signal. The appropriate operator for this index set is a two-dimensional function

Since is a linear operator, the action of the system on the input signal x[n] is a linear transformation represented by the following superposition sum

If the linear operator

If we let

then it follows that

is also time-invariant, then

We usually drop the zero second argument to h[n1,n2] for brevity of notation so that the superposition integral now becomes the familiar convolution sum used in filtering

Thus, the convolution sum represents the effect of a linear, time-invariant system on any input function. For a finite-dimensional analog, see the article on a circulant matrix.

Impulse response If we input a discrete delta function to this system, the result of the LTI transformation is known as the impulse response because the delta function is an ideal impulse. We illustrate this idea as follows:

(by the sifting property of the delta function). Note that

so that h[n] is the impulse response of the system. The impulse response can be used to find the response of any input in the following way. Again using the sifting property of the δ[n], we can write any input as a superposition of deltas:

Applying the system to the input,

(because

is linear and can pass inside the sum)

(because x[m] is constant in n and

is linear)

(by definition of h[n]) All information about the system is contained in the impulse response h[n].

Exponentials as eigenfunctions An eigenfunction is a function for which the output of the operator is the same function, just scaled by some amount. In symbols, , where f is the eigenfunction and λ is the eigenvalue, a constant. The exponential functions zn = esTn, where

, are eigenfunctions of a linear, time-

invariant operator. is the sampling interval, and simple proof illustrates this concept. Suppose the input is then

. The output of the system with impulse response h[n] is

which is equivalent to the following by the commutative property of convolution

, where

.A

is dependent only on the parameter z. So, zn is an eigenfunction of an LTI system because the system response is the same as the input times the constant H(z).

Z and discrete-time Fourier transforms The eigenfunction property of exponentials is very useful for both analysis and insight into LTI systems. The Z transform

is exactly the way to get the eigenvalues from the impuse response. Of particular interest are pure sinusoids, i.e. exponentials of the form ejωn, where . These can also be written as zn with z = ejω. These are generally called complex exponentials even though the argument is purely imaginary. The Discrete-time Fourier transform (DTFT) gives the eigenvalues of pure sinusoids. Both of H(z) and H(ejω) are called the system function, system response, or transfer function. The Z transform is usually used in the context of one-sided signals, i.e. signals that are zero for all values of t less than some value. Usually, this “start time” is set to zero, for convenience and without loss of generality. The Fourier transform is used for analyzing signals that are infinite in extent. Due to the convolution property of both of these transforms, the convolution that gives the output of the system can be transformed to a multiplication in the transform domain.

Not only is it often easier to do the transforms, multiplication, and inverse transform than the original convolution, one can gain insight into the behavior of the system from the system response. One can look at the modulus of the system function |H(z)| to see whether the input zn is passed (let through) by the system, or rejected or attenuated by the system (not let through).

Examples A simple example of an LTI operator is the delay operator D{x}[n]: = x[n − 1].

When the Z transform of the difference is taken, it transforms to a simple multiplication by z:

That the difference has such a simple Z transform partly explains the utility of the transform. Another simple LTI operator is an averaging operator

. It is linear because of the linearity of sums:

. It is time invariant too:

.

Important system properties Some of the most important properties of a system are causality and stability. Unlike CT systems, non-causal DT systems can be realized. It is trivial to make an acausal FIR system causal by adding delays. It is even possible to make acausal IIR systems (See Vaidyanathan and Chen, 1995). Non-stable systems can be built and can be useful in

many circumstances. Even non-real systems can be built and are very useful in many contexts. Causality Causal system

A system is causal if the output depends only on present and past inputs. A necessary and sufficient condition for causality is

where h[n] is the impulse response. It is not possible in general to determine causality from the Z transform, because the inverse transform is not unique. When a region of convergence is specified, then causality can be determined. Stability BIBO stability

A system is bounded input, bounded output stable (BIBO stable) if, for every bounded input, the output is finite. Mathematically, if

and

(i.e., the maximum absolute values of x[n] and y[n] are finite), then the system is stable. A necessary and sufficient condition is that h[n], the impulse response, satisfies

In the frequency domain, the region of convergence must contain the unit circle | z | = 1.

Fourier transform In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. Loosely speaking, the Fourier transform decomposes a function into a continuous spectrum of its frequency components, and the inverse transform synthesizes a function from its spectrum of frequency components. In mathematical physics, the Fourier transform of a signal can be thought of as that signal in the “frequency domain.” This is similar to the basic idea of the various other Fourier transforms including the Fourier series of a periodic function.

Definition Suppose is a complex-valued Lebesgue integrable function. The Fourier transform to the frequency domain, , is given by the function:

, for every real number

.

When the independent variable t represents time (with SI unit of seconds), the transform variable ω represents angular frequency (in radians per second).

Other notations for this same function are: and complex-valued in general. ( represents the imaginary unit.)

. The function is

If is defined as above, and is sufficiently smooth, then it can be reconstructed by the inverse transform:

, for every real number . The interpretation of

is aided by expressing it in polar coordinate form, , where: the amplitude the phase

Then the inverse transform can be written:

which is a recombination of all the frequency components of complex sinusoid of the form initial phase (at t = 0) is

. Each component is a

whose amplitude is proportional to

and whose

.

Normalization factors and alternative forms

The factors before each integral ensure that there is no net change in amplitude when one transforms from one domain to the other and back. The actual requirement is that their product be . When they are chosen to be equal, the transform is referred to as unitary. A common non-unitary convention is shown here:

As a rule of thumb, mathematicians generally prefer the unitary transform (for symmetry reasons), and physicists use either convention depending on the application.

The non-unitary form is preferred by some engineers as a special case of the bilateral Laplace transform. And the substitution: , where is ordinary frequency (hertz), results in another unitary transform that is popular in the field of signal processing and communications systems:

We note that the table below.

and

represent different, but related, functions, as shown in

Variations of all three forms can be created by conjugating the complex-exponential kernel of both the forward and the reverse transform. The signs must be opposites. Other than that, the choice is (again) a matter of convention.

Summary of popular forms of the Fourier transform

unitary angular frequency

(rad/s)

ordinary frequency

nonunitary

unitary

(hertz)

Generalization There are several ways to define the Fourier transform pair. The “forward” and “inverse” transforms are always defined so that the operation of both transforms in either order on a function will return the original function. In other words, the composition of the transform pair is defined to be the identity transformation. Using two arbitrary real constants a and b, the most general definition of the forward 1-dimensional Fourier transform is given by:

and the inverse is given by:

Note that the transform definitions are symmetric; they can be reversed by simply changing the signs of a and b.

The convention adopted in this article is (a,b) = (0,1). The choice of a and b is usually chosen so that it is geared towards the context in which the transform pairs are being used. The non-unitary convention above is (a,b) = (1,1). Another very common definition is (a,b) = (0,2π) which is often used in signal processing applications. In this case, the angular frequency ω becomes ordinary frequency f. If f (or ω) and t carry units, then their product must be dimensionless. For example, t may be in units of time, specifically seconds, and f (or ω) would be in hertz (or radian/s).

Properties In this section, all the results are derived for the following definition (normalization) of the Fourier transform:

Completeness We define the Fourier transform on the set of compactly-supported complex-valued functions of and then extend it by continuity to the Hilbert space of square-integrable functions with the usual inner-product. Then is a unitary operator. That is. and the transform preserves inner-products (see Parseval’s theorem, also described below). Note that, refers to adjoint of the Fourier Transform operator. Moreover we can check that,

where

and

is the Time-Reversal operator defined as,

is the Identity operator defined as,

Extensions The Fourier transform can also be extended to the space integrable functions defined on

where,

and

is the space of continuous functions on

.

In this case the definition usually appears as

where

and

is the inner product of the two vectors ω and x.

One may now use this to define the continuous Fourier transform for compactly supported smooth functions, which are dense in

The Plancherel theorem then

allows us to extend the definition of the Fourier transform to functions on (even those not compactly supported) by continuity arguments. All the properties and formulas listed on this page apply to the Fourier transform so defined. Unfortunately, further extensions become more technical. One may use the HausdorffYoung inequality to define the Fourier transform for

for

. The

p

Fourier transform of functions in L for the range requires the study of distributions, since the Fourier transform of some functions in these spaces is no longer a function, but rather a distribution.

The Plancherel theorem and Parseval’s theorem It should be noted that depending on the author either of these theorems might be referred to as the Plancherel theorem or as Parseval’s theorem. If f(x) and g(x) are square-integrable and F(ω) and G(ω) are their Fourier transforms, then we have the Parseval’s theorem:

where the bar denotes complex conjugation. Therefore, the Fourier transformation yields an isometric automorphism of the Hilbert space

.

The Plancherel theorem, a special case of the Parseval’s theorem, states that

This theorem is usually interpreted as asserting the unitary property of the Fourier transform. See Pontryagin duality for a general formulation of this concept in the context of locally compact abelian groups.

Localization property As a rule of thumb: the more concentrated f(x) is, the more spread out is F(ω). In particular, if we “squeeze” a function in x, it spreads out in ω and vice-versa; and we cannot arbitrarily concentrate both the function and its Fourier transform. Therefore a function which equals its Fourier transform strikes a precise balance between being concentrated and being spread out. It is easy in theory to construct examples of such functions (called self-dual functions) because the Fourier transform has order 4 (that is, iterating it four times on a function returns the original function). The sum of the four iterated Fourier transforms of any function will be self-dual. There are also some explicit examples of self-dual functions, the most important being constant multiples of the Gaussian function

This function is related to Gaussian distributions, and in fact, is an eigenfunction of the Fourier transform operators. The trade-off between the compaction of a function and its Fourier transform can be formalized. Suppose f(x) and F(ω) are a Fourier transform pair. Without loss of generality, we assume that f(x) is normalized:

It follows from Parseval’s theorem that F(ω) is also normalized. Define the expected value of a function A(x) as:

and the expectation value of a function B(ω) as:

Also define the variance of A(x) as:

and similarly define the variance of B(ω). Then it can be shown that

The equality is achieved for the Gaussian function listed above, which shows that the gaussian function is maximally concentrated in “time-frequency”. The most famous practical application of this property is found in quantum mechanics. The momentum and position wave functions are Fourier transform pairs to within a factor of and are normalized to unity. The above expression then becomes a statement of the Heisenberg uncertainty principle. The Fourier transform also translates between smoothness and decay: if f(x) is several times differentiable, then F(ω) decays rapidly towards zero for .

Analysis of differential equations Fourier transforms, and the closely related Laplace transforms are widely used in solving differential equations. The Fourier transform is compatible with differentiation in the following sense: if f(x) is a differentiable function with Fourier transform F(ω), then the Fourier transform of its derivative is given by iω F(ω). This can be used to transform differential equations into algebraic equations. Note that this technique only applies to problems whose domain is the whole set of real numbers. By extending the Fourier transform to functions of several variables (as outlined below), partial differential equations with domain ) can also be translated into algebraic equations.

Convolution theorem Convolution theorem The Fourier transform translates between convolution and multiplication of functions. If f(x) and h(x) are integrable functions with Fourier transforms F(ω) and H(ω) respectively, and if the convolution of f and h exists and is absolutely integrable, then the Fourier transform of the convolution is given by the product of the Fourier transforms F(ω)H(ω) (possibly multiplied by a constant factor depending on the Fourier normalization convention). In the current normalization convention, this means that if

where * denotes the convolution operation; then

The above formulas hold true for functions defined on both one- and multi-dimension real space. In linear time invariant (LTI) system theory, it is common to interpret h(x) as the impulse response of an LTI system with input f(x) and output g(x), since substituting the unit impulse for f(x) yields g(x) = h(x). In this case, H(ω) represents the frequency response of the system. Conversely, if f(x) can be decomposed as the product of two other functions p(x) and q(x) such that their product p(x)q(x) is integrable, then the Fourier transform of this product is given by the convolution of the respective Fourier transforms P(ω) and Q(ω), again with a constant scaling factor. In the current normalization convention, this means that if

then

Cross-correlation theorem In an analogous manner, it can be shown that if g(x) is the cross-correlation of f(x) and h(x):

then the Fourier transform of g(x) is:

where capital letters are again used to denote the Fourier transform.

Tempered distributions The most general and useful context for studying the continuous Fourier transform is given by the tempered distributions; these include all the integrable functions mentioned

above and have the added advantage that the Fourier transform of any tempered distribution is again a tempered distribution and the rule for the inverse of the Fourier transform is universally valid. Furthermore, the useful Dirac delta is a tempered distribution but not a function; its Fourier transform is the constant function . Distributions can be differentiated and the above mentioned compatibility of the Fourier transform with differentiation and convolution remains true for tempered distributions.

Table of important Fourier transforms The following table records some important Fourier transforms. G and H denote Fourier transforms of g(t) and h(t), respectively. g and h may be integrable functions or tempered distributions. Note that the two most common unitary conventions are included.

Functional relationships

Signal

Fourier transform unitary, angular frequency

Fourier transform unitary, ordinary frequency

Remarks

1

Linearity

2

Shift in time domain

3

Shift in frequency domain, dual of 2

If 4

is large, then

is concentrated around 0 and

spreads out and flattens. It is interesting to consider the limit of this as | a | tends to infinity - the delta function

5

Duality property of the Fourier transform. Results from swapping “dummy” variables of and .

6

Generalized derivative property of the Fourier transform

7

This is the dual of 6

8

denotes the convolution of and — this rule is the convolution theorem

9

This is the dual of 8

Square-integrable functions Signal

Fourier transform

Fourier transform

Remark

unitary, angular frequency

unitary, ordinary frequency

10

The rectangula pulse and t normalized sinc functi

11

Dual of rul 10. The rectangula function is idealized low-pass filter, and t sinc functi is the noncausal impulse response o such a filte

12

tri is the triangular function

13

Dual of rul 12.

14

Shows that the Gaussi function ex

− αt2) is its own Fouri transform. For this to integrable must have Re(α) > 0.

15

common in optics

16

17

18

a>0

19

the transfo is the function it

20

J0(t) is the Bessel function of first kind o order 0

21

it’s the generalizat of the previous

transform; (t) is the Chebyshev polynomia the first kin

Un (t) is th Chebyshev polynomia the second kind

22

Distributions

Signal

23

Fourier transform unitary, angular frequency

Fourier transform unitary, ordinary frequency

Remarks

δ(ω) denotes the Dirac delta distribution. This rule shows why the Dirac delta is important: it shows up as the Fourier transform of a constant function.

24

Dual of rule 23.

25

This follows from and 3 and 24.

26

Follows from rules 1 and 25 using Euler’s formula: cos(at) = (eiat + e − iat) / 2.

27

Also from 1 and 25.

28

Here, n is a natural number. δn(ω) is the n-th distribution derivative of the Dirac delta. This rule follows from rules 7 and 24. Combining this rule with 1, we can transform all polynomials.

29

Here sgn(ω) is the sign function; note that this is

consistent with rules 7 and 24.

30

Generalization of rule 29.

31

The dual of rule 29.

32

Here u(t) is the Heaviside unit step function; this follows from rules 1 and 31.

33

u(t) is the Heaviside unit step function and a > 0.

34

The Dirac comb — helpful for explaining or understanding the transition from continuous to discrete time.

Fourier transform properties Notation: Conjugation

denotes that f(x) and F(ω) are a Fourier transform pair.

Scaling

Time reversal Time shift Modulation (multiplication by complex exponential)

Multiplication by sin ω0t

Multiplication by cos ω0t

Integration

Parseval’s theorem

Signal (electrical engineering) In the fields of communications, signal processing, and in electrical engineering more generally, a signal is any time-varying quantity. Signals are often scalar-valued functions of time (waveforms), but may be vector valued and may be functions of any other relevant independent variable. The concept is broad, and hard to define precisely. Definitions specific to subfields are common. For example, in information theory, a signal is a codified message, ie, the sequence of states in a communications channel that encodes a message. In a communications system, a transmitter encodes a message into a signal, which is carried to a receiver by the communications channel. For example, the words “Mary had a little lamb” might be the message spoken into a telephone. The telephone transmitter converts the sounds into an electrical voltage signal. The signal is transmitted to the receiving telephone by wires; and at the receiver it is reconverted into sounds. Signals can be categorized in various ways. The most common distinction is between discrete and continuous spaces that the functions are defined over, for example discrete and continuous time domains. Discrete-time signals are often referred to as time series in other fields. Continuous-time signals are often referred to as continuous signals even when the signal functions are not continuous; an example is a square-wave signal. A second important distinction is between discrete-valued and continuous-valued. Digital signals are discrete-valued, but are often derived from an underlying continuous-valued physical process.

Discrete-time and continuous-time signals If for a signal, the quantities are defined only on a discrete set of times, we call it a discrete-time signal. In other words, a discrete-time real (or complex) signal can be seen as a function from the set of integers to the set of real (or complex) numbers. A continuous-time real (or complex) signal is any real-valued (or complex-valued) function which is defined for all time t in an interval, most commonly an infinite interval.

Analog and digital signals Less formally than the theoretical distinctions mentioned above, two main types of signals encountered in practice are analog and digital. In short, the difference between them is that digital signals are discrete and quantized, as defined below, while analog signals possess neither property.

Discretization Discrete signal One of the fundamental distinctions between different types of signals is between continuous and discrete time. In the mathematical abstraction, the domain of a continuous-time (CT) signal is the set of real numbers (or some interval thereof), whereas the domain of a discrete-time signal is the set of integers (or some interval). What these integers represent depends on the nature of the signal. DT signals often arise via of CT signals. For instance, sensors output data continuously, but since a continuous stream may be difficult to record, a discrete-time signal is often used as an approximation. Computers and other digital devices are restricted to discrete time.

Quantization Quantization (signal processing) If a signal is to be represented as a sequence of numbers, it is impossible to maintain arbitrarily high precision - each number in the sequence must have a finite number of digits. As a result, the values of such a signal are restricted to belong to a finite set; in other words, it is quantized.

Examples of signals •

•

•

•

•

Motion. The motion of a particle through some space can be considered to be a signal, or can be represented by a signal. The domain of a motion signal is onedimensional (time), and the range is generally three-dimensional. Position is thus a 3-vector signal; position and orientation is a 6-vector signal. Sound. Since a sound is a vibration of a medium (such as air), a sound signal associates a pressure value to every value of time and three space coordinates. A microphone converts sound pressure at some place to just a function of time, using a voltage signal as an analog of the sound signal. Compact discs (CDs). CDs contain discrete signals representing sound, recorded at 44,100 samples per second. Each sample contains data for a left and right channel, which may be considered to be a 2-vector (since CDs are recorded in stereo). Pictures. A picture assigns a color value to each of a set of points. Since the points lie on a plane, the domain is two-dimensional. If the picture is a physical object, such as a painting, it’s a continuous signal. If the picture a digital image, it’s a discrete signal. It’s often convenient to represent color as the sum of the intensities of three primary colors, so that the signal is vector-valued with dimension three. Videos. A video signal is a sequence of images. A point in a video is identified by its position (two-dimensional) and by the time at which it occurs, so a video signal has a three-dimensional domain. Analog video has one continuous domain dimension (across a scan line) and two discrete dimensions (frame and line).

•

Biological membrane potentials. The value of the signal is a straightforward electric potential (“voltage”). The domain is more difficult to establish. Some cells or organelles have the same membrane potential throughout; neurons generally have different potentials at different points. These signals have very low energies, but are enough to make nervous systems work; they can be measured in aggregate by the techniques of electrophysiology.

Frequency analysis Frequency domain It is often useful to analyze the frequency spectrum of a signal. This technique is applicable to all signals, both continuous-time and discrete-time. For instance, if a signal is passed through an LTI system, the frequency spectrum of the resulting output signal is the product of the frequency spectrum of the original input signal and the frequency response of the system.

Entropy Another important property of a signal (actually, of a statistically defined class of signals) is its entropy or information content.

Electrical Engineering

Electrical Engineering

The electrical engineering handbook

The electrical engineering handbook