Index

Symbols

δ-function, 5, 67, 78, 81, 83, 205, 242

discrete-time -, 82

δ-pulse, 51

a.c. component, 27

aliasing, 86

distortion, 52

amplifier

front-end -, 146

low-noise -, 146

optical -, 145

amplitude modulated (AM) signal, 101

amplitude shift keying (ASK), 112, 173, 190

analog-to-digital conversion, 54

analytic

function, 180, 181

signal, 102

antenna noise, 144

anti-causal function, 180, 181

antipodal signals, 222

attenuation, 229

autocorrelation function, 11, 69

properties of -, 13

autocorrelation of discrete-time process, 32, 57, 90

autocovariance of discrete-time process, 32

available

power gain, 138

spectral density, 134

spectral density in amplifiers, 138

average

time -, 14

band-limited process

definition of -, 74

band-limiting filter

definition of -, 74

bandpass

filter, 73

signal, 101

bandpass filtering

of white noise, 110

bandpass process, 44

conversion to baseband, 119

definition of -, 74

direct sampling of -, 119

properties of -, 107

bandwidth

of bandpass process, 44

of low pass process, 43

root-mean-squared (r.m.s.) -, 43

baseband process

definition of -, 74

Bayes' theorem, 155

Bessel function, 113

biorthogonal signals, 225

binary detection

in Gaussian noise, 158

binary phase shift keying (BPSK), 191

bipolar nonreturn-to-zero (NRZ) signal, 98

bit error probability, 155, 157

Boltzmann's constant, 130, 247

Brownian motion, 130, 133

Butterworth filter, 98

Campbell's theorem, 201

extension of -, 204, 206

CATV networks, 225

causality, 68

chain rule, 232

characteristic frequency, 104, 112

characteristic function, 194, 204

joint -, 202

of shot noise, 201

second -, 194

second joint -, 202, 204

clutter, 207

code division multiple access (CDMA), 49

coloured noise, 130

complementary error-function, 158, 243

complete orthonormal set, 216

complex envelope, 102, 105, 111

spectrum of -, 105, 111

complex impulse response, 105

complex processes, 30, 111

applied to current, 31

applied to voltage, 31

conditional probabilities, 154, 155, 159, 163

constant resistance network, 151

constants

mathematical -, 247

physical -, 247

convolution, 67, 68, 160, 245

discrete -, 82, 90

correlation, 240

coefficient, 28

receiver, 171

correlation function, 11

measurement of -, 24

cost matrix, 165

covariance function

auto-, 26

cross-, 26

cross-correlation, 220, 222

of discrete-time process, 32, 90

cross-correlation function, 19, 70

properties of -, 20

time averaged -, 21

cross-covariance of discrete-time process, 32

cross-power spectrum, 45

properties of -, 46

cumulant, 197

cumulative probability distribution function, 9

cyclo-stationary process, 16, 77

data signal, 19, 77, 154, 157

autocorrelation function of -, 78

spectrum of -, 78, 79

d.c. component, 27, 203

de-emphasis, 97

decibel, 118, 145, 229

list of - values, 230

decision

criterion, 165

regions, 155, 156

statistics, 162

threshold, 155, 160

decision rule

Bayes -, 165

maximum a posteriori (MAP) -, 165

maximum likelihood (ML) -, 165

minimax -, 165

Neyman-Pearson -, 165

demodulation

coherent -, 48

synchronous -, 48, 118

derivatives, 232

detection

of binary signals, 154

optimum -, 162

differentiating network, 96

differentiation, 232

digital signal processor (DSP), 50, 184

discrete Fourier transform (DFT), 84

discrete-time

process, 5, 54, 193

signal, 82, 86

system, 82, 86

discrete-time Fourier transform (DTFT), 83

discriminator, 103

distance

measurement, 21

metric, 162

distortionless transmission, 121

distribution function, 9

diversity, 188

doubly stochastic process, 205

duality, 245

dynamic range, 55

effective noise temperature, 139

eigenfunctions, 67

electrons, 193, 198, 207

ensemble, 3

mean, 3

envelope, 102

detection, 106, 111

distribution, 113

equalization, 99

equivalent

noise bandwidth, 75, 140

noise resistance, 134

noise temperature, 134, 138

equivalent baseband

system, 105

transfer function, 104

Erbium doped fiber amplifier (EDFA), 145

ergodic

jointly -, 21

ergodic process

definition of -, 14

ergodicity of discrete-time process, 33

error probability, 155, 164

estimation error, 176, 178

excess noise factor, 206

expansion

McLaurin -, 237

Taylor -, 237

expectation, 239

false alarm, 165

fast Fourier transform (FFT), 84

filter

non-recursive -, 88

recursive -, 89

tapped delay line -, 88

transversal -, 88

filtering

of processes, 68

of thermal noise, 135

finite impulse response (FIR) filter, 82, 89

FM

detection, 97

signal, 101

Fourier series, 57, 218

Fourier transform, 39, 57, 67, 71, 194

discrete -, 82, 84

properties of -, 245

table, 246

two-dimensional -, 202

frequency conversion, 49

frequency shift keying (FSK), 112, 161, 190

Friis' formulas, 143

front-end amplifier, 146

Gaussian noise, 158, 162, 163, 167, 219

Gaussian processes, 27, 72

bandpass -, 112

jointly -, 27

properties of -, 29

Gaussian random variables, 27

covariance matrix of -, 28

jointly -, 28

properties of -, 29

Gram-Schmidt orthogonalization, 218

group delay, 122

guard band, 62

harmonic signal, 218

Hermitian spectrum, 104

hypothesis testing, 154, 161

impulse response, 67, 68, 71, 87, 199, 201

complex -, 105

of matched filter, 161

of optimum filter, 167

independence, 239

independent processes, 10, 21

infinite impulse response (IIR) filter, 82, 89

information signal, 1, 205

inner product

of signals, 216

of vectors, 215

integrals

definite -, 236

indefinite -, 233

integrate-and-dump receiver, 175

interarrival time, 197

interpolation by sinc function, 52

intersymbol interference, 99

inverse discrete Fourier transform (IDFT), 84

inverse discrete-time Fourier transform (IDTFT), 83

inverse fast Fourier transform (IFFT), 84

irrelevant noise, 160, 221

jitter, 175

Kronecker delta, 216

Laplace transform, 179

light emitting diode (LED), 50

likelihood ratio, 155

linear time-invariant (LTI)

filter, 65, 160, 201

system, 66

logarithms, 238

Lorentz profile, 42

low-noise amplifier, 146

M-ary biorthogonal signals, 225

M-ary detection

in Gaussian noise, 161

M-ary phase modulation, 224

Manchester code, 98

matched filter, 161, 162, 167

for coloured noise, 167

matched impedances, 138

matching network, 149

mathematical

constants, 247

relations, 231

maximum ratio combining, 189

McLaurin expansion, 237

mean

ensemble -, 3

frequency, 44

of discrete-time process, 31

value, 10

mean-squared error, 52, 178

minimization of -, 176

miss, 165

mixer, 117

modems

cable -, 225

telephone -, 225

modulation, 47

amplitude -, 101

by random carrier, 49

frequency -, 101

phase -, 102

moment generating function, 196

moments of random variable, 196

multiamplitude signals, 224

multiphase signals, 224

multiple-input multiple-output (MIMO) systems, 65

narrowband

bandpass process definition of -, 75

system definition of -, 74

neper, 229

noise, 1

coloured -, 130

Gaussian bandpass -, 111

in optical amplifiers, 145

in systems, 137

multiplicative -, 201

presentation in signal space, 219

thermal -, 130

vector, 159, 219

noise bandwidth

equivalent -, 76

noise figure

average -, 140

definition of -, 140

of a cable, 141

of an attenuator, 141

of cascade, 143

spot -, 140

standard -, 140

noise in amplifiers, 138

noise temperature

effective -, 139

of cascade, 143

system -, 143

noisy amplifier model, 139

nonreturn-to-zero (NRZ) signal

bipolar -, 98

polar -, 79, 98

norm of a vector, 215

Norton equivalent circuit, 131

Nyquist frequency, 51

Nyquist's theorem, 136

optical amplifier

Erbium doped fiber -, 145

semiconductor -, 145

optical signal detection, 207

optimum filter characteristic, 177

optimum smoothing filter, 178

orthogonal

processes, 21

quadrature processes, 109

random variables, 109

vectors, 215

orthogonal signals, 223

M-ary -, 225

orthonormal set, 158, 216

complete -, 216

orthonormal signals, 216

oscillator spectrum, 42

Parseval's formula, 168, 245

periodically stationary process, 16

phase

delay, 122

distribution, 5, 114, 115

shift, 229

phase modulation

M-ary -, 224

phase reversal keying (PRK), 191

phase shift keying (PSK), 190

phasor, 102

photodetector, 198, 205

photodiode, 193, 194

avalanche -, 194, 207

photomultiplier tube, 194, 207

photons, 193, 198, 205

physical

constants, 247

interpretation, 27

Planck's constant, 130

Poisson

distribution, 193

impulse process, 194, 205

processes, 193

sum formula, 81

Poisson processes

homogeneous -, 193, 198

inhomogeneous -, 194, 204

sum of independent -, 196

polar nonreturn-to-zero (NRZ) signal, 79, 98

posterior probabilities, 165

power, 40

a.c. -, 27

electrical -, 133

in output process, 71

maximum transfer of -, 138, 147

of discrete-time process, 57

of stochastic process, 133

power spectrum, 39

cross-, 45

measurement of -, 116

properties of -, 40

pre-emphasis, 97

pre-envelope, 102

prediction, 175, 179

discrete-time -, 184

pure -, 184

pure -, 179

prior probabilities, 154, 163, 165

probability density function, 10, 239

Gaussian -, 27, 239

joint -, 10

Laplacian -, 186

Poisson -, 193

probability distribution function, 239

joint -

Nth-order -, 10

second-order -, 10

process

bandpass -, 48, 106

stationary -

Nth-order -, 11

first-order -, 10

second-order -, 11

Q-function, 158, 164, 243

quadrature

components, 102, 106, 118

measurement of -, 118

description of bandpass processes, 106

description of modulated signals, 101

processes, 107–109

signals, 218

quadrature amplitude modulated (QAM) signals, 225

quadrature phase shift keying (QPSK), 224

quantization, 54

characteristic, 55

error, 55

levels, 55

noise, 56

step size, 55

quantizer, 55

non-uniform -, 57

uniform -, 56

queuing, 197

radar, 207

detection, 165

ranging, 22

random

gain, 205

signal, 1

variable, 2

vector, 155

random data signal, 77, 154

spectrum of -, 78

random point processes, 193

random sequence

continuous -, 5

discrete -, 7

random-pulse process, 205

Rayleigh-distribution, 113

RC-network, 71, 76, 137

realization, 3

reconstruction

of bandpass processes, 119

of sampled processes, 52

of sampled signals, 51

rectangular

function, 241

pulse, 79, 84, 201

rectifier, 103

relevant noise, 159, 221

return-to-zero signal

unipolar -, 80

Rice-distribution, 113

root-mean-squared (r.m.s.)

bandwidth, 43

value, 27

sample function, 3

sampling, 5

direct -, 119

flat-top -, 62

ideal -, 51

of bandpass processes, 119

rate, 51, 52, 119

theorem

for deterministic signals, 51

for stochastic processes, 52

Schwarz's inequality, 166, 216

semi-invariant, 197

semiconductor optical amplifier (SOA), 145

series, 237

shot noise, 199

signal

constellation, 219, 222

energy, 159, 162, 218

harmonic -, 218

restoration, 179

space, 161, 163, 216

vector, 159, 216

signal-to-noise ratio, 140, 164, 166

matched filter output -, 168

signal-to-quantization-noise ratio, 56

signal-to-shot-noise ratio, 201, 206

signum function, 241

simplex signal set, 163, 164, 227

sinc function, 52, 81, 84, 242

single-input single-output (SISO) systems, 65

smoothing, 175, 176

discrete-time -, 183

spectrum

analyzer, 116

of data signal, 77

of discrete-time process, 57

of filter output, 71

spill-over, 63, 99

split-phase code, 98

spread spectrum, 50

stable system, 87

stationary

points, 233

processes, 9

stochastic processes, 2

continuous -, 4

discrete -, 4, 6

discrete-time -, 4, 5, 31

independent -, 10

spectra of -, 39

strict-sense stationary process, 11

sufficient statistic, 160

superheterodyne, 116

switching center, 208

synchronization, 175

system

causal -, 179

optimal -, 153

stable -, 180

synthesis, 153

Taylor expansion, 237

Thévenin equivalent circuit, 131

thermal noise, 130

current in passive networks, 137

in passive networks, 131

spectrum of current in a resistor, 131

spectrum of voltage across a resistor, 130

voltage in passive networks, 136

time average, 14

transfer function, 67, 89

equivalent baseband -, 104

of bandpass filter, 104

triangular function, 242

trigonometric relations, 231

uncorrelated processes, 27

unipolar return-to-zero, 80

unit-step function, 241

variance, 26, 55, 197, 201, 204, 240

vector spaces, 215

voice channel, 62

waiting time, 197

white noise, 70, 75, 91, 129, 158, 163, 167, 203, 205, 219

bandpass filtering of -, 110

whitening filter, 169

wide-sense stationary, 219

jointly -, 175

wide-sense stationary processes, 71, 175, 206

definition of -, 12

discrete-time -, 57, 90

jointly -, 20, 70

Wiener filter, 154, 175

discrete time -, 183

Wiener-Hopf equation, 179

Wiener-Khinchin relation, 39

z-transform, 57, 86