Index [assets.cambridge.org]assets.cambridge.org/97805218/87755/index/9780521887755... ·...

Index

abs, see Matlab, command, absadaptive filters, 442adder, 223aliasing effect, 38, 41, 49, 50, 52, 369, 372, 456, 458,

475, 480, 504, 505, 508, 518, 540, 822anti-, filter, 44, 50, 145, 466

allpass filtertransfer function, 824

alternation theorem, 322analog filters

all-pole, 351, 356bandpass, 350bandstop, 350Butterworth, 351, 356

computation, 363frequency transformation, 360in Matlab, 399transfer function, 353

Chebyshev, 353, 356, 828computation, 363frequency transformation, 360in Matlab, 400transfer function, 356

design, 361elliptic, 357, 828

computation, 363frequency transformation, 360in Matlab, 400transfer function, 358

frequency transformation, 359, 360highpass, 350inverse Chebyshev, 406

in Matlab, 400ladder form, 828, 849, 852lattice realization, 848lowpass, 43, 350, 351specification, 350

analog signal processing, 5analog-to-digital conversion, 46AR model, see modeling theory, autoregressive modelARMA model, see modeling theory, moving average

autoregressive modelASIC, see implementation, application-specific

integrated circuit

autocorrelation method, see spectral estimation,parametric, autocorrelation method

autoregressive model, see modeling theory,autoregressive model

averaged periodogram, see spectral estimation,nonparametric, averaged periodogram

Bartlett bisection theorem, 848, 849Bartlett window, 295, 338Bessel function, 301bias, see estimation theory, biasbilinear transformation, see digital filters, IIR, bilinear

transformation methodbiorthogonal lapped transform, 573bit-reversal ordering, 180, 183, 186, 203Blackman window, see window functions, BlackmanBlackman–Tukey, see spectral estimation,

nonparametric, Blackman-Tukeyblock transforms, 548butterfly computation, 181, 190, 192Butterworth filter, see analog filters, Butterworth and

digital filters, IIR, Butterworth approximation

canonic form networks, 223, 234, 235canonic signed-digit representation, see digital

representations, fixed-point, canonic signed-digitcascade realization, see digital filters, FIR, cascade

form and digital filters, IIR, cascade formCauer filters, see elliptic filterscausality

classification, 12, 13, 19, 20definition, 12

CDF, see random variable, cumulative distributionfunction

Chebyshev filter, see analog filters, Chebyshev anddigital filters, IIR, Chebyshev approximation

Chebyshev functions, 353Chebyshev optimization, see digital filters, FIR,

Chebyshev methodChebyshev polynomials, 306, 353Cholesky decomposition, 431circulant matrix, 481comb filter, 261

FIR, 276IIR, 262, 276

Cambridge University Press978-0-521-88775-5 - Digital Signal Processing System Analysis and Design: Second EditionPaulo S. R. Diniz, Eduardo A. B. da Silva and Sergio L. NettoIndexMore information

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878 Index

commutator models, 471, 509complementary filters, 343, 760convolution, continuous-time

time domain, 43convolution, discrete-time

circular, 212frequency domain, 293matrix form, 158time domain, 157, 158, 164, 166, 167, 205, 480

frequency domain, 123, 147in Matlab, see Matlab, command, convlinear, 14, 21, 101, 211

time domain, 157, 164, 165, 168, 169, 173notation, 15time domain, 14, 98, 123, 144using the discrete Fourier transform, 157z domain, 98, 100

cosine functiondiscrete-time, 7

cosine transformseven, 204

coupled formstate-space structure, 707

covariance method, see spectral estimation,parametric, covariance method

Cramer–Rao lower bound, see estimation theory,Cramer–Rao lower bound

CSD, see digital representations, fixed-point, canonicsigned-digit

DCT, see discrete cosine transformdecimation, 456, 458, 460, 465–467, 475, 480, 492,

494, 504, 505, 508, 624frequency response, 457in Matlab, see Matlab, command, decimateproperty

computational complexity, 459periodic time invariance, 459time variance, 459

decimation-in-frequency FFT algorithm, 184, 190decimation-in-time FFT algorithm, 179, 181, 190delay, 223DFT, see discrete Fourier transformDHT, see discrete Hartley transformdifference equations, 6, 62

annihilator polynomial, 26–28, 30auxiliary conditions, 17–20, 24, 29, 31, 32characteristic polynomial, 23–25, 27delay operator, 25Fibonacci equation, 23, 24homogeneous solution, 17–19, 22–27, 30nonrecursive, 21, 222particular solution, 17–19, 25–28recursive, 21representation, 17, 104

solution, 17, 18steady-state solution, 30

differential equations, 5, 17differentiator, 229, 231, 280, 283, 317, 333–336

in Matlab, see Matlab, command, diffdigital filters

allpass section, 259, 384bandpass, 229, 278, 283, 303, 317bandstop, 228, 278, 283, 303, 317building blocks

allpass, 257, 259bandpass, 257, 258comb filter, 261, 262highpass, 257highpass notch, 257, 259, 381lowpass, 257, 258lowpass notch, 257, 259oscillator, 260

chrominance, 274complementary, 343, 760decimator, 460, 461, 465, 475, 476, 492, 509, 510decimators, 518delay equalizer, 395doubly complementary, 822, 825, 826FIR

alternative direct form, 223cascade form, 224Chebyshev method, 321, 323, 325, 328, 331, 335,

341, 461, 476, 546, 559, 754, 755, 766, 778coefficients, 278constrained weighted least-squares method, 319definition, 21difference equation, 21, 222direct form, 223eigenfilter method, 319frequency response, 278frequency-response masking design, 477,

760–762, 765, 767frequency sampling design, 283, 286, 287,

290, 310frequency-domain form, 750interpolation and decimation design, 474,

475, 497interpolation design, 756, 759lattice form, 558, 740, 747lattice realization, 438linear-phase, 395maximally flat design, 309, 312minimum phase, 546modified-sinc filter, 753optimization methods, 313, 316polyphase form, 469, 507, 749prefilter design, 753, 755quadrature design, 771, 775recursive running sum, 750, 751, 754transfer function, 105






879 Index

weighted least-squares method, 317, 319, 320,328, 331, 340

window method, 291, 292, 297, 302, 304, 309,313, 337

WLS–Chebyshev method, 328–332, 561half-band, 545highpass, 228, 278, 283, 317IIR

analog prototype, 350, 368bilinear transformation method, 372–377, 397,

402, 828Butterworth approximation, 376, 399cascade form, 236, 239, 263, 703, 787, 790, 792,

796, 799, 812, 855characteristics, 349Chebyshev approximation, 376, 400Chebyshev or minimax criterion, 383definition, 21difference equation, 21direct form, 234, 263, 787, 817elliptic approximation, 113, 376, 400, 709,

711, 792impulse-invariance method, 368, 370–372,

377, 402inverse Chebyshev approximation, 400, 406lattice realization, 255, 815–820numerical optimization methods, 382,

387–390, 403parallel form, 237, 239, 240, 264, 703, 787, 788,

792, 799, 812prewarping procedure, 374, 376second-order section, 687, 788, 791, 792,

796, 798state-space description, 244, 245, 253, 265, 704,

705, 719, 733state-space section, 689, 728, 800, 801, 803, 806,

807, 810, 812time-domain approximation, 391, 394transfer function, 105, 232variable cutoff-frequency method, 381wave filters, see wave digital filters

implementationapplication-specific integrated circuit,

see implementation, application-specificintegrated circuit

digital signal processors, see implementation,digital signal processors

discrete hardware, see implementation, discretehardware

distributed arithmetic, see implementation,distributed arithmetic

field programmable gate array,see implementation, field programmablegate array

programmable logic devices, see implementation,programmable logic devices

very large scale integration, see implementation,very large scale integration

interpolator, 464, 465, 475, 476, 491, 509, 510, 756,757, 759

half-band filter, 465Lth-band filter, 464

linear-phase, 225, 226amplitude response, 285, 286forms, 227–229, 280, 283, 285, 286, 314, 315,

317, 323phase response, 285, 286properties, 231realizations, 232zeros, 231

lowpass, 113, 231, 278, 283, 317masking, 761, 762, 764, 773matrix representation, 241PCM, 325phase equalizer, 259, 384power complementary, 558prefilter, 753–755signal flowgraph, 223

digital representationsblock-floating-point, 697fixed-point, 673, 719

canonic signed-digit, 673, 732one’s-complement, 670–672, 697, 732sign-magnitude, 670, 732signed power-of-two, 672, 673two’s-complement, 670–676, 678, 683, 715, 732

floating-point, 673pseudo-floating-point, 711

digital signal processing, 5, 33, 46of continuous-time signals, 44

digital-to-analog conversion, 46practical case, 72

direct realizationFIR filters, 223IIR filters, 234, 263

discrete cosine transform, 548, 554, 563, 568application, 201, 206, 207butterfly, 202computational complexity, 202, 203definition, 199filter bank, 548, 550in Matlab, see Matlab, command, dctinverse, 200matrix notation, 554, 559, 566, 569, 570, 574

factorization, 202discrete Fourier transform, 143, 194

computation, 149, 150, 155, 159, 162computational complexity, 152, 175definition, 146, 148, 175, 210fast Fourier transform, see fast Fourier transformFFT, see fast Fourier transformfrequency resolution, 147






880 Index

discrete Fourier transform, (contd.)implementation

overlap-and-add, 168, 169, 480overlap-and-save, 171, 173

interpretation, 145, 149, 153inverse, 145, 148, 175, 182, 210, 290

computation, 155length, 144, 147, 153matrix notation, 151, 156, 183

factorization, 184period, 144property

circular convolution, 157complex conjugation, 159correlation, 158linearity, 153, 162modulation, 156of real and imaginary sequences, 159of symmetric and antisymmetric sequences, 160Parseval’s theorem, 163real sequences, 159time-reversal, 153time-shift, 153, 156

recursive computation, 220relation with

Fourier transform, 147z transform, 163

resolution, 147unitary, 196zero-padding, 147, 165, 168, 173, 750

discrete Hartley transformapplication, 206definition, 205inverse, 205property, 205

discrete transformscosine transforms, 204discrete cosine transform, 199discrete Hartley transform, 205Hadamard transform, 206Karhunen–Loève transform, 207unitary, 196, 199, 531Walsh–Hadamard transform, 206wavelet transform, 207

Dolph–Chebyshev window, see window functions,Dolph–Chebyshev

double sideband modulation, 777doubly complementary, 822downsampling, see decimationDSB, see double sideband modulationDSP, see implementation, digital signal processors

elliptic filters, see analog filters, elliptic and digitalfilters, IIR, elliptic approximation

entropy, 73, 437entropy, differential, 74

equiripple approximation for FIR filters, see digitalfilters, FIR, Chebyshev method

error spectrum shaping, 795ESS, see error spectrum shapingestimation theory, 410

Bayesian estimation, 410bias, 410, 411, 413, 414classic estimation, 410consistency, 410, 413Cramer–Rao lower bound, 411efficiency, 411mean squared error, 410standard deviation, 410variance, 410, 411, 414

even part of a sequence, 70exponential function

discrete-time, complex, 75discrete-time, real, 8

fast Fourier transform, 143, 175, 212, 213, 413basic cell, 181bit-reverse ordering, 180butterfly, 181, 186computational complexity, 175, 176, 178, 181,

190, 193decimation-in-frequency, 184, 190decimation-in-time, 179, 181, 190, 697FIR filtering, 750in Matlab, see Matlab, commands, fft, ifft,

fftshiftinverse, 182matrix notation

factorization, 184overlap-and-add, 750overlap-and-save, 750quantization effects, 697radix-2, 190, 697

algorithm, 176, 177butterfly, 179

radix-3butterfly, 190, 192

radix-4, 187, 190butterfly, 188

radix-N , 192Winograd, 193

FFT, see fast Fourier transformfilter banks, 503

wavelet transform relationship, 617, 628allpass decomposition, 822analysis

M -band, 517analysis filters, 507, 742biorthogonality, 518, 530, 531block transforms, 548, 551conjugate quadrature, 543, 544, 547, 548, 581, 585,

633, 638design, 544–547






881 Index

example, 587cosine-modulated, 543, 554, 559, 561, 563

design, 559implementation, 558

CQF, see filter banks, conjugate quadraturecritically decimated, 508, 599

M -band, 506using lattice form, 742

Daubechies 4, 657doubly complementary, 822extended lapped transform, 563Haar filter bank, 513, 638hierarchical decomposition, 599Johnston filter bank, 542lapped orthogonal transform, 563, 566–570,

588, 591fast algorithm, 568, 570, 573generalized, 563, 576, 577implementation, 567polyphase decomposition, 567

lapped transformbiorthogonal, 573, 574biorthogonal, generalized, 576

maximally decimatedM -band, 542

modulation matrix representation, 518, 5202-band, 540

near-perfect reconstruction, 540noncritically decimated, 508octave-band, 601, 605orthogonality, 518, 531, 533, 543, 544, 591, 620,

628, 633z domain, 533, 534

paraunitary, 534, 544, 548, 563, 566perfect reconstruction, 467, 507, 509–511,

516–518, 520, 527, 529, 531, 534, 536, 537,541, 543–545, 561, 563, 566, 574, 628, 640,742, 746

analysis filters, 513, 514, 536example, 513, 515, 516, 546M -band, 506, 509, 548, 556M -band, cascade, 534practical case, 507quadrature mirror, 542synthesis filters, 513, 514, 536transmultiplexer, 535two-band, 506, 535, 638two-band, design procedure, 537, 538using lattice form, 742, 744

polyphase decomposition, 513, 514, 517, 535, 536,541, 554, 556, 566, 567, 573, 742, 749

M -band, 507polyphase matrix, 743, 747power complementary, 822pseudo-quadrature mirror, 542QMF, see filter banks, quadrature mirror

quadrature mirror, 540–542design, 542implementation, 826in Matlab, see Matlab, command, qmfperfect reconstruction, 542two-band, 540

symmetric short kernel, 539, 638synthesis filters, 507, 742time-domain analysis, 521

finite impulsive-response filters, see digital filters, FIRFIR filters, see digital filters, FIRfloating-point representation, see digital

representations, floating-pointFortran, 325Fourier series, 36

coefficients, 34, 210convergence, 292definition, 34, 210discrete-time, 125periodic signal, 149truncation, 294

Fourier transform, 36, 46, 194continuous-time

definition, 34, 116, 209frequency sampling, 147inverse, 34, 43, 116, 209

definition, 607discrete-time, 115

computation, 117–119definition, 76, 117, 143, 209existence, 118inverse, 117, 145, 209, 288of impulse response, 43periodic signal, 123relation with the discrete Fourier transform, 147

of periodic sequence, 125property

complex conjugation, 121complex differentiation, 120convolution, 34frequency-domain convolution, 123linearity, 120modulation, 120, 777of real and imaginary sequences, 121of symmetric and antisymmetric sequences,

122, 123Parseval’s theorem, 123time-domain convolution, 123time-reversal, 120time-shift, 120, 124

sampling of, 144FPGA, see implementation, field programmable gate

arrayfrequency response

computation, 110, 111computation from zeros and poles, 114






882 Index

frequency response (contd.)definition, 109generalized Hamming window, 296in Matlab, see Matlab, command, freqzKaiser window, 301of decimated signal, 457, 458of ideal bandpass filter, 280of ideal bandstop filter, 279of ideal highpass filter, 280of ideal lowpass filter, 278of interpolated signal, 462rectangular window, 295

frequency sampling, see digital filters, FIR, frequencysampling design

frequency transformationcontinuous-time domain, 359, 360discrete-time domain, 378

lowpass-to-bandpass, 380lowpass-to-bandstop, 381lowpass-to-highpass, 380, 381lowpass-to-lowpass, 379

in Matlab, see Matlab, commands, lp2lp,lp2hp, lp2bp, lp2bs

Gibbs’ oscillations, 292, 319gradient vector, 387grid of poles, 707group delay, 109, 110, 225

cascade IIR filters, 385computation, 110in Matlab, see Matlab, command, grpdelay

Haar filter bank, 513, 638Haar wavelet, see wavelet transform, HaarHadamard transform

application, 206definition, 206fast algorithm, 207

half-band filter, see digital filters, interpolator,half-band filter

Hamming window, see window functions, HammingHann window, see window functions, HannHanning window, see window functions, HannHartley tranform, see discrete Hartley transformHessian matrix, 387, 388, 428Hilbert transformer, 229, 231, 281, 283, 317, 319,

321, 336, 546, 581, 582impulse response, 283in Matlab, see Matlab, command, hilbert

Hölder inequality, 701, 702Hölder regularity, see wavelet transform, regularityhyperbolic-sine function, 300

IDFT, see discrete Fourier transform, inverseIIR filters, see digital filters, IIRimplementation

application-specific integrated circuit, 668

digital signal processors, 669discrete hardware

parallel adder, 684parallel multiplier, 684serial adder, 674serial multiplier, 676, 678, 679, 681, 683serial subtractor, 675

distributed arithmetic, 685–687, 689, 728fixed-point, 693, 703, 709, 711floating-point, 696pipelining, 678, 681very large scale integration, 668

implementation, field programmable gate array, 668implementation, programmable logic devices, 668impulse function, 7

delayed, 7, 8, 14train of, 34, 35, 46, 47, 51, 52

Fourier series of, 36Fourier transform of, 36

impulse response, 14, 15, 21, 22, 31–33, 43, 44, 46,52, 53, 62, 84, 93, 126, 129, 211

analysis filter, 521antisymmetric, 228, 229, 286finite-duration, 21

linear phase, 226from state-space description, 244in Matlab, see Matlab, command, impzof differentiator, 334of ideal bandpass filter, 280of ideal bandstop filter, 279of ideal differentiator, 280of ideal highpass filter, 280of ideal Hilbert transformer, 283of ideal lowpass filter, 278symmetric, 227, 228, 285

impulse-invariance method, see digital filters, IIR,impulse-invariance method

infinite impulse-invariance filters, see digital filters,IIR

interpolation, 462, 465–468, 475, 491, 505, 761formulas, 44, 148, 324frequency response, 462in Matlab, see Matlab, command, interpLagrange, barycentric-form, 324property

computational complexity, 464time variance, 464

interreciprocity, 249, 250

Kaiser window, see window functions, KaiserKarhunen–Loève transform, 207Kuhn–Tucker conditions, 561

Lth-band filter, see digital filters, interpolator,Lth-band filter

Lagrange interpolator, 324






883 Index

Lagrange multiplier, 417, 561Laplace transform, 75, 208lapped orthogonal transform, see filter banks, lapped

orthogonal transformlattice realization, see digital filters, FIR, lattice form

and digital filters, IIR, lattice formLaurent series, 77Lawson algorithm, 328least-squares filter design, see digital filters, FIR,

weighted least-squares methodLevinson–Durbin algorithm, 431, 432, 434, 435, 437,

438, 449Lim–Lee–Chen–Yang algorithm, 328limit cycles, see quantization errors, limit cycleslinear prediction, see spectral estimation, parametric,

linear predictionlinearity

classification, 12, 13, 20definition, 11of the discrete Fourier transform, 153of the Fourier transform, 120z transform, 94

Lipschitz continuity, 633LU decomposition, 431Lyapunov function, 720, 725

MA model, see modeling theory, moving averagemodel

magnitude responsecascade IIR filters, 385computation, 110computation from zeros and poles, 114definition, 76, 109

magnitude truncation, see quantization errors,arithmetic quantization, magnitude truncation

Matlabcommandabs, 130, 135angle, 130, 135appcoef, 654, 663appcoef2, 664aryule, 449bartlett, 338bi2de, 735bilinear, 402bin2num, 735binvec2dec, 735biorfilt, 661biorwavf, 661blackman, 298, 339boxcar, 338burg, 450buttap, 400butter, 395, 399butterord, 395buttord, 399

ceil, 735cfirpm, 341cheb1ap, 400cheb1ord, 400cheb2ap, 400cheb2ord, 400chebwin, 339cheby, 395cheby1, 400cheby2, 400chebyord, 395coifwavf, 660conv, 67, 211, 212cov, 450cplxpair, 264, 268cremez, 341dbwavf, 660dct, 216dctmtx, 216de2bi, 735dec2bin, 735dec2binvec, 735decimate, 495, 496detcoef, 654, 664detcoef2, 664dftmtx, 215diff, 336dwt, 661dwt2, 664dwtmode, 662dyaddown, 594dyadup, 594ellip, 395, 400, 792ellipap, 401ellipord, 395, 401, 792fft, 215fftfilt, 215fftshift, 215filter, 211, 269fir1, 305, 337fir2, 338fircls, 340fircls1, 340firls, 321, 331, 340firpm, 325, 326, 331, 335, 336, 341, 778firpmord, 341, 778fix, 735floor, 735fminsearch, 591fminunc, 591freqspace, 136freqz, 130, 132, 135, 793grpdelay, 130, 136, 395hamming, 298, 338hanning, 298, 339hilbert, 337






884 Index

Matlab (contd.)idct, 216idwt, 662idwt2, 664ifft, 215impinvar, 402impz, 67interp, 495, 496intfilt, 497invfreqs, 403invfreqz, 403kaiser, 339kaiserord, 339latc2tf, 781, 857latcfilt, 781levinson, 449lp2bp, 401lp2bs, 402lp2hp, 401lp2lp, 401lpc, 403, 450maxflat, 403num2bin, 735orthfilt, 661periodogram, 450poly, 267prony, 403psd, 446, 450psdopts, 450qmf, 594quant, 735rand, 129randn, 65, 129real, 212remez, 341remezord, 341resample, 496residue, 129, 264residuez, 267, 792, 857roots, 131, 267round, 735sos2ss, 269, 857sos2tf, 267, 857sos2zp, 269spectrum, 446, 450sptool, 341, 403ss2sos, 269, 857ss2tf, 268, 857ss2zp, 269stem, 67stmcb, 403symwavf, 660tf2latc, 781, 857tf2sos, 794tf2ss, 268, 857tf2zp, 131, 264, 266

tf2zpk, 585triang, 338unwrap, 136upcoef, 664upcoef2, 664upfirdn, 495upwlev, 663upwlev2, 664wavedec, 662wavedec2, 664wavedemo, 664wavefun, 664waveinfo, 660wavemenu, 664waverec, 663waverec2, 664welch, 450wfilters, 660wrcoef, 664wrcoef2, 664yulear, 450yulewalk, 403zp2sos, 264, 268zp2ss, 269zp2tf, 131, 132, 266zplane, 137, 585

toolboxFixedpoint, 735Optimization, 591Signal Processing, 67, 135, 215, 266, 341, 495Spectral Estimation, 449Wavelet, 594, 656, 659, 660, 664

maximally flat design, see digital filters, FIR,maximally flat design

MIMO, see multiple-input, multiple-output systemminimax approach, see digital filters, IIR, Chebyshev

approximationminimum-phase stable ARMA model, 421minimum-variance, see spectral estimation,

nonparametric, minimum-variancemodeling theory, 419

autoregressive model, 419–422, 424, 426, 437, 442,589, 591

moving average autoregressive model, 419–422,424, 426, 442

moving average model, 419, 420, 423, 426, 442Wiener filter, see Wiener filterYule–Walker equations, 423, 426, 429, 432

autoregressive model, 424moving average autoregressive model, 424moving average model, 424

modified-sinc filter, see digital filters, FIR,modified-sinc filter

Moiré effect, 72moving average autoregressive model, see modeling

theory, moving average autoregressive model






885 Index

moving average model, see modeling theory, movingaverage model

MPEG2 standard, 548multiple-input, multiple-output system, 480, 486multiplier, 223

Newton method, 388noble identities, 467, 469, 601non-Toeplitx matrix, 431number theoretic transform, 194Nyquist frequency, 37, 64, 369

odd part of a sequence, 70one’s-complement representation, see digital

representations, fixed-point, one’s-complementorder estimate, 347overflow, 693, 697, 698, 700, 717, 718, 729overlapped block filtering, 479, 480, 483, 484, 487

parallel multiplier, 684parallel realization, see digital filters, IIR, parallel

formparaunitary matrix, 534, 544, 547, 548, 566Parks–McClellan algorithm, see digital filters, FIR,

Chebyshev methodParseval’s theorem, 195, 200, 610

discrete Fourier transform, 163discrete transforms, 195Fourier transform, 123, 608, 612z transform, 100

partial-fraction expansion, 88–90, 129, 238, 267Pascal matrix, 398PDF, see random variable, probability density functionperiod, 9, 144periodic signals, 9periodogram, 411, see spectral estimation,

nonparametric, periodogramphase response, 109

computation, 110computation from zeros and poles, 114definition, 76, 109digital filters, linear, 225, 395FIR filters, linear

conditions, 226pipelining, 678, 681PLD, see implementation, programmable logic

devicespoles, see transfer function, polespolynomial division, 90, 91polyphase decomposition, see digital filters, FIR,

polyphase formprewarping procedure, see digital filters, IIR,

prewarping procedurePSD, see random process, power spectral densitypseudo-circulant matrix, 481, 483, 485, 486, 488,

489, 491

quantization errorsanalog-to-digital conversion, 669arithmetic quantization, 669, 691, 694, 698,

789–791, 805, 810–812controlled rounding, 727magnitude truncation, 691, 692, 725, 727, 806overflow, 719rounding, 691, 698, 719saturation, 806truncation, 691, 719

coefficient quantization, 669, 706, 708, 709, 711,713, 788, 792, 812, 820

instability, 729limit cycles

constant-input, 725–728, 806, 807, 812granular, 715, 720overflow, 715, 717, 718zero-input, 719–721, 723, 726, 728

signal scaling, 697, 700, 702–704, 717, 788, 790,792, 806, 810, 818

saturation, 717quasi-Newton methods, 388

Broyden–Fletcher–Goldfarb–Shannon algorithm,389, 390

Davidson–Fletcher–Powell algorithm, 388

ramp function, 8random process, 58, 62, 125, 703

autocorrelation function, 58–60, 62, 125, 411, 413,426, 490, 589

autocorrelation matrix, 60, 428, 431, 440, 491, 492cross-correlation function, 59ensemble, 58ergodicity, 60joint wide-sense stationarity, 59, 429, 441, 589power spectral density, 125–128, 411, 589, 693realization, 58sample, 58strict-sense stationarity, 59white noise, 128, 129, 419, 424wide-sense cyclostationarity, 490–493wide-sense stationarity, 58–62, 125–127, 411, 419,

430, 491–493Wiener–Khinchin theorem, 126, 128

random processesautocorrelation function, 62

random signal, 53, 58, 60, 126random variable, 54, 58, 62

cross-correlation, 57cross-correlation, complex, 57cross-covariance, 57cumulative distribution function, 55

properties, 55energy, 56independence, 57joint cumulative distribution function, 57






886 Index

random variable, (contd.)joint probability density function, 57, 58, 491mean-squared value, 56probability density function, 55, 56, 57, 62

properties, 55continuous uniform distribution, 55, 59discrete uniform distribution, 55, 56Gaussian distribution, 55, 65

standard deviation, 56statistical mean, 56, 60–62variance, 56, 60, 62

reciprocity, 248rectangular window, see window functions,

rectangularrecursive filters, see digital filters, IIRrelative noise variance, 694relative power spectral density, 694relative power spectrum density, 797Remez exchange algorithm, see digital filters, FIR,

Chebyshev methodresidue, 84, 86, 87

definition, 83in Matlab, see Matlab, command, residuez

residue theorem, 83, 84, 86, 87, 705, 798rounding, see quantization errors, arithmetic

quantization, roundingRRS, see digital filters, FIR, recursive

running sum

sampling frequency, 37, 38, 41, 42, 49, 63, 64,145, 602

change, 455, 456, 465, 493in Matlab, see Matlab, command, resample

sampling rate, see sampling frequencysampling theorem, 38, 42, 64, 369saturation arithmetic, 717, 718, 732scaling function, 605Schwartz inequality, 610, 701sensitivity, 250, see transfer function, sensitivity

computation, 253, 255, 256serial adder, 674serial multiplier, 676serial subtractor, 675short-time Fourier transform, 608, 609, 612short-time spectrum, 220sign-magnitude representation, see digital

representations, fixed-point, sign-magnitudesignal flowgraph, 223, 241, 246

branch, 241delayless loop, 243, 828, 830, 832interreciprocity, 249, 250loop, 247matrix representation, 241node, 241

node ordering, 243reciprocity, 248source node, 241topology, 247transposition, 249, 250, 255

signal scaling, 735signals

random, see random signalcontinuous-time, 5, 33, 34, 44

band-limited, 38, 456discrete-time, 5, 6, 8, 33, 34

antisymmetric sequences, 121, 160causal, 81complex sine, 75, 109complex sinusoid, 333conjugate antisymmetric sequences, 121, 160conjugate symmetric sequences, 121, 160cosine, 7, 9critically decimated, 504decimated, 457, 505delayed impulse, 7, 8, 14even part, 70finite-length sequences, 80graphical representation, 7imaginary sequences, 97, 159impulse, 7interpolated, 462left-handed, one-sided sequences, 79, 82, 90, 91odd part, 70periodic, 9, 34, 148ramp, 8real exponential, 8real sequences, 97, 159right-handed, one-sided sequences, 79, 82, 90, 92step, 7, 8symmetric sequences, 121, 160two-sided sequences, 79, 82

signed power-of-two representation, see digitalrepresentations, fixed-point, signed power-of-two

sine functiondiscrete-time, complex, 75, 109

sine transformseven, 204

single sideband modulation, 780single-input, single-output system, 480, 486, 489SISO, see single-input, single-output systemspectral estimation, 409

nonparametric, 411averaged periodogram, 413, 414, 443Blackman–Tukey, 414, 415, 445, 447minimum-variance, 416, 418, 445periodogram, 411–414, 416, 426, 443windowed periodogram, 413, 414, 445

parametric, 426, 446autocorrelation method, 431, 432, 437, 446, 447autoregressive model, 426, 429, 431, 432, 434






887 Index

Burg’s method, 436, 437, 446, 447covariance method, 430, 431, 437, 442Itakura–Saito method, 442linear prediction, 426, 428, 429, 434, 439MUSIC algorithm, 442Prony’s method, 442subspace methods, 442

SPT, see digital representations, fixed-point, signedpower-of-two

SSB, see single sideband modulationSSS, see random process, strict-sense stationaritystability

classification, 369with overflow nonlinearity, 729

stability, BIBOclassification, 16, 93, 106–109definition, 16property, 81, 93

standard deviation, see estimation theory, standarddeviation

state-space descriptioncomputation, 245, 253definition, 244, 245

step function, 7, 8subsampling, see decimationsystem decomposition theorem, 420systems

continuous-time, 6sample-and-hold, 44, 46

discrete-time, 5, 10causality, 12, 13, 19, 20, 31, 44FIR, 21IIR, 21linearity, 11–14, 18, 20, 60, 486, 488nonrecursive, 21recursive, 21, 22representation, 17shift invariance, 11stability, 16, 30time invariance, 11–14, 16, 20, 60, 486, 488

Taylor series, 92, 638Tellegen’s theorem, 247, 250, 251time invariance

classification, 12, 13, 20definition, 11

time–frequency analysis, 607, 612, 613Toeplitx matrix, 431Toeplitz-plus-Hankel matrix, 330transfer function

allpass filter, 824allpass section, 259, 384Butterworth analog filter, 353Chebyshev analog filter, 356computation, 112

definition, 105delay equalizer, 395elliptic analog filter, 358FIR filters, 105from state-space description, 244, 801IIR filters, 105, 232phase equalizer, 259, 384poles, 81, 84, 105, 106, 114, 245rational, 81sensitivity, 250, 253, 255, 256, 706, 708, 709,

711, 713state space, 254zeros, 81, 105, 114, 231, 790

transmultiplexers, 534transposition, 249, 250, 255triangular window, see window functions,

triangulartruncation, see quantization errors, arithmetic

quantization, truncationtwo’s-complement representation, see digital

representations, fixed-point, two’s-complementtwo-scale difference equations, 660, 661

uncertainty principle, 610unitary transforms, see discrete transforms, unitaryupsampling, see interpolation

variance, see estimation theory, varianceVLSI, see implementation, very large scale

integration

Walsh–Hadamard transformapplication, 206definition, 206fast algorithm, 207

warping effect, 374wave digital filters, 828, 830

lattice form, 848lattice realization, 848, 849, 851, 852n-port elements

parallel adaptor, 839series adaptor, 844

one-port elementscapacitor, 832inductor, 833open-circuit, 833resistor, 833series connection, 833short-circuit, 833voltage source, 833

two-port elementscurrent generalized immittance converter, 836gyrator, 836parallel adaptor, 838, 839transformer, 836voltage generalized immittance converter, 833






888 Index

wavelet transform, 207, 599, 604, 612, 6389–7 wavelet, 654application, 641, 646, 653biorthogonality, 617, 620, 622, 623, 632–634,

639, 654computation of, 607continuous-time, 612Daubechies, 638, 640Daubechies 4, 657definition, 604discrete-time, 614–616, 619filter bank relationship, 617, 628, 635Haar, 638image processing, 641, 643, 646in Matlab, see Matlab, command,

wfiltersinverse, 605mother function, 599, 604, 605, 612, 619multiresolution representation, 617, 619, 620,

622, 635of discrete-time signal, 605orthogonality, 616, 619, 620, 622, 628, 632,

634, 639regularity, 633–636

estimation, 635examples, 636

scaling function, 605–607, 620, 633, 635, 636,638, 643

time–frequency analysis, 607, 612two-dimensinal, 641two-scale difference equations, 660, 661vanishing moments, 636, 637

weighted least-squares method, see digital filters, FIR,weighted least-squares method

white noise, 128, 129, 413Wiener filter, 439, 440

applications, 441computation, 440, 441

Wiener–Hopf equation, 428–430, 432, 434, 449Wiener–Khinchin theorem, 126, 411, 412, 426window functions, 608

Bartlett, 338, 412definition, 295

Blackman, 335, 339computation, 299definition, 297properties, 297

Dolph–Chebyshev, 339definition, 307properties, 309

Gaussian, 612generalized Hamming

definition, 296frequency response, 296properties, 296

Hamming, 338, 415, 431computation, 299definition, 296

Hann, 339computation, 299definition, 296

Kaiser, 339computation, 304definition, 301Fourier transform, 301frequency response, 301order estimation, 339

rectangular, 319, 334, 338, 412, 416computation, 299definition, 294frequency response, 295

triangular, 338definition, 295

windowed periodogram, see spectral estimation,nonparametric, windowed periodogram

Winograd Fourier transform, 193WLS method, see digital filters, FIR, weighted

least-squares methodWold decomposition, 419WSCS, see random process, wide-sense

cyclostationarityWSS, see random process, wide-sense

cyclostationarity

Yule–Walker equations, see modeling theory,Yule–Walker equations

z transformcomputation, 77, 80, 81definition, 76, 208inverse, 116, 208

computation, 84, 86–93definition, 83

Laurent series, 77of finite-length sequences, 163one-sided, 76property

complex conjugation, 96complex differentiation, 95initial value theorem, 97linearity, 94modulation, 95of real and imaginary sequences, 97time-domain convolution, 98, 105time-reversal, 94, 101time-shift, 95, 104, 130z-domain convolution, 98

region of convergence, 77, 85, 94–100, 106computation, 78, 80, 81of finite-length sequences, 80






889 Index

of left-handed, one-sided sequences, 79, 82of rational transfer functions, 82of right-handed, one-sided sequences, 79, 82of stable causal systems, 81of two-sided sequences, 79, 82

relation with the discrete Fourier transform, 164

table of, 101two-sided, 76

zero-padding, see discrete Fourier transform,zero-padding

zeros, see transfer function, zerosZolotarev filters, see elliptic filters






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