elliott, d. f. (1987) handbook of digital signal processing - engineering applications

Thisdescription is provided without the detailed derivations that get one "lost inthe trees." ac-Many new useful ideas are presented in this handbook, including new finiteimpulse respons

Handbook of Digital Signal Processing Engineering Applications

Handbook of Digital Signal Processing

Engineering Applications

Edited byDouglas F ElliottRockwell International Corporation

Anaheim, California

ACADEMIC PRESS, INC

Harcourt Brace Jovanovich, Publishers

San Diego New York Berkeley Boston London Sydney Tokyo Toronto

ALL RIGHTS RESERVED

NO PART OF THIS PUBLICATION MAY BE REPRODUCED ORTRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC

OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, ORANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUTPERMISSION IN WRITING FROM THE PUBLISHER

ACADEMIC PRESS, INC.

San Diego, California 92101

United Kingdom Edition published by

ACADEMIC PRESS LIMITED

24-28 Oval Road, London NW1 7DX

Library of Congress Cataloging in Publication Data

Handbook of digital signal processing.

Includes index.

1 Signal processing-Digital techniques-Handbooks, manuals, etc I Elliott, Douglas F.

TK5102.5.H32 1986 621.38'043 86-26490 ISBN 0-12-237075-9 (alk, paper)

PRINTED IN THE UNITED STATES OF AMERICA

89 9 0 9 1 9 8 7 6 5 4 3 2

Acronyms and Abbreviations

Notation

XI xiii xvii

Chapter 1 Transforms and Transform Properties

DOUGLAS F ELLIOTT

I Introduction

II Review of Fourier Series

III Discrete-Time Fourier Transform

IV z-Transform

V Laplace Transform

VI Table of z-Transforms and Laplace Transforms

VII Discrete Fourier Transform

VIII Discrete-Time Random Sequences

IX Correlation and Covariance Sequences

X Power Spectral Density

XI Summary

References

1 2 6 16 24 27 27 41 45 50 51 53

Chapter 2 Design and Implementation of Digital FIR Filters

P P VAIDYANATHAN

I Introduction 55

II FIR Digital Filter Preliminaries 56

HI FIR Filter Design Based on Windowing 61

IV Equiripple Approximations for FIR Filters 71

V Maximally Flat Approximations for FIR Filters 90

VI Linear Programming Approach for FIR Filter Designs 95 VII Frequency Transformations in FIR Filters 100 VIII Two-Dimensional Linear-Phase FIR Filter Design and Implementation 112

IX Recent Techniques for Efficient FIR Filter Design 118

X Other Useful Types of FIR Filters 136

XI Summary 146 Appendix A Design Charts for Digital FIR Differentiators and Hilbert Transformers 147 Appendix B Program Listings for Linear-Phase FIR Filter Design 150

HI Data Rate Reduction (Desampling) by 1/Af Filters 208

IV Heterodyne Processing 223

V Interpolating Filters 234

VI Architectural Models for FIR Filters 245 VII Summary 252 Appendix Windows as Narrowband Filters 253

IV Digital Filter Realizations 295

V Frequency Domain Design 300

VI Analog Filter Design and Filter Types 306 VII Frequency Transformations 331 VIII Digital Filter Design Based on Analog Transfer Functions 332

IX Spectral Transformations 343

X Digital Filters Based on Continuous-Time Ladder Filters 344

XI Summary 353 Appendix HR Digital Filter CAD Programs 355

IV Dynamic Range Constraints and Scaling 373

V Signal-to-Roundoff-Noise Ratio in Simple HR Filter Structures 378

VI Low-Noise HR Filter Sections Based on Error-Spectrum Shaping 387 VII Signal-to-Noise Ratio in General Digital Filter Structures 395 VIII Low-Noise Cascade-Form Digital Filter Implementation 396

IX Noise Reduction in the Cascade Form by ESS 399

X Low-Noise Designs via State-Space Optimization 402

XI Parameter Quantization and Low-Sensitivity Digital Filters 412 XII Low-Sensitivity Second-Order Sections 416 XIII Wave Digital Filters 419 XIV The Lossless Bounded Real Approach for the Design of Low-

Sensitivity Filter Structures 434

XV Structural Losslessness and Passivity 443 XVI Low-Sensitivity All-Pass-Based Digital Filter Structures 444 XVII Digital All-Pass Functions 453 XVIII Orthogonal Digital Filters 458 XIX Quantization Effects in FIR Digital Filters 460

XX Low-Sensitive FIR Filters Based on Structural Passivity 465 XXI Limit Cycles in HR Digital Filters 469 References 475

Contents vii

Chapter 6 Fast Discrete Transforms

PAT YIP AND K RAMAMOHAN RAO

I Introduction

II Unitary Discrete Transforms

III The Optimum Karhunen Loeve Transform

IV Sinusoidal Discrete Transforms

V Nonsinusoidal Discrete Transforms

VI Performance Criteria

VII Computational Complexity and Summary

Appendix A Fast Implementation of DCT via FFT

Appendix B DCT Calculation Using an FFT

Appendix C Walsh-Hadamard Computer Program

References

481 482 483 485 499 510 516 517 521 523 523

Chapter 7 Fast Fourier Transforms

DOUGLAS F ELLIOTT

I Introduction

II DFTs and DFT Representations

III FFTs Derived from the MIR

IV Radix-2 FFTs

V Radix-3 and Radix-6 FFTs

VI Radix-4 FFTs

VII Small-JVDFTs

VIII FFTs Derived from the Ruritanian Correspondence (RC)

IX FFTs Derived from the Chinese Remainder Theorem

X Good's FFT

XL Kronecker Product Representation of Good's FFT

XII Polynomial Transforms

XIII Comparison of Algorithms

XIV FFT Word Lengths

XV Summary

Appendix A Small-AT DFT Algorithms

Appendix B FFT Computer Programs

Appendix C Radix-2 FFT Program

Appendix D Prime Factor Algorithm (PFA)

Appendix E Highly Efficient PFA Assembly Language Computer Program

References

527 528 532 553 558 564 565 567 571 573 574 579 580 587 595 596 600 602 605 621 630

Chapter 8 Time Domain Signal Processing with the DFT

FREDERIC J HARRIS

I Introduction

II The DFT as a Bank of Narrowband Filters

III Fast Convolution and Correlation

IV The DFT as an Interpolator and Signal Generator

V Summary

References

633 639 666 683 698 698

Chapter 9 Spectral Analysis

JAMES A CADZOW

I Introduction

II Rational Spectral Models

HI Rational Modeling: Exact Autocorrelation Knowledge

IV Overdetermined Equation Modeling Approach

V Detection of Multiple Sinusoids in White Noise

VI MA Modeling: Time Series Observations

VII AR Modeling: Time Series Observations

VIII ARMA Modeling: Time Series Observations

IX ARMA Modeling: A Singular Value Decomposition Approach

X Numerical Examples

XI Conclusions

References

701 702 707 714 716 721 723 724 726 731 739 739

Chapter 10 Deconvolution

MANUEL T SILVIA

I Introduction

II Deconvolution and LTI Systems with No Measurement Noise

III Deconvolution and the Identification of DTLTI Systems with

Measurement Noise

IV Fast Algorithms for Deconvolution Problems

V Some Practical Applications of Deconvolution

VI Summary

Appendix A References for Obtaining Computational Algorithms

Appendix B Implementing the Levinson or Toeplitz Recursion

Appendix C Implementing the Lattice Form of the Levinson Recursion

References

741 746

760 766 777 784 785 786 787 787

Chapter 11 Time Delay Estimation

V The Implementation of Some Time Delay Estimation Algorithms

Using the Fast Fourier Transform (FFT) 837

VI Algorithm Performance 844 VII Summary 853 References 853

Chapter 12 Adaptive Filtering

NASIR AHMED

I Introduction

II Some Matrix Operations

III A Class of Optimal Filters

IV Least-Mean-Squares (LMS) Algorithm

857

858 860 866

Contents IX

V LMS Lattice Algorithms

VI Concluding Remarks

Appendix Four FORTRAN-77 Programs

References

882 888 889 896

Chapter 13 Recursive Estimation

GENE H HOSTETTER

I Introduction

II Least Squares Estimation

III Linear Minimum Mean Square Estimation

IV Discrete Kalman Filtering Examples

II Digital Machine Fundamentals

III The Essence of Digital Signal Processing

IV Number Representations

V Hardware Components

VI Microprogramming

VII Keeping Things in Perspective

VIII Distributed Arithmetic

IX Summary

References

899 900 908 915 922 929 938 938

941 942 947 947 950 959 963 964 972 972

When Academic Press approached me with the proposal that I serve aseditor of a handbook for digital signal processing, I was aware of the need forsuch a book in my work in the aerospace industry Specifically, I wanted basicdigital signal processing principles and approaches described in a book that aperson with a standard engineering background could understand Also, Iwanted the book to cover the more advanced approaches, to outline the advan-tages and disadvantages of each approach, and to list references in which Icould find detailed derivations and descriptions of the approaches that might

be most applicable to given implementation problems

The various authors in this volume have done an outstanding job of complishing these goals Coverage of the fundamentals alone makes the bookself-sufficient, yet many advanced techniques are described in readable,descriptive prose without formal proofs Detailing fundamental approachesand describing other available techniques provide an easily understandablebook containing information on a wide range of approaches For example, thechapter on adaptive filters derives basic adaptive filter structures and providesthe reader with a background to "see the forest" of adaptive filtering Thechapter then describes various alternatives, including adaptive lattice struc-tures that might be applicable to particular engineering problems Thisdescription is provided without the detailed derivations that get one "lost inthe trees."

ac-Many new useful ideas are presented in this handbook, including new finiteimpulse response (FIR) filter design techniques, half-band and multiplierlessFIR filters, interpolated FIR (IFIR) structures, and error spectrum shaping.The advanced digital filter design techniques provide for low-noise, low-sensitivity, state-space, and limit-cycle free filters Filters for decimation andinterpolation are described from an intuitive and easily understandable view-point New fast Fourier transform (FFT) ideas include in-place and in-ordermixed-radix FFTs, FFTs computed in nonorthogonal coordinates, and primefactor and Winograd Fourier transform algorithms Transmultiplexing discus-sions carefully describe how to control crosstalk, how to satisfy dynamic rangerequirements, and how to avoid aliasing when resampling Using an over-determined set of Yule-Walker equations is a key concept described for reduc-ing data-induced hypersensitivities of parameters in model-based spectralestimation Tools are provided for understanding the basic theory, physics,

Preface

and computational algorithms associated with deconvolution and time delayestimation Recursive least squares adaptive filter algorithms for both latticeand transversal structures are compared to other approaches, and their advan-tage in terms of rapid convergence at the expense of a modest computationalincrease is discussed Extensions of Kalman filtering include square-root filter-ing The simplicity and regularity of distributed arithmetic are lucidly describedand are shown to be attractive for VLSI implementation.

There is some overlap in the material covered in various chapters, butreaders will find the overlap helpful For example* in Chapter 2 there is an ex-cellent derivation of FIR digital filters that provides the necessarymathematical framework, and in the first part of Chapter 3 there is an intuitiveexplanation of how various FIR filter parameters, such as impulse responselength, affect the filter performance Similarly, in Chapter 9 the Yule-Walkerequations are discussed in the context of spectral analysis, whereas in Chapter

10 these equations appear from a different viewpoint in the context of volution

decon-Many applications in digital signal processing involve the use of computerprograms After many discussions the chapter authors decided to includeuseful programs and to give references to publications in which related pro-gram listings can be found For example, Chapter 7 points out that a largepercentage of FFT applications are probably best accomplished with a radix-2FFT, and such an FFT is found in Appendix 7-C However, Appendixes 7-Dand 7-E present prime factor algorithms designed for IBM ATs and XTs Thelisting in Appendix 7-E is a highly efficient 1008-point assembly language pro-gram Other sources for FFTs are also listed in Appendix 7-B

The encouragement of Academic Press was crucial to the development ofthis book, and I would like to thank the editors for their support and advice Iwould also like to express my appreciation to Stanley A White for his behind-the-scenes contribution as an advisor, and to thank all of the chapter authorsfor their diligent efforts in developing the book Finally, I would like to thank

my wife, Carol, for her patience regarding time I spent compiling, editing, andwriting several chapters for the book

Isb Least significant bit

msb Most significant bit

ADC Analog-to-digital converter

AGC Automatic gain control

ALE Adaptive line enhancer

BRO Bit-reversed order

CAD Computer-aided design

CCW Counterclockwise

CO Coherent gain

CMOS Complementary metal-on-silicon

CMT C-matrix transform

CRT Chinese remainder theorem

CSD Canonic sign digit

DA Distributed arithmetic

DAC Digital-to-analog converter

DCT Discrete cosine transform

DFT Discrete Fourier transform

DF2 Direct-form 2

DIP Decimation-in-frequency

DIT Decimation-in-time

DPCM Differential pulse code modulation

DRO Digit-reversed order

DSP Digital signal processing

DST Discrete sine transform

DTFT Discrete-time Fourier transform

DTLTI Discrete-time linear time-invariant

DTRS Discrete-time random sequence

DWT Discrete Walsh transform

EFB Error feedback

ENBW Equivalent noise bandwidth

EPE Energy packing efficiency

ESS Error-spectrum shaping

FDM Frequency-division (domain) multiplexing

FDST Fast discrete sine transform

FFT Fast Fourier transform

FIR Finite impulse response

GT General orthogonal transform

HHT Hadamard-Haar transform

Acronyms and Abbreviations

HPF Highpass filter

HT Haar transform

IDFT Inverse discrete Fourier transform

IDTFT Inverse discrete-time Fourier transform

IFFT Inverse fast Fourier transform

IFIR Interpolated finite impulse response

IIR Infinite-duration impulse response

IQ In-phase and quadrature

IT Inverse transform; identity transform

LSA Least squares analysis

LSI Large-scale integration

LTI Linear time-invariant

MA Moving average

MAC Multiplier-accumulator

MFIR Multiplicative finite impulse response

MIR Mixed-radix integer representation

PFA Prime factor algorithm

PROM Programmable read-only memory

PSD Power spectrum density

PSR Parallel-to-serial register

QMF Quadrature mirror filter

RAM Random-access memory

RC Ruritanian correspondence

RCFA Recursive cyclotomic factorization algorithm

RHT Rationalized Haar transform

RLS Recursive least squares

ROM Read-only memory

RRS Recursive running sum

RT Rapid transform

SD Sign digit

SDSLSI Silicon-on-sapphire large-scale integration

SER Sequential regression

SFG Signal-flow graph

SNR Signal-to-noise ratio

Acronyms and Abbreviations xv

SPR Serial-to-parallel register

SR Shift register

SRFFT Split-register fast Fourier transform

SSBFDM Single-sideband frequency-division multiplexing

ST Slant transform

SVD Singular value decomposition

TDM Time division (domain) multiplexed

VLSI Very large-scale integration

WDF Wave digital filter

WFTA Winograd Fourier transform algorithm

WHT Walsh-Hadamard transform

WSS Wide-sense stationary

Symbol Meaning

a—b Give variable a the value of expression b (or replace a by b)

a, x, Lowercase denotes scalars

a, x, Underbar denotes a random variable

a*, x*, The complex conjugate of a x,

a k , b k , c k ,

Filter coefficients

a k Coefficients for the numerator polynomial of a transfer function, coefficients

of corresponding difference equation

b t , c it di, y, Elements of Jury's array for stability testing

b Number of bits used to represent the value of a number (does not include the

sign bit)

b k Coefficients of the denominator for polynomial of a transfer function,

coeffi-cients in the corresponding difference equation

c Recursive least squares scalar divisor, initial state mean

c, Scale factor given by

Cj

~ UA/2 i f / = O o r N cj(m) Autocovariance sequence for the discrete-time random sequence xj(n) where

d(n), g(n) Input output sequences

d(n) Hilbert transform of d(ri)

[d{n), jd(n)] Analytic signal

d(s,n) Data sequence where s is slow time index (identifies groups) and n is fast time

index (identifies position in a group)

e' UT Steady-state frequency domain contour in the z-plane

e(n) Error sequence

/ Frequency in hertz (Hz)

/„ Passband upper edge frequency in hertz

/, Stopband lower edge frequency in hertz

f r Stopband (rejection band) edge frequency in hertz

/, Sampling frequency in hertz;/, = \/T

/,' Resampling frequency

/(z) A linear factor (z - re 3 *)

f ' ( z ) A linear factor (rz - e^)

g(n), h(ri), Time domain scalars

h(n) Filter impulse response, filter coefficient, data sequence window

h,(t) Impulse response of an analog prototype filter

Notation

Transform sequence number, integer step index, selectivity parameter

Logarithm to the base e

Logarithm to the base 10 Logarithm to the base 2 rth multiplier coefficient Data sequence number (time index), system dynamic order

Data sequence from filter bank where k is the filter index and s is the time

index Magnitude of a complex number (pole, zero)

Autocorrelation sequence for the discrete-time random sequence x(n) where r x ,(m) = E[x(n)x*(n - m)]

Cross-correlation sequence for the discrete-time random sequences x(n) and y_(ri) where r ty (m) = E[x(ri)y*(n - m)]

Laplace transform variable, 5 = a + jui

Zeros of the inverse Chebyshev filters

10 otherwise

Unit step sequence defined by u(ri) =

Value of inductance or capacitance Input sequence; «th data sample Discrete-time random sequences

Time domain scalar-valued function at time t

z-transform independent variable, z = e sT , but used in this book for a

nor-malized sampling period of T = 1 unless otherwise indicated

Minimum stopband attenuation

Filter passband attenuation in decibels, A p = - 20 log, 0 5,

Minimum acceptable filter stopband attenuation in decibels, A r = 20 log, 0

5 2

Bit by bit addition of the binary numbers A and B

Steady-state frequency domain phase response Maximum allowable specified passband ripple in decibels BPF bandwidth (rad s' 1 )

Chebyshev polynomial of degree n (Chapter 4)

Distortion function The desired frequency response of a digital filter Denominator polynomial of a transfer function

The discrete Fourier transform of the sequence x(n) The discrete-time Fourier transform of the sequence x(n)

Expected value, expectation Approximation error spectrum

s

Analog frequency in hertz

A causal approximant to predictor z

Transform domain scalars

An intermediate complex variable Intermediate complex variable in cascade description Steady-state frequency response function for an analog prototype filter Steady-state frequency response function of a digital filter

Steady-state frequency domain magnitude response Spectral response (Chapter 3)

Transfer function of individual quadratic blocks in a parallel realization of

a digital filter Transfer function of a digital filter

Zero-phase part of linear-phase filter with (N + l)-point impulse response, H(z) = z-"'2 // 0 (z)

Wiener filter transfer function Complex conjugate of the point spread transfer function Window (filter) spectrum (Chapter 3)

Integer indices Discrete integration operator (1 + z")/(l - *-') The imaginary part of the quantity in brackets

The TV x N identity matrix

Modified zeroth-order Bessel function of the first kind Performance measure

Attenuation-related scale factor

The highest power of z in the numerator polynomial of a transfer function

7/(z), number of filter weights (coefficients)

Transform dimension order of a digital filter (the highest power of z in the

characteristic polynomial) Numerator polynomial of a transfer function Filter quality factor defined by ratio of center frequency to bandwidth

Quantized value of x(ri) where Q[x(ri)\ = x(ri) + e(n)

The real part of the quantity in brackets Chebyshev rational function

Spectrum of the autocorrelation sequence r f ,(m) where

Coefficient number A: in a series expansion of a peroidic sequence

The z-transform domain representation of the sequence x(n)

dX(z)/dz dX(e J ")/du (compare with above)

Symbol Meaning

a, b, x, Vectors are designated by lowercase boldfaced letters

a Measurement noise mean vector

b State noise mean, constant measurement bias

u Arbitrary vector

v Measurement noise vector, arbitrary vector

w State noise vector

x State vector

y Arbitrary vector

z Measurement vector

A, B, X, Matrices are designated by capital boldfaced letters

A Arbitrary matrix, noise-shaping filter state coupling matrix

A" 1 The inverse of matrix A

A* Complex conjugate of matrix A

At (AT

A o B The M x N matrix formed from element by element multiplication of the

elements in the M x N matrices A and B; i.e., A oB = (A(k,n)B(k,ri))

A ® B The Kronecker product of A and B

B Arbitrary matrix, noise-shaping filter input coupling matrix

C Noise-shaping filter output coupling matrix

D Composite system input coupling matrix

F State coupling matrix

G Deterministic input coupling matrix

H Equation coefficient matrix, output coupling matrix

Ha(k) Haar transform of size 2*

IK Identity matrix of size R x R

K Gain matrix, Kalman gain

L Input noise coupling matrix

M Measurement noise coupling matrix, state error covariance square root

N Square root of inverse state error covariance matrix

O Null matrix

P Covariance matrix, state covariance matrix

P, Steady state prediction error covariance matrix

P* Permutation matrix

Po Initial state covariance matrix

Q State noise covariance matrix

R Measurement noise covariance matrix

RH(&) Rationalized Haar transform of size 2*

S(0 Slant transform of size 2 l

W Symmetric weighting matrix

X A transform domain vector resulting from the data vector x

X, DCT of x(n)

X* Coefficients of kth basis function

w(n) Data sequence window function; also called a weighting function (Chapters 1 and

2)

y[x(t)\ The Laplace transform of the function x(t)

M(i/ni) The remainder when / is divided by m

(e ju ) Window function spectrum (Chapters 1 and 2)

The z-transform of the sequence x(n) where X(z) = 2 [x(n)]

Chain parameters of digital two-pair

Transfer matrix of digital two-pair

Chebyshev polynomial of degree M (Chapter 5)

Ratio of 6-dB bandwidth to sample rate

6-dB bandwidth referred to sample rate

Digital filter coefficients

Peak error in Arth filter band, where 25* is the peak-to-peak error

Unit impulse (also called discrete-time impulse, impulse, or unit sample), defined by

1, n = 0

0, n * 0

Mean-squared error ripple factor

Quantization error at sample number n

The mean value of the random variable x given by 77 = E[x]

Argument (phase) of a complex number (pole, zero)

Eigenvalue

Covergence parameter

Noise-shaping filter state coupling

Adjacent correlation coefficient

Real part of 5 (the Laplace transform variable)

E[(x - fi) 2 ]

Group delay

Signal power spectra

Noise power spectra

Frequency in radians per second, w = 2wf, where/is usually normalized to/, =

1 Hz in discrete-time systems

Cutoff frequency of a filter, the - 3dB cutoff frequency

The lower cutoff frequency of a bandpass or bandstop filter

Geometric mean frequencey for bandpass transformation

Center frequency (elliptic filters)

Passband edge frequency

Specified passband edge frequency

Stopband (rejection band) edge frequency

Sampling radian frequency given by

The upper cutoff frequency of a bandpass or bandstop filter

Transition bandwidth of a filter, A/ = (u, - w r )/2ir

/th element of #th basis vector

Arbitrary square matrix, noise-shaping filter initial covariance matrix

Covariance matrix, composite system state coupling matrix

Inverse of state error covariance matrix

Walsh-Hadamard matrix of size (21 x 2 L )

Product of sparse matrices

DST of type N

Magnitude of (•)

The /,2-norm (Euclidean norm) of (•)

Without subscript means || X(e JW ) ||2

f 2T

! 0

The L p-norm of X(e J< *)

Largest integer <(•); e.g., p.5J = 3, 1-2.5J = -3

Smallest integer >(•); e.g., [3.51 = 4, [-2.51 = ~2

Chapter 1 Transforms and Transform Properties

DOUGLAS F ELLIOTTRockwell International Corporation

Continuous waveforms are not alone in being amenable to analysis bytransforms and transform properties Data sequences that result from samplingwaveforms likewise may be studied in terms of their frequency content Sampling,however, introduces a new problem: analog waveforms that do not look anythingalike before sampling yield exactly the same sampled data; one sampledwaveform "aliases" as the other

This chapter briefly reviews the nature of sampled data and developstransforms and transform properties for the analysis of data sequences We start

by reviewing Fourier series that represent periodic waveforms We note that thealiasing phenomenon leads to a periodic spectrum for data sequences so that thespectrum has a Fourier representation in terms of the data We can find thisrepresentation from the data by using the discrete-time Fourier transform(DTFT)

The (DTFT) is generalized to the z-transform, which is a powerful tool for datasequence analysis We also review the discrete Fourier transform (DFT) andrecall the Laplace transform We review discrete-time random sequences beforediscussing correlation and covariance sequences and their power spectraldensities Tables of properties are presented for each transform

HANDBOOK OF DIGITAL SIGNAL PROCESSING Copyright ©1987 by Academic Press, Inc.

II REVIEW OF FOURIER SERIES

Fourier series have been a fundamental engineering tool since J Fourierannounced in 1807 that an arbitrary periodic function could be represented as thesummation of scaled cosine and sine waveforms We shall use Fourier series as abasis for developing the DTFT in the next section We show that the integralsdefining the series coefficients correspond to the inverse discrete-time Fouriertransform (IDTFT)

This section simply recalls for the reader's convenience the definition ofFourier series We consider one- and two-dimensional series

A One-Dimensional Fourier Series

Let X (a) have period P and be the function to be represented by a dimensional (1-D) series Let X(cc) be such that

one-'P/2

'-P/2 Then X(ot) has the 1-D Fourier series representation

oo

At a point of discontinuity, «0, the series converges to [.X"(ao) + X(a 0 )~]/2, where X(cto) and X(%o) are tne function's values at the left and right sides of the

discontinuity, respectively The x(n), n = 0, ±1, ±2, , are Fourier series

coefficients given by

p/2

X(a)e J2nanlP da (1.3) -P/2

We can easily derive Eq (1.3) from Eq (1.2) by using the orthogonalityproperty for exponential functions:

1 Transforms and Transform Properties 3

For most engineering applications the function X(a) is bounded and

con-tinuous, except, possibly, at a finite number of points In this case the Fourierseries holds for very general integrability conditions The orthogonality con-dition, Eq (1.4), makes the Fourier series useful by allowing a function to beconverted from one domain (frequency, etc.) to another (time, etc.) Other

-1

(a)

3P/4

Fig 1.1 A periodic waveform and its Fourier series representation, (a) One period of the

waveform; (b) One-term approximation, (c) Two term-approximation, (d) Three-term mation, (e) Ten-term approximation.

approxi-( e )

Fig 1.1 (Continued)

ransforms (Walsh, etc.; see Chapter 6) also have orthogonality conditions andmay be considered for the analysis of periodic functions

Figure l.l(a) shows one period of a square wave of period P Figure l.l(b)-(e)

shows Fourier series representations using 1, 2, 3, or 10 terms of the series The

reader may verify that the AT-term approximation, XN (OL), to the square wave

reduces to

If we let x(n) = 2ajn, we note that a0 = 0, an = (— 1)(" 1)/2/n when the index n is

an odd integer, and an = 0 when n is even The series coefficients a n are plotted versus both n and n/P in Fig 1.2.

Figure 1.1 (e) illustrates an advantage and a disadvantage of the Fourier seriesrepresentation of the square wave An advantage is that only 10 terms of theseries give a fairly accurate approximation to the waveform A disadvantage is theovershoot, or Gibbs phenomenon, at the points of discontinuity of the waveform.Further discussion of this phenomenon and Fourier series in general is in [1]

We have illustrated the representation of a periodic continuous function

X(ct) by a sequence of coefficients x(n) Given the sequence x(n), we can find the

1 Transforms and Transform Properties

Fig 1.2 Scaled Fourier series coefficients for the waveform in Fig 1.1.

function X(y,), and, indeed, the procedure of taking a data sequence and finding the corresponding X(a) is that of the DTFT, discussed in Section III.

Two-Dimensional Fourier Series B

Let X((x, P) be an image with period Pt along the a axis and period P2 along the/? axis (see Fig 1.3) Note that the periodic image is generated by simply repeating

a single image in both the horizontal and vertical directions Let

Paralleling the derivation of the Fourier coefficients for the 1-D series, we obtainthe Fourier coefficients for the 2-D series:

The reader will doubtless see a pattern emerging from the 1-D and 2-D seriesdevelopment This pattern leads to series representations for JV-D functions,

N = 3, 4, We will not present these representations but will exploit a similar

pattern in a later section to develop JV-D discrete Fourier transforms

Ill DISCRETE-TIME FOURIER TRANSFORM

The periodic waveforms, discussed in the previous section have Fourier seriesrepresentations determined, in general, by an infinite number of coefficients.Given the waveform, we can determine the sequence of coefficients Conversely,given a sequence, we can find the continuous waveform It is this latter procedurethat yields the DTFT

The DTFT provides a frequency domain representation of a data sequence

that might result, for example, from sampling an analog waveform every T

seconds (s) The distinct difference between the frequency spectrum of the analogsignal and the discrete-time sequence derived from it is that the sampling process

causes the analog spectrum to repeat periodically at intervals of f s) where f s =1/r is the sampling frequency This section reviews the reason for the period-icity of the discrete-time spectrum, derives the DTFT and IDTFT, and presents

a table of DTFT properties

A Reason for Periodicity in Discrete-Time Spectra

Figure 1.4 shows cosine waveforms with frequencies of 1 and 9 Hz There is nochance of mistaking one of these analog waveforms for the other However, whenthey are sampled every ^ s, the situation changes dramatically because the cosinefunctions intersect at ^ s, f s,

cos[27r(i)] = cos[2jr9(i)], cos[27r(f)] = cos[2w9(t)],

respectively; the sampled data from one is exactly the same as the sampled datafrom the other, and we say that sampled data from one "aliases" as sampled data

1 Transforms and Transform Properties

f = 8 SAMPLES/SECOND9

cos(2?rt) ^ cos (27T9t)

t (s)

Fig 1.4 Cosine waveforms yielding the same data at sampling instants.

from the other It is easy to verify cosines of frequencies 1 + kf s , f s = 1/T, k —

±1, ±2,.,,, go through the same points of intersection Although Fig 1.4depicts cosine waveforms, aliasing will occur for any sinusoid

We have shown that sampled sinusoids of frequency 1 Hz are indistinguishable

from those of 1 + kf s Hz, where k is any integer Likewise, sampled sinusoids of frequencies / and / -f kf s are indistinguishable:

where $ is an arbitrary phase angle Consequently, a spectrum analyzer would get the same value at / as at / + kf s We conclude that if by some means we

determine the frequency spectrum of a discrete-time data sequence, the aliasing

feature causes the spectrum to repeat at intervals of f s, as shown in Fig 1.5 In

general, the frequency spectrum X(f) is complex, so only the magnitude is

plotted in the figure The nonsymmetry of the spectrum about 0 Hz is due to acomplex-valued data sequence that might result, for example, from frequencyshifting (i.e., complex demodulation), which is described later

Fourier Series Representation of Periodic Spectra B

We have found that the spectrum of a data sequence is periodic If the dataresults from sampling a continuous-time signal every T s, then the period of thespectrum is /s = 1/T Hz Since periodic functions can be represented by Fourier

f (Hz)

-N 2

3N 2

Fig 1.5 Magnitude spectrum for a complex data sequence.

series under relatively mild conditions, we can use Eq (1.2) to represent the

spectrum by the series

X 1 ( f ) = £ x(n)e~ j2nfn/f « (1.10) where the series coefficient x(n) is given by

x(n) = | I XM^w-df (1.11)

JsJ-f s /2

The series coefficient x(n) is the data sequence giving rise to the spectrum We use

x(n) for samples of the continuous-time function x(t) sampled at t = nT and for

data sequences in general We know that x(n) has a periodic spectrum.

Substituting / + kfs , where k is an integer, for f s in (1.10) shows that Xt (f)

is the same for / as for / + kfs Thus, X v (f) has period / Hz, as required.

Note that in the Fourier series development we assumed a periodic function

was given, and we found the sequence of coefficients for the Fourier series

representation, using Eq (1.11) If we are given a sequence of coefficients instead

of the spectrum, we can use the coefficients to find the spectrum by using

Eq (1.10) When dealing with sequences, we are more likely to be given data

that corresponds to the coefficients If the data is the sequence jc(n), we find

its spectrum using Eq (1.10) We recover the data sequence from its spectrum

by using Eq (1.11) In any case Eqs (1.2) and (1.3) or Eqs (1.10) and (1.11)

are a transform pair.

Another transform pair is the continuous-time Fourier transform and its

inverse defined, respectively, by

in the sense that it derives y(nT) from y(t) through Eq (1.15) If we let Eq (1.14)

be the integrand of Eq (1.12), then Eq (1.15) yields

y(nT) = x(nT)e\p(—j2nfnT) = x(n)Qxp(—j2nfn/f s )

1, Transforms and Transform Properties 9

which is a term in Eq (1.10) Thus, Eq (1.10) is the Fourier transform of Eq (1.14)

Whereas Eq (1.12) yields the same answer as Eq (1.10) if x(t) is sampled with

delta functions, Eqs (1.11) and (1.13) do not correspond directly because

Eq (1.11) applies to a sequence and Eq (1.13) applies to a continuous-timefunction Since the spectrum given by Eq (1.10) is periodic, only one period isrequired to obtain the sample x(«), as Eq (1.11) shows This is in contrast to

Eq (1.13), where the entire spectrum is used to obtain x(t).

One-Dimensional DTFT and IDTFT C

We will now simplify the notation by using a normalized sampling interval of

T = 1 s and radian frequency CD = 2nf Let X ^ f ) = X(e i<aT ) Then rewriting Eqs (1.10) and (1.11) for T= 1 s gives

00

X(e j<a )= £ x(n)e' iton (1.16)

X(e jm )e*° n d<o (1.17)

Equations (1.16) and (1.17) are defined as the 1-D DTFT and 1-D IDTFT,

respectively The DTFT yields a periodic spectrum X(e iia ) for a given data sequence x(n) The IDTFT recovers the data sequence from the spectrum We will

also use the notation

X(e J<0 ) = DTFT[x(n)] (1.18) x(n) = IDTFTrxV")] (1.19)

for Eqs (1.16) and (1.17), respectively Let Q be the analog radian frequency Then

conversion from the radian frequency co normalized for a sampling interval of 1 s

to analog radian frequency Q for an arbitrary sampling interval T requires only the substitution co — QT Figure 1.6 indicates corresponding points on the

0 0 0 0

1

/2 _ TT

frequency axes for the variables /, a> = 2nf, Q = 2nF, and F, where F is the analog

frequency in hertz

D DTFT Properties

Table I summarizes properties of the 1-D DTFT A property is described by atransform pair consisting of a data sequence representation and a transform

sequence representation For example, x(n) and X(e jf °) constitute a transform

pair We will illustrate derivation of the pairs with several examples For furtherdetails see [2, 3]

e~ j(a ° n x(n) is left-shifted in frequency so that e ±jto °"x(n) and X(ej(0)T'ao)) constitute

a pair

2 Data Sequence Convolution

Convolution of the sequence x(n) with y(n) is represented by x(n) * y(n) and is

3 Frequency Domain Convolution

Frequency domain convolution is defined by

1 P*

X(e j<0 ) * Y(e jca ) = — X(e je ) Y(e i(m ~ 9)) dO ( 1 ,24}

Using the IDTFT definition, Eq ( 1 1 7 ), interchanging integrations, and making achange of variables yield

IDTFT lX(e j(a ) * Y(e jm )'] = x(ri)y(n) (1 25)

as stated in Table I

4 Symmetry Properties

Several properties in Table I deal with conjugate symmetric sequences

sat-isfying JC(M) = x*( — n) and conjugate antisymmetric sequences satsat-isfying x(n) =

— x*( — n) If a sequence is real, then conjugate symmetric or antisymmetriccorrespond to even or odd, respectively

5 Sampling Frequency Change

As an example of the utility of transform properties, consider the samplingfrequency change properties (the two entries before Parseval's theorem at the end

of Table I) Let the periodic repetitions of a spectrum of a sequence jc1(«) be

widely spaced so that the signal bandwidth (BW) satisfies BW < f s/M Then the

sequence may be desampled by M : 1 ; that is, only 1 of every M samples is retained[see Fig 1.7(a), (b)] This reduces the spectral amplitude by 1/M and causes the

spectrum to repeat at the new sampling frequency fJM [Fig 1.7(c); the curve for X$(e j2nf ) applies to X2(e j2nf ) after frequency units are changed to Hz/3].

Desampling is used, for example, to more efficiently analyze a signal with a DFT.Before going to the DFT, the signal is desampled as much as possible withoutintroducing aliasing, and, as a consequence of the desampling, the DFT can berun at a lower rate

A signal can be interpolated by a 1 : M upsampling that adds M — 1 zeros toevery sample (padding with zeros by 1 : M) Although the upsampling increases

the sampling frequency, it does not effect the spectrum, which still repeats at fJM (Fig 1.7(c)] When we remove the spectral replicas at integer multiplies of fJM

by filtering, the zero values introduced by padding disappear and we obtain the

original sequence x^w) If we start with the signal x 2(n) and wish to interpolate to

find intermediate sample values, we simply pad with zeros by 1 : M and use alowpass filter with a zero frequency gain of M to get a sequence x^n) such thatevery Mth value matches x2(n) Another interesting application of upsampling is

to effect a sampling frequency change (see Chapter 3)

1 Transforms and Transform Properties 15

PAD WITH ZEROS BY

r

0**3**6 * "

1, DIGITAL FILTER IltTHt, r

*

I (Hi) (b)

DIGITAL FILTER GAIN

f(Hz)

1: M interpolates the signal, (a) Block diagram showing desampling, upsampling, and filter to remove replicas, (b) Spectral magnitude for Xj(n) (c) Spectral magnitude of x 3 (n) for M = 3.

Two-Dimensional DTFT E

Let an image x(r, s) be sampled at intervals of Tt and T2 along the r and s axes, respectively, yielding the 2-D sequence x(m, n) The spectrum will be 2-D with periods 1/Ti and 1/T2 along the ft and /2 axes, respectively, for the same reasonthat a 1-D spectrum is periodic Since the 2-D spectrum is periodic, we canrepresent it by a 2-D Fourier series Paralleling the steps for the 1-D DTF andIDTFT leads to

respec-IV z-TRANSFORM

The z-transform generalizes the DTFT and gives additional information onsystem stability This section discusses the z-transform, the inverse z-transform,and a table of properties

A One-Dimensional z-Transform

Equation (1.16) defines the 1-D DTFT:

We can generalize this equation by replacing e jton by e an jmn , letting z — e a+jta ,

and defining the resulting summation as the two-sided, 1-D z-transform of x(n)

or, simply, the z-transform of x(«), denoted by

00

.A\zj == =£|_x(w^J == y x\n)z fl Zo)

n = — oo

For cr = 0, z = ejw, and Eq (1.28) is the same as Eq (1.16) In this case |z| =

\e j<a \ = |cos o> + jsma)\, which defines the unit circle (a circle with unity radius

centered at the origin) Evaluating the z-transform on the unit circle in the z-planecorresponds to the DTFT

B Region of Convergence

The infinite series in Eq (1.28) is meaningful only if it converges One test ofconvergence is the ratio test: a series converges if the magnitude of the ratio of

term n 4- 1 to term n (term — n — 1 to term — n on the negative axis) is less than 1

as n -* oo For n > 0 we require that

The region where Eqs (1.29) and (1.30) are satisfied is called the region of

, v fa", n > 0

x(n) =

1- Transforms and Transform Properties

Applying the geometric series summation formula

]JT a"z " converges for |z| > a and

From Eq (1.34) we conclude that the region of convergence for Eq (1.33) is the

annulus defined by jz| > a and \z\ < b, as shown in Fig 1.8 As is evident from

Eq (1.33), the function X(z) diverges at z = a and z — b Such points are called

poles of the function Similarly, X'(z) = 0 at z = (a 4- b)/2 and z = 0 Such points

are called zeros of the function If b < a, there is no region of convergence for

(1.33) because the z-transform diverges everywhere.

Fig 1.8 Region of convergence for x(n) — (a", n > 0; —b~",n< 0).