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CuuDuongThanCong.com CuuDuongThanCong.com FAST TRANSFORMS A l g o r i t h m s , A n a l y s e s , Applications Douglas F Elliott Electronics Research Center Rockwell International A n a h e i m , California K Ramamohan Rao Department of Electrical Engineering The University of Texas at A r l i n g t o n A r l i n g t o n , Texas A C A D E M I C PRESS, INC (Harcourt Brace Jovanovich, Publishers) Orlando San Diego Toronto Montreal CuuDuongThanCong.com San Francisco Sydney Tokyo New York Sao Paulo London COPYRIGHT © , BY ACADEMIC PRESS, INC ALL RIGHTS R E S E R V E D N O P A R T O F T H I S P U B L I C A T I O N M A Y B E R E P R O D U C E D OR T R A N S M I T T E D I N A N Y F O R M OR B Y A N Y M E A N S , E L E C T R O N I C OR M E C H A N I C A L , I N C L U D I N G P H O T O C O P Y , RECORDING, OR A N Y I N F O R M A T I O N STORAGE A N D RETRIEVAL S Y S T E M , W I T H O U T PERMISSION I N WRITING F R O M THE PUBLISHER A C A D E M I C PRESS, Orlando, Florida 32887 United Kingdom Edition INC published by A C A D E M I C PRESS, INC ( L O N D O N ) 24/28 Oval R o a d , L o n d o n N W 7DX LTD Library of Congress Cataloging in Publication Data Elliott, Douglas F Fast transforms: algorithms, analyses, applications Includes bibliographical references and index Fourier transformations—Data processing Algorithms I Rao, K Ramamohan (Kamisetty Ramamohan) II Title III Series QA403.5.E4 515.7'23 79-8852 ISBN 0-12-237080-5 AACR2 AMS (MOS) Subject Classifications: C , C , C , C P R I N T E D I N T H E U N I T E D S T A T E S O F AMERICA 83 84 85 CuuDuongThanCong.com To Caroiyn and Karuna CuuDuongThanCong.com CuuDuongThanCong.com CONTENTS Preface Acknowledgments List of Acronyms Notation Chapter 1.0 1.1 1.2 1.3 1.4 Chapter 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 xiii xv xvii xix Introduction Transform D o m a i n Representations F a s t Transform Algorithms F a s t Transform Analyses F a s t Transform Applications Organization of the B o o k 4 Fourier Series and the Fourier Transform Introduction Fourier Series with Real Coefficients F o u r i e r Series with Complex Coefficients E x i s t e n c e of F o u r i e r Series T h e F o u r i e r Transform S o m e F o u r i e r Transforms and Transform Pairs Applications of Convolution Table of F o u r i e r Transform Properties Summary Problems 6 10 12 18 23 25 25 vii CuuDuongThanCong.com vlii CONTENTS Chapter 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 Chapter 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 Chapter 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 CuuDuongThanCong.com Discrete Fourier Transforms Introduction D F T Derivation Periodic P r o p e r t y of the D F T Folding P r o p e r t y for Discrete Time Systems with Real Inputs Aliased Signals Generating kn Tables for the D F T D F T Matrix Representation D F T Inversion—the IDFT T h e D F T and I D F T — U n i t a r y Matrices Factorization of W Shorthand Notation Table of D F T Properties Summary Problems , E 33 34 36 37 38 39 41 43 44 46 47 49 52 53 Fast Fourier Transform Algorithms Introduction Power-of-2 F F T Algorithms Matrix Representation of a Power-of-2 F F T Bit R e v e r s a l to Obtain F r e q u e n c y O r d e r e d Outputs Arithmetic Operations for a Power-of-2 F F T Digit R e v e r s a l for Mixed Radix Transforms M o r e F F T s b y M e a n s of Matrix T r a n s p o s e M o r e F F T s b y M e a n s of Matrix I n v e r s i o n — t h e I F F T Still M o r e F F T s by M e a n s of F a c t o r e d Identity Matrix Summary Problems 58 59 63 70 71 72 81 84 88 90 90 FFT Algorithms That Reduce Multiplications Introduction Results from N u m b e r T h e o r y Properties of Polynomials Convolution Evaluation Circular Convolution Evaluation of Circular Convolution through the C R T C o m p u t a t i o n of Small N D F T Algorithms Matrix Representation of Small N D F T s K r o n e c k e r Product E x p a n s i o n s 99 100 108 115 119 121 122 131 132 fx CONTENTS 5.9 5.10 5.11 5.12 5.13 5.14 5.15 Chapter 6.0 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 Chapter T h e G o o d F F T Algorithm T h e Winograd Fourier Transform Algorithm Multidimensional Processing Multidimensional Convolution by Polynomial Transforms Still M o r e F F T s by M e a n s of Polynomial Transforms C o m p a r i s o n of Algorithms Summary Problems D F T Filter Shapes and Shaping Introduction D F T Filter R e s p o n s e I m p a c t of the D F T Filter R e s p o n s e Changing the D F T Filter Shape Triangular Weighting H a n n i n g Weighting and H a n n i n g W i n d o w Proportional Filters S u m m a r y of Weightings and W i n d o w s Shaped Filter Performance Summary Problems - 7.0 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 Chapter 8.0 8.1 8.2 CuuDuongThanCong.com 136 138 139 145 154 162 168 169 178 179 188 191 196 202 205 212 232 241 242 Spectral Analysis Using the FFT Introduction Analog and Digital S y s t e m s for Spectral Analysis Complex D e m o d u l a t i o n and M o r e Efficient U s e of the F F T Spectral Relationships Digital Filter Mechanizations Simplifications of F I R Filters D e m o d u l a t o r Mechanizations O c t a v e Spectral Analysis Dynamic Range Summary Problems 252 253 256 260 263 268 271 272 281 289 290 Walsh-Hadamard Transforms Introduction Rademacher Functions Properties of Walsh F u n c t i o n s 301 302 303 CONTENTS X 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 Chapter 9.0 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 9.11 9.12 9.13 Chapter 10 10.0 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 10.11 10.12 CuuDuongThanCong.com Walsh or S e q u e n c y Ordered Transform ( W H T ) H a d a m a r d or N a t u r a l O r d e r e d Transform ( W H T ) Paley or Dyadic Ordered Transform ( W H T ) C a l - S a l O r d e r e d Transform ( W H T ) W H T Generation Using Bilinear F o r m s , Shift Invariant P o w e r Spectra Multidimensional W H T Summary Problems W P CS h 310 313 317 318 321 322 327 329 329 The Generalized Transform Introduction Generalized Transform Definition E x p o n e n t Generation Basis F u n c t i o n F r e q u e n c y A v e r a g e Value of the Basis F u n c t i o n s Orthonormality of the Basis F u n c t i o n s Linearity Property of the Continuous Transform Inversion of the Continuous Transform Shifting T h e o r e m for the Continuous Transform Generalized Convolution Limiting Transform Discrete Transforms Circular Shift Invariant P o w e r S p e c t r a Summary Problems 334 335 338 340 341 343 344 344 345 347 347 348 353 353 353 Discrete Orthogonal Transforms Introduction Classification of Discrete Orthogonal Transforms M o r e Generalized Transforms Generalized P o w e r Spectra Generalized P h a s e or Position S p e c t r a Modified Generalized Discrete Transform ( M G T ) P o w e r Spectra T h e Optimal Transform: K a r h u n e n - 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Wiley, N e w Y o r k , 1976 A G Tescher, Transform image coding, in " I m a g e Transmission T e c h n i q u e s " (W J Pratt, ed.), Academic Press, N e w Y o r k , 1979 Y T a d o k o r o a n d T Higuchi, C o m m e n t s o n "Discrete Fourier transform via Walsh t r a n s f o r m , " IEEE Trans Acoust Speech Signal Process A S S P - , 295-296 (1979) Y T a d o k o r o a n d T Higuchi, A n o t h e r discrete Fourier transform c o m p u t a t i o n with small Atlanta, multiplications via the Walsh transform, Int Conf Acoust Speech Signal Process., Georgia, p p 306-309 (1981) J L U l m a n , C o m p u t a t i o n of the H a d a m a r d transform a n d the R-transform in ordered form, IEEE Trans Comput C-19, 359-360 (1970) V Vlasenko, K R R a o , a n d V Devarajan, Unified matrix treatment of discrete transforms, Proc Ann Southeast Symp Syst Theory, 10th, Mississippi State, Mississippi, p p I I B - II.B-29 (1978); also published in IEEE Trans Comput C O M - , 934-938 (1979) R L Veenkant, A serial minded F F T , IEEE Trans Audio Electroacoust AU-20, 180-185 (1972) J L Vernet, Real signals fast Fourier transform storage capacity a n d step n u m b e r reduction by m e a n s of an o d d discrete Fourier transform, Proc IEEE 59, 1531-1532 (1971) M C V a n w o r m h o u d t , O n n u m b e r theoretic Fourier transforms in residue class rings, IEEE Trans Acoust Speech Signal Process A S S P - , 585-586 (1977) M C V a n w o r m h o u d t , Structural properties of complex residue rings applied to n u m b e r theoretic Fourier transforms, IEEE Trans Acoust Speech Signal Process A S S P - , 99-104 (1978) U A von der Embse a n d M C Austin, A n efficient multichannel F F T demodulator, Proc Int Telemeter Conf., Los Angeles, California, p p 499-508 (1978) P Varg a n d U Heute, A short-time spectrum analyzer with polyphase-network a n d D F T , Signal Process 2, 55-65 (1980) J E Whelchel, Jr a n d D F Guinn, T h e fast F o u r i e r - H a d a m a r d transform a n d its use in signal representation a n d classification, Aerosp Electron Conf (EASCON), Rec, Washington, B.C., p p 561-573 ( I E E E P u b 68 C 3-AES (1968) J L Walsh, A closed set of n o r m a l orthogonal functions, Am J Math 55, 5-24 (1923) P P W a n g a n d R C Shaiau, M a c h i n e recognition of printed Chinese characters via transformation algorithms, Pattern Recognit 5, 303-321 (1973) CuuDuongThanCong.com 480 REFERENCES W-4 H D Wishner, Designing a special-purpose digital image processor, Comput Design 1 , 71-76 (1972) S Wendling a n d G Stamon, H a d a m a r d a n d H a a r transforms a n d their power spectra in character recognition, Joint W o r k s h o p in Pattern Recognition a n d Artificial Intelligence, Hyannis, Massachusetts, p p 103-112 (1976) S Winograd, A new m e t h o d for c o m p u t i n g D F T , Proc IEEE Int Conf Acoust Speech Signal Process Hartford, Connecticut, p p 366-368 (1977) S Winograd, Some bilinear forms whose multiplicative complexity depends o n the field of constants, Math Syst Theory 10, 169-180 (1977) S Winograd, O n c o m p u t i n g the discrete Fourier transform, Proc Nat Acad Sci U.S.A 73, 1005-1006 (1976)/ S Winograd, O n multiplication of x matrices, Linear Algebra and Appl , 381-388 (1971) S Winograd, O n the n u m b e r of multiplications necessary t o c o m p u t e certain functions, Commun Pure Appl Math , 165-179 (1970) S Winograd, O n the n u m b e r of multiplications required to c o m p u t e certain functions, Proc Nat Acad Sci U.S.A 58, 1840-1842 (1967) S A White, I n t r o d u c t i o n to Implementation of Digital Filters Electronics Research Division, Rockwell International, Anaheim, California, Technical R e p XI3-311/501 (1973) S A White, Recursive Digital Filter Design Autonetics Division, Rockwell International, Anaheim, California, Technical R e p X8-2725/501 (1968) C J Weinstein, R o u n d o f f noise in floating point fast Fourier transform c o m p u t a t i o n , IEEE Trans Audio Elecroacoust AU-17, 209-215 (1969) P P Welch, A fixed point fast F o u r i e r transform error analysis, IEEE Trans Audio Electroacoust AU-17, 151-157 (1969) Proc Symp Appl Walsh Functions, Washington, D.C (1970), Order N o A D 707431; Proc Symp Appl Walsh Functions, Washington, D.C (1971), Order N o A D 727000; Proc Symp Appl Walsh Functions, Washington, D.C (1972), Order N o A D 744650; Proc Symp Appl Walsh Functions, Washington, D.C (1973), Order N o A D 763000, N a t i o n a l Technical Information Services, Springfield, Virginia; Proc Symp Appl Walsh Functions, Washington, D.C (1974), Order N o C H E M C , I E E E Service Center, Piscataway, N e w Jersey H Whitehouse et al, A digital real-time intraframe video b a n d w i d t h compression system, SPIE Int Tech Symp., 21st, San Diego, California, p p 64-78 (1977) M D W a g h a n d S V K a n e t k a r , A multiplexing theorem a n d generalization of R-transform, Int J Comput Math 5, Sect A, 163-171 (1975) M D Wagh, Periodicity in R-transformation, Inst Electron Telecom Eng 21, 560-561 (1975) M D W a g h , A n extension of R-transform to patterns of arbitrary lengths, Int J Comput Math 1, Sect B, 1-12 (1977) M D W a g h a n d S V K a n e t k a r , A class of translation invariant transforms, IEEE Trans Acoust Speech Signal Process ASSP-25, 203-205 (1977) M D W a g h , Translational invariant transforms, P h D Dissertation, Indian Institute of Technology, Bombay, India (1977) S Wendling, G G a g n e u x , a n d G Stamon, Use of the H a a r transform a n d some of its properties in character recognition, Proc Int Conf Pattern Recognit., 3rd, Coronado, California, p p 844-848 (1976) C Watari, A generalization of H a a r functions, Tohoku Math J 8, 286-290 (1956) P A Wintz, Transform picture coding, Proc IEEE 60, 809-820 (1972) J Whitehouse, R W M e a n s , a n d J M Speiser, Signal processing architectures using transversal filter technology, Proc IEEE Adv Solid-State Compon Signal Process., Newton, p p 5-29 (1975) Massachusetts, P M W o o d w a r d , "Probability and Information Theory, with Applications to R a d a r " Pergamon, Oxford, 1953 L C W o o d , Seismic d a t a compression methods, Geophysics 39, 499-525 (1974) W-5 W-6 W-7 W-8 W-9 W-10 W-l W-l2 W-l3 W-l4 W-l5 W-16 W-l7 W-18 W-19 W-20 W-21 W-22 W-23 W-24 W-25 W-26 W-27 W-28 CuuDuongThanCong.com REFERENCES 481 W-29 W G Wee a n d S Hsieh, An application of projection transform technique in image transmission, Proc Nat Telecommun Conf., New Orleans, Louisiana, p p 22-1-22-9 (1975) W-30 D M Walsh, Design considerations for digital Walsh filters, Proc IEEE Fall Electron Conf., Chicago, Illinois, p p 372-377 (1971) W-31 L C Wilkins a n d P A Wintz, Bibliography on d a t a compression, picture properties and picture coding, IEEE Trans Informat Theory IT-17, 180-197 (1971) W-32 M D W a g h , R-transform amplitude b o u n d s and transform volume, J Inst Electron Telecom Eng , 501-502 (1975) W-33 H W i d o m , Toeplitz matrices, in "Studies in Real a n d Complex Analysis" (I I H i r s c h m a n n , Jr., ed.) Prentice-Hall, Englewood Cliffs, N e w Jersey, 1965 W-34 S A White, O n mechanization of vector multiplication, Proc IEEE 63, 730-731 (1975) W-35 S W i n o g r a d , On computing the discrete Fourier transform, Math Comput 32, 175-199 (1978) W-36 P D Welch, The use of fast Fourier transform for the estimation of power spectra: A m e t h o d based on time averaging over short, modified p e r i o d o g r a m s , IEEE Trans Audio Electroacoust AU-15, 70-73 (1967) W-37 S Wendling, G Gagneux, and G Stamon, A set of invariants within the power spectrum of unitary transformations, IEEE Trans Comput C-27, 1213-1216 (1978) Y-l C K Yuen, Walsh functions and gray code, IEEE Trans Electromagn Compat EMC-13, 68-73 (1971) Y-2 C K Yuen, N e w Walsh function generator, Electron Lett 1, 605-607 (1971) C K Yuen, C o m m e n t s on " A Hazard-free Walsh function generator, IEEE Trans Instrum Y-3 Measurement IM-22, 99-100 (1973) Y-4 P Y i p a n d K R R a o , Energy packing efficiency of generalized discrete transforms, Proc Midwest Symp Circuits Syst., 20th, Texas Tech Univ., Lubbock, Texas, p p 711-712 (1977); also published in IEEE Trans Commun COM-26, 1257-1262 (1978) Y-5 R Y a r l a g a d d a , A note on the eigenvectors of D F T matrices, IEEE Trans Acoust Speech Signal Process, p p 586-589 (1977) Y-6 L S Y o u n g , C o m p u t e r p r o g r a m s for m i n i m u m cost F F T algorithms, Rockwell International, A n a h e i m , California, R e p T77-986/501 (1977) Y-7 P Y i p , Some aspects of the z o o m transform, IEEE Trans Comput C-25, 287-296 (1976) Y-8 P Y i p , Concerning the block-diagonal structure of the cyclic shift m a t r i x u n d e r generalized discrete o r t h o g o n a l transforms, IEEE Trans Circuits Syst CAS-25, 48-50 (1978) Y-9 C K Yuen, Analysis of Walsh transforms using integration by-parts, SIAMJ Math Anal 4, 574-584 (1973) Y-10 C K Yuen, A n algorithm for computing t h e correlation functions of Walsh functions, IEEE Trans Electr omagn Compat EMC-17, 177-180 (1975) Y - l P Y i p , T h e z o o m Walsh transform, Proc Midwest Symp Circuits and Syst., 18th, Montreal, Canada, p p 21-24 (1975) Y - l C K Yuen, C o m p u t i n g robust W a l s h - F o u r i e r transform by error p r o d u c t minimization, IEEE Trans Comput C - , 313-317 (1975) Y-13 P Y i p , T h e z o o m Walsh transform, IEEE Trans Electromagn Compat EMC-18, 79-83 (1976) Y-14 M Yanagida, Discrete Fourier transform based o n a double sampling a n d its applications, Proc IEEE Int Conf Acoust Speech Signal Process., Philadelphia, Pennsylvania, p p 141-144 (1976) Y - l P Yip a n d K R R a o , Sparse-matrix factorization of discrete sine transform, Proc Ann Asilomar Conf Circuits Syst Comput., 12th, Pacific Grove, California, p p 549-555 (1978); also published in IEEE Trans Commun C O M - , 304-307 (1980) Y~-16 P Y i p a n d K R R a o , O n the c o m p u t a t i o n a n d effectiveness of discrete sine transform, Proc Midwest Symp Circuits Syst., 22nd, Philadelphia, Pennsylvania, p p 151-155 (1979); also published in Comput Electr Eng 7, 45-55 (1980) CuuDuongThanCong.com 482 Z-1 Z-2 Z-3 REFERENCES R Zelinski and P Noll, Adaptive transform coding of speech signals, IEEE Trans Speech Signal Process A S S P - , 299-309 (1977) S Z o h a r , Toeplitz matrix inversion, the algorithm of W F Trench, J Assoc Comput 16, 592-601 (1969) S Z o h a r , A prescription of W i n o g r a d ' s discrete Fourier transform algorithm, IEEE Acoust Speech Signal Process A S S P - , 409-421 (1979) CuuDuongThanCong.com Acoust Mach Trans INDEX A Aliased signals, 38, 290 Analog-to-digital converter (ADC), 180 Autocorrelation arithmetic, 449 continuous function, 32 B Bandwidth, 236, see also N o i s e bandwidth B I F O R E Transform, 315 complex, 366 modified, 378 modified complex, 378 Bit reversal, 70, 445 Bit reversed order (BRO), 312 Butterfly, 63 C Chebyshev polynomials, 228, 386 Chinese remainder theorem (CRT) expansion of polynomials, 147 for integers, 105 for polynomials, 112 Circulant matrix, 452 block, 452 Circular convolution, see Convolution Circular or periodic shift, 446 C matrix transform, 416 Coherent gain, 232 Coherent processing, 238 Comb function definition, 23 Fourier transform, 29 Congruence modulo an integer, 27 modulo a polynomial, 108 modulo a product, 101 Constant Q filters, 205 Convolution circular (periodic), 50, 123, 450 evaluation using the CRT, 121 evaluation using polynomial transforms, 152 property, 419 dyadic or logical, 450 frequency domain, 18 noncircular (aperiodic), 50 minimum number of multiplications for, 116 time domain, 17 C o o k - T o o m algorithm, 115 Correlation arithmetic, 50, 449 dyadic, 448 Cross-correlation, 31 Cyclotomic polynomial, 109 D Data sequence definition, 34 number, 35 Decimation in frequency (DIF) F F T , see Fast Fourier transform Decimation in time, 255 Decimation in time (DIT) F F T , see also Fast Fourier transform radix-2, 82 483 CuuDuongThanCong.com 484 Delta function Dirac, 14, 26 Kronecker, Demodulation, c o m p l e x analog, 290 digital, 290 Demodulator, 256, see also Single sideband modulation mechanization, 271 D F T , see Discrete Fourier transform D F T filter figure of merit (FOM), 232 nonperiodic, 185 nonperiodic shaped, 195 performance, 232 periodic, 180 periodic shaped, 192 response, 179, 182 shaping, 191 shaping approximation, 244 shaping by means of FIR filters, 195, 297 Digital filter characteristics, 267 elliptic, 268 finite impulse response (FIR): definition, 266 use in design of D F T w i n d o w s , 297 infinite impulse response (IIR), 266 mechanizations, 263 order, 266 recursive, 263 transversal, 263 Digital word length, 281 Digit reversal, 72, 77 Discrete cosine transform, 386 e v e n , 393, 412 odd, 413 Discrete D transform, 414 Discrete Fourier transform ( D F T ) , see also Fast Fourier transform calculation using an (M2)-point F F T , 56 definition, 35 of D F T output definition, 249 filter shaping for, 250 frequence response for, 251 equivalence of 1-D and L - D , 135 equivalent representations, 181, 192 evaluation by circular convolution, 127 folding property, 37 matrix factorization, 46 multidimensional, 51 CuuDuongThanCong.com INDEX of an JV-point e v e n (odd) sequence using an (M4)-point F F T , 56 periodic property, 36 with power of a prime number dimension, 125 with prime number dimension, 124 properties summarized, 49 reduced, 157 of sequence padded with zeros, 56, 92, 248 small N algorithms, 127 matrix representation of, 131 structure, 420 of two real TV-point s e q u e n c e s , 55 Discrete sine transform, 412 Discrete time system, 34 Discrete transform classification, 365 Discrete transform comparison, 410 Double sideband modulation, 16, 28 Dyadic matrix, 451 Dyadic or Paley ordering, 453 Dyadic shift, 446 matrix, 322 Dyadic time shift, 359 Dynamic range, 281 E Effective noise bandwidth ratio, 248 Eigenvector transform, 383 Elliptic filters, 268 Energy packing efficiency (EPE), 327 Equivalent noise bandwidth, 235, 286 Euclid's algorithm, 111 Euler's phi function, 102 Euler's theorem, 104 External frequencies, 269 F Fast Fourier transform (FFT), see also Discrete Fourier transform algorithms, algorithm comparison, 165 arithmetic requirements, , 164 bit reversal, 70 decimation in frequency (DIF) mixed radix, 173 radix-2, 60, 69 decimation in time (DIT) mixed radix, 173 radix-2, 82, 91 485 I N D E X derivation using factored identity matrix, 88 matrix inversion, 84 matrix transpose, 81 digit reversal, 72 in-place computation, 90 mixed radix, 58, 72 polynomial transform method of computation, 154 power-of-2 (radix-2), 59, 81, 173 minimum multiplications for, 90 radix-3, -4, e t c , 81, 94 scaling, 283 Fermat numbers, 422 Fermat number transform, 166, 422 complex, 427 complex p s e u d o , 432 fast, 442 matrix, 423 pseudo, 430, 443 Fermat's theorem, 103 F F T , see Fast Fourier transform Field of integers, 102, 418 of polynomials, 108 Filter equiband, 270, 294 passband, 269 stopband, 269 transition interval, 269 Filters, see Digital filters Fit function, 208 Fourier series with complex coefficients, existence of, with real coefficients, Fourier transform derivation, 10 inverse, 11 multidimensional, 23 pair, 12 properties, 24 Frequency bin number, 36 tags, 77 G Gauss's theorem, 103 Generalized continuous transform basis function average value, 341 frequency, 340 CuuDuongThanCong.com generation, 338 orthonormality, 343 period,341 convolution, 347 definition, 335 discrete transforms derived from, 348 inversion, 344 linearity property, 344 shifting theorem, 345 Generalized transform (GT) definition, 366 modified, 374 phase or position spectra, 373 power spectra, 370 Good algorithm, 99, 136 arithmetic requirements, 164 Gray c o d e , 95, 307, 447 to binary conversion (GCBC), 313, 447 Greatest c o m m o n divisor (gcd) for integers, 10 for polynomials, 110 r H Haar functions, 399 matrices, 400 transform, 400 complex, 415 rationalized, 403 Hadamard-Haar transform, 414 rationalized, 414 Hotelling transform, 383 Hybrid ( D I T - D I F ) F F T , 98 I Impulse response, 19 In-place computation, 90, 313 Index of an integer relative to a primitive root, 105 Infinite impulse response (IIR), see Digital filter Integer representation constraint, 420 mixed radix integer representation (MIR), 73, 171 second integer representation (SIR), 106 Inverse discrete Fourier transform, 44 Inverse fast Fourier transform (IFFT), 85, 95 486 INDEX K K a r h u n e n - L o e v e transform, 382 asymptotic equivalence, 414 Kronecker product, 132, 444 algebra, 444 power, 445 Number theory, 100 Nyquist frequency, 38 rate, 38 for sampling D F T output, 251 L Lagrange interpolation formula, 114 Leakage, spectral, 183 O Order of an integer modulo N, Orthogonal functions, 7, Overlap correlation, 238 M Matrix permutation, 88 sparse, 47, 67 unitary, 44 Mersenne numbers, 425 Mersenne number transform, 425 complex, 429 complex p s e u d o , 443 pseudo, 434, 443 Mixed radix integer representation (MIR), 73, 171 Modified generalized transform, 374 Modified W a l s h - H a d a m a r d transform, 329 Modulo (mod) arithmetic, 418 an integer, 27, 40, 447 a polynomial, 108 Multidimensional D F T , 51, 139 W H T , 327 Multiplicative inverse, 102, 418 N Noise bandwidth, 208 equivalent noise bandwidth, 235, 247 normalization, 208 power spectral density, 208 Noncoherent processing, 238 Nonrecursive filters, see Digital filters Number theoretic transforms ( N T T ) , 411 computational complexity, 439 constraints, 421 figure of merit, 438 properties, 440 relative evaluation, 436 selection of parameters, 421 CuuDuongThanCong.com 104 P Padding with zero-valued samples added at end of s e q u e n c e , 56, 92, 248 inserted b e t w e e n c o n s e c u t i v e samples, 56 Parseval's theorem for continuous functions, 31 for D F T , 53 for W H T , 322 Passband, see Filter Permutation values, 163 Picket-fence effect, 236 Polynomial transforms, 99 definition, 149 evaluation of circular convolution b y , 175 generalized, 150 inverse, 149 multidimensional convolution by, 145 plus nesting, 177 Power-of-2 F F T algorithms, 59 arithmetic operations, 71 Power spectral density (PSD) definition, 32 estimation, 252 Power spectrum circular shift invariant for generalized continuous transform, 353 W H T , 323 D F T , 53 (GT) , 370 (MGT)„ 378 W H T , 322 Prime factor algorithm, 137 Prime number, 102 Primitive root of an integer, 104, 418 of unity, 109 Principal component transform, 383 Processing l o s s , worst c a s e , 236 r 487 I N D E X Proportional filters, 205 Pruning, 92 Pure tone, 236 Q Sign(/c), Q of a filter, 205 R R a d e m a c h e r f u n c t i o n s , 302 Rader-Brenner algorithm, 161 Rader transform, 426 Rapid transform, 405 properties, 405 Rate distortion, 385 Rationalized Haar transform, 403 rect function definition, 12 Fourier transform, 13 Reduced multiplications F F T algorithms, see Fast Fourier transform Redundancy, 238 Relatively prime integers, 102 polynomials, 111 R e m e z exchange algorithm, 268 Residue, 40 Residue number system, 107 rep operator, 23 Ring of integers, 101, 418 of polynomials, 108 Ripple in filter frequency response, 269 Roundoff noise, 285 S Sampled-data system, 34 Sampling theorem frequency domain, 30 sequency, 307 time domain, 29, 38 Scaling in the F F T , 281 Scalloping loss, 236 Second integer representation (SIR), 106 Sequency definition, 304 power spectrum, 313 Shorthand notation D F T matrix, 47 F G T matrix, 349 CuuDuongThanCong.com Sidelobe, fall off, 232,-246 highest level, 232 maximum level v s worst-case processing loss, 238 spurious, 287 Signal level detection, 240 sine function definition, 13 Fourier transform, 13 Single sideband modulation, 15, 28, 256 Slant-Haar transform, 414 Slant transform, 393 Spectral analysis, 178, 252 equivalent systems for, 297 octave, 272 system using A G C , 282 Spectrum, demodulated, 258, see also Power spectrum Stage, 63, 311 Stepping variable, 335 Stopband, see Filter T Time sample number, 35 Toeplitz matrix, 411, 452 Totient, 103 Transfer functions, 18 Transform sequence definition, 36 number, 36 Transforms using other transforms, 416 Twiddle factor, 60, 134, 171 U Unit step function, 23, 28 V Variance distribution for a first-order Mark o v process, 384 W Walsh-Fourier transform, 337 Walsh functions, 301 generation of, 356, 357 Walsh-Hadamard matrices, 309 INDEX 488 Walsh-Hadamard transform c a l - s a l ordered, 318 generation using bilinear forms, 321 Hadamard or natural ordered, 313 Paley or dyadic ordered, 317 Walsh or sequency ordered, 310 Walsh ordering, 307 Weighting Abel, 226 B a r c i l o n - T e m e s , 231 Bartlet, 213 Blackman, 215 Blackman-Harris, 217 Bochner, 220 Bohman, 223 Cauchy, 226 202, 213 cos (mrlN), cosine cubed, 246 cosine squared, 202, 213, 246 cosine tapered, 222 cubic, 221, 245 D e L a V a l l e - P o u s s i n , 221 Dirichlet, 213 D o l p h - C h e b y s h e v , 228 Frejer, 213 Gaussian, 227 Hamming, 214 generalized, 246 Hanning, 202, 246 a CuuDuongThanCong.com Hanning-Poisson, 225 Jackson, 221 K a i s e r - B e s s e l , 230 K a i s e r - B e s s e l approximation to Blackman-Harris, 219 parabolic, 220 Parzen, 221 Poisson, 224, 226 raised cosine, 222 rectangular, 213 Riemann, 221 Riesz, 220 time domain, 191, 194 triangular, 196 Tukey, 222 Weierstrass, 227 White noise, 207 Windowing, frequency domain, 191, 194, see also Weighting Winograd Fourier transform algorithm, 99, 138 arithmetic requirements, 162 Worst-case processing l o s s , 236, 238 Z Zero padding, see Padding with zero-valued samples Z o o m transform, 251 CuuDuongThanCong.com ... X{t) Qsinc(tQ) m Tl * (-/ ) rect(//g) esinc(/0 rect(f/e) W+/o) +W - / o ) cos(27c/o0 sin(27i/o0 3(t - t ) ij5(f ~j2nft e + f )-yS(f-f ) 0 J2%fot S(f-fo) f
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