zeevi, coifman - signal and image representation in combined spaces

Such expansions are more suitable for the analysis and processing of natural signals and images than expansion by the traditional application of Fourier series, polynomials, and other fu

Wavelets are a generic name for a collection of self similar localized forms suitable for signal and image processing The first set of such func-tions that constituted an orthonormal basis for L^(R) was introduced in

wave-1910 by Haar However the Haar functions do not have good localization

in the combined time-frequency space and, therefore, in many cases do not satisfy the properties required in signal and image processing and anal-ysis The problem of how to construct functions that are well localized

in both time and frequency was confronted by communication engineers dealing with the analysis of speech in the 1920s and 1930s About half a century ago Gabor introduced the optimally localized function, obtained

by windowing a complex exponential with a Gaussian window The main advantage of this localized waveform is in achieving the lowest bound on the joint entropy, defined as the product of effective temporal or spatial ex-tent and frequency bandwidth However, the Gabor elementary functions, which span L^(R), are not orthogonal

The subject of representation in combined spaces refers to type and Gabor-type expansions Such expansions are more suitable for the analysis and processing of natural signals and images than expansion by the traditional application of Fourier series, polynomials, and other functions

wavelet-of infinite support, since the nonstationarity wavelet-of natural signals calls for localization in both time (or spatial variables in the case of images) and frequency (or scale) in their representation While global transforms such

as the Fourier transform, which is the most widely used in engineering, describe the spectrum of the entire signal as a whole, the wavelet-type and Gabor-type transforms allow for extraction of the local signatures of the signal as they vary in time, or along the spatial coordinates in the case of images By correlating signals with appropriately chosen wavelets, certain analysis tasks such as feature extraction, signal compression, and recognition can be facilitated The ability of wavelets to localize signals in time, or spatial variables in the case of images, allows for a multiresolution approach in signal processing In fact, since the wavelet transform is defined

by either its basic time-scale, position-scale, or decomposition structure,

ix

it naturally lends itself to multiresolution analysis Yet, a great deal of

freedom is left for the exact choice of the transform's kernel and various

parameters Thus, the wavelet approach provides us with a wide range of

powerful tools for signal processing and analysis These are described in

this volume

The general interrelated topics involving multiscale analysis, wavelet

and Gabor analysis, can all be viewed as enhancing the traditional Fourier

analysis by enabling an adaptation of combined time and frequency

local-ization procedures to various tasks The simple and basic transition from

the global Fourier transform to the localized (windowed) Fourier

analy-sis, consists of segmenting the signal into windows of fixed length, each of

which is expanded by a Fast Fourier Transform (FFT) or Discrete Cosine

Transform (DCT) This type of procedure corresponds to spectrograms, to

Gabor transform, as well as to localized trigonometric transforms A dual

version of this procedure corresponds to filtering the signal, or windowing

its Fourier transform, usually referred to as wavelet, wavelet packets, or

subband coding transforms

Wavelet analysis and more generally adapted waveform analysis has

provided a simple comprehensive mathematical and algorithmic

infrastruc-ture for the localized signal processing tools, as well as many new tools

which evolved as a result of the cross-fertilization of ideas originated in

many fields, such as the Calderon-Zygmund theory in mathematics,

multi-scale ideas from geophysical seismic prospecting, mathematical physics of

coherent states and wave packets, pyramid structures in image processing,

band and subband filtering in signal processing, music, numerical analysis,

etc In this volume we don't intend to elaborate on the origin of these ideas,

but rather on the current state of this elaborate toolkit and the relative

advantages it brings to the scene

While to some extent most of the qualitative analytical aspects of

wavelet analysis, and of the windowed Fourier transform, have been well

understood by mathematicians for at least 30 years, the recent explosion of

activity and algorithms is due to the discovery of the orthogonal wavelets

by Stromberg and Meyer, and the connection to Quadrature Mirror Filter

(QMF) by Mallat and Daubechies More fundamental yet is our better

understanding of structures permitting construction of a multitude of

or-thogonal and nonoror-thogonal expansions customized to tasks at hand, and

enabling the introduction of fast computational methods and realtime

pro-cessing The role and usefulness of redundancy in providing stability in

signal representation, as opposed to efficiency, has also become clear by

means of the application of frame analysis and the Zak transform

Some of the main tasks that can be accomplished by the application

of wavelet-based tools are related to feature extraction and efficient

de-scription of large data sets for processing and computations This is the

point where, instead of using algebraic or analytic formulas, functions or

measured data are described efficiently by adapted waveforms which are, in

turn, described algorithmically and designed specifically to optimize

vari-ous tasks Perhaps the most natural analogy to the new modes of analysis

(or signal transcription) is provided by musical scores and orchestration;

an overlay of time frequency analysis The musical score is somewhat more

general and abstract than the alphabet and corresponds roughly to a

de-scription of a piece of music by specifying which notes are being played, i.e.,

the note's characteristic pitch, amplitude, duration, and location in time

While traditional windowed Fourier analysis considers a Fourier

represen-tation of the signal in each window of space (or time), wavelets, wavelet

packets, and their variants provide a description in which notes of different

duration (or resolution) are superimposed For images, this corresponds

to an overlay of patterns of different size and scale This multiscale

rep-resentation allows for a better separation of textures and structures, and

of decomposition of the textures into their basic elements The

comple-mentary procedure introduces a new approach to speech, music, and image

synthesis, yet to be further explored

Most of the chapters in this book are based on the lectures delivered at

the Neaman Workshop on Signal and Image Representation in Combined

Spaces, held at Technion Additional chapters were contributed by invitees

who could not attend the workshop The material presented in this volume

brings together a rich variety of ideas that blend most aspects of analysis

mentioned above These papers can be clustered into affinity groups as

follows:

Variations on the windowed Fourier transform and its applications,

re-lating Fourier analysis to analysis on the Heisenberg group, are provided in

the following group of papers: M An, A Bordzik, I Gertner, and R

Tolim-ieri: "Weyl-Heisenberg System and the Finite Zak Transform;" M

Basti-aans: "Gabor's Expansion and the Zak Transform for Continuous-Time and

Discrete-Time Signals;" W Schempp: "Non-Commutative Affine

Geome-try and Symbol Calculus: Fourier Transform Magnetic Resonance Imaging

and Wavelets;" M Zibulski and Y Y Zeevi: "The Generalized Gabor

Scheme and Its Application in Signal and Image Representation."

Constructions of special waveforms suitable for specific tasks are given

in: J S Byrnes: "A Low Complexity Energy Spreading Transform Coder;"

A Coheu and N Dyn: "Nonstationary Subdivision Schemes,

Multiresolu-tion Analysis and Wavelet Packets."

The use of redundant representations in reconstruction and

enhance-ment is provided in: J J Benedetto: "Noise Reduction in Terms of the

Theory of Frames;" Z Cvetkovic and M Vetterli: "Overcomplete

Expan-sions and Robustness;" F Bergeaud and S Mallat: "Matching Pursuit of

Images."

Applications of efBcient numerical compression as a tool for fast

nu-merical analysis are described in: A Averbuch, G Beylkin, R Coifman,

and M Israeli: "Multiscale Inversion of Elliptic Operators;" A Harten:

"Multiresolution Representation of Cell-Averaged Data: A Promotional

Review."

Approximation properties of various waveforms in diflFerent contexts

are described in the following series of papers: A J E M Janssen: "A

Density Theorem for Time-Continuous Filter Banks;" V E Katsnelson:

"Sampling and Interpolation for Functions with Multi-Band Spectrum:

The Mean Periodic Continuation Method;" M A Kon and L A Raphael:

"Characterizing Convergence Rates for Multiresolution Approximations;"

C Chui and Chun Li: "Characterization of Smoothness via Functional

Wavelet Transforms;" R Lenz and J Svanberg: "Group Theoretical

Trans-forms, Statistical Properties of Image Spaces and Image Coding;" J Prestin

and K Selig: "Interpolatory and Orthonormal Trigonometric Wavelets;"

B Rubin: "On Calderon's Reproducing Formula;" and "Continuous Wavelet

Transforms on a Sphere;" V A Zheludev: "Periodic Splines, Harmonic

Analysis and Wavelets."

A c k n o w l e d g m e n t s

The Neaman Workshop was organized under the auspices of The Israel

Academy of Sciences and Humanites and co-sponsored by The Neaman

Institute for Advanced Studies in Science and Technology; The Institute

of Advanced Studies in Mathematics; The Institute of Theoretical Physics;

and The Ollendorff Center of the Department of Electrical Engineering,

Technion—Israel Institute of Technology

Several people helped in the preparation of this manuscript We wish

to thank in particular Ms Lesley Price for her editorial assistance and

word-processing of the manuscripts provided by the authors, Ms Margaret

Chui for her editing and overall guidance in the preparation of the book,

and Ms Katy Tynan of Academic Press for her communications assistance

Haifa, Israel Yehoshua Y Zeevi

New Haven, Connecticut Ronald Coifman

June 1997

Contributors

Numbers in parentheses indicate where the authors^ contributions begin

M A N (1), Prometheus Inc., 52 Ashford Street, AUston, MA 02134

[myoung@ccs.neu.edu]

A M I R AVERBUCH (341), School of Mathematical Sciences, Tel Aviv

Uni-versity, Tel Aviv 69978, Israel

[amir@math.tau.ac.il]

MARTIN J. BASTIAANS (23), Technische Universiteit Eindhoven, Faculteit

Elektrotechniek, Postbus 513, 5600 MB Eindhoven, Netherlands

[M.J.Bastiaans@ele.tue.nl]

J O H N J. B E N E D E T T O (259), Department of Mathematics, University of

Maryland, College Park, Maryland 29742

[jjb@math.umd.edu]

FRANgois BERGEAUD (285), Ecole Centrale Paris, Applied Mathematics

Laboratory, Grande Voie des Vignes, F-92290 Chatenay-Malabry, France

[francois@mas.ecp.fr]

GREGORY BEYLKIN (341), Program in Applied Mathematics, University

of Colorado at Boulder, Boulder, CO 80309-0526

CHARLES K CHUI (395), Center for Approximation Theory, Texas A&;M

University, College Station, TX 77843

[cchui@tamu.edu]

A L B E R T C O H E N (189), Laboratoire d^AnalyseNumerique, UniversitePierre

et Marie Curie, 4 Place Jussieu, 75005 Paris, France

[cohen@ann j ussieu fr]

R O N A L D COIFMAN (341), Department of Mathematics, P.O Box 208283,

Yale University New Haven, CT 06520-8283

[coifman@j ules mat h yale edu]

ZORAN CVETKOVIC (301), Department of Electrical Engineering and

Com-puter Sciences, University of California at Berkeley, Berkeley, CA

94720-1772

[zor an@eecs berkeley.edu]

NiRA D Y N (189), School of Mathematical Sciences, Sackler Faculty of

Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel

[niradyn@math.tau.ac.il]

I. G E R T N E R (1), Computer Science Department, The City College of New

York, Convent Avenue at 138th Street, New York, NY 10031

[csicg@csfaculty.engr.ccny.cuny.edu]

A M I HARTEN (361), School of Mathematical Sciences, Tel-Aviv University,

Tel-Aviv, 69978 Israel

MosHE ISRAELI (341), Faculty of Computer Science, Technion-Israel

In-stitute of Technology, Haifa 32000, Israel

[isr aeli@ cs t echnion ac il]

A J E M. JANSSEN (513), Philips Research Laboratories, WL-01, 5656

AA Eindhoven, The Netherlands

V I C T O R E KATSNELSON (525), Department of Theoretical Mathematics,

The Weizmann Institute of Science, Rehovot 76100, Israel

[katze@wisdom.weizmann.ac.il]

MARK A K O N (415), Department of Mathematics, Boston University,

Boston, MA 02215

R E I N E R LENZ (553), Department of Electrical Engineering, Linkoping

University, S-58183 Linkoping, Sweden

[reiner@isy.liu.se]

CHUN L I (395), Institute of Mathematics, Academia Sinica, Beijing 100080,

China

S T E P H A N E MALLAT (285), Ecole Poly technique, CMAP, 91128 Palaiseau

BORIS RUBIN (439, 457), Department of Mathematics, Hebrew University

of Jerusalem, Givat Ram 91904, Jerusalem, Israel

J O N A S SVANBERG (553), Department of Electrical Engineering, Linkoping

University, S-58183 Linkoping, Sweden

[svan@isy.liu.se]

R TOLIMIERI (1), Electrical Engineering Department, The City College

of New York, Convent Avenue at 138th Street, New York, NY

10031

M A R T I N V E T T E R L I (301), Department d'Electricite EPFL, CH'1015

Lau-sanne, Switzerland

[martin.vetterli@de.epfl.ch]

YEHOSHUA Y. ZEEVI (121), Department of Electrical Engineering,

Technion-Israel Institute of Technology, Haifa 32000, Technion-Israel

[zeevi@ee technion ac il]

VALERY A ZHELUDEV (477), School of Mathematical Sciences, Tel Aviv

University, 69978 Tel Aviv, Israel

[zhel@math.tau.ac.il]

M E I R ZIBULSKI (121), Multimedia Department, IBM Science and

Tech-nology, MATAM, Haifa 31905, Israel

[meir z @ vnet ibm com]

Finite Zak Transform

M An, A Brodzik, I Gertner, a n d R Tolimieri

A b s t r a c t Previously, a theoretical foundation for designing

algo-rithms for computing Weyl-Heisenberg (W-H) coefficients at critical

sampling was established by applying the finite Zak transform This

theory established clear and easily computable conditions for the

ex-istence of W-H expansion and for stability of computations The

main computational task in the resulting algorithm was a 2-D finite

Fourier transform

In this work we extend the applicability of the approach to

rationally over-sampled W-H systems by developing a deeper

un-derstanding of the relationship established by the finite Zak

trans-form between linear algebra properties of W-H systems and function

theory in Zak space This relationship will impact on questions of

existence, parameterization, and computation of W-H expansions

Implementation results on single RISC processor of i860 and

the PARAGON parallel multiprocessor system are given The

algo-rithms described in this paper possess highly parallel structure and

are especially suited in a distributed memory, parallel-processing

en-vironment Timing results show that real-time computation of W-H

expansions is realizable

§1 I n t r o d u c t i o n

D u r i n g t h e last four years powerful new m e t h o d s have been introduced for

analyzing Wigner transforms of discrete and periodic signals [10, 11, 13]

based on finite W - H expansions [2, 5, 6, 12] A recent work [10] a d a p t e d

these m e t h o d s t o gain control over t h e cross-term interference problem

[9] by constructing signal systems in t i m e frequency space for expanding

Wigner trg,nsforms from W-H systems based on Gaussian-like signals

T h e c o m p u t a t i o n a l feasibility of t h e m e t h o d in [10] d e p e n d s strongly

on t h e availability of eflScient and stable algorithms for c o m p u t i n g W-H

expansion coefficients Since W - H systems are not orthogonal, s t a n d a r d

Hilbert space inner-product m e t h o d s do not generally apply Moreover,

Signal and Image Representation in Combined Spaces O

Y Y Zeevi and R R Coifman (Eds.), PP- 3 - 2 1

Copyright © 1 9 9 8 by Academic Press

All rights of reproduction in any form reserved

since critically sampled W-H systems may not form a basis, over-sampling

in time frequency is necessary for the existence of arbitrary signal sions In fact, this is usually the case for systems based on the Gaussian

expan-In [10, 11, 12, 13, 15], the concept of biorthogonals was applied to the lem of W-H coefficient computation In [15], the Zak transform provided the framework for computing biorthogonals for rationally over-sampled W-

prob-H systems forming frames A similar approach for critically and integer over-sampled W-H systems can be found in [3, 4] The goal in this work is somewhat different in that major emphasis is placed on describing linear spans of W-H systems that are not necessarily complete and on establish-ing, in a form suitable for RISC and parallel processing, algorithms for computing W-H coefficients of signals in such linear spans For the most part, our approach extends on that developed in [3, 14] and frame theory, although an important part in [15] plays no role in this work However,

as in these previous works [7, 8], the finite Zak transform will be lished as a fundamental and powerful tool for studying critically sampled and rationally over-sampled W-H systems and for designing algorithms for computing W-H coefficients for discrete and periodic signals The role of the finite Zak transform is analogous to that played by the Fourier trans-form in replacing complex convolution computations by simple pointwise multiplication In this new setting, properties of W-H systems, such as their spanning space and dimension, can be determined by simple opera-tions on functions in Zak space This relationship will impact on questions

estab-of existence, parameterization, and computation estab-of W-H expansions

In the over-sampled case, both integer and rational over-sampling are investigated Implementation results on single RISC processor of i860 and the PARAGON parallel multiprocessor system are given for sample sizes both of powers of 2 and mixed sizes with factors 2, 3, 4, 5, 6, 7, 8, and 9 The algorithms described in this paper possess highly parallel structure and are especially suited in a distributed memory, parallel-processing environment Timing results on single i860 processor and on 4- and 8-node computing systems show that real-time computation of W-H expansions is realizable

In Section 2, the basic preliminaries will be established Algorithms will

be described in Section 3 for critically sampled W-H systems, in Section

4 for integer sampled systems, and in Section 5 for rationally sampled systems Implementation results will be given in Sections 6, 7, and 8

over-§2 Preliminaries

2.1 Weyl-Heisenberg systems

Choose an integer AT > 0 A discrete function / ( a ) , a e Z is called

N-periodic if

f{a + N) = f{a), aeZ

Denote by L{N) the Hilbert space of all AT-periodic functions with inner

wavelets having generator g

Suppose N = KM with positive integers K and M The collection of

N functions

{QkM^mK :0<k<K, 0 < m < M } denoted by {g, M, K) is called a critically sampled Weyl-Heisenberg (W-H)

system Critically sampled W-H systems have been extensively studied in

several works including [3] where the finite Zak transform was used to

establish conditions for such systems to be a basis of L{N) The extension

to 2-D for applications to image representation and image analysis can be

found in [14]

Suppose N = K'M' is a second factorization of N into two positive

integers The collection of K'M functions

{gk'M'^mK :0<k' <K\ 0 < m < M } denoted by {g, M', K) is called a general W-H system The system (^, M', K)

is called over-sampled \i M' < M and under-sampled if M' > M

Over-sampled systems are necessarily redundant having finer time-resolution

as compared with associated critically sampled systems A dual theory

can easily be developed which introduces redundancies by having finer

frequency-resolution

An expansion of a signal / G L{N) as a linear combination over a W-H

system is called a W-H expansion, with the corresponding coefficients called

W-H coefficients In general, except for critically sampled W-H systems

forming a basis of L{N), W-H coefficients are not uniquely determined

W-H basis were studied in [3, 14]

An over-sampled W-H system (^, M ' , K) is called an m^e^er over-sampled

system if i? = M/M' is an integer and a rationally over-sampled system

otherwise

2.2 Finite Zak t r a n s f o r m (FZT)

Fundamental properties of FZT have been described in several works [3,

14] with applications of FZT to algorithms for computing W-H expansion

coefficients for a critically sampled W-H basis We will briefly outline the

one-dimensional case

Suppose N = KM For / G L{N) define the finite Zak Transform

(FZT), Z ( / ^ ) / ( a , 6 ) , a , 6 G Z b y

K-l Z{K)f{a,b)=J2 / ( ^ + Mfc)e2^^^'^/^, a.beZ (2.2.1)

k=0

The functional equations

Z{K)f{a -^M,b) = e-2^^^/^Z(X)/(a, 6), a, 6 G Z (2.2.2)

Z{K)f{a, b + K) = Z{K)f{a, 6), a, 6 G Z (2.2.3) imply Z{K)f is A/'-periodic in each variable and is completely determined

by its values

Z{K)f{a,b), 0 < a < M , 0<b<K (2.2.4) Denote by L{M^K) the Hilbert space of all functions F{a,b), 0 < a <

M, 0 < b < K, with inner product

M-lK-l {F,G)= Y^ ^ F ( a , 6 ) G * ( a , 6 ) , F,GeL{M,K) (2.2.5)

a=0 6=0

Define Zo{K)f G L{M,K) by

ZQ{K)f{a, b) = Z{K)f{a, 6), 0 < a < M, 0 < 6 < K (2.2.6) The mapping K~'^^'^Zo{K) is an isometry from L{N) onto L(M, K) If

F G L(M, X) and / G L(iV) is defined by

K-l f{a + Mk) = K-^ Yl ^(^'6)6-2^'^^^/^, 0<a<K, 0<b<K

6=0

(2.2.7)

T h e n F = : Z o ( i ^ ) /

An important relationship exists between the FZT of W-H wavelets

corresponding to a fixed generator g given by

Z{K)gmA<^^ b) = e-2^^^^/^Z(i^)^(a + m, 6 - n), a, 6 G Z (2.2.8)

In particular,

0<k<K, 0<m<M (2.2.9)

From these relationships we can prove two fundamental results which

gov-ern the application of FZT to the analysis of W-H expansions Set F =

Zo{K)f and G = Zo{K)g

First fundamental result

K-lM-l F{a,b)G''{a,b) = ^ E E < f'9kM,n.K > e'^^^^^'^l'^^^'l^^

k=0 m=0

(2.2.10) The second result uses the FZT to unravel a W-H expansion as a prod-

uct in Zak space

Second fundamental result For / G L{N),

K-lM-l f=Y.Yl c{kM,mK)gkM,mK (2.2.11)

k=0 m=0

For critically sampled W-H systems, we have the following result:

Theorem 1 The critically sampled W-H system

{g,M,K) = {gkM^mK :0<k<K, 0<m<M} (2.2.13)

is a basis of L{N) if and only if G never vanishes

§3 Critically sampled W - H systems

Generalizations of the results in the previous section depend on an analysis

of the zero-sets of FZT Consider a critically sampled system {g, M, K) and

set G = Zo{K)g Denote the zero-set of G by

C-Theorem 2 A function f e L{N) is in the linear span of [g, M, K) if and

only if F — Zo{K)f vanishes on ( The dimension of the Unear span of {g, M, K) is N — J where J is the number of points in (

Proof: By the second fundamental result we can identify the linear span

of {g, M, K) with the space of all F of the form F =^ GP with P € L{M, K)

In particular, if / is in the linear span of (^, M, K)^ then F must vanish on C, Conversely, suppose F vanishes on ( Define P G I/(M,K) by

P{a,b) = [ na,b)/G{a,b), {a,b) ^ C

otherwise

Then F = GP and F is in the linear span of {g^M^K) Since we have shown that the linear span of {g, M, K) can be identified with the space of all F e L{M, K) which vanish on C, the theorem follows •

Denote by L{Q the space of all functions a G I/(M, K) which vanish

on the complement, C^, of ( Theorem 2 leads to the following algorithm

for computing expansion coefficients of / relative to {g,M,K) To each

a G L(C), define the function P ^ G L{M, K) by

^ ' ^ \ a(a,6), {a,b)e(:

By the second fundamental result, a collection of W-H coefficients is given

by the 2-D M x K FT of P'^ia, b)

If C is not empty, the expansion coefficients are not uniquely determined

In fact, every / in the linear span of {g, M, K) has a J-dimensional space

of W-H expansions over (^, M, K) parameterized by L{(^)

§4 Integer over-sampled W - H systems

Consider an integer over-sampled W-H system g = (^, M', K) Since K' =

RK with R an integer, each ^ <k' < K' can be written uniquely as

It is just as simple to consider the more general case where g is the

union of arbitrary critically sampled W-H systems

For any subset 7 of {(a, 6) : 0 < a < M, 0 < b < K} define F\y £

L{M,K) by

^i,(M)={„^(-')' i-l\;:

Theorem 3 Suppose g is the union of critically sampled W-H systems,

g^ = [g^^M^K), 0 < r < R Denote the zero-set of Gr = Zo{K)gr by

Cr and set C = H^^QCV Then f is in the linear span of g if and only if

F = Zo(i^)/ vanishes on ( The dimension of the linear span of g is N - J

where J is the number of points in (

Proof: If / is in the linear span of g, then we can write / = Xlr^To fr

where fr is in the linear span of g^ Theorem 2 implies F^ = Zo{K)fr

vanishes on (^r- Since F = X^^JQ ^r^ f vanishes on ^

Conversely, suppose F vanishes on C If we can write F = ^^ZQ Fr

where Fr vanishes on (r, then Theorem 2 implies that / is in the linear

span of g The following construction determines one such decomposition

Define Fr G L(M, K),0<r <Rhy

Fo = F\CS

Fi = F\ConC,

FR.I - F | C o n - - - n C i ? - 2

By definition, Fr vanishes on (^r and since F vanishes on ^, F = Ylr=o

^r-Since the linear span of g can be identified with the space of all F e

L{M, K), which vanish on C, the theorem is proved •

Prom the construction in the theorem we have the following:

Corollary 1 If / is in the linear span of g, then we can write F =

Y^r=o Fr, where FrFg = 0 whenever r ^ s, 0 <r,s < R

Choose / G L{N) in the linear span of g An algorithm for computing

a W-H expansion of / over g is given as follows

• Compute the collection of 2D M x K FT of

Pr{a,b) = ^ ^ ^ , 0<r<R

This stage is understood to be taken as in the critically sampled

case, with arbitrary values assigned to the quotient at points where

the functions Gr, 0 < r < R vanish

If we assume that T l o g T computations are needed for the T-point FT,

then the complexity of one W-H expansion computation is

NlogK-h R{N\ogK + NlogM) -f RN (4.2)

but advantage can be taken of the large number of zero data values

The coefficient set of W-H expansions of / G L{N) over g is

param-eterized by the collection of decompositions of F and by the arbitrarily

assigned values to the quotients at the points (r^ 0 <r < R

§5 Rationally over-sampled W - H s y s t e m s

Consider the rationally over-sampled W-H system g' = {g^M'^K) where

A^ = MK = M'K' R = M/M' is no longer an integer Denote the

least common multiple of M and M' by M and set M = MS = M'S'

S and S' are positive integers such that S divides K, S' divides K' ^ and

N = 'Mf

=^$ Arguing as in the integer over-sampled case, we have that g' is the union

of the under-sampled W-H systems

g;, = {QS'.'M.K), QS' = 5's'M',o, 0 < s' < 5'

Since M = MS with 5 a positive integer, the under-sampled W-H

sys-tem g^, is contained in the critically sampled W-H syssys-tem gg/ = {gs> ^M,K)

Denote the union of these critically sampled W-H systems by g and set

Gs'=Zo{K)gs'

Theorem 4 A function f G L{N) is in the lineax span of g' if and only if

F = ZQ{K)f has the form

s'-i F= Y^Gs'Ps' (5.1)

s'=0

where Pg' G L{M, K) satisfies

Ps'(a,b+^j =Ps'{a,b), 0<s'<S', 0 < a < M, 0 < 6 < ^

(5.2)

Proof: Since

the theorem follows from the second fundamental result •

If an expansion of the form given in the theorem can be found, then arguing as before, a collection of W-H expansion coefficients of / over g' is

given by the collection of 2-D Mf FTs of

Ps/(a,6), 0 < a < M , 0 < 6 < —, 0 < 5 ' < 5 '

In [1], an algorithm was given for computing W-H coefficients for rationally over-sampled W-H systems based on pseudo-matrix inversion of the matrix function

G ( a , 6 ) = \GS' ( a , 6 - h 5 — )

L \ '^ / Jo<s<5,0<s'<5' Implementation results for this case will be given below An alternate

approach will be taken in this work which presents an iterative algorithm

more in line with the philosophy of the preceding sections We will describe

an algorithm which for any / G L{N) computes a W-H expansion for the

orthogonal projection of / onto the linear span of g'

Denote by L(M, ^ ) the subspace of all P e L{M, K) satisfying P{a,

6-1-^) = P(a,6) 0 < a < M , 0 < 6 < / ^ The following result describes

an algorithm for computing orthogonal projections onto the subspace G • L{M,f) = {GP:PeL{M,f}

Theorem 5 Suppose F e L{M, K) has the form F = GP with P € L{M, K) Then there exists P' G L{M, ^) satisfying the condition

J2\G(^a,b + Sj^\ P'{a,h) = ^ | G L 6 + 5 | ' ) | pLb + s^Y

0<a<M,0<b<

~S' (5.3) and F' — GP' is the orthogonal projection of F onto G • L(M, ^ )

Proof: Since the right-hand side of (5.3) vanishes at any point (a, 6),

0 < a < M , 0 < 6 < f, at which

s - i

s=0

G [a,b + s K = 0

and we can solve (5.3) for some P ' G L{M, ^ ) Define Q € L{M, K) by

The computation of P ' requires N additions and multiplications

Algorithm for computing W - H coefficients

• For each 0 < s' < 5', compute Pg' G L{M, K) such that

P | 0 = G , P , , , Ps'£L{M,K)

• Compute the orthogonal decomposition

Gs'Ps' = Gs'Pg' +

Gs'Pg'-• If Pj, = 0 for all 0 < s' < 5', then / i s orthogonal to the linear span

of g', and we are done

• Otherwise, choose 0 < SQ < 5 ' such that

and at some point of the iteration, we will arrive at P = P ' -f P " , with

F' = Zo{K)f with f in the linear span of g' and F" = Zo{K)f', f"

orthogonal to g' A W-H expansion of / ' over g' can be given by a collection

of 2-D M X f FTs as before

§6 Implementation results

In this section we describe implementation issues and present timing results

for the implementation of the algorithms presented in the previous sections

Implementations on a single Intel i860 RISC microprocessor as well as on

the Paragon multi-processor parallel platform are reported

6.1 Critical sampling (C.S.)

We have tested three basic analysis functions:

• Gaussian function

When K and M are both even integers, the FZT of Gaussian window

function has a zero at {K/2,M/2) Set Q{K/2,M/2) = 0.0 The

to-tal energy of Gabor coefficients will be minimum

When either K or M is an odd integer, or both of them are odd

integers, the FZT of Gaussian window function has no zeros

• Rectangular function

A small-size rectangular window will result in FZT with no zeros

For example, N = K x M = 1200, a window of width 90 centered at

600, has no zeros in Zak space

A rectangular window of width 150 centered at 600 has zeros in Zak

space located at: (j,8), (j,16), (j,24), (j,32), where j=0 to 39

• Triangular function

When either K or M is an odd integer, or both of them are odd

integers, there are no zeros in Zak space

A relatively small triangular window will result in a single zero at

the center of Zak space For example, iV = 40 x 30 = 1200, a window

of 61 non-zero values centered at 600, has one zero in Zak space at

(20, 15)

We have implemented the computation for Critical Sampling case: the

main program is in FORTRAN and the FFT modules are fine-tuned i860

assembly with mixed sizes Timing results are given in Tables 1 and 2

Complexity

For a real input signal / , the FZT of / is Hermitian symmetric along dimension If the analysis signal is also real, then the 2-D MxK Q{a,b) has the same symmetry The inverses of the FZT of g{a^ b) are pre-computed

K-and stored in memory The complexity of the computation (F(n) denotes the complexity of n-point FFT):

Z{K)f (FZT of / )

Z{K)f/Z{K)g 2-D FT of Q Herm Symm along K

M X real F{K) K/2 X M multiplications

Table 1 Timing results (in milliseconds) on the Intel i860 RISC microprocessor (critical sampling - 2^)

§7 Integer over-sampling

We choose the decomposition F = Z{K)f — Yl,r=^ ^r such that F i , , FR-I each has only one non-zero point, so that the computation of the 2-D

F T of Qi(a, 6 ) , Q/?-i(a, b) is trivial The codes are similar to a critically

sampled case with data rearrangement at the end

7.1 Rational over-sampling

In [12], the authors point out that a Gaussian window function

over-sampled by more than 20 percent (5/4), does not have significant ence We have implemented the computation for over-sampling rates 3/2 and 5/4 Again, the main routine is coded in FORTRAN, and the DFT

128 X 24

128 X 48 64x96

Table 2 Timing results (in milliseconds) on the Intel i860 RISC Microprocessor (critical sampling - mixed sizes)

routines are fine-tuned i860 assembly codes for mixed sizes For the plex singular value decomposition (SVD), we used the LINPACK routine

com-We have tested three basis functions:

• Gaussian basis function

Rational over-sampling of 3/2 and 5/4 were tested If the rank

(G(a,6)) equals to 2 or 4 correspondingly, then g is complete and every / has a W-H expansion over g

• Rectangular basis function

Rational over-sampling by 3/2 and 5/4 are tested Rectangular

win-dow sizes have to be chosen such that it is not a factor of K along

X-dimension to have every / expandable in the W-H system

• Triangular basis function

An example of size AT = 40 x 30 = 1200 has been tested with tional over-sampling by 3/2 The experimental results are:

ra-A window of size 101 centered at 600 results in an expandable W-H system

A window of size 151 centered at 600 results in an expandable W-H system

A window of size 201 results in point (20,10) being a zero singular value in Zak transform space

C o m p l e x i t y

In the case of real input and real analysis signals, the FZT is Hermitian

symmetric along X-dimension We can show that the S' 2-D M^ Ps{cii b)

has Hermitian symmetry along ^-dimension The complexity of real-time computation is:

FZT of /

G+{a,b)F{a,b) S' 2-D FT of Ps with

Hermitian Symmetry along ^

M x real FiK)

M X ^ matrix 1 S' X S multiply a vector S

Table 4 Timing results (in milliseconds) on the Intel i860 microprocessor

The algorithms described in Sections 3, 4 and 5 possess highly parallel

structure They are particularly suitable in a distributed memory

multipro-cessor system For example, in the critically sampled case, the algorithm

can be implemented as follows:

• Each processor receives Ki X-point input data

• Compute Ki K-point real F F T

• Point-wise multiplication of the pre-calculated Zak transform of the

basis function l/Z{K)g{a, b)

• Compute Ki /T-point Hermitian FFT

• Data permutation between processors (matrix transpose)

• Compute K2 M-point real FFT

Implementation of an integer over-sampled case has a similar structure

to the critically sampled case, and the rationally over-sampled case has

a better parallel structure, since it has 5 ' relatively small 2-D ^ x M

FFT's, and they might be carried out locally in each processor without interprocessor data permutation Timing results of critical sampling on the Intel 4-nodes and 8-nodes Paragon are given in Tables 5 and 6 The parallel flow diagram is given in Figure 1

M-pt

real FT

M-pt real FT

\ M-pt

Table 5 Timing results (in milliseconds) on the Intel Paragon (4-nodes)

Table 6 Timing results (in milliseconds) on the Intel Paragon (8-nodes)

§9 Conclusions

Algorithms for the computation of Weyl-Heisenberg (W-H) coefficients for the cases of critical sampling, integer over-sampling, and rational over-sampling have been presented, and easily computable conditions for the existence of W-H expansions have been derived in terms of the Zak trans-form of the signal and the analysis function We have shown that the al-gorithms described lead to very efficient FFT-based implementations both for single DSP processor systems as well as for parallel multi-processor configurations

Acknowledgments The research of M An was supported by ARPA

F49620-C-91-0098, and research of R Tolimieri is supported by AFOSR RF#447323

2] Auslander, L and R Tolimieri, On finite Gabor expansion of signals,

IMA Pvoc Signal Processing, Springer-Verlag, New York, 1988

3] Auslander, L., I Gertner, and R Tolimieri, Finite Zak transforms

and the finite Fourier transforms, IMA on Radar and Sonar, Part II

39 Springer-Verlag, New York, 1991, 21-36

4] Auslander, L., I Gertner, and R Tolimieri, The discrete Zak transform application to time-frequency analysis and synthesis of nonstationary

signals, IEEE Trans Signal Processing 39(4) (1991), 825-835

5] Bastiaans, M J., Gabor signal expansion and degree of freedom of a

signal Optica Acta 29 (1982), 1223-1229

6] Gabor, D., Theory of communications, Proceedings lEE III 93 (1946),

429-457

7] Gertner, I and G A Geri, Image representation using Hermite

func-tions Biological Cybernetics 71 (1994), 147-151

8] Gertner, I and G A Geri, J Optical Soc Amer 11 (8) (1994),

2215-2219

9] Hlawatsch, F., Interference terms in the Wigner distribution, in Proc

1984 Int Conf on DSP, Florence, Italy, 1984, pp 363-367

[10] Qian, S and J M Morris, Wigner distribution decomposition and

cross-term deleted representation Signal Processing 27 (1992),

125-144

[11] Raz, S., Synthesis of signals from Wigner distributions:

Representa-tion on biorthogonal basis, Signal Processing 20(4) (1990), 303-314

[12] Wexler, J and S Raz, Discrete Gabor expansions, Signal Processing

21(3) (1990), 207-220

[13] Wexler, J and S Raz, Wigner-space synthesis of discrete-time periodic

signals, IEEE Trans Signal Processing 40{8) (1990)

[14] Zeevi, Y Y and I Gertner, The finite Zak transform: An efficient tool

for image representation and analysis, J Visual Comm and Image

Representation 3(1) (1992), 13-23

[15] Zibulski, M and Y Y Zeevi, Oversampling in the Gabor scheme,

IEEE Trans Signal Processing 41(8) (1993)

Computer Science Department

The City College of New York

Convent Avenue at 138th Street

New York, NY 10031

csicg@csfaculty.engr.ccny.cuny.edu

R Tolimieri

Electrical Engineering Department

The City College of New York

Convent Avenue at 138th Street

New York, NY 10031

Continuous-Time and Discrete-Time Signals

M a r t i n J Bastiaans

A b s t r a c t Gabor's expansion of a signal into a discrete set of shifted

and modulated versions of an elementary signal is introduced and its

relation to sampling of the sHding-window spectrum is shown It is

shown how Gabor's expansion coefficients can be found as samples

of the sliding-window spectrum, where, at least in the case of critical

sampling, the window function is related to the elementary signal in

such a way that the set of shifted and modulated elementary signals

is bi-orthonormal to the corresponding set of window functions

The Zak transform is introduced and its intimate relationship

to Gabor's signal expansion is demonstrated It is shown how the

Zak transform can be helpful in determining the window function

that corresponds to a given elementary signal and how it can be

used to find Gabor's expansion coefficients

The continuous-time as well as the discrete-time case are

con-sidered, and, by sampling the continuous frequency variable that

still occurs in the discrete-time case, the discrete Zak transform and

the discrete Gabor transform are introduced It is shown how the

discrete transforms enable us to determine Gabor's expansion

coef-ficients via a fast computer algorithm, analogous to the well-known

fast Fourier transform algorithm

Not only Gabor's critical sampling is considered, but also, for

continuous-time signals, the case of oversampling by an integer

fac-tor It is shown again how, in this case, the Zak transform can be

helpful in determining a (no longer unique) window function

corre-sponding to a given elementary signal An arrangement is described

which is able to generate Gabor's expansion coefficients of a rastered,

one-dimensional signal by coherent-optical means

§1 I n t r o d u c t i o n

It is sometimes convenient t o describe a t i m e signal (/?(t), say, not in t h e t i m e

domain, b u t in t h e frequency domain by means of its frequency spectrum^

Signal and Image R e p r e s e n t a t i o n in Combined Spaces 23

Y Y Zeevi and R R Coifman ( E d s ) , PP- 2 3 - 6 9

Copyright © 1 9 9 8 by Academic P r e s s

All rights of r e p r o d u c t i o n in any form reserved

i.e the Fourier transform (p{uj) of the function ^{t), which is defined by

^{uj)= f ip{t)e-^'^'dt; (1.1)

a bar on top of a symbol will mean throughout that we are dealing with a

function in the frequency domain (Unless otherwise stated, all integrations

and summations in this paper extend from — oo to +00.) The inverse

Fourier transformation takes the form

^{t) = ^jip{^)e^^'dw (1.2)

The frequency spectrum shows us the global distribution of the energy of the

signal as a function of frequency However, one is often more interested in

the momentary or local distribution of the energy as a function of frequency

The need for a local frequency spectrum arises in several disciplines It

arises in music, for instance, where a signal is usually described not by

a time function nor by the Fourier transform of that function, but by its

musical score] indeed, when a composer writes a score, he prescribes the

frequencies of the tones that should be present at a certain moment It

arises in optics: geometrical optics is usually treated in terms of rays^ and

the signal is described by giving the directions (see frequencies) of the rays

(see tones) that should be present at a certain position (see time moment)

It arises also in mechanics, where the position and the momentum of a

particle are given simultaneously, leading to a description of mechanical

phenomena in a phase space

A candidate for a local frequency spectrum is Gaborts signal expansion

In 1946 Gabor [14] suggested the expansion of a signal into a discrete set

of properly shifted and modulated Gaussian elementary signals [5, 6, 7, 14,

15, 16] A quotation from Gabor's original paper might be useful Gabor

writes in the summary:

Hitherto communication theory was based on two alternative

methods of signal analysis One is the description of the signal

as a function of time; the other is Fourier analysis But

our everyday experiences insist on a description in terms of

both time and frequency Signals are represented in two

dimensions, with time and frequency as co-ordinates Such

two-dimensional representations can be called 'information

di-agrams,' as areas in them are proportional to the number of

independent data which they can convey There are certain

'elementary signals' which occupy the smallest possible area in

the information diagram They are harmonic oscillations

mod-ulated by a probability pulse Each elementary signal can be

considered as conveying exactly one datum, or one 'quantum of

information.' Any signal can be expanded in terms of these by

a process which includes time analysis and Fourier analysis as

extreme cases

Although Gabor restricted himself to an elementary signal that had a

Gaussian shape, his signal expansion holds for rather arbitrarily shaped

elementary signals [5, 6, 7]

We will restrict ourselves to one-dimensional time signals; the extension

to two or more dimensions, however, is rather straightforward Most of the

results can be applied to continuous-time as well as discrete-time signals

We will treat continuous-time signals in Sections 2, 3, and 4 (and also in

Sections 7 and 8), and we will transfer the concepts to the discrete-time

case in Sections 5 and 6 To distinguish continuous-time from discrete-time

signals, we will denote the former with curved brackets and the latter with

square brackets; thus ip{t) is a continuous-time and (p[n] a discrete-time

signal We will use the variables in a consistent manner: in the

continuous-time case, the variables t, m and T have something to do with continuous-time, the

variables a;, k and fi have something to do with frequency, and the relation

Q>T — 2n holds throughout; in the discrete-time case, the variables n, m and

A'' have something to do with time, the variables ^, k and © have something

to do with frequency, and the relation QN = 27r holds throughout

In his original paper, Gabor restricted himself to a critical sampling of

the time-frequency domain; this is the case that we consider in Sections 2-6

In Section 2 we introduce Gabor's signal expansion, we introduce a window

function with the help of which the expansion coefficients can be found, and

we show a way in which—at least in principle—this window function can be

determined In Section 3 we introduce the Zak transform and we use this

transform to determine the window function that corresponds to a given

elementary signal in a mathematically more attractive way A more general

application of the Zak transform to Gabor's signal expansion is described

in Section 4 We translate the concepts of Gabor's signal expansion and the

Zak transform to discrete-time signals in Section 5 Finally, in Section 6,

we introduce, on the analogy of the well-known discrete Fourier transform,

a discrete version of the Zak transform; the discrete versions of the Fourier

and the Zak transform enable us to determine Gabor's expansion

coeffi-cients by computer via a fast computer algorithm

In Section 7 we extend Gabor's concepts to the case of oversampling^

in particular to oversampling by an integer factor We use the Fourier

transform and the Zak transform again to transform Gabor's signal

expan-sion into a mathematically more attractive form and we show how a (no

longer unique) window function can be determined in this case of integer

oversampling Finally, in Section 8, we introduce an optical arrangement

which is able to generate Gabor's expansion coefficients of a rastered,

one-dimensional signal by coherent-optical means

§2 G a b o r ' s signal e x p a n s i o n

Let us consider an elementary signal g{t)^ which may or may not have a

Gaussian shape Gabor's original choice was a Gaussian [14],

g ( t ) = 2 i / V - ( * / ^ ' ' , (2.1)

where we have added the factor 24 to normalize (1/T) / \g{t)\'^dt to unity,

but in this paper the elementary signal may have a rather arbitrary shape;

we will use Gabor's choice of a Gaussian-shaped elementary signal as an

example only Prom the elementary signal g{t), we construct a discrete set

of shifted and modulated versions gmk (0 defined by

gmk{t)=g{t-m.T)e^^''', (2.2) where the time shift T and the frequency shift Q satisfy the relationship

Q>T = 27r, and where m and k may take all integer values Gabor stated in

1946 that any reasonably well-behaved signal (p{t) can be expressed in the

form

^{t) = 5Z 5Z ^rnkgmk{t), (2.3)

m k

with properly chosen coefficients amk- Thus Gabor's signal expansion

rep-resents a signal (p{t) as a superposition of properly shifted (over discrete

distances mT) and modulated (with discrete frequencies kO.) versions of an

elementary signal g{t) We note that there exists a completely dual

expres-sion in the frequency domain; in this paper, however, we will concentrate

on the time-domain description

Gabor's signal expansion is related to the degrees of freedom of a signal:

each expansion coefficient Omk represents one complex degree of freedom

[8, 14] If a signal is, roughly, limited to the space interval \t\ < \a and to

the frequency interval \UJ\ < \h^ the number of complex degrees of freedom

equals the number of Gabor coefficients in the time-frequency rectangle

with area ab, this number being about equal to the time-bandwidth product

ab/2TT The reason for Gabor to choose a Gaussian-shaped elementary

signal was that for such a signal, each shifted and modulated version, which

conveys exactly one degree of freedom, occupies the smallest possible area

in the time-frequency domain Indeed, if we choose the elementary signal

according to (2.1), the 'duration' of such a signal and the 'duration' of its

Fourier transform—defined as the square roots of their normalized

second-order moments (see [23, Sect 8-2])—read T/2y/Ti and fl/2^/7^, respectively,

and their product takes the minimum value ^

Two special choices of the elementary signal might be instructive If

we choose a rectangular-shaped elementary signal such that g{t) = 1 for

-^T < t < ^T and g{t) = 0 outside that time interval, then Gabor's

signal expansion has an easy interpretation: we simply consider the signal

ip{t) in successive time intervals of length T and describe the signal in

each time interval by means of a Fourier series In the case of a

sine-shaped elementary signal g{t) = sin(7rt/T)/(7rt/T)—and hence g{uj) = T

for - | f i < a; < ^fi and g{uj) = 0 outside that frequency interval—Gabor's

signal expansion has again an easy interpretation: we simply consider the

signal in successive frequency intervals of length Q and describe the signal

in each frequency interval by means of the well-known sampling theorem

for band-limited signals

For the rectangular- or sine-shaped elementary signals considered in

the previous paragraph, the discrete set of shifted and modulated versions

of the elementary signal gmk{i) Is orthonormal\ in general, however, this

need not be the case, which implies that Gabor's expansion coefficients

amk cannot be determined in the usual way Let us consider two elements

9mk(t) and gni{t) from the (possibly non-orthonormal) set of shifted and

modulated versions of the elementary signal, and let their inner product

be denoted by dn-m,i-k\ hence

j 9nlii)9mk{t)dt = dn-m,l-k = d*m-n,k-l • (2-4)

It is easy to see that for Gabor's choice of a Gaussian elementary signal,

the array dmk takes the form

dmk = T{-ir^e-h(^"^^"\ (2.5) which does not have the form of a product of two Kronecker deltas TSm^k]

therefore, the set of shifted and modulated versions of a Gaussian

elemen-tary signal is not orthonormal

Gabor's expansion coefficients can easily be found, even in the case of a

non-orthonormal set gmk{t)i if we could find a window function w{t) such

and

^ 5 3 wl,k{h)gmk{t2) = 6{h -12); (2.8)

m k

we will show later that the first bi-orthonormality condition implies the

sec-ond one, so we can concentrate on the first one The first bi-orthonormality

condition guarantees that if we start with an array of coefficients amk^

construct a signal (p{t) via (2.3) and subsequently substitute this signal

into (2.6), we end up with the original coefficients array; the second

bi-orthonormality condition guarantees that if we start with a certain signal

(p{t), construct its Gabor coefficients amk via (2.6) and subsequently

sub-stitute these coefficients into (2.3), we end up with the original signal We

thus conclude that the two equations (2.3) and (2.6) form a transform pair

We remark that (2.6), with the help of which we can determine Gabor's

expansion coefficients, is, in fact, a sampled version of the sliding-window

spectrum [7, 10] (or complex spectrogram, or windowed Fourier transform,

or short-time Fourier transform), where the sampling appears on the

time-frequency lattice (mT, fcfi) with QT = 27r In quantum mechanics this

lattice is known as the Von Neumann lattice [4, 20], but for obvious

rea-sons we prefer to call it the Gabor lattice in the context of this paper

Hence, whereas sampling the sliding-window spectrum yields the Gabor

coefficients, Gabor's signal expansion itself can be considered as a way to

reconstruct a signal from its sampled sliding-window spectrum The name

window function for the function w{t) that corresponds to a given

elemen-tary signal g{t) will thus be clear

It is easy to see that the window function w{t) is proportional to the

elementary signal g{t) if the set gmk{i) is orthonormal In the remainder of

this section we show a first way in which a window function can be found

if the set gmk{t) is non-orthonormal We therefore express the window

function by means of its Gabor expansion (2.3) with expansion coefficients

in which the left-hand side has the form of a convolution of the given array

dmk with the array Cmk that we have to determine Equation (2.10) can be

solved, in principle, when we introduce the Fourier transform of the arrays

according to

m k

and a similar expression for c(t,a;;T) Note that these Fourier transforms

are periodic in the time variable t and the frequency variable a; with periods

T and Q, respectively:

d{t + mT.uj + kft;T) = d{t,uj;T) (2.12)

Hence, in considering such Fourier transforms we can restrict ourselves to

the fundamental Fourier interval {—-^T < t < | T , —^fi < cu < ^O) The

inverse Fourier transformation reads

dmk = ^ [ I d(t,u-T)e^^'^^^-^^'^dtduj (2.13)

^^ JT JQ

and a similar expression for Cmk] / p 'dt and /^^ -du denote integrations over

one period T and Q, respectively After Fourier transforming both sides

of (2.10), the convolution transforms into a product, and (2.10) takes the

form

c{t,u;T)d{t,u;T) = l (2.14)

The function c(^, a;; T) can easily be found from the latter relationship,

pro-vided that the inverse of d{t, UJ; T) exists, and inverse Fourier transforming

c{t,(j\ T) (see (2.13)) then results in the array Cmk that we are looking for

e2{x) = e2{x; e-2^) = Yl e-2^(^+2)%^(2^+i)^ (2.17)

The functions ^(x;e~2^) are known as theta functions [1, 30] with nome e~2^ The Fourier transform {l/T)d{t^uj]T) has been depicted in Figure 1,

where we have restricted ourselves to the fundamental Fourier interval

Note that the values of {l/T)d{t,(jj\ T) for {t = \T^-mT, uj = | f i + fcf])

zeros for (t = ^T + mT, uj = \^ •\- kVt) Inversion of d{t,uj-^T) in order to

find c{t,uo\T) may thus be difficult

Zeros of d{t,u] T) not only prohibit an easy determination of the window

function w{t), but they lead to another unwanted property: they enable us

to construct a (not identically zero) function z{t,uj]T) such that the

prod-uct z{t,uj\T)d{t,u\T) (see (2.14)) vanishes For a Gaussian elementary

signal, with zeros for (t = ^T 4- mT,u = ^D^ -h fcfi), we might choose

z{t,i^\T) = YlY^6{t - \ T - mT)27r6{uj - ^n - k9)z (2.18)

m k

Inverse Fourier transforming this function yields the array

Zmk = ( - i r + ' ^ , (2.19) which is a homogeneous solution of (2.10) Hence, Gabor coefficients might

not be unique: if Cmk are Gabor coefficients that determine the window

function w{t)^ then Cmk + ^mk are valid Gabor coefficients, as well!

In the next section we present a different and mathematically more

attractive way to find the window function w{t) that corresponds to a

given elementary signal g{t)

§3 Zak transform

In this section we introduce the Zak transform [31, 32, 33] and we show

its intimate relationship to Gabor's signal expansion The Zak transform

(p(t^ uj\ T) [31, 32, 33] of a signal ip(t) is defined as a one-dimensional Fourier

transformation of the sequence ip{t -f mr) (with m taking on all integer

values and t being a mere parameter), hence

(p(t, u;; r ) = ^ ^{t + m T ) e - ^ - - - ; (3.1)

m

throughout we will denote the Zak transform of a signal by the same symbol

as the signal itself, but marked by a tilde on top of it We remark that the

Zak transform (p(t, a;; r ) is periodic in the frequency variable UJ with period

2'K/T and quasi-periodic in the time variable t with quasi-period r:

( p U + mT,a;4-fc—;rj = (p{t,uj]T)ế^'^'' (3.2)

Hence, in considering the Zak transform we can restrict ourselves to the

fundamental Zak interval (—^r < ^ < ^r, - T T / T < a; < TT/T). The inverse

relationship of the Zak transform has the form

ip{t -h m r ) = ^ / (p{t, a;; r)e-^'^^'^da;; (3.3)

2 ^ 727r/T

it will be clear that the variable t in the latter equation can be restricted

to an interval of length r, with m taking on all integer values From the

properties of the Zak transform we mention Parseval's energy theorem,

which leads to the relationship

^ j l \^{t,iJ;T)\'dtd^ = ^l\ip{t)\'dt (3.4)

The Zak transform (p{t^uj\T) provides a means to represent an

arbi-trarily long one-dimensional time function (or one-dimensional frequency

function) by a two-dimensional time-frequency function on a rectangle with

finite area 27r This two-dimensional function (p{t^uj]T) is known as the Zak

transform, because Zak was the first who systematically studied this

trans-formation in connection with solid state physics [31, 32, 33] Some of its

properties were known long before Zak's work, however The same

trans-form is called Weil-Brezin map and it is claimed that the transtrans-form was

already known to Gauss [28] It was also used by Gel'fand (see, for instance,

[27, Chap XIII]); Zak seems, however, to have been the first to recognize it

as the versatile tool it is The Zak transform hats many interesting

proper-ties and also interesting applications to signal analysis, for which we refer

to [17, 18] In this section we will show how the Zak transform can be

applied to Gabor's signal expansion

We want to make an observation to which we will return later on in

this paper Suppose that, for small r for instance, we can approximate a

function g{i) by the piecewise constant function

n ^ -^

where rect(x) = l f o r - ^ < x < ^ and rect(x) =^ 0 outside that interval

In the time interval —^r < ^ < ^r, the Zak transform g{t^u\T) then takes

the form

9{t^uj',T) = ^ ^ n e - ^ " " " = giur); (3.6)

n

note that this Zak transform does not depend on the time variable t, and

that the one-dimensional Fourier transform g{ujT) of the sequence gn arises

We remark that Parseval's energy theorem (3.4) now leads to the relation

i - / / \g{t^u;;r)\'dtdu; = ^ f \-g{u;T)\'cL; = J2\gn\' (3.7)

We still have to solve the problem of finding the window function

w{t) that corresponds to a given elementary signal g{t) such that the

bi-orthonormality conditions (2.7) and (2.8) are satisfied We consider again

the first bi-orthonormality condition (2.7)

and apply a Fourier transformation (see (2.11)) to both sides of this

in which expression we recognize (see (3.1)) the definitions for the Zak

transforms g(t,uo]T) and w{t,uo\T) of the two functions g{t) and w{t),

respectively; hence

Tg(t,u-T)w\t,uj',T) = l (3.8) The first bi-orthonormality condition (2.7) thus transforms into a product^

enabling us to find the window function w{t) that corresponds to a given

elementary signal g{t) in an easy way:

• from the elementary signal g{t) we derive its Zak transform g{t^ cj; T)

via definition (3.1);

• under the assumption that division by g{t^uo\T) is allowed, the

func-tion w{t^uj]T) can be found with the help of relafunc-tion (3.8);

• finally, the window function w{t) follows from its Zak transform

w{t^Lj;T) by means of the inversion formula (3.3)

It is shown in Appendix A that the window function w{t) found in this way

also satisfies the second bi-orthonormality condition (2.8)

Let us consider Gabor's original choice of a Gaussian elementary signal

again The Zak transform g{t,uj;aT) of the Gaussian signal (2.1) reads

g{t,uj;aT) = 2^e-^(*/^^'^3 (OCT:

n ^T ; e - - - , (3.9)

where

Os ( z ; e - ^ ^ ' ) =^^-^a'm\j2mz ( 3 ^ Q )

is a theta function again, in this case with nome e"^^ This Zak transform

has been depicted in Figure 2 for several values of the parameter r = aT^

where we have restricted ourselves to the fundamental Zak interval; note

that for a < | , the Zak transform becomes almost independent of ^, as we

have mentioned before We remark that the Zak transform of a Gaussian

signal has zeros for {t = ^aT -f maT^u = ^Ct/a + kCl/a)

In the case of a Gaussian elementary signal and choosing r = T (Gabor's

original choice), the Zak transform of the window function takes the form

Tw{t,uj;T) = — - ^ — r = 2 - ^ e ^ ( * / ^ ) ' - — ^ -, (3.11)

in which expression we have set, for convenience, ( = uj/Q + jt/T In the

fundamental Zak interval, the function l/6s{7r(; e~^) can be expressed as

(see, for instance, [30, p 489, Example 14]); the constant KQ is the complete

eUiptic integral for the modulus ^ \ / 2 : KQ = 1.85407468 (see, for instance

Figure 2 The Zak transform g(t,uj;aT) in the case of a Gaussian elementary

signal for different values of a: (a) a = 2, (b) a = 1, (c) a = | , and (d) a = |

[30, p 524]) It is now easy to determine the window function w[t) via the

inversion formula (3.3), yielding

where m is the nonnegative integer defined by ( m - \)T < \t\ <{m-\- \)T

Since the summation in the latter expression yields a result which is close