zeevi, coifman - signal and image representation in combined spaces

Preface
Wavelets are a generic name for a collection of self similar localized wave-
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
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 wavelet-
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
of infinite support, since the nonstationarity 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
X Preface
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 nonorthogonal 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
Preface xi
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."
xii Preface
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."
Acknowledgments
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. AN (1), Prometheus Inc., 52 Ashford Street, AUston, MA 02134
[]
AMIR AVERBUCH (341), School of Mathematical Sciences, Tel Aviv Uni-
versity, Tel Aviv 69978, Israel
[]
MARTIN J. BASTIAANS (23), Technische Universiteit Eindhoven, Faculteit
Elektrotechniek, Postbus 513, 5600 MB Eindhoven, Netherlands
[]
JOHN J. BENEDETTO (259), Department of Mathematics, University of
Maryland,
College
Park,
Maryland 29742
[]
FRANgois BERGEAUD (285), Ecole Centrale Paris, Applied Mathematics

Laboratory, Grande Voie des Vignes, F-92290 Chatenay-Malabry,
France
[]
GREGORY BEYLKIN (341), Program in Applied Mathematics, University
of Colorado at Boulder, Boulder, CO 80309-0526
[]
A. BRODZIK (1), Rome Laboratory/EROI, Hanscom AFB, MA 01731-
2909
J. S. BYRNES (167), Prometheus Inc., 52 Ashford St., Allston, MA 02134
[]
xni
xiv Contributors
CHARLES K. CHUI (395), Center for Approximation Theory, Texas
A&;M
University, College Station, TX 77843
[]
ALBERT COHEN (189), Laboratoire d^AnalyseNumerique, UniversitePierre
et Marie Curie, 4 Place Jussieu, 75005 Paris, France
[cohen@ann. j ussieu. fr]
RONALD 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 DYN (189), School of Mathematical Sciences, Sackler Faculty of
Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel
[]
I. GERTNER (1), Computer Science Department, The City College of New
York,
Convent Avenue at 138th Street, New
York,
NY 10031
[]
AMI
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
VICTOR E. KATSNELSON (525), Department of Theoretical Mathematics,
The Weizmann Institute of Science, Rehovot 76100, Israel
[]
MARK
A. KON (415), Department of Mathematics, Boston University,
Boston, MA 02215
REINER LENZ (553), Department of Electrical Engineering, Linkoping
University, S-58183 Linkoping, Sweden

[]
CHUN
LI (395), Institute of Mathematics, Academia Sinica, Beijing 100080,
China.
Contributors xv
STEPHANE MALLAT (285), Ecole Poly technique, CMAP, 91128 Palaiseau
Cedex, France
[]
JURGEN PRESTIN (201), FB Mathematik, Universitat Rostock, D-18051
Rostock, Germany
[]
LOUISE
A. RAPHAEL (415), Department of Mathematics, Howard Uni-
versity, Washington, DC 20059
BORIS
RUBIN
(439, 457), Department of Mathematics, Hebrew University
of Jerusalem, Givat Ram 91904, Jerusalem, Israel
[]
WALTER SCHEMPP (71), Lehrstuhl fuer Mathematik I, University of Siegen,
D-57068 Siegen, Germany
[]
KATHI SELIG (201), FB Mathematik, Universitat Rostock, D-18051 Ros-
tock,
Germany
[]
JONAS SVANBERG (553), Department of Electrical Engineering, Linkoping
University, S-58183 Linkoping, Sweden
[]
R. TOLIMIERI (1), Electrical Engineering Department, The City College

of New
York,
Convent Avenue at 138th Street, New
York,
NY
10031
MARTIN VETTERLI (301), Department d'Electricite EPFL, CH'1015 Lau-
sanne, Switzerland
[]
YEHOSHUA Y. ZEEVI (121), Department of Electrical Engineering, Technion-
Israel Institute of Technology, Haifa 32000, Israel
[zeevi@ee. technion.
ac.
il]
VALERY A. ZHELUDEV (477), School of Mathematical Sciences, Tel Aviv
University, 69978 Tel Aviv, Israel
[]
MEIR ZIBULSKI (121), Multimedia Department, IBM Science and Tech-
nology,
MATAM,
Haifa 31905, Israel
[meir z @ vnet. ibm. com]
Weyl-Heisenberg Systems and the
Finite Zak Transform
M. An, A. Brodzik, I. Gertner, and R. Tolimieri
Abstract. 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 Introduction
During the last four years powerful new methods 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] adapted
these methods to gain control over the cross-term interference problem
[9] by constructing signal systems in time frequency space for expanding
Wigner trg,nsforms from W-H systems based on Gaussian-like signals.
The computational feasibility of the method in [10] depends strongly
on the availability of eflScient and stable algorithms for computing W-H
expansion coefficients. Since W-H systems are not orthogonal, standard
Hilbert space inner-product methods do not generally apply. Moreover,
Signal and Image Representation in Combined Spaces O
Y. Y. Zeevi and R. R. Coifman (Eds.), PP- 3-21.
Copyright ©1998 by Academic Press.
All rights of reproduction in any form reserved.
ISBN 0-12-777830-6

4 M. An et al.
since critically sampled W-H systems may not form a basis, over-sampling
in time frequency is necessary for the existence of arbitrary signal expan-
sions.
In fact, this is usually the case for systems based on the Gaussian.
In [10, 11, 12, 13, 15], the concept of biorthogonals was applied to the prob-
lem of W-H coefficient computation. In [15], the Zak transform provided
the framework for computing biorthogonals for rationally over-sampled W-
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 estab-
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
of existence, parameterization, and computation 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 over-sampled systems, and in Section 5 for rationally over-
sampled systems. Implementation results will be given in Sections 6, 7,
and 8.
Weyl-Heisenberg Systems and the Finite Zak Transform 5
§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
product
N-l
a=0
For g G L{N) and 0 < m, n< iV define gm,n ^ H^) by
QmAc^) = 9{a + m)e-2 /^, a e Z (2.1.1)
The functions in the set {gm,n : 0 < m, n < N} are called Weyl-Heisenberg
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.
6 M. An et al.
2.2 Finite Zak transform (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,6GZby
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)
ThenF=:Zo(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)
Weyl-Heisenberg Systems and the Finite Zak Transform 7
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)
A:=0 m=0
if and only if
F = GP, PeL{M,K),
where
K-\M-1
P{a,b) =J2Y1 c(A:M,mX)e-2^^(^^/^+^'^/^\ (2.2.12)
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-
8 M. An et ai
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
k' = r-\-kR, 0 < r < i?, 0 < k < K.
Since
9k'M',mK — {9rM\o)kM,mKi
g is a union of critically sampled W-H systems
R-l
g = U S^' S^ ^ (^^' ^' ^)' ^^ " ^rM',0, 0<r <R. (4.1)
r=0
Weyl-Heisenberg Systems and the Finite Zak Transform 9
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|Con nCi?-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.
• Decompose F = Zo{K)f
R-l
F=J2Fr. FreL{M,K)
r=0
where Fr vanishes on the zero set (r oi Gr^ 0 <r < R.
10 M. An et ai
• 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 TlogT 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 systems
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 system 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)
Weyl-Heisenberg Systems
and the
Finite
Zak
Transform
11
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-h5—)
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)
=
^|GL6
+
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
12 M. An et al.
and we can solve (5.3) for some P' G L{M, ^). Define Q € L{M, K) by
Q =
\G\\P-P').
(5.4)
By (5.3) we have
^Qfa,6-hs—1=0, 0<a<M, 0<6< —
which, since P' G I/(M, y), impUes
< G{P - P'), GP' > = <Q,P' > = 0.
Defining P" e L{M, K) by P = P' 4- P", we have that
GP = GP' + GP"
is an orthogonal decomposition in L{M,K), completing the proof of the
theorem. •
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 /is orthogonal to the linear span

of g', and we are done.
• Otherwise, choose 0 <
SQ
< 5' such that
iiG.„p;ii>iiG,-p;ii, o<s'<5',
and iterate the previous steps with F - Gs^Pg^ replacing F.
Since
is an orthogonal decomposition, we have
iii^-G,„p;ji<iiF||,
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.
Weyl-Heisenberg Systems and the Finite Zak Transform 13
§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.
14
M. An et al
Complexity
For a real input signal /, the FZT of / is Hermitian symmetric along K-
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
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
M X Herm. F{K)
K X real F{M)
Size N
256
512
1024
2048
4096
8192
16384
32768
65536
131072
1 262144
2-D X
X
M
16
X
16
16
X
32
32
X

32
32 x64
64
X
64
64
X
128
128
X
128
128
X
256
256
X
256
256
X
512
512
X
512
Time 1
0.67
1.20
2.02
3.98
7.41
14.96

29.82
60.89
125.55
264.60
566.99
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 Fi, ,
FR-I
each has only one non-zero point, so that the computation of the 2-D
FT 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 influ-
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
Weyl-Heisenberg Systems and the Finite Zak Transform 15
Size
AT
384
768
1536
3072

3072
6144
6144
12288
12288
24576
49152
98304
98304
196608
1 393216
2-D K xM
8x48
16x48
32x48
64x48
128 X 24
128
X 48
64x96
512 X 24
128 X 96
256 X 96
256 X 192
256 X 384
512 X 192
512 X 384
1024
X 384
Time 1

1.47
1.99
3.12
5.91
6.15
12.07
12.48
26.07
24.05
48.70
98.71
203.52
209.12
433.41
1011.61
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 com-
plex singular value decomposition (SVD), we used the LINPACK routine.
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 ra-

tional over-sampling by 3/2. The experimental results are:
A window of size 101 centered at 600 results in an expandable W-H
system.
16
M. An et al.
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.
Complexity
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 ^
Mx real FiK)
M X ^ matrix 1
S' X S multiply a
vector S
S' xMx
Hermitian F{^),
5'
X f real F{M)
Timing results of various sizes are given in Tables 3 and 4.

Size
TV
384
768
1536
3072
3072
6144
6144
12288
24576
49152
98304
98304
196608
2-D KxM
16x24
32x24
64x24
64x48
128 x 24
128 x 48
64x96
128
X
96
256
X
96
256

X
192
256
X
384
512 X 192
512 X 384
Time 1
2.06
2.97
5.31
10.79
10.05
20.85
22.86
43.15
84.71
171.39
412.12
413.50
840.02
Table 3. Timing results (in milliseconds) in the Intel i860 RISC microprocessor
(rational over-sampling (3/2)).
Weyl-Heisenberg Systems and the Finite Zak Transform
17
Size
N
320
640
1280

2560
5120
5120
10240
10240
20480
40960
81920
163840
1 327680
2-D K xM
8x40
16
X 40
32
x40
64
X 40
128
X 40
64
X 80
128
X 80
64
X 160
128
X 160
128
X 320

256
X 320
512
X 320
512
X 640
Time
1
2.82
3.85
5.66
9.65
16.42
18.32
32.09
37.99
67.65
134.08
258.40
522.19
1149.76
Table 4. Timing results (in milliseconds) on the Intel i860 microprocessor (ratio-
nal over-sampling (5/4)).
§8 Parallel implementation
Assume that a distributed memory parallel computer has p (< min(i^, M))
processors. Set
P = K/Ki =M/K2. (8.1)
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 FFT
• 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
18
M. An et al.
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.
Input Data /
receive
/(O + Mr)
K-pt
real FT
Multiply
i^-pt
Herm. FT
\
receive
/(I + Mr)
K-pt

real FT
Multiply
K-pt
Herm. FT
\
receive
f{M -
1 +
Mr)
K-pt
real FT
Multiply
K-pt
Herm. FT
Data Permutation
/
M-pt
real FT
M-pt
real FT
\
M-pt
FT
c{a, 0)
c{a, 1)
c{a,K-l)
0<r<K-l, 0<a<M-l
Figure 1. Parallel implementation flow diagram.
Weyl-Heisenberg Systems and the Finite Zak Transform
19

SizeiV
16384
32768
1
65536
131072
262144
524288
1048576
2097152
2-D KxM
128
X 128
128
X
256
256
X
256
256
X
512
512
X
512
512
X
1024
1024
X

1024
1024
X
2048
Time
1
10.06
19.66
39.31
80.24
163.10
368.99
801.82
1661.96
Table 5. Timing results (in milliseconds) on the Intel Paragon (4-nodes).
Size
AT
65536
131072
262144
524288
1048576
2097152
8388608
2-D KxM
256
X
256
256
X

512
512
X
512
512
X
1024
1024
X
1024
1024
X
2048
2048
X
2048
Time
22.18
42.45
86.32
189.54
404.32
840.17
1716.03
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.
20
M. An et al.
References
1]
An, A., G.
Kechriotis,
C. Lu, and R.
Tolimieri,
The
computation
of
Weyl-Heisenberg coefficients
for
critically sampled
and
over-sampled
signals, Proceedings
of the Int. Conf.
SPAT ICSPAT94
2,
October
1994,
pp.

824-829.
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).