Shift-Invariant and Sampling Spaces Associated with the Special Affine Fourier Transform

0.77 MB·

by  Ayush Bhandari

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Shift–Invariant and Sampling Spaces Associated with the
Special Affine Fourier Transform
Ayush Bhandari† and Ahmed I. Zayed‡
†Media Laboratory, Massachusetts Institute of Technology
Cambridge, MA 02139–4307 USA.
‡Department of Mathematical Sciences,
DePaul University, Chicago, IL 60614-3250
Email: [email protected] • [email protected]
Abstract: The Special Affine Fourier Transformation or the SAFT generalizes a number of well known unitary
transformations as well as signal processing and optics related mathematical operations. Shift-invariant spaces also
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play an important role in sampling theory, multiresolution analysis, and many other areas of signal and image
1
processing. Shannon’s sampling theorem, which is at the heart of modern digital communications, is a special case
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ofsamplinginshift-invariantspaces.Furthermore,itiswellknownthatthePoissonsummationformulaisequivalent
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to the sampling theorem and that the Zak transform is closely connected to the sampling theorem and the Poisson

n summation formula. These results have been known to hold in the Fourier transform domain for decades and were
a recently shown to hold in the Fractional Fourier transform domain by A. Bhandari and A. Zayed.
J The main goal of this article is to show that these results also hold true in the SAFT domain. We provide a short,

1 self–contained proof of Shannon’s theorem for functions bandlimited in the SAFT domain and then show that
2 sampling in the SAFT domain is equivalent to orthogonal projection of functions onto a subspace of bandlimited
basis associated with the SAFT domain. This interpretation of sampling leads to least–squares optimal sampling

] theorem. Furthermore, we show that this approximation procedure is linked with convolution and semi–discrete
T convolution operators that are associated with the SAFT domain. We conclude the article with an application of
I fractional delay filtering of SAFT bandlimited functions.
.
s
c
[ CONTENTS

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v I Introduction 2
3
9 II Preliminaries 4
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5 III Convolution Structures 6
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1. IV The Zak Transform Associated with the SAFT 10
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6 V Poisson Summation Formula for SAFT 12
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: VI Shannon’s Sampling Theorem and the SAFT:
v
i Reinterpretation, Extension and Applications 13
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VI-A Revisiting Shannon’s Sampling Theorem for SAFT Domain . . . . . . . . . . . . . . . . 13
r VI-B Reconstruction with Arbitrary Basis Functions . . . . . . . . . . . . . . . . . . . . . . . 17
a
VI-C Application: Fractional Delay Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
VI-C1 Power Cosine Filters for FDF in SAFT Domain . . . . . . . . . . . . . . . . 19
VI-C2 Experimental Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
VII Conclusions 20
References 21
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I. INTRODUCTION
TheSpecialAffineFourierTransformation(SAFT),whichwasintroducedin[1],isanintegraltransformation
associatedwithageneralinhomogeneouslosslesslinearmappinginphase-spacethatdependsonsixparameters
independent of the phase-space coordinates. It maps the position x and the wave number k into
      
x(cid:48) a b x p
 =  +  (1)
k(cid:48) c d k q
with
ad−bc=1. (2)
This transformation, which can model many general optical systems [1], maps any convex body into another
convexbodyand(2)guaranteesthattheareaofthebodyispreservedbythetransformation.Suchtransformations
form the inhomogeneous special linear group ISL(2,R). The SAFT offers a unified viewpoint of known signal
processing transformations as well as optical operations on light waves. We have parametrically summarized
these operations in Table I.
The integral representation of the wave-function transformation linked with the transformation (1) and (2)
TABLEI
SAFT,UNITARYTRANSFORMATIONSANDOPERATIONS
SAFT Parameters (A) Corresponding Unitary Transform
(cid:2) (cid:3)
a b 0 =A Linear Canonical Transform
c d 0 LCT
(cid:2) cosθ sinθ p(cid:3)=AO Offset Fractional Fourier Transform
− sinθ cosθ q θ
(cid:2) cosθ sinθ 0(cid:3)=A Fractional Fourier Transform
− sinθ cosθ 0 θ
(cid:2) 01 p(cid:3)=AO Offset Fourier Transform (FT)
−10 q FT
(cid:2) 01 0(cid:3)=A Fourier Transform (FT)
−10 0 FT
(cid:2)0  0(cid:3)=A Laplace Transform (LT)
 0 0 LT
(cid:2)cosθ sinθ 0(cid:3) Fractional Laplace Transform
sinθ −cosθ 0
(cid:2) (cid:3)
1b 0 Fresnel Transform
01 0
(cid:2)1b 0(cid:3) Bilateral Laplace Transform
 1 0
(cid:2)1−b 0(cid:3), b≥0 Gauss–Weierstrass Transform
0 1 0
√1 (cid:2) 0 e−π/2 0(cid:3) Bargmann Transform
2 −e−π/2 1 0
SAFT Parameters (A) Corresponding Signal Operation
(cid:2)1/α 0 0(cid:3)=A Time Scaling
0 α 0 α
(cid:2) (cid:3)
10 τ =A Time Shift
01 0 τ
(cid:2)10 0(cid:3)=A Frequency Shift
01 ξ ξ
SAFT Parameters (A) Corresponding Optical Operation
(cid:2) cosθ sinθ 0(cid:3)=A Rotation
− sinθ cosθ 0 θ
(cid:2) (cid:3)
1 0 0 =A Lens Transformation
τ 1 0 τ
(cid:2)1η 0(cid:3)=A Free Space Propagation
01 0 η
(cid:2)eβ 0 0(cid:3)=A Magnification
0 e−β 0 β
(cid:2) (cid:3)
coshα sinhα 0 =A Hyperbolic Transformation
sinhα coshα 0 η
3
is given by,
(cid:90)
F(ω)=fˆ (ω)= k(t,ω)f(t)dt (SAFT of f(t)) (3)
A
R
= 1 (cid:90) exp(cid:26) j (cid:0)at2+dω2−2tω+2pt+2(bq−dp)ω(cid:1)(cid:27)f(t)dt,
(cid:112)
2π|b| R 2b
where A stands for the six parameters (a,b,c,d,p,q), and
(cid:26) (cid:27)
k(t,ω)= 1 exp j (cid:0)at2+dω2−2tω+2pt+2(bq−dp)ω(cid:1) .
(cid:112)
2π|b| 2b
The inversion formula for the SAFT is easily shown to be
f(t)= 1 (cid:90) F(ω)exp(cid:26)−j (cid:0)dω2+at2−2tω+2ω(bq−dp)+2pt(cid:1)(cid:27)dtω, (4)
(cid:112)
2π|b| R 2b
which may be considered as the SAFT evaluated using A−1 where1,
 
+d −b bq−dp
A−1 d=ef (cid:2)A−1|λ−1(cid:3)≡ 
−c +a cp−aq
and to be precise,
   
+d −b bq−dp
A−1 =  and λ−1 d=ef  .
−c +a cp−aq
We also have
(cid:90) (cid:90)
(cid:104)f,g(cid:105)= f(t)g(t)dt= F(ω)G(ω)dω =(cid:104)F,G(cid:105),
R R
from which we obtain (cid:107)f(cid:107) = (cid:107)F(cid:107). When p = 0 = q, we obtain the homogeneous special group SL(2,R),
which is represented by the unimodular matrix
 
a b
M= .
c d
The associated integral transform, which is called the Linear Canonical Transform (LCT), is given by
F (ω)= 1 (cid:90) exp(cid:26) j (cid:0)at2+dω2−2tω(cid:1)(cid:27)f(t)dt.
LCT (cid:112)2π|b| R 2b
The linear canonical transform has been used to solve problems in physics and quantum mechanics; see [2].
It includes several known transforms as special cases. For example, for a = 0 = d,b = −1,c = 1, we obtain
the Fourier transform and for a = cosθ = d,b = sinθ = −c, we obtain the fractional Fourier transform. The
Laplace,Gauss-Weierstrass,andBargmanntransformsarealsospecialcases.TheinversionformulafortheLCT
is given by
f(t)= 1 (cid:90) exp(cid:26)−j (cid:0)dω2+at2−2tω(cid:1)(cid:27)F(ω)dω. (5)
(cid:112)
2π|b| R 2b
1Withalittleabuseofnotation,weuseλ−1 whichshouldbeunderstoodasaparametervectorcorrespondingtotheinverse–SAFT.
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If the LCT of f and g are denoted by F and G, respectively, then Parseval’s relation holds
(cid:90) (cid:90)
(cid:104)f,g(cid:105)= f(x)g(x)dx= F(t)G(t)dt=(cid:104)F,G(cid:105).
R R
Let
   
a b a b
1 1 2 2
M1 = , M2 = ,
c d c d
1 1 2 2
so that
    
a b a b a a +b c a b +b d
2 2 1 1 2 1 2 1 2 1 2 1
M21 =  = .
c d c d c a +d c c b +d d
2 2 1 1 2 1 2 1 2 1 2 1
If the LCT corresponding to M ,M ,M are denoted by L ,L ,L , respectively, it can be shown that the
1 2 21 1 2 21
composition relation L L =CL , holds, where C is a constant. On the other hand, the composition relations
2 1 21
is associative, that is,
(L L )L =L (L L ).
3 2 1 3 2 1
The analogue of Shannon sampling theorem for the fractional Fourier transform, the linear canonical
transform, and the SAFT were obtained in [3], [4], [5], [6], [7], [8], [9]. Although the sampling theorem
can be easily obtained in a direct way, we will obtain it as a special case of more general results.
Our goal is to extend key harmonic analysis results to the SAFT analogous to those for the FT and FrFT
and obtain the sampling theorem as a by-product. For example, in the Fourier transform domain, it is known
that Shannon sampling theorem is a special case of sampling in shift-invariant spaces, as well as, sampling in
reproducing-kernel Hilbert spaces. Moreover, it is also known that the Poisson summation formula is equivalent
to the sampling theorem and the Zak transform is closely connected to the sampling theorem and the Poisson
summation formula. These results have been known to hold in the Fourier transform domain for decades and
were recently shown to hold in the FrFT domain [10]. The main goal of this article is to extend these results
to the SAFT domain.
Inthenextsection,wewillintroducesomeoftheseclassicalresultsthatwewillextendtotheSAFTdomain.
II. PRELIMINARIES
Shift-invariant spaces have been the focus of many research papers in recent years because of their close
connectionwithsamplingtheory[11],[12]andwaveletsandmultiresolutionanalysis[13],[14],[15].Theyhave
many applications in signal and image processing [10]. For example, in many signal processing applications, it
is of interest to represent a signal as a linear combination of shifted versions of some basis function ϕ, called
the generators of the space, that generates a stable basis for a space. More precisely, we consider spaces of the
form
(cid:40) +∞ (cid:41)
(cid:88)
V(ϕ)= f(t)= c[n]ϕ(t−n),ϕ∈L2(R),{c[n]}∈(cid:96) . (6)
2
n=−∞
The closure of V(ϕ) is a closed subspace of L2. Furthermore, it is shift-invariant in the sense that for all
f ∈V(ϕ), its shifted version, f(·−k)∈V(ϕ), k ∈Z, where Z denotes the set of integers.
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For the basis functions to be stable, it is required that the family of functions {ϕ(t−n)}∞ forms a
n=−∞
Riesz basis or equivalently, there exists two positive constants 0<η ,η <+∞, such that
1 2
∀c∈(cid:96) , η (cid:107)c(cid:107)2 (cid:54)(cid:13)(cid:13)(cid:13) (cid:88)∞ c[k]ϕ(t−k)(cid:13)(cid:13)(cid:13)2 (cid:54)η (cid:107)c(cid:107)2 (7)
2 1 (cid:96)2 (cid:13)(cid:13) (cid:13)(cid:13) 2 (cid:96)2
n=−∞ L2
where (cid:96) is the space of all square-summable sequences and (cid:107)c(cid:107)2 is the squared (cid:96) -norm of the sequence. We
2 (cid:96)2 2
define the Fourier transform of h(t) by (cid:98)h(ω) = √1 (cid:82)+∞h(t)e−jωtdt. Recall, the Fourier domain equivalent
2π −∞
of (7) is
η (cid:54)(cid:88)+∞ |ϕ(ω+2πn)|2 (cid:54)η . (8)
1 (cid:98) 2
n=−∞
The ratio ρ = η /η is called the condition number of the Riesz basis. The basis is shift-orthonormal 2 or a
2 1
tight frame if ρ=1.
One of the important tools used in the study of sampling spaces is the Zak transform [16], [14]. The Zak
transform, which was introduced in quantum mechanics by J. Zak [17] to solve Schrödinger’s equation for an
electron subject to a periodic potential in a constant magnetic field, may be defined as
(cid:88)+∞
Z (t,ω)= f(t+k)e−2πjkω,f ∈L1(R)
f
k=−∞
It is easy to see that
Z (t,ω+1)=Z (t,ω) and Z (t+1,ω)=e2πjωZ (t,ω)
f f f f
and that the Zak transform is a unitary transformation from L2(R) onto L2(Q), with (cid:107)Z (cid:107) =(cid:107)f(cid:107) ,
f L2(Q) L2(R)
where Q is the unit square Q=[0,1]×[0,1].
To see the connection between the Zak transform and sampling spaces, let us for the sake of convenience
define the Fourier transform of f as
(cid:90) ∞
fˆ(w)= f(t)e−2πjwtdt
−∞
so that the inverse transform, whenever it exists, is given by
(cid:90) ∞
f(t)= fˆ(w)e2πjwtdw.
−∞
Now if f belongs to a sampling space with sampling function ψ ∈ L2 ∩ L1, it is easy to see that since
f(t)=(cid:80)+∞ f(k)ψ(t−k), then
k=−∞
fˆ(w)=Fˆ(w)ψˆ(w), where Fˆ(w)=(cid:88)+∞ f(k)e−2πkw.
k=−∞
Clearly Fˆ(w) is periodic with period one; hence,
(cid:12) (cid:12) (cid:12) (cid:12)(cid:12) (cid:12)
(cid:12)fˆ(w+k)(cid:12)=(cid:12)Fˆ(w)(cid:12)(cid:12)ψˆ(w+k)(cid:12),
(cid:12) (cid:12) (cid:12) (cid:12)(cid:12) (cid:12)
(cid:26)2 Shift-orthonormality means that (cid:104)ϕ,ϕ(·−k)(cid:105) = δk where (cid:104)x,y(cid:105) = (cid:82)−+∞∞x(t)y∗(t)dt is the L2-inner product and δk =
1, ifk=0
denotestheKroneckerdelta.
0, ifk(cid:54)=0
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which, in view of the fact that Fˆ(w)=Z (0,w), implies that
f
G (w)=|Z (0,w)|2G (w),
f f ψ
where G denotes the Grammian of g ∈ L2(R), defined by G (w) = (cid:80) |gˆ(w+k)|2. Therefore, for such
g g k∈Z
a function f, we have
A|Z (0,w)|2 ≤G (w)≤B|Z (0,w)|2, (9)
f f f
for some A,B >0. From this, we obtain
(cid:90) 1
A(cid:88)+∞ |f(k)|2 ≤ G (w)dw ≤B(cid:88)+∞ |f(k)|2, (10)
f
k=−∞ k=−∞
0
and
(cid:12) (cid:12)
(cid:90) 1 (cid:80)k∈Z(cid:12)(cid:12)fˆ(w+k)(cid:12)(cid:12)dw =(cid:90) (cid:12)(cid:12)ψˆ(w)(cid:12)(cid:12)dw <∞, (11)
|Z (0,w)| (cid:12) (cid:12)
0 f R
whenever ψˆ ∈ L1(R). The Shannon sampling theorem is also known to follow from the Poisson summation
formula
(cid:88)+∞ f(t+k)=(cid:88)+∞ fˆ(k)e2πjkt, (12)
k=−∞ k=−∞
or equivalently
(cid:88)+∞ fˆ(w+k)=(cid:88)+∞ f(k)e−2πjkw, (13)
k=−∞ k=−∞
which, when f is band-limited to (−1/2,1/2), leads to
(cid:90)
(cid:88)+∞
f(k)= f(t)dt. (14)
k=−∞ R
Moreover, we have in view of (13), that Z (0,w)=(cid:80)+∞ fˆ(w+k).
f k=−∞
III. CONVOLUTIONSTRUCTURES
In this section we introduce convolution operations associated with the SAFT, one for functions and one for
a sequence of numbers and a function. Furthermore, we introduce a definition for the discrete SAFT that will
be useful for deriving Poisson summation formula and Zak transform for SAFT.
In order define convolution operators associated with the SAFT , let us first define unitary, modulation
(cid:16) (cid:17)
operation. Let λ (t)=exp at2 be the chirp–modulation function and let us define,
A 2b
(cid:42)f (t)d=ef λA(t)f(t)=ea2tb2f(t)
(cid:40)f (t)d=ef λA(t)f(t)=e−a2tb2f(t)
where z means the conjugate of z.
Definition 1 (SAFT–Convolution). We define the convolution ∗ of two functions f and g as
A
(cid:18) (cid:19)
(f ∗ g)(t)= λA(t) (cid:42)f (t)∗(cid:42)g(t) (t) (15)
A (cid:112)
|b|
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where ∗ stands for the standard convolution, that is,
1 (cid:90)
(f ∗g)(t)= √ f(t−x)g(x)dx.
2π R
We will show that the convolution operator in (15) leads to the well-known duality property of the Fourier
transform. A similar definition was presented in [8] but our proof relies on [18], [10], [3].
Theorem 1. Let h(t)=(f ∗ g)(t). Then the SAFT H(ω) of h is given by
A
H(ω)=η (ω)F(ω)G(ω),
A
where η (ω)=exp(cid:0)  (cid:0)dω2+Ωω(cid:1)(cid:1) and Ω=2(bq−dp).
A 2b
Proof:
(cid:90)
H(ω) = h(t)k(t,ω)dt
R
(cid:32) (cid:33)
= (cid:90) λA(t) (cid:90) (cid:42)f (t−x)(cid:42)g(x)dx k(t,ω)dt
(cid:112)
R 2π|b| R
(cid:124) (cid:123)(cid:122) (cid:125)
h(t)
= (cid:90) k(t,ω) λA(t) (cid:90) f(t−x)exp(cid:16)  (cid:0)a(t−x)2(cid:1)(cid:17)g(x)exp(cid:16)  (cid:0)ax2(cid:1)(cid:17)dxdt
R (cid:112)2π|b| R 2b 2b
= η (ω) 1 (cid:90) f(t−x)exp(cid:16)  (cid:0)a(t−x)2+dω2−2ω(t−x)+Ωω+2p(t−x)(cid:1)(cid:17)dt
A 2π|b| 2b
R
× (cid:90) g(x)exp(cid:16)  (cid:0)ax2+dω2−2ωx+Ωω+2px(cid:1)(cid:17)dx
2b
R
= η (ω) 1 (cid:90) f(u)exp(cid:16)  (cid:0)au2+dω2−2ωu+Ωω+2pu(cid:1)(cid:17)du
A 2π|b| 2b
R
× (cid:90) g(x)exp(cid:16)  (cid:0)ax2+dω2−2ωx+Ωω+2px(cid:1)(cid:17)dx
2b
R
= η (ω)F(ω)G(ω).
A
The following definition of the convolution of a sequence and a function is a generalization of that given in
[10].
Definition2(DiscreteTimeSAFT(DT–SAFT)). LetP ={p(k)}beasequencein(cid:96)2,thatis,(cid:80) |p(k)|2 <∞.
k
We define the discrete time SAFT of P as
(cid:26) (cid:27)
P(cid:98)A(ω)= (cid:112)21π|b|(cid:88)p(k)exp 2jb(cid:0)ak2+dω2−2ωk+Ωω+2pk(cid:1) , (16)
k
and define the convolution of a sequence P and a function φ∈L2(R) as
h(t)=(P ∗ φ)(t)= 1 e−jat2/2b(cid:88)ejak2/2bp(k)eja(t−k)2/bφ(t−k).
A (cid:112)
2π|b|
k
Lemma 1. Let P and φ be as above and h(t)=(P ∗ φ)(t). Then
A
H(ω)=(cid:98)hA(ω)=ηA(ω)P(cid:98)A(ω)ΦA(ω),
(cid:12) (cid:12)
where ΦA is the SAFT of φ. Moreover, (cid:12)(cid:12)P(cid:98)A(ω)(cid:12)(cid:12) is periodic with period ∆=2πb.
8
Proof: From the definition of SAFT, we have
H(ω) = (cid:90) h(t)k(t,ω)dt= 1 (cid:88)ejak2/2bp(k)(cid:90) e−jat2/2bφ(t−k)eja(t−k)2/2b
2π|b|
R R
k
(cid:26) (cid:27)
× exp j (cid:0)at2+dω2−2ωt+Ωω+2pt(cid:1) dt
2b
= 1 (cid:88)ejak2/2bp(k)(cid:90) φ(u)exp(cid:26) j (cid:0)au2+dω2−2ω(u+k)+Ωω+2p(u+k)(cid:1)(cid:27)du
2π|b| 2b
R
k
= ηA(ω)(cid:88)exp(cid:26) j (cid:0)ak2−2kω+2pk(cid:1)(cid:27)p(k)×(cid:90) φ(u)exp(cid:26) j (cid:0)au2+dω2−2ωu+Ωω+2pu(cid:1)(cid:27)du
2π|b| 2b 2b
R
k
(cid:34) (cid:26) (cid:27) (cid:35)
= ηA(ω) (cid:88)exp j (cid:0)ak2+dω2−2kω+Ωω+2pk(cid:1) p(k) Φ (ω)
(cid:112)2π|b| 2b A
k
= η (ω)Pˆ (ω)Φ (ω).
A A A
Furthermore, since e−jk∆/b =e−2jkπ =1, we have
(cid:26) (cid:27)
Pˆ (ω+∆)(ω) = 1 (cid:88)p(k)exp j (cid:0)ak2+d(ω+∆)2−2k(ω+∆)+Ω(ω+∆)+2pk(cid:1)
A (cid:112)2π|b| 2b
k
(cid:26) (cid:27)
= 1 (cid:88)p(k)exp j (cid:0)ak2+dω2−2kω+Ωω+2pk(cid:1)
(cid:112)
2π|b| 2b
k
(cid:26) (cid:27)
× exp j (cid:0)d∆2+2dω∆−2k∆+Ω∆(cid:1)
2b
(cid:26) (cid:27)
= Pˆ (ω)exp j (cid:0)d∆2+2d(ω)∆+Ω∆(cid:1)
A 2b
Thus,
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12)Pˆ (ω+∆)(cid:12)=(cid:12)Pˆ (ω)(cid:12).
(cid:12) A (cid:12) (cid:12) A (cid:12)
In the next theorem we give a necessary and sufficient condition for a function φ(t) to be a generator for a
shift-invariant space in terms of its SAFT.
Theorem 2. Let P = {p(n)} ∈ (cid:96) , φ ∈ L2(R) and consider the chirp-modulated shift-invariant subspaces of
2
L2(R)
(cid:40) (cid:41)
V(φ)=closure f ∈L2 :f(t)=(P (cid:63) φ)(t) .
A
(cid:110) (cid:111)
Then ej(t−k)2/2bφ(t−k) isaRieszbasisforV(φ)ifandonlyifthereexisttwopositiveconstantsη ,η >0
1 2
such that
+∞
η ≤ (cid:88) |Φ (w+k)|2 ≤η (17)
1 A 2
k=−∞
for all w ∈[0,∆], and Φ is SAFT of φ.
A
Proof: Since f(t)=(P (cid:63) φ)(t), we have by the previous lemma,
A
F (w)=λ (w)Pˆ (w)Φ (w);
A A A A
9
and hence
(cid:12) (cid:12)2
|F (w)|2 =(cid:12)Pˆ (w)(cid:12) |Φ (w)|2,
A (cid:12) A (cid:12) A
(cid:12) (cid:12)
where Pˆ (w) is given by Definition (2). Thus, because (cid:12)Pˆ (ω)(cid:12) is periodic with period ∆
A (cid:12) A (cid:12)
∞
(cid:90) (cid:12) (cid:12)2
(cid:107)F (ω)(cid:107)2 = (cid:12)Pˆ (ω)(cid:12) |Φ (ω)|2dω
A L2(R) (cid:12) A (cid:12) A
−∞
(k+1)∆
= (cid:88)∞ (cid:90) (cid:12)(cid:12)Pˆ (ω)(cid:12)(cid:12)2|Φ (ω)|2dω
(cid:12) A (cid:12) A
k=−∞
k∆
= (cid:88)∞ (cid:90) ∆(cid:12)(cid:12)Pˆ (ω+k∆)(cid:12)(cid:12)2|Φ (ω+k∆)|2dω.
(cid:12) A (cid:12) A
k=−∞ 0
=(cid:90) ∆(cid:12)(cid:12)Pˆ (ω)(cid:12)(cid:12)2 (cid:88)∞ |Φ (ω+k∆)|2dω
(cid:12) A (cid:12) A
0 k=−∞
(cid:90) ∆(cid:12) (cid:12)2
= (cid:12)Pˆ (ω)(cid:12) G (ω)dω,
(cid:12) A (cid:12) φ,A
0
where G (ω)=(cid:80) |Φ (ω+k∆)|2 is the Grammian of φ. But
φ,A k A
(cid:90) ∆(cid:12)(cid:12)Pˆ (ω)(cid:12)(cid:12)2dω = 1 (cid:90) ∆(cid:88)p(k)p(l)exp(cid:26) j (cid:2)a(k2−l2)−2ω(k−l)+2m(k−l)(cid:3)(cid:27)dω
(cid:12) A (cid:12) 2πb 2b
0 0 k,l
and since
(cid:90) ∆ (cid:90) ∆ (cid:90) 2π
ej(−2ω)(k−l)/2bdω = ej(ω)(l−k/b)dω =b eju(l−k)du=2πbδ ,
k,l
0 0 0
it follows that
(cid:13)(cid:13)Pˆ (cid:13)(cid:13)2 =(cid:90) ∆(cid:12)(cid:12)Pˆ (ω)(cid:12)(cid:12)2dω =(cid:88)|p(k)|2 =(cid:107)p(k)(cid:107)2 .
(cid:13) A(cid:13) (cid:12) A (cid:12) (cid:96)2
L2[0,∆] 0 k
Since
0<η ≤G (ω)≤η <∞
1 φ,A 2
and
(cid:13) (cid:13)2
(cid:107)p(k)(cid:107)2 =(cid:13)ej(ak2)/2bp(k)(cid:13)
(cid:96)2 (cid:13) (cid:13)
(cid:96)2
we have
(cid:13) (cid:13)2 (cid:13) (cid:13)2
η (cid:13)Pˆ (cid:13) =η (cid:107)p(k)(cid:107)2 ≤(cid:107)F (cid:107)2 ≤η (cid:107)p(k)(cid:107)2 ≤η (cid:13)Pˆ (cid:13)
1(cid:13) A(cid:13) 1 A 2 2(cid:13) A(cid:13)
which completes the proof.
To get orthonormal basis for the V(φ) we use the standard trick of putting
Φ (ω)
H (ω)= A
A (cid:112)
G (ω)
φ,A
so that
(cid:88)|H (ω+k∆)|2 = 1 (cid:88)|Φ (ω+k∆)|2 =1.
A G (ω) A
φ,A
k k
10
IV. THEZAKTRANSFORMASSOCIATEDWITHTHESAFT
Definition 3. We define the Zak transform associated with the SAFT of a signal f as
(cid:26) (cid:27)
Z (t,ω)= √1 (cid:88)f(t+k)exp j (cid:2)dω2+ak2−2kω+Ωω+2pk(cid:3)
A 2πb 2b
k
We have the following theorem
Theorem 3. The Zak transform given by definition 3 is an isometry between L2(R) and L2(B), where B =
[0,1]×[0,∆], that is there is a one-to-one correspondence between f ∈ L2(R) and Z ∈ L2(B) such that
A
(cid:107)f(cid:107)2L2(R) =(cid:107)ZA(cid:107)2L2(B).
Proof: First, let us observe that since
e−jk∆/b =e−2jkπ =1,
it follows that
1 (cid:88)
Z (t,ω+∆)= √ f(t+k)
A
2πb
k
(cid:26) (cid:27)
×exp j (cid:2)d(ω+∆)2+ak2−2k(ω+∆)+Ω(ω+∆)+2pk(cid:3)
2b
(cid:26) (cid:27)
j∆
=exp [d∆+2dω++Ω] Z (t,ω)
2b A
Thus |Z (t,ω+∆)|2 =|Z (t,ω)|2, and we have
A A
(cid:90) ∆|Z (t,ω)|2dω = 1 (cid:88)f(t+k)f(t+l)
A 2πb
0 k,l
× (cid:90) ∆exp(cid:26) j (cid:2)a(k−l)2−2ω(k−l)+2m(k−l)(cid:3)(cid:27)dω
2b
0
(cid:26) (cid:27)
= 1 (cid:88)f(t+k)f(t+l)exp j (cid:2)a(k−l)2+2m(k−l)(cid:3)
2πb 2b
k,l
(cid:90) ∆
× e−jω(k−l)/bdω
0
(cid:26) (cid:27)
= 1 (cid:88)f(t+k)f(t+l)exp j (cid:2)a(k−l)2+2m(k−l)(cid:3)
2πb 2b
k,l
(cid:90) 2π
× e−jx(k−l)bdx
0
= (cid:88)|f(t+k)|2.
k
Therefore
(cid:90) 1(cid:90) ∆ (cid:90) 1
(cid:107)Z (cid:107)2 = |Z (t,ω)|2dωdt= (cid:88)|f(t+k)|2dt
A L2(B) A
0 0 0 k
(cid:90) ∞
= |f(t)|2dt=(cid:107)f(cid:107)2 .
L2(R)
−∞