Gabor’s signal expansionbased on a non-orthogonal sampling geometry

Martin J. Bastiaans

Technische Universiteit Eindhoven, Faculteit ElektrotechniekPostbus 513, 5600 MB Eindhoven, Netherlands

tel: +31 40 247 3319 – fax: +31 40 2448375email:M.J.Bastiaans@ele.tue.nl

Abstract

Gabor’s signal expansion and the Gabor transform are formulated on a non-orthogonal time-frequency lattice instead of on the traditional rectangular lattice.The reason for doing so is that a non-orthogonal sampling geometry might be bet-ter adapted to the form of the window functions (in the time-frequency domain)than an orthogonal one: the set of shifted and modulated versions of the usualGaussian synthesis window, for instance, corresponding to circular contour lines inthe time-frequency domain, can be arranged more tightly in a hexagonal geometrythan in a rectangular one. Oversampling in the Gabor scheme, which is required tohave mathematically more attractive properties for the analysis window, then leadsto better results in combination with less oversampling. The procedure presentedin this paper is based on considering the non-orthogonal lattice as a sub-lattice ofa denser orthogonal lattice that is oversampled by a rational factor. In doing so,Gabor’s signal expansion on a non-orthogonal lattice can be related to the expan-sion on an orthogonal lattice (restricting ourselves, of course, to only those sam-pling points that are part of the non-orthogonal sub-lattice), and all the techniquesthat have been derived for rectangular sampling – including an optical means ofgenerating Gabor’s expansion coefficients via the Zak transform in the case of in-teger oversampling – can be used, albeit in a slightly modified form.

Keywords: Gabor’s signal expansion, Gabor transform, Zak transform, time-

frequency signal analysis

1. Introduction

In 1946,2 Gabor suggested the representation of a time signal in a combined time-frequency domain; in particular he proposed to represent the signal as a superposi-tion of shifted and modulated versions of a so-called elementary signal or synthesis

This paper is an expanded version of the paper1 that has been presented at the SPIEConference 4392, Optical Processing and Computing: A Tribute to Adolf Lohmann,Orlando, FL, 17 April 2001.

windowg(t). Moreover, as a synthesis windowg(t) he chose a Gaussian signal,

g(t) = 21/4 e−π(t/σt)2 (1)

with Fourier transform

g(ω) =∫ ∞

−∞g(t)e−jωtdt = 21/4 σt e−π(ω/σω)2 (σtσω = 2π), (2)

because such a signal has a good localization, both in the time domain and in thefrequency domain. The other choices that Gabor made were (i) that his signalexpansion was formulated on a rectangular lattice in the time-frequency domain,(mT,kΩ), (ii) that the sampling distancesT andΩ satisfied the relationΩT = 2π,and (iii) that they were in like manners proportional to the widthsσt andσω of thesynthesis window and its Fourier transform, respectively:T/σt = Ω/σω.

The coefficients in Gabor’s signal expansion can be determined by using ananalysis windoww(t). In the case of critical sampling, i.e.,ΩT = 2π, the anal-ysis windoww(t) follows uniquely from the given synthesis windowg(t). How-ever, such a unique analysis window appears to have some mathematically veryunattractive properties. For this reason, the expansion should be formulated on adenser lattice,ΩT < 2π. This makes the analysis window no longer unique andthus allows for finding an analysis window that is optimal in some way. We can,for instance, look for the ‘optimum’ analysis window that resembles best the syn-thesis window; a better resemblance can then be reached for a higher degree ofoversampling. Some examples of optimum analysis windowsw(t) that correspondto a Gaussian synthesis windowg(t) for different degrees of oversampling havebeen presented in Fig. 1.

A better resemblance can also be reached if we adapt the structure of the latticeto the form of the window as it is represented in the time-frequency domain. Thetime-frequency representation of a Gaussian window, for instance, for which itsWigner distribution3,4 reads

∫ ∞

−∞g(t + 1

2 t′)g∗(t− 12 t′)e−jωt′dt′ = 2 σt e−2π[(t/σt)2 + (ω/σω)2],

has circular contour lines in the time-frequency domain; and it is well known thatcircles are better packed on a hexagonal lattice than on a rectangular lattice, seeFig. 2. Gabor’s signal expansion on such a hexagonal, non-orthogonal lattice thenleads to a better resemblance between the window functionsg(t) andw(t) than theexpansion on a rectangular, orthogonal lattice does.

In this paper we consider the case of a non-orthogonal sampling geometry. Af-ter a short introduction of Gabor’s signal expansion and the Gabor transform on arectangular lattice, we will introduce the Fourier transform of the array of expan-sion coefficients and the Zak transforms of the signal and the window functions

2

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t/σt(a)

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1.2

t/σt(b)

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t/σt(c)

-2 -1 0 1 2 3 4 5-0.2

0

0.2

0.4

0.6

0.8

1

1.2

t/σt(d)

Figure 1. A Gaussian synthesis windowg(t) = 21/4 exp[−π(t/σt)2] (dashed line)and its corresponding optimum analysis window(T/q)w(t) (solid line), for differ-ent values of rational oversampling2π/ΩT = p/q, while maintaining the propor-tionality conditionσt/T = σω/Ω =

√2π/ΩT : (a) no oversampling2π/ΩT = 1,

(b) 2π/ΩT = 7/6, (c) 2π/ΩT = 3/2, and (d)2π/ΩT = 3.

to formulate the Gabor expansion and the Gabor transform in product forms.5,6

These product forms hold in the case of critical sampling on a rectangular lattice(2π/ΩT = 1), but – in a modified form7,8 – product forms can also be formulatedin the case of rational oversampling (2π/ΩT = p/q > 1); the use of an orthogo-nal lattice, however, is crucial! We do not restrict ourselves to the continuous-timecase, but we consider the discrete-time case, as well. However, since the generalresults hold for both cases, we concentrate mainly on the case of continuous-timesignals.

After the introductory sections, we will introduce a non-orthogonal samplinggeometry, and we will consider the non-orthogonal sampling lattice as a sub-latticeof a denser lattice that is orthogonal. In doing so, Gabor’s signal expansion on anon-orthogonal lattice can be related to the expansion on an orthogonal lattice, andall the techniques that have been developed for rectangular sampling – not only for

3

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µ´¶³µ´¶³

µ´¶³µ´¶³

µ´¶³µ´¶³

µ´¶³

µ´¶³

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Figure 2. A rectangular and a hexagonal packing of circles, with filling factorπ/4 = 0.7854 andπ/2

√3 = 0.9069, respectively.

continuous-time signals as described here, but for discrete-time signals, as well –can be used,6–9 albeit in a slightly modified form.

2. Gabor’s signal expansion on a rectangular lattice

We start with the usual Gabor expansion2,5–7,9 on a rectangular time-frequency lat-tice, in which case a signalϕ(t) can be expressed as a linear combination of prop-erly shifted and modulated versionsgmk(t) = g(t−mT ) exp(jkΩt) of a synthesiswindowg(t):

ϕ(t) =∞∑

m=−∞

∞∑

k=−∞amkgmk(t) =

∞∑m=−∞

∞∑

k=−∞amkg(t−mT )ejkΩt. (3)

The time stepT and the frequency stepΩ satisfy the relationshipΩT ≤ 2π; notethat the factor2π/ΩT represents the degree of oversampling, and that in his orig-inal paper2 Gabor considered the case of critical sampling, i.e.,ΩT = 2π. Theexpansion coefficientsamk follow from sampling the windowed Fourier transformwith analysis windoww(t),

∫ ∞

−∞ϕ(t)w∗(t− τ)e−jωtdt,

on the rectangular lattice(τ = mT,ω = kΩ):

amk =∫ ∞

−∞ϕ(t)w∗mk(t)dt =

∫ ∞

−∞ϕ(t)w∗(t−mT )e−jkΩtdt. (4)

This relationship is known as the Gabor transform.

The synthesis windowg(t) and the analysis windoww(t) are related to eachother in such a way that their shifted and modulated versions constitute two sets offunctions that are biorthogonal:

∞∑m=−∞

∞∑

k=−∞gmk(t1)w∗mk(t2) = δ(t1 − t2), (5)

4

with δ(t) the Dirac delta function. If the biorthogonality condition (5) is satisfied,the Gabor transform (4) and Gabor’s signal expansion (3) form a transform pairin the following sense: if we start with an arbitrary signalϕ(t) and determine itsexpansion coefficientsamk via the Gabor transform (4), the signal can be recon-structed via the Gabor expansion (3).

The biorthogonality relation (5) leads immediately to the equivalent but simplerexpression

2π

Ω

∞∑m=−∞

g(t−mT )w∗(

t−[mT + n

2π

Ω

])= δn, (6)

whereδn is the Kronecker delta. In the case of critical sampling, i.e.,2π/Ω = T ,the biorthogonality relation (6) reduces to

T

∞∑m=−∞

g(t−mT )w∗(t− [m + n]T ) = δn (7)

and the analysis windoww(t) follows uniquely from a given synthesis windowg(t), or vice versa. An elegant way to find the analysis window if the synthesiswindow is given, is presented in the next section.

3. Fourier transform and Zak transform

It is well known,5–7,9 that in the case of critical sampling,ΩT = 2π, Gabor’s signalexpansion (3) and the Gabor transform (4) can be transformed into product form.We therefore need the Fourier transforma(ξ, η) of the two-dimensional array ofGabor coefficientsamk, defined by

a(ξ, η) =∞∑

m=−∞

∞∑

k=−∞amke

−j2π(mη− kξ), (8)

and the Zak transformsϕT (ξT, η2π/T ), gT (ξT, η2π/T ), andwT (ξT, η2π/T ) ofthe signalϕ(t) and the window functionsg(t) andw(t), respectively, where theZak transformfτ (t,ω) of a functionf(t) is defined as,6,7,9–12

fτ (t,ω) =∞∑

n=−∞f(t + nτ)e−jnτω. (9)

Note that the Fourier transforma(ξ, η) is periodic inξ andη with period 1, and thatthe Zak transformfτ (t,ω) is periodic inω with period2π/τ and quasi-periodic int with periodτ :

fτ (t + mτ,ω + k2π/τ) = fτ (t,ω)ejmτω.

5

Upon substituting from the Fourier transform (8) and the Zak transforms[cf. Eq. (9)] into Eqs. (3) and (4), it is not too difficult to show that Gabor’s signalexpansion (3) can be transformed into the product form

ϕT

(ξT, η

2π

T

)= a(ξ, η)gT

(ξT, η

2π

T

), (10)

while the Gabor transform (4) can be transformed into the product form

a(ξ, η) = T ϕT

(ξT, η

2π

T

)w∗T

(ξT, η

2π

T

). (11)

In particular the product form (11) is useful for determining Gabor’s expansion co-efficients. Since a Zak transform is merely a Fourier transform [cf. Eq. (9)], theexpansion coefficients can be determined by Fourier transformations and multipli-cations; and if things are formulated for discrete-time signals, see Sections 6 and7, we can use thefastFourier transform to formulate a fast algorithm for the Gabortransform.7,8

The relationship between the Zak transforms of the analysis windoww(t)and the synthesis windowg(t) now follows from substituting from Eq. (11) intoEq. (10) and reads

T gT

(ξT, η

2π

T

)w∗T

(ξT, η

2π

T

)= 1. (12)

From the latter relationship we conclude that (the Zak transform of) the analysiswindow w(t) follows uniquely from (the Zak transform of) the given synthesiswindowg(t); see Fig. 1a for the analysis window that corresponds to the Gaussiansynthesis window and Fig. 3a for the Zak transform of that Gaussian window. Un-fortunately, in general, the unique analysis windoww(t) has some very unattractivemathematical properties. We are therefore urged to consider Gabor’s signal expan-sion on a denser lattice, in which case the analysis window is no longer unique.This enables us to choose an analysis window that is better suited to our purpose ofdetermining Gabor’s expansion coefficients.

4. Rational oversampling

In the case of oversampling by a rational factor,2π/ΩT = p/q ≥ 1, with p andq relatively prime, positive integers,p > q ≥ 1, Gabor’s expansion (3) and theGabor transform (4) can be transformed (see Appendices A and B) into the sum-of-products forms7,8 [cf. Eqs. (10) and (11)]

ϕs(ξ, η) =1p

p−1∑

r=0

gs,r(ξ, η)ar(ξ, η) (s = 0,1, . . . , q− 1), (13)

ar(ξ, η) =pT

q

q−1∑

s=0

w∗s,r(ξ, η)ϕs(ξ, η) (r = 0,1, . . . , p− 1), (14)

6

respectively, where we have introduced the shorthand notations

ar(ξ, η) = a (ξ, η + r/p) ,

ϕs(ξ, η) = ϕpT ([ξ + s]pT/q, η2π/T ) ,

gs,r(ξ, η) = gT ([ξ + s]pT/q, [η + r/p] 2π/T ) ,

ws,r(ξ, η) = wT ([ξ + s]pT/q, [η + r/p] 2π/T ) ,

with 0≤ ξ < 1 ands = 0,1, . . . , q− 1 [and hence0≤ (ξ + s)/q < 1], and0≤ η <1/p andr = 0,1, . . . , p−1 [and hence0≤ η + r/p < 1]. The relationship betweenthe Zak transforms of the analysis windoww(t) and the synthesis windowg(t) thenfollows from substituting from Eq. (14) into Eq. (13) and reads [cf. Eq. (12)]

T

q

p−1∑

r=0

gs1,r(ξ, η)w∗s2,r(ξ, η) = δs1−s2 , (15)

with s1, s2 = 0,1, . . . , q − 1. For each(ξ, η) and given the synthesis windowg(t), the latter relationship represents a set ofq2 equations for thepq unknownsw∗s,r(ξ, η), which set of equations is underdetermined sincep > q, and we concludethat the analysis window does not follow uniquely from the synthesis window. Notethat forq = 1, i.e., for integer oversampling by a factorp, the sum-of-products form(14) reduces to a product again [cf. Eq. (11)]:

a(ξ, η) = pT ϕpT

(ξpT, η

2π

T

)w∗T

(ξpT, η

2π

T

). (16)

After combining thep functionsar(ξ, η) into a p-dimensional column vectora(ξ, η), theq functionsϕs(ξ, η) into a q-dimensional column vectorφφφ(ξ, η), andthe q × p functionsgs,r(ξ, η) andws,r(ξ, η) into theq × p-dimensional matricesG(ξ, η) andW(ξ, η), respectively, the sum of products forms can be expressed asmatrix-vector and matrix-matrix multiplications:

φφφ(ξ, η) =1p

G(ξ, η)a(ξ, η), (17)

a(ξ, η) =pT

qW∗(ξ, η)φφφ(ξ, η), (18)

I q =T

qG(ξ, η)W∗(ξ, η), (19)

whereI q denotes theq × q-dimensional identity matrix and where, as usual, theasterisk in connection with vectors and matrices denotes complex conjugationandtransposition.

The latter relationship again representsq2 equations forpq unknowns, and thep× q matrix W∗(ξ, η) cannot be found by a simple inversion of theq × p matrix

7

G(ξ, η). An optimum solution that is often used, is based on the generalized inverseand reads

W∗opt(ξ, η) =

q

TG∗(ξ, η)[G(ξ, η)G∗(ξ, η)]−1.

This solution forW(ξ, η) is optimal in the sense that (i) it yields the analysis win-doww(t) with the lowestL2 norm, (ii) it yields the Gabor coefficientsamk with thelowestL2 norm, and (iii) it yields the analysis window that – in anL2 sense, again –best resembles the synthesis window. As examples we have represented in Fig. 3the absolute values of the Zak transforms that correspond to the Gaussian synthesiswindow g(t) and to the optimum analysis windowsw(t) depicted in Figs. 1b, c,and d.

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ξη

(d)

Figure 3. (a) The absolute value|gT (ξT, η2π/T )| of the Zak transform that cor-responds to the Gaussian synthesis windowg(t) = 21/4 exp[−π(t/T )2]; and theabsolute values|wT (ξT, η2π/T )| of the Zak transforms that correspond to the op-timum analysis windowsw(t) in Figs. 1b, c, and d, respectively, i.e., for differentvalues of oversampling: (b)2π/ΩT = 7/6, (c)2π/ΩT = 3/2, and (d)2π/ΩT = 3.

The optimum solution gets better if the degree of oversamplingp/q becomeshigher. However, there is another way of finding a better solution, based on the

8

structure of the lattice. If the lattice structure is adapted to the form of the windowfunction as it is represented in the time-frequency domain, the optimum solutionwill be better, even for a lower degree of oversampling. In Section 8 we will there-fore consider the case of a non-orthogonal sampling geometry, but we will do that insuch a way that we can relate this non-orthogonal sampling to orthogonal sampling.In that case we will still be able to use product forms of Gabor’s expansion and theGabor transform, and benefit from all the techniques that have been developed forthem.

5. Coherent-optical generation of the Gabor coefficientsvia the Zak transform

The product form (16) of the Gabor transform in the case of integer oversampling,suggests a generation of this transform by coherent-optical means.6,7,13,14 Apartfrom being able to realize ideal imaging, coherent optics is well suited for twospecific signal operations, viz., multiplication by a constant function (like in a slideprojector, for instance) and Fourier transformation (between the back and the frontfocal plane of a lens, for instance). These two operations are exactly the ones thatare needed for generation of the Gabor transform.

Let a plane wave of monochromatic laser light be normally incident upon atransparency situated in the input plane of a coherent-optical system, see Fig. 4.The transparency contains the time signalϕ(t) in a rastered format. WithXo ∼ pTbeing the width of this raster andpµXo (with µ > 0) being the spacing between theraster lines, the light amplitudeϕi(xi, yi) just behind the transparency reads

ϕi(xi, yi) = rect

(xi

Xo

) ∞∑n=−∞

ϕ

(xi + nXo

XopT

)δ(yi − npµXo), (20)

where rect(z) represents a rectangular window function: rect(z) = 1 for−12 < z ≤

12 and rect(z) = 0 for z outside this range.

An anamorphic optical system between the input plane and an intermediatemiddle plane performs a Fourier transformation in they-direction and an idealimaging (possibly with inversion) in thex-direction. Such an anamorphic systemcan be realized, for instance, using a combination of a spherical and a cylindricallens. The anamorphic operation results in the light amplitude

ϕ1(x, y) =∫ ∞

−∞

∫ ∞

−∞ϕi(xi, yi)e−jβiyyiδ(x− xi)dxidyi

= rect

(x

Xo

)ϕpT

(x

XopT,

y

Yo

2π

T

)(21)

just in front of the intermediate plane. The parameterβi contains the effect ofthe wavelengthλ of the laser light and the focal lengthfi of the spherical and the

9

Figure 4. Coherent-optical setup for generation of the Gabor transform.

cylindrical lens:βi = 2π/λfi; moreover, the vertical distanceYo is defined throughβiµXoYo = 2π.

A transparency with amplitude transmittance

m(x, y) = rect

(x

Xo

)rect

(y

Yo

)pT w∗T

(x

XopT,

y

Yo

2π

T

)(22)

is situated in the intermediate plane. Just behind this transparency, the light ampli-tude takes the form

ϕ2(x, y) = m(x, y)ϕ1(x, y) = rect

(x

Xo

)rect

(y

Yo

)a

(x

Xo,

y

Yo

), (23)

where use has been made of the product form (16). Note that, withξ = x/Xo

and η = y/Yo, the aperture rect(ξ)rect(η) containsone period of the periodicFourier transforma(ξ, η), p horizontal periods of the (periodic) Zak transformϕpT (ξpT, η2π/T ), andp vertical quasi-periods of the (quasi-periodic) Zak trans-form w∗T (ξpT, η2π/T ).

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Finally, a two-dimensional Fourier transformation is performed between theintermediate plane and the output plane. Such a Fourier transformation can berealized, for instance, using a spherical lens. The light amplitude in the outputplane then takes the form

ϕo(xo, yo) =1

XoYo

∫ ∞

−∞

∫ ∞

−∞ϕ2(x, y)e−jβo(xox− yoy)dxdy

=∞∑

m=−∞

∞∑

k=−∞amk sinc

(βo

βi

yo

µXo−m

)sinc

(βo

βi

xo

µYo− k

),(24)

where the sinc-function sinc(z) = sin(πz)/(πz) has been introduced; the param-eterβo, again, contains the effects of the wave lengthλ of the laser light and thefocal lengthfo of the spherical lens:βo = 2π/λfo. We conclude that the Gabortransform appears on a rectangular lattice of points

amk = ϕo

(k

βi

βoµYo,m

βi

βoµXo

)(25)

in the output plane. Note that relationship (24) represents the output signal as aninterpolated version of the Gabor transform, where the interpolation kernel consistsof two sinc-functions, in accord with the rectangular aperture in the intermediate(Fourier) plane.

We remark that it is not an essential requirement that the input transparencyconsists of Dirac functions. When we replace the practically unrealizable Diracfunctionsδ(yi−npµXo) by realizable functionsd(yi−npµXo), say, then equation(20) reads

ϕi(xi, yi) = rect

(xi

Xo

) ∞∑n=−∞

ϕ

(xi + nXo

XopT

)d(yi − npµXo)

and the light amplitudeϕ1(x, y) just in front of the middle plane takes the form[cf. Eq. (21)]

ϕ1(x, y) = rect

(x

Xo

)ϕpT

(x

XopT,

y

Yo

2π

T

)d(βiy).

The additional factord(βiy) – the Fourier transform ofd(yi) – can easily be com-pensated for by means of a transparency in the middle plane.

The technique described in this section to generate the Gabor transform, fullyutilizes the two-dimensional nature of the optical system, its parallel processingfeatures, and the large space-bandwidth product possible in optical processing.The technique exhibits a resemblance to folded spectrum techniques, where space-bandwidth products in the order of 300 000 are reported.15

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6. Discrete Gabor expansion and discrete Gabor transform

In this section we consider the case of discrete-time signalsϕ[n] instead ofcontinuous-time signalsϕ(t); we shall use square brackets[ ] instead of normalbrackets ( ) to distinguish discrete-time functions from continuous-time functions.The importance of having discrete versions of Gabor’s signal expansion and theGabor transform lies in the fact that we can formulate fast algorithms to computethem.

The Gabor expansion for a discrete-time signalϕ[n] takes the form [cf. Eq. (3)]

ϕ[n] =∞∑

m=−∞

∑

k=<K>

amkgmk[n] =∞∑

m=−∞

∑

k=<K>

amkg[n−mN ]ej2πkn/K,

(26)

where the expressionk =<K> throughout denotes a summation over a finite in-terval ofK successive integersk, and the Gabor transform reads [cf. Eq. (4)]

amk =∞∑

n=−∞ϕ[n]w∗mk[n] =

∞∑n=−∞

ϕ[n]w∗[n−mN ]e−j2πkn/K. (27)

These formulas are similar to the expressions for the Gabor expansion (3) and theGabor transform (4) for continuous-time signals, where nowN instead ofT takesthe role of the time step and2π/K instead ofΩ takes the role of the frequencystep. Note, moreover, thatwmk[n] andgmk[n], and thus the array of coefficientsamk, are periodic ink with periodK, and that we can restrict ourselves in theGabor expansion (26) to one period ofk. We introduce again two integersp andq (p ≥ q ≥ 1) that do not have common factors and for which the relationshippN = qK holds; note thatK/N = p/q ≥ 1 represents the degree of oversampling,which is always arational oversampling, of course.

We now require that the analysis windoww[n] has a finite supportNw. Forconvenience, we consider signalsϕ[n] that have a finite supportNϕ, too; in the casethat we are dealing with signals of longer (or even infinite) support, we can alwayssplit the signal in parts that do have a finite supportNϕ and treat all these partsseparately,7,8 similar to the overlap-add technique used in digital signal processing.Under these conditions of finite support, the arrayamk, which is periodic ink, hasa finite supportM in m, where the supportM satisfies the condition

MN ≥ Nϕ + Nw − 1. (28)

We now introduce the periodized versionAmk of the arrayamk according to

Amk =∞∑

r=−∞am+rM,k; (29)

12

note that the periodized arrayAmk is periodic inm andk with periodsM andK, respectively, and that we can identifyamk (which, by itself, does not show aperiodicity with respect tom) as one period ofAmk. Furthermore we introduce theperiodized versionW [n] of the (finite support) windoww[n] according to

W [n] =∞∑

r=−∞w[n− rMN ]; (30)

note that the periodized window functionW [n] is periodic with periodMN andthat we can identifyw[n] as one period ofW [n]. It is easy to see that the periodizedarrayAmk can be expressed in terms of the periodized window functionW [n] viaa kind of Gabor transform [cf. Eq. (27)]:

Amk =∞∑

n=−∞ϕ[n]W ∗

mk[n] =∞∑

n=−∞ϕ[n]W ∗[n−mN ]e−j2πkn/K.

We could as well periodize the (finite support) signalϕ[n] according to

Φ[n] =∞∑

r=−∞ϕ[n− rMN ]; (31)

note that the periodized signalΦ[n] is periodic with periodMN and that we canidentify ϕ[n] as one period ofΦ[n]. If we substitute from Gabor’s signal ex-pansion (26) into Eq. (31) and require thatK is a divisor ofMN (and henceexp[j2πkr(MN/K)] = 1), we can write

Φ[n] =∞∑

m=−∞

∑

k=<K>

amk

( ∞∑r=−∞

g[n− rMN −mN ]

)ej2πkn/K.

After introducing the periodized versionG[n] of the synthesis windowg[n][cf. Eq. (30)], we can write

Φ[n] =∞∑

m=−∞

∑

k=<K>

amkGmk[n] =∞∑

m=−∞

∑

k=<K>

amkG[n−mN ]ej2πkn/K,

which relationship has the form of a Gabor expansion [cf. Eq. (26)].

It is not difficult to show that we also have the relationships

Φ[n] =∑

m=<M>

∑

k=<K>

AmkGmk[n] (32)

and

Amk =∑

n=<MN>

Φ[n]W ∗mk[n], (33)

which are fully periodized versions of Gabor’s signal expansion (26) and the Gabortransform (27), respectively. Equation (32) is known as the discrete Gabor expan-sion, while Eq. (33) is known as the discrete Gabor transform.

13

7. Discrete Fourier transform and discrete Zak transform

We will now formulate sum-of-products forms for the discrete Gabor expansion andthe discrete Gabor transform, similar to the ones derived in Section 4. We there-fore introduce discrete versions of the Fourier transform and the Zak transform,cf. Section 3.

The discrete Fourier transformaKM [n, l] of the periodic arrayAmk is definedaccording to [cf. Eq. (8)]

aKM [n, l] =∑

m=<M>

∑

k=<K>

Amke−j2π(ml/M − kn/K); (34)

note that the discrete Fourier transformaKM [n, l] is periodic in the variablesn andl with periodsK andM , respectively.

The discrete Zak transformwNM [n, l] of the periodized window functionW [n]is defined as a one-dimensional discrete Fourier transform of the sequenceW [n +mN ] (with m =<M> andn being a mere parameter), hence [cf. Eq. (9)]

wNM [n, l] =∑

m=<M>

W [n + mN ]e−j2πml/M . (35)

We remark that the discrete Zak transformwNM [n, l] is periodic in the frequencyvariablel with periodM and quasi-periodic in the time variablen with quasi-periodN :

wNM [n + mN, l + kM ] = wNM [n, l]ej2πml/M .

We also introduce the discrete Zak transform ofΦ[n]

ϕpN,M/p[n, l] =∑

m=<M/p>

Φ[n + mpN ]e−j2πmpl/M , (36)

where the condition thatp is a divisor ofM should hold.

From the condition thatp is a divisor ofM , we can writeM = pL, whereLis an integer. FrompN = qK andM = pL we conclude thatMN = qKL, whichimplies thatK is a divisor ofMN . Moreover, frompN = qK and knowing thatpandq do not have common factors, we also conclude thatp is a divisor ofK; hencewe can writeK = pJ , whereJ is an integer. We thus conclude thatK, M , andN can be expressed in terms of the integersp, q (with p ≥ q ≥ 1, andp andq nothaving common factors),J andL: K = pJ , M = pL, andN = qJ .

Using the discrete Fourier transform and the discrete Zak transform, it can beshown7,8 that the discrete Gabor expansion (32) and the discrete Gabor transform

14

(33) can be transformed into the sum-of-products forms

ϕs[n, l] =1p

p−1∑

r=0

gs,r[n, l]ar[n, l] (s = 0,1, . . . , q− 1), (37)

ar[n, l] = K

q−1∑

s=0

w∗s,r[n, l]ϕs[n, l] (r = 0,1, . . . , p− 1), (38)

respectively, [cf. Eqs. (13) and (14), respectively], where we have introduced theshorthand notations

ar[n, l] = aKM [n, l + rM/p],ϕs[n, l] = ϕpN,M/p[n + sK, l],

gs,r[n, l] = gNM [n + sK, l + rM/p],ws,r[n, l] = wNM [n + sK, l + rM/p],

with 0 ≤ n < K and s = 0,1, . . . , q − 1 [and hence0 ≤ n + sK < qK], and0 ≤ l < M/p andr = 0,1, . . . , p− 1 [and hence0 ≤ l + rM/p < M ]. Note thatqK = pN .

After combining thep functionsar[n, l] into a p-dimensional column vectora[n, l], theq functionsϕs[n, l] into aq-dimensional column vectorφφφ[n, l], and theq× p functionsgs,r[n, l] andws,r[n, l]) into theq× p-dimensional matricesG[n, l]andW[n, l], respectively, the sum of products forms can be expressed as matrix-vector multiplications,

φφφ[n, l] =1p

G[n, l]a[n, l], (39)

a[n, l] =pN

qW∗[n, l]φφφ[n, l], (40)

which are similar to the matrix-vector multiplications (17) and (18) that have beenderived in Section 4 for the case of continuous-time signals.

Since the discrete-time case leads to formulas that are very similar to the onesthat arise in the continuous-time case, we shall concentrate in the remaining sec-tions on the continuous-time case and only mention the differences, if any, thatwould occur in the discrete-time case.

8. Non-orthogonal sampling

The rectangular (or orthogonal) lattice that we considered in the previous sections,where sampling occurred on the lattice points(τ = mT,ω = kΩ), can be obtained

15

by integer combinations of two orthogonal vectors[T,0]t and[0,Ω]t, see Fig. 5a,which vectors constitute the lattice generator matrix

[T 00 Ω

].

We now consider a time-frequency lattice that is no longer orthogonal. Such a lat-tice is obtained by integer combinations of two linearly independent, but no longerorthogonal vectors, which we express in the forms[aT, cΩ]t and [bT, dΩ]t, witha, b, c andd integers, and which constitute the lattice generator matrix

[aT bTcΩ dΩ

]=

[T 00 Ω

][a bc d

].

Without loss of generality, we may assume that the integersa andb have no com-mon divisors, and that the same holds for the integersc andd; possible commondivisors can be absorbed inT and Ω. Note that we only consider lattices thathave samples on the time and frequency axes and that are therefore suitable for adiscrete-time approach, as well.

The area of a cell (a parallelogram) in the time-frequency plane, spanned bythe two vectors[aT, cΩ]t and [bT, dΩ]t, is equal to the determinant of the latticegenerator matrix, which determinant is equal toΩTD, with D = |ad− bc|. To beusable as a proper Gabor sampling lattice, this area should satisfy the conditionD ≤ 2π/ΩT .

There are a lot of lattice generator matrices that generate the same lattice. Wewill use the one that is based on the Hermite normal form, unique for any lattice,

[T 0

RΩ DΩ

]=

[T 00 Ω

][1 0R D

],

whereR andD are relatively prime integers and0 ≤ |R| < D. Sampling thenoccurs on the lattice points(τ = mT,ω = [mR+nD]Ω), and it is evident that thesepoints of the non-orthogonal lattice form a subset of the points(τ = mT,ω = kΩ)of the orthogonal lattice. To be more specific: the non-orthogonal lattice is formedby those points of the rectangular (orthogonal) lattice for whichk −mR is aninteger multiple ofD. Note that the original rectangular lattice arises forR = 0andD = 1, see Fig. 5a, and that a hexagonal lattice occurs forR = 1 andD = 2,see Fig. 5b.

Non-orthogonal sampling in the discrete-time case yields exactly the same re-sults. Of course, the time stepT is now replaced by the integerN and the frequencystepΩ by 2π/K, with K an integer, but that does not affect the procedure of non-orthogonal sampling.

16

-

6

(a)

rrrrr

rrrrr

rrrrr

rrrrr

rrrrr

rrrrr

rrrrrω

t

-¾T

6?Ω

-

6

(b)

bbbbb

bbbbb

bbbbb

bbbbb

bbbbb

bbbbb

bbbbb

r

r

r

r

r

r

r

r

r

r

r

r

r

r

r

r

rω

t

-¾T

6?Ω

Figure 5. (a) A rectangular lattice with lattice vectors[T,0]t and[0,Ω]t, and thusR = 0 and D = 1; and (b) a hexagonal lattice with lattice vectors[T,Ω]t and[0,2Ω]t, and thusR = 1 andD = 2.

9. Gabor’s signal expansion on a non-orthogonal lattice

If we define the two-dimensional arrayλmk as

λmk =∞∑

n=−∞δk−mR−nD, (41)

Gabor’s signal expansion on a non-orthogonal lattice can be expressed as[cf. Eq. (3)]

ϕ(t) =∞∑

m=−∞

∞∑

k=−∞λmkamkgmk(t) =

∞∑m=−∞

∞∑

k=−∞as

mkgmk(t), (42)

while – with a different analysis windoww(t), though! – the expansion coefficientsamk are still determined by the Gabor transform (4). Of course, since we only needthe limited arrayas

mk = λmkamk (which is, in fact, a properly sampled version ofthe full arrayamk) we need only calculate the coefficientsamk for those values ofmandk for whichk−mR is an integer multiple ofD. We note that the Fourier trans-form as(ξ, η) of the limited arrayas

mk is related to the Fourier transforma(ξ, η) ofthe full arrayamk via the periodization relation

as(ξ, η) =1D

D−1∑

n=0

a

(ξ − n

D,η− nR

D

)(43)

and thus

asr(ξ, η) =

1D

D−1∑

n=0

ar

(ξ − n

D,η− nR

D

). (44)

17

In the non-orthogonal case, the biorthogonality condition takes the form[cf. Eq. (5)]

∞∑m=−∞

∞∑

k=−∞λmkgmk(t1)w∗mk(t2) = δ(t1 − t2) (45)

and leads to the equivalent but simpler expression [cf. Eq. (6)]

2π

DΩ

∞∑m=−∞

g(t−mT )w∗(

t−[mT + n

2π

DΩ

])ej2πmnR/D = δn. (46)

Note that forR = 0 andD = 1, for which we have a rectangular lattice (see Fig. 5a),Eq. (46) reduces to Eq. (6), and that forR = 1 andD = 2, for which we have ahexagonal lattice (see Fig. 5b), Eq. (46) takes the form

π

Ω

∞∑m=−∞

g(t−mT )w∗(t−

[mT + n

π

Ω

])(−1)mn = δn. (47)

The biorthogonality condition expressed in terms of the Zak transforms of thewindow functions now takes the form [cf. Eq. (15)]

T

Dq

p−1∑

r=0

gs1,r(ξ, η)w∗s2,r

(ξ − n

D,η− nR

D

)= δnδs1−s2 , (48)

with s1, s2 = 0,1, . . . , q − 1 andn = 0,1, . . . ,D − 1, and allows an easy de-termination of the analysis windoww(t) for a given synthesis windowg(t). ForR = 0 andD = 1, for instance, relation (48) reduces to Eq. (15), while forR = 1,D = 2, q = 1, andp an even integer – which corresponds to the integer (1

2p-times)oversampled hexagonal case – it reduces to

T

2

p−1∑

r=0

g0,r(ξ, η)w∗0,r−np/2(ξ, η)(−1)nr = δn (n = 0,1; p even), (49)

from which the Zak transformwT (t,ω) and hence the window functionw(t) caneasily be determined.

Since we have related Gabor’s signal expansion on a non-orthogonal lattice tosampling on a denser but orthogonal lattice, followed by restriction to a sub-latticethat corresponds to the non-orthogonal lattice, we can still use all the techniquesthat are developed for rectangular lattices, in particular the technique of determin-ing Gabor’s expansion coefficients via the Zak transform, cf. Eqs. (14) and (16).Hence, the optical setup described in Section 5 can still be applied; the only differ-ence is that the sampling in the output plane is now on the non-orthogonal lattice.

The discrete-time case, again, does not yield essentially new results. Weshould only be aware that the periodicity that arises in the periodized arrayAmk

should also appear in the sampling arrayλmk; hence, the periodicity conditionλm+rM,k+sK = λmk should hold.

18

10. From non-orthogonal to rectangular sampling via shearing

If we eliminate the arrayλmk from Gabor’s signal expansion (42), we can directlywrite

ϕ(t) =∞∑

m=−∞

∞∑n=−∞

am,mR+nDg(t−mT )ej(mR + nD)Ωt. (50)

We now write the shifted and modulated version of the synthesis windowg(t) inthe form

g(t−mT )ej(mR + nD)Ωt = g′mn(t)ejRΩt2/2T ejRm2ΩT/2,

where we have introduced the sheared versiong′(t) of g(t):

g′(t) = g(t)e−jRΩt2/2T . (51)

If we apply the same shear operation to the signalϕ(t) to get

ϕ′(t) = ϕ(t)e−jRΩt2/2T (52)

and introduce the sheared array

a′mn = am,mR+nDejRm2ΩT/2, (53)

we get Gabor’s signal expansion in the original, rectangular form, again:

ϕ′(t) =∞∑

m=−∞

∞∑n=−∞

a′mng′mn(t). (54)

And, after introducing the sheared versionw′(t) of the analysis windoww(t), thesame holds for the Gabor transform:

a′mn =∫ ∞

−∞ϕ′(t)w′∗mn(t)dt. (55)

We conclude that multiplication of the signal, the window functions, and theexpansion coefficients with proper quadratic phase factors, corresponds to shear-ing of the Gabor lattice; this shearing transforms a non-orthogonal lattice into arectangular one. Consequently, in calculating Gabor’s expansion coefficients in thenon-orthogonal case, algorithms for the rectangular case can still be used, but thesignalϕ(t) and the window functionsw(t) andg(t) have to be pre-multiplied bya quadratic phase factor, see Eqs. (51) and (52). After calculating the Gabor co-efficientsa′mn, these coefficients have to be post-multiplied by a quadratic phasefactor to obtain the actual coefficientsamn, see Eq. (53).

19

Note that shearing is not the only way to transform a non-orthogonal latticeinto an orthogonal one. In some recent papers,16,17 a fractional Fourier transformcombined with a proper scaling was used to do this.

The procedure described above, when applied to discrete-time signals andcombined with the Zak transform, leads to fast algorithms for the calculation ofGabor’s expansion coefficients by means of a digital computer.17 It is unclear,however, whether an optical setup based upon this procedure – i.e., pre- and post-multiplication by quadratic-phase factors to shear a non-orthogonal lattice into anorthogonal one – can be found.

11. Conclusions

Gabor’s signal expansion and the Gabor transform on a rectangular lattice havebeen introduced, along with the Fourier transform of the array of expansion coef-ficients and the Zak transforms of the signal and the window functions. Based onthese Fourier and Zak transforms, the sum-of-products forms for the Gabor expan-sion and the Gabor transform, which hold in the rationally oversampled case, havebeen derived. An optical setup for the generation of Gabor’s expansion coefficientsin the case of integer oversampling has been presented.

The discrete-time case – with finite-length signals and window functions – hasbeen treated, as well. It has been shown that this case leads to results that are verysimilar to the results that arise in the continuous-time case.

We have then studied Gabor’s signal expansion and the Gabor transform basedon a non-orthogonal sampling geometry. We have done this by considering thenon-orthogonal lattice as a sub-lattice of an orthogonal lattice. This procedure al-lows us to use all the formulas that hold for the orthogonal sampling geometry. Inparticular we can use the sum-of-products forms that hold in the case of a rationallyoversampled rectangular lattice. Moreover, in the case of integer oversampling, theoptical setup presented before can still be used.

We finally note that if everything remains to be based on a rectangular samplinggeometry, it will be easier to extend the theory of the Gabor scheme to higher-dimensional signals; see, for instance, Ref. 17, where the multi-dimensional caseis treated for continuous-time as well as discrete-time signals.

Appendix A. Derivation of Eq. (13)

In Gabor’s signal expansion (3) we substitute from the inverse Fourier transform[cf. Eq. (8)]

ϕ(t) =∞∑

m=−∞

∞∑

k=−∞

[∫ 1

0

∫ 1

0a(ξ, η)ej2π(mη− kξ)dξdη

]g(t−mT )ejkΩt

20

and rearrange factors

∫ 1

0

∫ 1

0a(ξ, η)

[ ∞∑m=−∞

g(t−mT )ej2πmη

][ ∞∑

k=−∞e−jk(2πξ −Ωt)

]dξdη.

We replace the sum of exponentials by a sum of Dirac functions and recognize theZak transform of the synthesis windowg(t) [cf. Eq. (9)]:

∫ 1

0

∫ 1

0a(ξ, η)gT

(t, η

2π

T

)[ ∞∑

k=−∞δ

(ξ − Ω

2πt + k

)]dξdη.

We rearrange factors again and substitute from the periodicity property ofa(ξ, η)

∫ 1

0

[ ∞∑

k=−∞

∫ 1

0a(ξ + k, η)δ

(ξ + k− Ω

2πt

)dξ

]gT

(t, η

2π

T

)dη

and we replace the summation overk together with the integral over the finiteξ-interval by an integral over the entireξ-axis

∫ 1

0

[∫ ∞

−∞a(ξ, η)δ

(ξ − Ω

2πt

)dξ

]gT

(t, η

2π

Ω

)dη.

Evaluation of the resulting integral overξ yields the intermediate result

ϕ(t) =∫ 1

0a

(Ω2π

t, η

)gT

(t, η

2π

T

)dη.

We now write down the definition of the Zak transform [cf. Eq. (9)]

ϕpT

(ξpT

q, ηo

2π

T

)=

∞∑n=−∞

ϕ

([ξ + nq]

pT

q

)e−jnq(pT/q)ηo(2π/T )

and substitute from the intermediate result above

∞∑n=−∞

[∫ 1

0a

(Ω2π

[ξ + nq]pT

q, η

)gT

([ξ + nq]

pT

q, η

2π

T

)dη

]

× e−jnq(pT/q)ηo(2π/T ).

We rearrange things, using the relation2π/Ω = pT/q,

∞∑n=−∞

[∫ 1

0a(ξ + nq, η)gT

(ξpT

q+ pnT, η

2π

T

)dη

]e−j2πpnηo

21

and use the periodicity property ofa(ξ, η) and the quasi-periodicity property ofgT (ξpT/q, y2π/T )

∞∑n=−∞

[∫ 1

0a(ξ, η)gT

(ξpT

q, η

2π

T

)ejpnTη(2π/T )dη

]e−j2πpnηo .

We rearrange factors∫ 1

0a(ξ, η)gT

(ξpT

q, η

2π

T

)[ ∞∑n=−∞

ej2πnp(η− ηo)]

dη

and replace the sum of exponentials by a sum of Dirac functions∫ 1

0a(ξ, η)gT

(ξpT

q, η

2π

T

)[1p

∞∑n=−∞

δ

(η− ηo − n

p

)]dη.

We replace the summation overn by a double summation overr andk through thesubstitutionn = kp + r, wherer extends over an interval of lengthp

∫ 1

0a(ξ, η)gT

(ξpT

q, η

2π

T

)[1p

∞∑

k=−∞

p−1∑

r=0

δ

(η− ηo − k− r

p

)]dη

and rearrange factors

1p

p−1∑

r=0

[ ∞∑

k=−∞

∫ 1

0a(ξ, η)gT

(ξpT

q, η

2π

T

)δ

(η− k− ηo − r

p

)dη

].

We use the periodicity ofa(ξ, η) andgT (ξpT/q, η2π/T )

1p

p−1∑

r=0

[ ∞∑

k=−∞

∫ 1

0a(ξ, η− k)gT

(ξpT

q, [η− k]

2π

T

)δ

(η− k− ηo − r

p

)dη

]

and replace the summation overk together with the integral over the finiteη-interval by an integral over the entireη-axis

1p

p−1∑

r=0

∫ ∞

−∞a(ξ, η)gT

(ξpT

q, η

2π

T

)δ

(η− ηo − r

p

)dη.

Evaluation of the integral and replacingηo by η results in

ϕpT

(ξpT

q, η

2π

T

)=

1p

p−1∑

r=0

a

(ξ, η +

r

p

)gT

(ξpT

q,

[η +

r

p

]2π

T

).

We finally replaceξ by ξ + s and use the periodicity of the Fourier transforma(ξ, η), which leads to the required result, Eq. (13):

ϕpT

([ξ + s]

pT

q, η

2π

T

)=

1p

p−1∑

r=0

a

(ξ, η +

r

p

)gT

([ξ + s]

pT

q,

[η +

r

p

]2π

T

).

22

Appendix B. Derivation of Eq. (14)

In the Fourier transform (8) of the arrayamk, we substitute from the Gabor trans-form (4)

a(ξ, η) =∞∑

m=−∞

∞∑

k=−∞

[∫ ∞

−∞ϕ(t)w∗(t−mT )e−jkΩtdt

]e−j2π(mη− kξ)

and rearrange factors

∞∑m=−∞

[∫ ∞

−∞ϕ(t)w∗(t−mT )

∞∑

k=−∞e−jk(Ωt− 2πξ)

dt

]e−j2πmη.

We replace the sum of exponentials by a sum of Dirac functions

∞∑m=−∞

[∫ ∞

−∞ϕ(t)w∗(t−mT )

2π

Ω

∞∑

k=−∞δ

(t− [ξ + k]

2π

Ω

)dt

]e−j2πmη

and rearrange factors again

2π

Ω

∞∑m=−∞

[ ∞∑

k=−∞

∫ ∞

−∞ϕ(t)w∗(t−mT )δ

(t− [ξ + k]

2π

Ω

)dt

]e−j2πmη.

We evaluate the integral

2π

Ω

∞∑m=−∞

[ ∞∑

k=−∞ϕ

([ξ + k]

2π

Ω

)w∗

([ξ + k]

2π

Ω−mT

)]e−j2πmη

and rearrange factors again

2π

Ω

∞∑

k=−∞ϕ

(ξ2π

Ω+ k

2π

Ω

)e−jk(2π/Ω)η(2π/T )

×[ ∞∑

m=−∞w∗

(ξ2π

Ω−

[m− k

2π

ΩT

]T

)e−j[m− k(2π/ΩT )]Tη(2π/T )

].

We replace2π/ΩT by p/q

2π

Ω

∞∑

k=−∞ϕ

(ξ2π

Ω+ k

2π

Ω

)e−jk(2π/Ω)η(2π/T )

×[ ∞∑

m=−∞w

(ξ2π

Ω−

[m− k

p

q

]T

)ej[m− k(p/q)]Tη(2π/T )

]∗

23

and replace the summation overk by a double summation overs andn through thesubstitutionk = nq + s, wheres extends over an interval of lengthq

2π

Ω

∞∑n=−∞

q−1∑

s=0

ϕ

(ξ2π

Ω+ [nq + s]

2π

Ω

)e−j(nq + s)(2π/Ω)η(2π/T )

×[ ∞∑

m=−∞w

(ξ2π

Ω−

[m− (nq + s)

p

q

]T

)ej[m− (nq + s)(p/q)]Tη(2π/T )

]∗.

We substitutem− np by−k

2π

Ω

∞∑n=−∞

q−1∑

s=0

ϕ

(ξ2π

Ω+

[n +

s

q

]q2π

Ω

)e−j[n + (s/q)]q(2π/Ω)η(2π/T )

×[ ∞∑

k=−∞w

(ξ2π

Ω+

[k + s

p

q

]T

)e−j[k + s(p/q)]Tη(2π/T )

]∗

and rearrange factors, while using the relationpT = q(2π/Ω)

2π

Ω

q−1∑

s=0

∞∑n=−∞

ϕ

([ξ + s]

2π

Ω+ npT

)e−jnpTη(2π/T )

×[ ∞∑

k=−∞w

([ξ + s]

2π

Ω+ kT

)e−jkTη(2π/T )

]∗.

In the last expression we recognize the definitions [cf. Eq. (9)] for the Zak trans-forms ϕpT (ξ2π/Ω, η2π/T ) and wT (x2π/Ω, y2π/T ) of the signalϕ(t) and thewindow functionw(t), respectively, and we can write

a(ξ, η) =2π

Ω

q−1∑

s=0

ϕpT

([ξ + s]

2π

Ω, η

2π

T

)w∗T

([ξ + s]

2π

Ω, η

2π

T

)

or, with 2π/Ω = pT/q,

a(ξ, η) =pT

q

q−1∑

s=0

ϕpT

([ξ + s]

pT

q, η

2π

T

)w∗T

([ξ + s]

pT

q, η

2π

T

).

We finally replaceη by η + r/p and use the periodicity of the Zak transformϕpT (ξpT/q, η2π/T ), which leads to the required result, Eq. (14):

a

(ξ, η +

r

p

)=

pT

q

q−1∑

s=0

ϕpT

([ξ + s]

pT

q, η

2π

T

)w∗T

([ξ + s]

pT

q,

[η +

r

p

]2π

T

).

24

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