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TIME-FREQUENCY ANALYSIS ON FLAT TORI AND GABOR

FRAMES IN FINITE DIMENSIONS

L. D. ABREU, P. BALAZS, N. HOLIGHAUS, F. LUEF, AND M. SPECKBACHER

Abstract. We provide the foundations of a Hilbert space theory for the short-

time Fourier transform (STFT) where the ﬂat tori

T2

N=R2/(Z×NZ) = [0,1] ×[0, N ]

act as phase spaces. We work on an N-dimensional subspace SNof distri-

butions periodic in time and frequency in the dual S0

0(R) of the Feichtinger

algebra S0(R) and equip it with an inner product. To construct the Hilbert

space SNwe apply a suitable double periodization operator to S0(R). On SN,

the STFT is applied as the usual STFT deﬁned on S0

0(R). This STFT is a

continuous extension of the ﬁnite discrete Gabor transform from the lattice

onto the entire ﬂat torus. As such, sampling theorems on ﬂat tori lead to Ga-

bor frames in ﬁnite dimensions. For Gaussian windows, one is lead to spaces

of analytic functions and the construction allows to prove a necessary and suf-

ﬁcient Nyquist rate type result, which is the analogue, for Gabor frames in

ﬁnite dimensions, of a well known result of Lyubarskii and Seip-Wallst´en for

Gabor frames with Gaussian windows.

1. Introduction

The short-time Fourier transform (STFT) is the central instrument of time-

frequency analysis. The most classical setting considers the analysis of functions f

with respect to windows g, both contained in L2(R), deﬁned as

(1) Vgf(x, ξ) = ZR

f(t)g(t−x)e−2πiξt dt =hf, MξTxgi=hf , π(x, ξ)gi,

where Txf(t) = f(t−x), Mξf(t) = e2πiξt f(t), and π(x, ξ) = MξTxdeﬁne the

translation,modulation and time-frequency shift operators, respectively. By inter-

preting the brackets as a duality pairing, this deﬁnition also holds for pairs of test

function and distribution spaces, like the Schwartz space and tempered distribu-

tions S(R),S0(R) [11] and, in particular, the Feichtinger algebra S0(R) and its dual

S0

0(R) [7].

In this paper, we consider the STFT acting on the N-dimensional space SNof

time and frequency periodic distributions in S0

0(R), see deﬁnitions in Section 2 and

[10, Chapter 16.3] or [4, Chapter 6]. This will lead to new phase spaces for the

joint time and frequency values: the ﬂat tori T2

N= [0,1] ×[0, N ], providing, as

2010 Mathematics Subject Classiﬁcation. 42C40, 46E15, 42C30, 46E22, 42C15.

Key words and phrases. short-time Fourier transform, ﬂat torus, ﬁnite Gabor frames, Fe-

ichtinger algebra, sampling theory.

The authors would like to thank Hans Georg Feichtinger for valuable discussions and comments,

and Antti Haimi for his input during the early stages of this work. This research was supported

by the Austrian Science Fund (FWF) through the projects P-31225-N32 (L.D.A.), P 34624 (P.B.)

Y-1199, J-4254 (M.S.), as well as I 3067-N30 (N.H.).

1

2 L. D. ABREU, P. BALAZS, N. HOLIGHAUS, F. LUEF, AND M. SPECKBACHER

we will show, a continuous extension of the coeﬃcient space of the ﬁnite Gabor

transform. The space SNis isometrically isomorphic to CNequipped with the

Euclidean norm and can similarly be obtained by sampling and periodization of

S0(R) [15]. This connection implies that results for the STFT on SNhave impli-

cations for the discrete Gabor transform (DGT) on CNand vice-versa. However,

we will demonstrate that the STFT on the distribution space SN, embedded into

S0

0(R), has much stronger structural properties, similar to those enjoyed by the

STFT on L2(R). As a continuous phase space extension of the DGT, the STFT on

ﬂat tori provides a natural way of deﬁning oﬀ-the-grid values, oﬀering ﬂexibility in

applications and the chance of using continuous variable methods in ﬁnite Gabor

analysis. In the case of Gaussian windows, we obtain spaces of analytic functions.

The resulting possibility of using analytic complex variable tools will allow us to

prove a necessary and suﬃcient Nyquist rate type result for Gabor frames with

Gaussian windows in ﬁnite dimensions, which can be seen as the ﬁnite-dimensional

analogue of the celebrated result of Seip-Wallst´en [22] and Lyubarskii [17] for Ga-

bor frames with Gaussian windows. The suﬃcient condition provides theoretical

support to numerical procedures for increasing grid resolution, due to the principle

of stable reconstruction using frames above the Nyquist rate. As a step in the

proof of the suﬃcient Nyquist rate, we show that the STFT of any signal in SN

with Gaussian window has exactly Nzeros, thereby making precise and proving

the claim in [9].

Our methods are innovative in the sense that they allow to obtain results for ﬁnite

sequences merely as a byproduct of the theory on SN. But it must be noted that

the relation between the continuous STFT and the discrete Gabor transform has

been studied by several authors over the last 30 years, in particular by Janssen [14],

and later by Kaiblinger [15] and Søndergaard [23, 24]. Where the works of Janssen

and Søndergaard are concerned with the construction of discrete Gabor frames

and dual windows from Gabor systems on S0(R), Kaiblinger’s work is concerned

with ﬁnite dimensional approximation of dual windows for Gabor frames on S0(R).

The sampling-periodization duality of the Fourier transform, succintly expressed in

(generalizations of) Poisson’s summation formula and considered in many works,

including [2, 5, 13, 15], is central to these contributions. In essence, the transition

between S0(R) and CNis achieved by studying a composition of periodization op-

erators P(1),P(2) on certain intervals with forward and inverse Fourier transforms

F(1),F−1

(2) as F−1

(2) P(2)F(1) P(1). Here, F(1) is the Fourier transform of L2(R) of a

ﬁnite interval and F−1

(2) is the inverse discrete Fourier transform. From this an-

gle, the central deviation of the present paper from these prior works is that we

consider F(1) and F−1

(2) to be distributional Fourier transforms on S0

0(R), such that

F−1

(2) P(2)F(1) P(1)f,f∈S0(R), yields a doubly periodic distribution in S0

0(R) in-

stead of a ﬁnite sequence in CN, enabling the subsequent application of the STFT

on S0

0(R) instead of the ﬁnite Gabor transform.

2. Overview

We consider functions, distributions, and ﬁnite sequences, denoted by lower case

latin letters f, g, greek letters φ, ψ, and sans font latin letters f,g, respectively.

For the latter, the discrete nature of the domain of f,gis emphasized by using

square brackets for indexing, e.g. f[l]. Operators are denoted by upper case letters

TIME-FREQUENCY ANALYSIS ON FLAT TORI 3

V,Σ. Exceptions from this convention are time-frequency shifts π, the Jacobi theta

function ϑ, and the Fourier transform F, for which we adopt established notation.

With the Gaussian window h0(t) = e−πt2, the Feichtinger algebra S0(R) [6, 12]

is the space

S0(R) := f∈L2(R): Vh0f∈L1(R2),equipped with the norm

kfkS0:= ZR2|Vh0f(x, ξ)|dxdξ =kVh0fkL1(R2).

We deﬁne the space SNas the span of {n}N−1

n=0 , the sequence of periodic delta

trains [10]

(2) n:= X

k∈Z

δn

N+k⊂S0

0(R), n = 0, ..., N −1,

and will show that SNcan be characterized as the image of S0(R) under the double

periodization operator

(3) ΣNf:= X

k1,k2∈Z

MNk2Tk1f=X

k1,k2∈Z

e2πiN k2·f(· − k1).

It can be directly observed that Vg(ΣNf) is quasiperiodic, i.e.

(4) Vg(ΣNf)(x+ 1, ξ) = e−2πiξ Vg(ΣNf)(x, ξ ),

Vg(ΣNf)(x, ξ +N) = Vg(ΣNf)(x, ξ).

Thus, the phase spaces of Vg◦ΣNare the ﬂat tori T2

N= [0,1] ×[0, N ]. As we

will see, Vg◦ΣN:S0(R)→L2(T2

N) and Vg:SN→L2(T2

N) have the same

range in phase-space. It will often be convenient to jump from one to the other

representation to simplify proofs.

The STFT on SNnaturally introduces the compact phase space T2

Nfor time-

frequency analysis on ﬁnite, N-dimensional Hilbert spaces. Thereby, it provides a

continuous model that, by construction, eliminates the truncation, or alternatively

aliasing, errors usually associated with the transition from the STFT on L2(R) to

numerical implementations by means of the ﬁnite Gabor transform. That is not to

say that these errors are removed: They are instead separated from the continuous

model to the double periodization operator ΣN, i.e., the mapping from S0(R) onto

SN⊂S0

0(R).

As discussed in [14, 15, 24] in a slightly diﬀerent formal framework, the compo-

sition Vg◦ΣNrelates to the ﬁnite Gabor transform on CN, deﬁned as

Vgf[k, l] =

N−1

X

m=0

f[m]g[m−k]e−2πilm

N,f,g∈CN,

We will show that Vgmaps SNinto L2(T2

N) and that Vgcan be viewed as a

continuous extension of Vg:CN→CN×Nto T2

Nin the sense of the following

result.

Theorem 1. Let f, g ∈S0(R)and let fN=PNf , gN=PNg, with the periodization

operator

PN:S0(R)→CN,deﬁned by f7→ X

j∈Z

f(n/N −j)N−1

n=0 .

4 L. D. ABREU, P. BALAZS, N. HOLIGHAUS, F. LUEF, AND M. SPECKBACHER

Then, for k, l ∈0, ...., N −1,

Vg(ΣNf)k

N, l=N−1·VgNfN[k, l].

Speciﬁcally, the ﬁnite discrete STFT can be obtained by sampling the phase space

T2

Nof the continuous STFT restricted to SNon the grid points (k/N, l), providing

a direct link between continuous and ﬁnite discrete time-frequency analysis.

Remark 2. We chose the periods (1, N )in equation (3) for notational convenience.

Any other pair (c, d)∈R2

+, with cd =N, leads to equivalent results on the phase

space f

T2

N= [0, c]×[0, d]. When studying the approximation of the STFT by ﬁnite

Gabor transforms, as in [15], it is usually more convenient to consider the sym-

metric convention (c, d)=(√N , √N), such that an increase in Nsymmetrically

expands the considered phase space area and the sampling density within.

We will study the Hilbert space properties of the map

Vg:SN→L2(T2

N),

and derive the Moyal-type orthogonality relation

ZT2

N

Vg1ϕ1(x, ξ)Vg2ϕ2(x, ξ)dxdξ =Nhϕ1, ϕ2iSNhg2, g1iL2,

as well as inversion and reproducing formulas similar to those of the continuous

STFT.

For the STFT with dilated Gaussian windows hλ

0(t) = e−πλt2we will obtain a

sampling theorem on the torus which leads to a full description of the frame set for

ﬁnite Gabor frames with Gaussian windows in CN. The proof uses a Bargmann-

type transform, (which up to a weight is the STFT with hλ

0) whose action on

the space SNhas previously been considered in a slightly diﬀerent form in [16].

Finally, combining the sampling theorem on the torus with Theorem 1, we are lead

to the following full description of the frame set for ﬁnite Gabor expansions using a

periodized, dilated Gaussian window. As far as we could check, this is a completely

new result.

Theorem 3. Let λ > 0,hλ

N=PNhλ

0, and {(jk, lk)}k=1,...,K be a collection of

distinct pairs of integers jk, lk∈0, ...., N −1. The following are equivalent:

1). The set {(jk, lk)}k=1,...,K gives rise to a ﬁnite Gabor frame with window

hλ

N, i.e., there are constants A, B > 0such that, for every f∈CN,

Akfk2

CN≤

K

X

k=1 Vhλ

Nf[jk, lk]

2≤Bkfk2

CN.

2). One of the three following conditions is satisﬁed:

(i) N2≥K≥N+ 1,

(ii) K=Nis odd,

(iii) K=Nis even and PN

k=1(jk, lk)/∈NN2.

We emphasize that this result has been possible to prove only thanks to our

Hilbert space theory for the STFT on ﬂat tori, and that it strongly depends on

the use of complex variable methods for almost periodic analytic functions. This

reinforces the suggestion that time-frequency analysis on the torus provides a rich

TIME-FREQUENCY ANALYSIS ON FLAT TORI 5

theory which encompasses the theory of ﬁnite Gabor frames and leads to new

insights, potential in applications and proof of results which were out of reach

without the toric phase space.

The paper is organized as follows. Some required properties of the Hilbert space

SNand the operator ΣNare presented in Section 3. Section 4 contains the proof

of Theorem 1 above, explicit computations with dilated Gaussian windows hλ

0, and

derivations of the Moyal-type formula, together with the inversion and reproducing

kernel formulas. In the last section, the window is specialized to be the Gaussian.

The resulting Bargmann-type transform is deﬁned, and several properties of its

range space of entire functions with periodic constraints (a toric analogue of the

Fock space) are studied in detail. All these properties are then used in the proof

of the main result of the section: the sampling theorem on the torus. Finally,

combining this result with Theorem 1, we derive a full characterization of ﬁnite

Gabor frames with periodized and sampled Gaussian windows.

3. Properties of SNand ΣN

3.1. The Hilbert space SNof time-frequency periodic distributions. The

space SNappears in theoretical physics in coherent state approaches [4, 10]. By

deﬁnition of SNit is clear that the family {n}N−1

n=0 deﬁned in (2) forms a basis.

Therefore, expanding ϕ, ψ ∈SNwith respect to this basis

ϕ=

N−1

X

n=0

ann, ψ =

N−1

X

n=0

bnn,

we can deﬁne an inner product on SNby

hϕ, ψiSN=

N−1

X

n=0

anbn.

Clearly, SNcan be identiﬁed with CNequipped with the standard inner product,

and {n}N−1

n=0 forms an orthonormal basis of SNas

hn, miSN=δn,m.

Note that for every ϕ∈SN

(5) T1ϕ=ϕ, and MNϕ=ϕ.

Therefore, SNis a space of distributions that are periodic in time and frequency.

Actually, SNcontains all distributions in S0

0(R) that satisfy (5), see [10, page 262,

(16.12)].

3.2. The double periodization operator ΣN.We formally deﬁne the double

periodization operator as

f7→ ΣNf=X

k1,k2∈Z

MNk2Tk1f .

The next lemma shows when and in which sense this object is well-deﬁned.

Lemma 4. The operator ΣNis well-deﬁned from S0(R)into S0

0(R)with uncondi-

tional weak-∗convergence in S0

0(R).

6 L. D. ABREU, P. BALAZS, N. HOLIGHAUS, F. LUEF, AND M. SPECKBACHER

Proof: If f, g ∈S0(R), then

|Vg(ΣNf)(x, ξ)| ≤ X

k1,k2∈Z|e−2πik1ξVgf(x−k1, ξ −Nk2)|

≤X

k1,k2∈Z|Vg(T−xM−ξf)(k1, N k2)|

≤CkT−xM−ξfkS0kgkS0

≤CkfkS0kgkS0≤ ∞,

by [11, Lemma 3.1.3 and Corollary 12.1.12]. This implies that ΣNf∈S0

0(R) is

well-deﬁned. If we choose x=ξ= 0, then absolute weak-∗convergence of the

series in S0

0(R) follows which in turn implies unconditional convergence.

As ΣNfis periodic in time and frequency, we can expand it with respect to the

orthonormal basis {n}N−1

n=0 . In the next lemma, this expansion is obtained explic-

itly. We will also show that ΣNis surjective as a mapping from S0(R) to SN. To

do so, we need to deﬁne the periodization operator Pf(t) = Pk∈Zf(t−k).

Lemma 5. For every f∈S0(R)

(6) ΣNf=1

NPf·X

k∈Z

δk

N=1

N

N−1

X

n=0

Pfn

Nn∈SN,

and

(7) hΣNf, giS0

0×S0=1

N

N−1

X

n=0

Pfn

NPgn

N=N· hΣNf, ΣNgiSN.

Moreover, ΣN:S0(R)→SNis surjective.

Proof: Let f, g ∈S0(R). Since the Poisson summation formula holds for functions

in S0(R) (see e.g. [11, Corollary 12.1.5]), we have that the following equality holds

in the distributional sense

X

k∈Z

e2πiktN =1

NX

k∈Z

δk

N(t)∈SN.

This shows that

ΣNf=X

k∈Z

MNk X

l∈Z

Tlf=X

k∈Z

MNk Pf=1

NX

k∈Z

δk

NPf

with unconditional weak-∗convergence in S0

0(R). Hence,

hΣNf, giS0

0×S0=1

N*X

k∈Z

δk

NPf, g+S0

0×S0

.

Let us write

X

k∈Z

δk

N=

N−1

X

n=0 X

k∈Z

δn

N+k=

N−1

X

n=0

n

TIME-FREQUENCY ANALYSIS ON FLAT TORI 7

where the change of summation order is justiﬁed by e.g. [11, Corollary 12.1.5].

Using the periodicity of Pfthen yields

hΣNf, giS0

0×S0=1

N

N−1

X

n=0 *X

k∈Z

δn

N+kPf, g+S0

0×S0

=1

N

N−1

X

n=0 X

k∈Z

Pfn

N+kgn

N+k

=1

N

N−1

X

n=0

Pfn

NX

k∈Z

gn

N+k

=1

N

N−1

X

n=0

Pfn

Nhn, giS0

0×S0.

Hence, (6) holds. The ﬁrst equality of (7) follows from the second to last equality

above. Finally, the second equality of (7) results from combining (6) and the ﬁrst

equality of (7).

It thus remains to show that Σn:S0(R)→SNis surjective. By (6) it suﬃces

to show that there exists a family of function fn∈S0(R), n= 0, . . . , N −1, sat-

isfying Pfnk

N=δn(k), k = 0, . . . , N −1. Such functions obviously exists. Take

for instance fn(t) := sinc(Nt−n)·e−π(t−n/N)2which is even a Schwartz function.

4. Time-frequency analysis on flat tori

4.1. Basic properties of Vgon SN.The STFT deﬁned on SNis, as we we

subsequently show, closely connected to the Zak transform which is deﬁned as

Zf(x, ξ) = X

k∈Z

f(x−k)e2πikξ .

For later reference we state here some elementary facts about the Zak transform

(see e.g. [11]):

Quasiperiodicity:

(8) Zf(x, ξ +k) = Zf(x, ξ),and Zf(x+k, ξ) = e2πikξZ(x, ξ),

Action on time-frequency shifts:

(9) Z(MωTyf)(x, ξ) = e2πiω xZf(x−y, ξ −ω),

Unitarity: for f1, f2∈L2(R) it holds

(10) Z1

0Z1

0

Zf1(x, ξ)Zf2(x, ξ)dxdξ =hf1, f2iL2.

Lemma 6. Let f, g ∈S0(R)and ϕ∈SN. Then

(11) Vg(ΣNf)(x, ξ) =

N−1

X

n=0

Pfn

Ne−2πiξ n

NZgn

N−x, ξ,

8 L. D. ABREU, P. BALAZS, N. HOLIGHAUS, F. LUEF, AND M. SPECKBACHER

and for ϕ=PN

n=1 ann

(12) Vgϕ(x, ξ) =

N−1

X

n=0

ane−2πiξ n

NZgn

N−x, ξ.

Proof: First, let us compute the STFT of the basis functions n

Vgn(x, ξ) = hn,MξTxgiS0

0×S0=X

k∈Zδn

N+k,MξTxgS0

0×S0

=X

k∈Z

gn

N+k−xe−2πiξ(n

N+k)=e−2πiξ n

NZgn

N−x, ξ,(13)

For general ϕ=PN−1

n=0 ann∈SNone thus gets (12). Applying Vgto (6) from

Lemma 5 gives

Vg(ΣNf)(x, ξ) = 1

N

N−1

X

n=0

Pfn

NVgn(x, ξ)

which combined with (13) yields (11).

With these basic observations, it is now straightforward to show Theorem 1. For

convenience, we repeat the statement here.

Theorem 1. Let f, g ∈S0(R)and let fN=PNf, gN=PNg. Then, for l, k ∈

0, ...., N −1,

Vg(ΣNf)k

N, l=N−1·VgNfN[k, l].

Proof: Setting ξ=l∈0, ...., N −1 and x=k

N, k ∈0, . . . , N −1,yields

Vg(ΣNf)k

N, l=1

N

N−1

X

n=0

Pfn

Ne−2πi nl

NZgn−k

N, l

=1

N

N−1

X

n=0

Pfn

Ne−2πi nl

NPgn−k

N

This shows that by deﬁnition of the ﬁnite Gabor transform

Vg(ΣNf)k

N, l=1

NVgNfN[k, l].

4.2. Moyal-type and inversion formulas for the STFT on ﬂat tori. One

of the most fundamental identities in time-frequency analysis is Moyal’s formula

ensuring that the STFT is an isometry from L2(R) to L2(R2)

ZR2

Vg1f1(x, ξ)Vg2f2(x, ξ)dxdξ =hf1, f2iL2hg2, g1iL2, f1, f2, g1, g2∈L2(R).

We will now show the toric equivalent of this identity ensuring that the STFT on

the ﬂat tori is a multiple of an isometry.

TIME-FREQUENCY ANALYSIS ON FLAT TORI 9

Theorem 7. If ϕ1, ϕ2∈SN, and g1, g2∈S0(R), then

(14) ZT2

N

Vg1ϕ1(x, ξ)Vg2ϕ2(x, ξ)dxdξ =Nhϕ1, ϕ2iSNhg2, g1iL2.

Proof: For ϕ1=PN−1

n=0 ann, and ϕ2=PN−1

n=0 bnn, we use Lemma 6 and the

periodicity of the Zak transform in the frequency variable (8) to obtain

ZT2

N

Vg1ϕ1(x, ξ)Vg2ϕ2(x, ξ)dxdξ

=ZN

0Z1

0

N−1

X

n,k=0

anbke−2πiξ n−k

NZg1n

N−x, ξZg2k

N−x, ξdxdξ

=

N−1

X

l=0 Z1

0Z1

0

N−1

X

n,k=0

anbke−2πi(ξ+l)n−k

N·

Zg1n

N−x, ξ +lZg2k

N−x, ξ +ldxdξ

=Z1

0Z1

0

N−1

X

n,k=0

anbk N−1

X

l=0

e−2πil n−k

N!e−2πiξ n−k

N·

Zg1n

N−x, ξZg2k

N−x, ξdxdξ

=N

N−1

X

n=0

anbnZ1

0Z1

0

Zg1n

N−x, ξZg2n

N−x, ξdxdξ,

where we used the basic fact PN−1

l=0 e−2πil k

N=Nδk,0to obtain the ﬁnal equality.

Applying consecutively (9), (8) and (10) to the integral above yields

Z1

0Z1

0

Zg1n

N−x, ξZg2n

N−x, ξdxdξ

=Z1

0Z1

0

Z(T−n

Ng1) (−x, ξ)Z(T−n

Ng2) (−x, ξ)dxdξ

=Z1

0Z1

0

e−2πiξ Z(T−n

Ng1) (1 −x, ξ )e−2πiξZ(T−n

Ng2) (1 −x, ξ )dxdξ

=Z1

0Z1

0

Z(T−n

Ng1) (x, ξ)Z(T−n

Ng2) (x, ξ)dxdξ

=hT−n

Ng1,T−n

Ng2iL2=hg2, g1iL2,

which concludes the proof.

Remark 8. Using (7) we thus have shown that ΣN(π(x, ξ)g)(x,ξ)∈T2

N

is a con-

tinuous tight frame for SN.

From Theorem 7 and (7) we can now derive two inversion formulas.

10 L. D. ABREU, P. BALAZS, N. HOLIGHAUS, F. LUEF, AND M. SPECKBACHER

Theorem 9. Let g∈S0(R)\{0}. For every ϕ∈SNand every f∈S0(R), it holds

(15) ϕ=1

Nkgk2

2

N−1

X

n=0 ZT2

N

Vgϕ(x, ξ)e2πiξ n

NZgn

N−x, −ξdxdξ!n,

and

(16) ΣNf=1

kgk2

2ZT2

N

Vg(ΣNf)(x, ξ)ΣNπ(x, ξ)gdxdξ,

where the integral is understood in the weak-∗sense.

Proof: Let ϕ=PN−1

n=0 ann. Then by Theorem 7 we know

an=hϕ, niSN=1

Nkgk2

2ZT2

N

Vgϕ(x, ξ)Vgn(x, ξ)dxdξ

=1

Nkgk2

2ZT2

N

Vgϕ(x, ξ)e2πiξ n

NZgn

N−x, ξdxdξ

=1

Nkgk2

2ZT2

N

Vgϕ(x, ξ)e2πiξ n

NZgn

N−x, −ξdxdξ,

where we used (12) and the fact that Zg(x, ξ) = Zg(x, −ξ). To show the second

identity we ﬁrst observe that ΣNsatisﬁes

hΣNf, giS0

0×S0=hf, ΣNgiS0×S0

0, f, g ∈S0(R).

Let f, h ∈ S0(R). The result thus follows from (7) and (14) as

hΣNf, hiS0

0×S0=1

kgk2

2ZTN2

Vg(ΣNf)(x, ξ)hπ(x, ξ)g, ΣNhiS0×S0

0dxdξ

=1

kgk2

2ZTN2

Vg(ΣNf)(x, ξ)hΣNπ(x, ξ)g, hiS0

0×S0dxdξ.

Remark 10. Note that coorbit theory [7, 8] also guarantees an inversion of the

short-time Fourier transform on S0

0(R)as V∗

gVg=kgk2IS0

0. The beneﬁt of our

point of view is however that the coeﬃcients in (15) can directly be calculated without

resorting to weakly deﬁned integrals.

4.3. Reproducing kernel. We now prove that the range of the STFT restricted

to SNis an N-dimensional reproducing kernel Hilbert space (RKHS) of L2(T2

N)

and give several expressions for its reproducing kernel.

Proposition 11. The space Vg(SN)⊂L2(T2

N)is a RKHS and its kernel is given

by

Kg(x0, ξ0),(x, ξ)=1

Nkgk2

2

N−1

X

n=0

e2πi(ξ−ξ0)n

NZgn

N−x, ξZgn

N−x0, ξ0

=N

kgk2

2hΣNπ(x, ξ)g,ΣNπ(x0, ξ0)giSN

(17)

=1

kgk2

2X

k1,k2∈Z

e−2πik1ξhπ(x+k1, ξ +N k2)g, π(x0, ξ0)giL2.

TIME-FREQUENCY ANALYSIS ON FLAT TORI 11

Proof: Using consecutively (12), Cauchy-Schwarz inequality, [11, Lemma 8.2.1]

and (14) yields

|Vgϕ(x, ξ)|≤kϕkSN N−1

X

n=0 Zgn

N−x, ξ

2!1/2

≤ kϕkSN√NkZgk∞=kZgk∞

kgk2kVgϕkL2(TN),

meaning that point evaluation is continuous and Vg(SN) is a RKHS.

By Theorem 9 and Lemma 6, it follows

Vgϕ(x0, ξ0) = 1

Nkgk2

2

N−1

X

n=0 ZT2

N

Vgϕ(x, ξ)e2πiξ n

NZgn

N−x, −ξVgn(x0, ξ0)dxdξ

=ZT2

N

Vgϕ(x, ξ)1

Nkgk2

2

N−1

X

n=0

e2πi(ξ−ξ0)n

NZgn

N−x, ξZgn

N−x0, ξ0dxdξ,

as well as by (16) and (7)

Vg(ΣNf)(x0, ξ0)

=1

kgk2

2ZT2

N

VgΣNf(x, ξ)hΣNπ(x, ξ)g, π(x0, ξ0)giS0

0(R)×S(R)dxdξ

=N

kgk2

2ZT2

N

VgΣNf(x, ξ)hΣNπ(x, ξ)g,ΣNπ(x0, ξ0)giSNdxdξ.

Therefore, the ﬁrst two identities of (17) hold. Moreover,

Kg((x0, ξ0),(x, ξ)) = N

kgk2

2hΣN(π(x, ξ)g),ΣN(π(x0, ξ0)g)iSN

=1

kgk2

2hΣN(π(x, ξ)g), π(x0, ξ0)giS0

0×S0

=1

kgk2

2X

k1,k2∈Z

e−2πik1ξhπ(x+k1, ξ +N k2)g, π(x0, ξ0)giL2.

4.4. Examples of the STFT on ﬂat tori. In this section, we consider ex-

plicit calculations of the objects discussed in the previous sections using the non-

normalized dilated Gaussian

hλ

0(t) := e−πλt2, λ > 0.

The reason we introduce the additional dilation is that, having the connection to

ﬁnite Gabor systems (Theorem 1) in mind, we would like to impose a certain degree

of localization of Phλ

0. This is only guaranteed if λ > 0 is chosen large enough, see

Figure 1.

Following (12), the STFT of a basis function nis given by

Vhλ

0n(x, ξ) = e−2πi n

NξZhλ

0n

N−x, ξ

=e−2πi n

NξX

k∈Z

e−πλ(x−n

N+k)2e2πiξk

12 L. D. ABREU, P. BALAZS, N. HOLIGHAUS, F. LUEF, AND M. SPECKBACHER

Figure 1. Phλ

0, for λ= 1, (left) λ= 16 (middle), and λ= 64 (right).

=e−πλ(x−n

N)2−2πiξ n

NX

k∈Z

e−πλk2e2πik(ξ+iλ(x−n

N))

=e−πλx2e−πλ(n

N)2

e2πzλn

Nϑizλ−λn

N, iλ,(18)

where zλ:= λx +iξ and the Jacobi theta function ϑis deﬁned as

ϑ(z, τ ) := X

k∈Z

eπik2τ+2πikz,Im(τ)>0.

Moreover, we explicitly calculate the reproducing kernel Khλ

0,T2

N. For this purpose,

the third equation of (17) is the most convenient representation of the kernel. Let

us write wλ=λx0+iξ0. By [11, Lemma 1.5.2]

hπ(x, ξ)hλ

0, π(x0, ξ0)hλ

0i= (2λ)−1

2eπi(ξ−ξ0)(x+x0)e−π

2[λ(x−x0)2+1

λ(ξ−ξ0)2].

from which we deduce

Khλ

0(wλ, zλ) = 1

khλ

0k2X

k1,k2∈Z

e−2πik1ξhπ(x+k1, ξ +N k2)hλ

0, π(x0, ξ0)hλ

0i

=eπi(ξ−ξ0)(x+x0)X

k1,k2∈Z

eπ

2[−2i(ξ+ξ0)k1+2i(x+x0)Nk2−λ(x−x0+k1)2−1

λ(ξ−ξ0+Nk2)2]

=eπ[i(ξ−ξ0)(x+x0)−λ

2(x−x0)2−1

2λ(ξ−ξ0)2]X

k1∈Z

e−πk1[λ(x−x0)+i(ξ+ξ0)]−πλk2

1

2

X

k2∈Z

eπi N k2

λ[λ(x+x0)+i(ξ−ξ0)]−πN2k2

2

2λ

=e−πλ(x2+(x0)2)eπ

2λ(zλ+wλ)2ϑizλ−wλ

2,iλ

2ϑ(zλ+wλ)N

2λ,iN2

2λ.

5. A Bargmann-type transform and finite Gabor Frames

Given ϕ=PN−1

n=0 ann∈SNwe can deﬁne a Bargmann-type transform by

B(λ,N)ϕ(z) = Vhλ

0ϕx

λ,−ξeπx2/λ,z=x+iξ, or equivalently using (18)

B(λ,N)ϕ(z) =

N−1

X

n=0

ane−πλ(n

N)2

e2πz n

Nϑiz−λn

N, iλ.

TIME-FREQUENCY ANALYSIS ON FLAT TORI 13

By the quasi-periodicity of Vhλ

0ϕ(4), we then get that

(19) B(λ,N)ϕ(z+λn) = eπλn2+2πz nB(λ,N )ϕ(z), n ∈Z,

as well as

(20) B(λ,N)ϕ(z+iN m) = B(λ,N)ϕ(z), m ∈Z.

This can also be derived using the quasi-periodicity of the Jacobi theta function

(21) ϑ(z+n+τ m, τ) = e−πiτm2e−2π imzϑ(z , τ), n, m ∈Z.

We will now show that the range of the Bargmann-type transform B(λ,N)is precisely

the space of analytic functions satisfying (19) and (20). This follows from the

following result.

Lemma 12. The space of entire functions satisfying the periodicity conditions (19)

and (20) is N-dimensional. Thus, any such function Fcan be written as the linear

combination of Northogonal functions

F(z) =

N−1

X

n=0

ane−πλ(n

N)2

e2πz n

Nϑiz−λn

N, iλ

and F(z) = B(λ,N)ϕ(z), for some ϕ=PN−1

n=0 ann.

Proof: Let Fbe analytic and satisfy (19) and (20). Since Fis N-periodic with

respect to purely imaginary shifts, we can write Fas a Fourier series

F(z) = X

k∈Z

cke2πkz/N .

Plugging this expression into (19) yields

X

k∈Z

cke2πk(z+λn)/N =eπλn2+2πzn X

k∈Z

cke2πkz/N =X

k∈Z

ckeπλn2e2π(k+nN )z/N

=X

k∈Z

ck−nN eπλn2e2πkz /N,

which shows that the coeﬃcients cksatisfy

(22) ck+nN =cke−πλ(n2+2kn/N), k ∈ {0, ..., N −1}, n ∈Z.

There are therefore exactly Ncoeﬃcients c0, ..., cN−1that can be chosen freely and

the other coeﬃcients are determined by (22). The orthogonality of the functions

e−πλ(n

N)2

e2πz n

Nϑiz−λn

N, iλ=Vhλ

0nx

λ,−ξeπx2/λ

follows from hn, miSN=δn,m and Moyal’s formula (14).

The periodicity conditions (19) and (20) will also allow us to count the number

of zeros of B(λ,N)ϕin the torus T2

λ,N := R2/(λZ×NZ), which coincides with the

number or zeros of Vhλ

0ϕin T2

N, and to obtain a constraint these zeros must satisfy.

14 L. D. ABREU, P. BALAZS, N. HOLIGHAUS, F. LUEF, AND M. SPECKBACHER

Proposition 13. Let ϕ∈SN\{0}. The function B(λ,N )ϕhas exactly Nze-

ros (counted with their multiplicities) on the torus T2

λ,N . Moreover, the Nzeros

z1, ..., zN∈T2

λ,N satisfy

N

X

k=1

zk=Nλ

2+λn +iN2

2+Nm,for some n, m ∈Z.(23)

Proof: Let the curve Γ = ∂T2

λ,N be positively oriented and assume for now that

Γ contains no zero of B(λ,N)ϕ. As B(λ,N)ϕis an analytic function on C, it follows

by Cauchy’s argument principle (see, e.g. [1, Sec. 5.2]) that the number of zeros of

B(λ,N)ϕ(counted with their multiplicities) is given by

1

2πi ZΓ

(B(λ,N)ϕ)0

B(λ,N)ϕdγ =1

2πi "Zλ

0

(B(λ,N)ϕ)0(t)

B(λ,N)ϕ(t)dt +iZN

0

(B(λ,N)ϕ)0(λ+it)

B(λ,N)ϕ(λ+it)dt

−Zλ

0

(B(λ,N)ϕ)0(t+iN )

B(λ,N)ϕ(t+iN )dt −iZN

0

(B(λ,N)ϕ)0(it)

B(λ,N)ϕ(it)dt#

=1

2π"ZN

0

(B(λ,N)ϕ)0(λ+it)

B(λ,N)ϕ(λ+it)dt −ZN

0

(B(λ,N)ϕ)0(it)

B(λ,N)ϕ(it)dt#,

where we used (20). By (19) we have

(B(λ,N)ϕ)0(z+λ)=2πeπ λ+2πz B(λ,N)ϕ(z) + eπ λ+2πz (B(λ,N)ϕ)0(z),

and consequently

1

2πi ZΓ

(B(λ,N)ϕ)0

B(λ,N)ϕdγ =1

2π"ZN

0

eπλ+2πit 2πB(λ,N)ϕ(it)+(B(λ,N)ϕ)0(it)

eπλ+2πit B(λ,N)ϕ(it)dt

−ZN

0

(B(λ,N)ϕ)0(it)

B(λ,N)ϕ(it)dt#=1

2πZN

0

2πdt =N.

Let z1, ..., zNbe the Nzeros of B(λ,N )ϕ. Then, using again (19), (20) and Cauchy’s

argument principle (see, e.g. [1, Sec. 5.2, (49)]) , we get with id: C→C, z 7→ z

N

X

k=1

zk=1

2πi ZΓ

id ·(B(λ,N)ϕ)0

B(λ,N)ϕdγ

=1

2πi "Zλ

0

t(B(λ,N)ϕ)0(t)

B(λ,N)ϕ(t)dt +iZN

0

(λ+it)(B(λ,N)ϕ)0(λ+it)

B(λ,N)ϕ(λ+it)dt

−Zλ

0

(t+iN)(B(λ,N)ϕ)0(t+iN)

B(λ,N)ϕ(t+iN )dt −iZN

0

it(B(λ,N)ϕ)0(it)

B(λ,N)ϕ(it)dt#

=1

2π"−NZλ

0

(B(λ,N)ϕ)0(t)

B(λ,N)ϕ(t)dt −ZN

0

it(B(λ,N)ϕ)0(it)

B(λ,N)ϕ(it)dt

+ZN

0

(λ+it)(B(λ,N)ϕ)0(λ+it)

B(λ,N)ϕ(λ+it)dt#

=1

2π"−NZλ

0

(B(λ,N)ϕ)0(t)

B(λ,N)ϕ(t)dt −ZN

0

it(B(λ,N)ϕ)0(it)

B(λ,N)ϕ(it)dt

TIME-FREQUENCY ANALYSIS ON FLAT TORI 15

+ZN

0

(λ+it)2πB(λ,N)ϕ(it)+(B(λ,N)ϕ)0(it)

B(λ,N)ϕ(it)dt#

=1

2π"λZN

0

(B(λ,N)ϕ)0(it)

B(λ,N)ϕ(it)dt −NZλ

0

(B(λ,N)ϕ)0(t)

B(λ,N)ϕ(t)dt + 2πZN

0

(λ+it)dt#.

Let us now think of t7→ B(λ,N)ϕ(t) as a parametrization of the curve Γ1=

{B(λ,N)ϕ(t)}t∈[0,λ], i.e.

Zλ

0

(B(λ,N)ϕ)0(t)

B(λ,N)ϕ(t)dt =ZΓ1

1

zdγ.

Let Γ2=B(λ,N)ϕ(0)(eπ λ(1 −t) + t)t∈[0,1] be the line segment that connects the

endpoints of Γ1. Therefore, Γ0= Γ1∪Γ2is a closed continuous curve. By the

residue theorem, it follows that

ZΓ0

1

zdγ = 2πik,

for some k∈Z. Therefore

Zλ

0

(B(λ,N)ϕ)0(t)

B(λ,N)ϕ(t)dt =−ZΓ2

1

zdγ + 2πik =Z1

0

eπλ −1

eπλ −t(eπλ −1) dt + 2πik

=Zeπλ−1

0

1

eπλ −tdt + 2πik =Zeπ λ

1

1

tdt + 2πik =πλ + 2πik.

Similarly, Γ3={B(λ,N )ϕ(it)}t∈[0,N]is a closed and continuous curve. Let n∈Z

be the winding number of Γ3around the origin. Then

ZN

0

(B(λ,N)ϕ)0(it)

B(λ,N)ϕ(it)dt =−iZΓ3

1

zdγ = 2πn.

Thus,

N

X

k=1

zk=1

2π2πλn −πλN −2πiNk + 2πλN +πiN2

=Nλ

2+λn +iN2

2+Nk, k, n ∈Z.

Finally, if Γ contains at least one zero of B(λ,N)ϕ, then there exists z∗∈Csuch

that z∗+ Γ does not contain any zero as every nonzero analytic function can only

have ﬁnitely many zeros on any compact set. The previous arguments can then

be repeated for the curve z∗+ Γ yielding the same number of zeros which are con-

strained by the same condition.

The next result is a full characterization of frames obtained from the STFT on SN

with Gaussian windows via sampling points in T2

N.

Theorem 14. Let λ > 0, and z1, ..., zK∈T2

Nbe a collection of Kdistinct points.

The following are equivalent:

16 L. D. ABREU, P. BALAZS, N. HOLIGHAUS, F. LUEF, AND M. SPECKBACHER

1). The points z1, ..., zK∈T2

Ngive rise to a frame for SN, i.e., there are

constants A, B > 0such that, for every ϕ∈SN,

Akϕk2

SN≤

K

X

k=1 |Vhλ

0ϕ(zk)|2≤Bkϕk2

SN.

2). One of the following two conditions is satisﬁed:

(i) K≥N+ 1,

(ii) K=Nand 1

NPN

k=1 zk6= (1/2 + n/N, N/2 + m),for every n, m ∈Z.

Proof: If the family is a frame for SN, then it has to include at least Nvectors,

i.e., K≥N.

Let K≥N+ 1. It follows from Proposition 13 and the relation between B(λ,N)

and Vhλ

0that Vhλ

0ϕhas at most Ndistinct zeros. Consequently, PK

k=1 |Vhλ

0ϕ(zk)|2

is always positive. Moreover, this expression depends continuously on the basis

coeﬃcients anof ϕby (12). Therefore, as the unit sphere in CNis compact, it

follows that

A= inf

kϕk2

SN=1

K

X

k=1 |Vhλ

0ϕ(zk)|2>0.

Now, let K=N, and suppose we are given a collection of Ndistinct points

z1, ..., zN∈T2

N. Moreover, we set zλ,k =λxk−iξk∈T2

λ,N ,k= 1, ..., N . Let

z0=λ/2 + iN/2 be the single zero of ϑ(iz /N, iλ/N) in [0, λ]×i[0, N] (see [19,

20.2(iv)]) and deﬁne the function F(z) as

F(z) =

N

Y

k=1

e2πRe(zλ,k−z0)z /λN ϑi(z−zλ,k +z0)/N, iλ/N .

Using (21) one can directly show that Fsatisﬁes the periodicity conditions

F(z+λn) =

N

Y

k=1

e2πRe(zλ,k−z0)(z+λn)/λN ϑi(z−zλ,k +z0)/N +iλn/N, iλ/N

=eπλn2F(z)

N

Y

k=1

e2πRe(zλ,k−z0)n/N e2πn(z−zλ,k +z0)/N

=eπλn2e2πnz e−2πinImN z0−PN

k=1 zλ,k/N F(z),

and

F(z+iNm) =

N

Y

k=1

e2πRe(zλ,k−z0)(z+N m)/λN ϑi(z−zλ,k +z0)/N −m, iλ/N

=e−2πimReN z0−PN

k=1 zλ,k/λ F(z).

Therefore, if PN

k=1 zλ,k =Nz0+λm +iN n =λN/2 +λm +i(N2/2 +N n),for some

n, m ∈Z,it follows that Fsatisﬁes (19) and (20) and is analytic in T2

λ,N . Thus,

by Lemma 12 there exists ϕ∈SNsuch that Vhλ

0ϕx

λ,−ξ=F(z)e−πx2/λ and, by

construction, Vhλ

0ϕ(z) = 0, for z=zλ,1, ..., zλ,K . Consequently,

K

X

k=1 |Vhλ

0ϕ(zλ,k)|2= 0,

TIME-FREQUENCY ANALYSIS ON FLAT TORI 17

and the lower frame bound is violated.

If PN

k=1 zλ,k 6=Nz0+λm +Nn, for every n, m ∈Z,it follows by Proposition 13

that {zλ,1, ..., zλ,N }is a uniqueness set for the space of entire functions satisfying

the periodicity conditions (19) and (20). It thus follows again by compactness of

the unit ball in SN, that the points z1, ..., zNgenerate a frame for SN

Theorem 14 implies that if one chooses Npoints in T2

Nuniformly at random, then

one obtains a frame with probability 1. While the frame set for the Gaussian

window STFT on L2(R) is known from [22] and [17], it seems to be a folklore

result, backed up by substantial experience through numerical computations, that

the ﬁnite Gabor transform with sampled, periodized Gaussian yields a frame when

sampled on any lattice within ZN×ZNwith cardinality larger than N, but we

found no proof in the literature. On the other hand, a result by Søndergaard [23],

adapted from a remarkable observation by Benedetto et al [3] for the STFT on

`2(Z), demonstrates that half-point shifts of sampled, periodized Gaussians yield

frames on lattices of cardinality N. The result below, a direct consequence of

Theorem 14, demonstrates that, in fact, any K > N distinct points from the set

{(j, k) : j, k ∈0, ..., N }yield a ﬁnite Gabor frame for CNwith the sampled,

periodized Gaussian window hλ

N:= PNhλ

0. For K=N, additional arithmetic

conditions ensure the frame property. This is the result for Gabor frames in ﬁnite

dimensions stated in the introduction as one of the main achievements of our theory.

We state it again for convenience.

Theorem 3. Let λ > 0, and {(jk, lk)}k=1,...,K a collection of distinct pairs of

integers jk, lk∈0, ...., N −1. The following are equivalent:

1). The set {(jk, lk)}k=1,...,K gives rise to a ﬁnite Gabor frame with window

hλ

N, i.e., there are constants A, B > 0such that, for every f∈CN,

Akfk2

CN≤

K

X

k=1 Vhλ

Nf[jk, lk]

2≤Bkfk2

CN.

2). One of the three following conditions is satisﬁed:

(i) N2≥K≥N+ 1,

(ii) K=Nis odd,

(iii) K=Nis even and PN

k=1(jk, lk)/∈NN2.

Proof: Select a collection of Kdistinct points z1, ..., zK∈TNof the form jk

N, lk∈

TN,jk, lk∈0, ...., N −1 and rewrite Corollary 14 as

AkΣNfk2

SN≤

K

X

k=1 Vhλ

0ΣNfjk

N, lk

2

≤BkΣNfk2

SN.

Lemma 4 ensures that for every f∈CNthere exists f∈S0(R) such that PNf=f.

By (6)

kΣNfkSN=

1

NPNf

CN

=1

NkfkCN.

Finally, by Theorem 1,

Vhλ

0ΣNfjk

N, lk=N−1Vhλ

Nf[jk, lk].

18 L. D. ABREU, P. BALAZS, N. HOLIGHAUS, F. LUEF, AND M. SPECKBACHER

By Theorem 14, it follows that the frame conditions are satisﬁed if and only if

either K≥N+ 1, or K=Nand PN

k=1(jk, lk)6= (N2/2 + N n, N 2/2 + Nm), for

all n, m ∈Z.

If N=Kis odd, then N2/2 + nN =N k + 1/2, for some k∈Z. Therefore, the

condition (ii) is automatically satisﬁed as PN

k=1(jk, lk)∈N2.

If N=Kis even, then for every k∈Zthere exist n∈Zsuch that N2/2+ nN =

Nk. Therefore, we get a frame if and only if PN

k=1(jk, lk)/∈NN2.

This result can be reformulated in the following sense: the discrete Gabor system

generated by the sampled, periodized Gaussian is in general linear position if Nis

odd (and almost in general linear position if Nis even). If any Npoints from the

set {(j, k) : j, k ∈0, ..., N }yield a ﬁnite Gabor frame for CNwith window g, then g

is said to be in general linear position, see for example [20]. In [18] it is shown that

almost every vector in CNis in general linear position for every N∈N. However,

explicitly known examples of vectors in general linear position are not localized and

thus of no use for practical purposes of ﬁnite Gabor analysis. Our result on the

other hand allows to use localized windows (for appropriate choices of λ) for the

prize that one needs to use one additional sampling point if the number of points

N∈Nis even and enjoys a particular arithmetic structure.

Theorem 14 and Theorem 3 can also be seen as ’Nyquist-type’ necessary and

suﬃcient results: with less than Nsamples the system is never a frame, but by

increasing the number of samples above N, one is assured to have a frame with

higher redundancy. Such a property comes in handy for situations where one is

given a signal representation sampled on a grid with more than Npoints (allowing

for perfect reconstruction by our result) and wishes to increase the grid resolution

for some numerical purpose. Then sampling again at a higher density is possible,

and it still leads to perfect reconstruction. This can be done until all the possible

N2points of the grid are used. By resorting to the extension to the torus and to

Theorem 14 one can further use oﬀ-grid points and increase the resolution to ar-

bitrary levels. The analogue necessary and suﬃcient result for inﬁnite-dimensional

Gabor frames with Gaussian window is stated in terms of Beurling density and was

proved by Seip and Wallst´en [22, 21] and independently by Lyubarskii [17].

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373, 10 2007.

Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090

Vienna, Austria

Email address:abreuluisdaniel@gmail.com

Email address:michael.speckbacher@univie.ac.at

Acoustics Research Institute, Austrian Academy of Sciences, Wohllebengasse 12-14,

1040 Vienna, Austria

Email address:peter.balazs@oeaw.ac.at

Email address:nicki.holighaus@oeaw.ac.at

Department of Mathematics, NTNU Trondheim, 7041 Trondheim, Norway

Email address:franz.luef@ntnu.no