Spikes, Roots, and Aliasing: Recovering Bandlimited Signals from Roots of the Short-Time Fourier Transform

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

SIAM J. APPL. MATH. c© 2012 Society for Industrial and Applied MathematicsVol. 72, No. 5, pp. 1449–1473

SPIKES, ROOTS, AND ALIASING: RECOVERING BANDLIMITEDSIGNALS FROM ROOTS OF THE SHORT-TIME FOURIER

TRANSFORM∗

BERNHARD G. BODMANN† AND CHRISTOPHER L. LINER‡

Dedicated to Hans Feichtinger on the occasion of his 60th birthday

Abstract. This paper exploits analyticity of time-frequency representations for the recovery ofbandlimited signals. The main goal is to recover the signal, up to an overall multiplicative constant,from the roots of its short-time Fourier transform. To ensure a sufficient number of roots in thetransform, we use a type of aliasing, a time-frequency quasi-periodization of the signal. We alsoconsider sparse bandlimited signals, meaning ones with few nonzero sample values. Under certainstatistical assumptions on the signal, the number of roots needed to recover the signal depends onlyon the number of nonzero sample values. We conclude with an experimental part that validates therecovery for sparse signals corrupted by noise.

Key words. signal recovery, sampling theorem, bandlimited signal, short-time Fourier trans-form, aliasing, sparse signal, roots, denoising, deconvolution

AMS subject classifications. 94A12, 94A20, 42C20, 11F30

DOI. 10.1137/110848578

1. Introduction. The recovery of signals from specific features has been anactive subject for many decades. Such features could be peak locations, singularities,significant components in linear or nonlinear transforms (see, e.g., [18, 16, 37, 6, 31]),or even random features such as measurements in compressive sensing [10, 17]. Quiteoften, the features are believed to be characteristic of the signal and less prone tocorruption. The main motivation for this paper was to find a simple procedure for therecovery of certain sparse signals based upon knowledge of their roots in a transformdomain. For example, one might have hoped to identify a function consisting of asmall number of well-localized spikes solely by the roots of its Fourier transform. If thiswere possible, then it would provide a nonadaptive strategy for deconvolution if theunknown filter does not introduce additional roots in the transform. Unfortunately,this idea is not feasible in such generality. A strategy for recovery from the rootscould be realized by restricting the class of signals to special types of pass-bandsignals [30]. Another work on recovery based on roots, also called zero-crossings,does not require restricting the signal class but uses a redundant representation, thesemidiscrete wavelet transform [32]. However, the effect of filtering on the roots inthese transforms is not as easily understood as in the Fourier transform, so it doesnot seem to be as convenient for deconvolution purposes. We refer the reader toadditional works with interesting uses of zero-crossings for highly accurate analog-digital conversion [13, 24, 14].

∗Received by the editors September 20, 2011; accepted for publication (in revised form) July 5,2012; published electronically September 25, 2012. This work was partially supported by NationalScience Foundation grant DMS-1109545, by Air Force Office of Scientific Research grant FA9550-11-1-0245, and by a grant from Petroleum Geo-Services.

http://www.siam.org/journals/siap/72-5/84857.html†Department of Mathematics, University of Houston, Houston, TX 77204-3008 ([email protected].

edu).‡Department of Earth and Atmospherical Sciences, University of Houston, Houston, TX 77204-

5007 ([email protected]).

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1450 BERNHARD G. BODMANN AND CHRISTOPHER L. LINER

In this paper, we stay close to the idea of recovery from the roots of the Fouriertransform and present a version of it which makes this feasible. We also discussresults for sparse signals and present an algorithm for the recovery of sparse signalsthat have been corrupted by additive noise as well as by a class of low-pass filters. Wefollow in the footsteps of Mallat and rely heavily on the redundancy inherent in theshort-time Fourier transform to recover a sparse signal. Redundancy has been usefulfor recovery from magnitudes of frame coefficients [1]. Here, we go one step furtherand establish recovery solely based on the roots of the transform. Specific motivationfor our work comes from the potential application to reflection seismology. In thisfield, deconvolution has long been used to restore data spectra to the full bandwidthemitted by the seismic source. This process partially compensates for attenuationand layer scattering experienced by waves propagating in the earth; see, e.g., [35]or [29, Chapter 20]. Efforts to reconstruct the full-bandwidth reflection coefficientseries include sparse spike inversion [28] and, more recently, spectral inversion [34, 36]by signal adaptive methods. Our goal was to find a method which does not requirestatistical estimation.

Bandlimited functions and the sampling theorem are central to our results. Fora treatment of sparse, possibly nonbandlimited functions, see the papers by Vetterli,Marziliano, and Blu [38] and Berent, Dragotti, and Blu [5]. These works use a signalmodel which is patched together from pieces that are annihilated by a given filter,similar to the local vanishing of low-pass components for the Daubechies wavelettransform of a piecewise smooth signal. Vetterli et al. [38] and Berent et al. [5] thenuse the response of the filter to the discontinuities for recovery. In brief, the roots ofthe filter are fixed and match the (local) frequency content of the signal. Here, weexploit the roots of the signal, which vary in a way that characterizes the signal, up toan overall multiplicative constant. A conceptual difference with both references is thatthey require sampling, linear measurements of the signal, whereas our reconstructionstrategy depends on the roots, which are in a nonlinear relationship with the signal.On the other hand, we have to impose a global sparseness assumption—the numberof nonvanishing samples—whereas their assumption is local, assuming the signal hasa finite rate of innovation.

Our main results are as follows: As a first goal, we focus on signals which arebandlimited, but have only finitely many nonzero sample values when evaluated atsuitable, uniformly spaced sample points. A main difficulty for recovery from theroots is to ensure that they provide enough information. For the short-time Fouriertransform with a Gaussian window, it is known that transforming any Gaussian doesnot yield any roots in the time-frequency domain, so in an extreme case we know thereis insufficient information. To avoid this problem we use aliasing, a quasi-periodizationin time and frequency, and adjust the space of input signals appropriately. For thespaces under consideration, the aliasing does not lead to any loss of information. Aproperly periodized short-time Fourier transform then has a guaranteed number ofroots which characterize the signal, up to an overall multiplicative constant. A moreprecise statement of the first main result is as follows (see the following section fordefinitions): If f is a square-integrable, continuous, and Ω-bandlimited function withnonzero sample values only at tn = nπ/Ω, n ∈ {0, 1, 2, . . . , k − 1}, then we define thequasi-periodized short-time Fourier transform by

Pk,0VΩf(t, ω) =∑

n,m∈Z

(−1)(k+1)n(VΩTknπ/ΩM2mΩ)p(t, ω),

where VΩ is the short-time Fourier transform with a Gaussian window function of

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SPIKES, ROOTS, AND ALIASING 1451

width Ω−1, Ta and Mω denote translation by a ∈ R and modulation by ω ∈ R,respectively, and p is the preperiodized function p = (f + (−1)k+1Tkπ/Ωf)/2. The

function Pk,0VΩf has (counting multiplicities) k roots {xj , yj}kj=1 in the domain D ={(t, ω) : 0 ≤ t < kπ/Ω,−Ω ≤ ω < Ω}. If we let zj = Ωxj − iyj/Ω for j ≤ k − 1,

z0 = kπ/2 + i, zk = z0 −∑k−1

j=1 zj , and Ω′ = Ω/k if k is even and otherwise Ω′ = 0,then there exists a constant C ∈ C such that

f(t) = C

k−1∑n=0

sin(Ω(t− tn))

Ω(t− tn)

∫ Ω

−Ω

e−ω2/2Ω2k∏

j=1

h(Ωtn − i(ω − Ω′)/Ω− zj + z0)dω

for all t ∈ R, where h(z) =∏∞

n=1(1 + e−4n+2 cos(2z/k) + e−8n+4). The constant Cdepends on f because rescaling f by some constant factor does not change the rootsof the transform.

The second main goal of this paper is to treat signals which have few nonzerosample values at points that are widely spaced among a much larger, finite sequenceof consecutive sample points. In this case, the number of roots needed for identifyingthe signal depends only on the number of nonzero sample values, not on the length ofthe sequence. This result is very analogous to the literature on compressive sensing,where the number of measurements needed to determine a signal depends only on thesize of its support; see, e.g., [19, 10, 17, 11, 12]. If f is as described above, but onlys + 1 ≤ k − 2 of its sample values are nonzero, and if any s roots of Pk,0VΩ chosenuniformly at random are continuously distributed in Ds, then with probability one, fcan be recovered as above, up to an overall multiplicative constant, from the locationof such s roots. If f is deterministic, then the same result is true if a randomizationprocedure is applied to it before the quasi-periodized short-time Fourier transformis computed. Finally, we examine by numerical experiments whether the recoveryprocedure is sensitive to noise in the signal. To this end, we use the equivalence ofreconstructing f , up to an overall scalar multiple, with reconstructing the orthogonalprojection onto the one-dimensional subspace Cf . This strategy has already beenapplied for reconstruction from magnitudes of frame coefficients [1], and here wecombine it with an �1-minimization procedure to propose an algorithm for sparserecovery in this context. More explicitly, we convert the problem of recovering thesubspace to a matrix recovery problem and then minimize the �1-norm of the matrixentries.

We have explored two applications of this recovery method: the removal of addi-tive noise for sparse signals presented here, and a nonadaptive deconvolution strategy,which is the topic of another paper [7].

This paper is organized as follows. In the next section, we introduce the essentialnotation and terminology. Properties of the quasi-periodized short-time Fourier trans-form are discussed in section 3. The general results on recovery from the roots areestablished in section 4, and the special case of sparse signals is treated in section 5.We conclude with the experimental evaluation of the robustness of this recovery pro-cedure against noise in the signal.

2. Preliminaries.

2.1. Reconstruction formulas for the short-time Fourier transform.Givena complex-valued square-integrable function f ∈ L2(R) which is also integrable on R,

we define its Fourier transform f by

f(ω) =

∫R

e−iωtf(t)dt, ω ∈ R .

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Plancherel’s theorem states the L2-norm identity for the Fourier transform, ‖f‖ =√2π‖f‖. Thus, the bounded linear map f �→ f specified by this integral transform

can be extended to all of L2(R) by taking Cauchy sequences in the dense subspaceL2(R)∩L1(R). If one identifies, as usual, functions in L2(R) that differ only on a setof Lebesgue measure zero, then the Fourier transform defines a bounded linear mapwhich is surjective and invertible.

In contrast to the extension process required to define the Fourier transform ofsquare-integrable functions, the short-time Fourier transform is an integral transformdefined on all of L2(R).

Definition 2.1. Let {Ta}a∈R be the family of translation operators, acting onf ∈ L2(R) by Taf(x) = f(x − a) for Lebesgue-almost every x ∈ R. The short-timeFourier transform of a function f ∈ L2(R) with a window g ∈ L2(R) is given by thefunction Vgf , defined on the time-frequency plane R2, with values

Vgf(t, ω) = (fTtg) (ω) =

∫R

g(s− t)e−iωsf(s)ds , (t, ω) ∈ R2,

where the bar over the window function g denotes complex conjugation. Henceforth,we exclusively use Gaussian windows, g(s) = e−Λ2s2/2, and replace the subscript gwith the parameter Λ > 0,

VΛf(t, ω) =

∫R

e−Λ2(s−t)2/2−iωsf(s)ds , (t, ω) ∈ R2.

We also define a transform AΛ which maps f to an analytic function in the complexplane. We map a point (t, ω) in the time-frequency plane to z = Λt+ iω/Λ and let

AΛf(z) = e−iz1z2ez22/2VΛf(z1/Λ,−z2Λ) =

∫R

e−(z−Λs)2/2f(s)ds, z ∈ C.

Up to multiplication with a complex Gaussian and a rescaling of the arguments withthe factor Λ, AΛ is the Bargmann transform [2, 3].

There is a convenient inversion formula for the short-time Fourier transform,reminiscent of the usual inverse of the Fourier transform, when f is integrable onR. In this case, after changing the values of f on a set of Lebesgue measure zero ifnecessary, it can be assumed to be a continuous function.

Proposition 2.2. If f ∈ L2(R) is continuous and has an integrable Fouriertransform, then f is obtained from its short-time Fourier transform by

f(t) =1

2π

∫R

VΛf(t, ω)eitωdω , t ∈ R .

In terms of its image under AΛ, it is recovered by

f(t) =1

2π

∫R

AΛf(Λt+ iω/Λ)e−ω2/(2Λ2)dω , t ∈ R.

Proof. After inserting the definition of VΛf(t, ω) and exchanging the order ofintegration, we substitute u = s− t and obtain∫

R

VΛf(t, ω)eitωdω =

∫R2

f(u+ t)e−Λ2u2/2eiuωdudω .

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Plancherel’s theorem for the inner product of g(u)eiuω, g(u) = f(u + t), with theGaussian window gives∫

R

g(u)e−Λ2u2/2eiuωdu =1√2πΛ

∫R

g(ξ − ω)e−ξ2/(2Λ2)dξ .

Inserting this identity and reversing the order of integration again shows the claimedreconstruction formula, because

∫Rg(ω)dω = 2πg(0) = 2πf(t) and

∫Re−ξ2/(2Λ2)dξ =√

2πΛ.The result for recovery fromAΛf instead of VΛf follows from inserting VΛf(t, ω) =

e−itωe−ω2/(2Λ2)AΛf(Λt− iω/Λ) and by a change of variables.A special class of functions with integrable Fourier transform is that of the band-

limited functions.Definition 2.3. A function f ∈ L2(R) is called Ω-bandlimited if f(ω) = 0 on

{ω : |ω| ≥ Ω}, apart from a set of Lebesgue measure zero.We recall the sampling theorem, the basis of many results in digital signal pro-

cessing.Theorem 2.4. Let 0 ≤ α < 1, Ω > 0 and tn = (n+α)π/Ω for n ∈ Z. The family

{sn}n∈Z defined by sn(t) = sin(Ω(t− tn))/(Ω(t− tn)) is an orthonormal basis for thesubspace of Ω-bandlimited functions in L2(R), and if f ∈ L2(R) is continuous andΩ-bandlimited, then the sample values {f(tn)}n∈Z form a square-summable sequenceand

f =

∞∑n=−∞

f(tn)sn .

The convergence of the series is with respect to the norms in L2(R) and L∞(R) [33, 9].

It is instructive to know that the proof of this theorem uses a Fourier series for fon the interval [−Ω,Ω]. The Fourier coefficients are related to the sample values by

1

2Ω

∫ Ω

−Ω

f(ω)eiπωn/Ωdω =π

Ωf

(nπ

Ω

).

3. Signal recovery and periodization. We denote the modulations by {Mξ}ξ∈R,Mξf(t) = eiξtf(t) and recall the intertwining relation of modulations and translations

with the Fourier transform, Mξf(ω) = f(ω− ξ) = Tξf(ω) and Taf(ω) = e−iaω f(ω) =

M−af(ω) for f ∈ L2(R), ξ, a ∈ R and almost every ω ∈ R. From the definition of theshort-time Fourier transform we have the intertwining relations

(VΛMξf)(t, ω) = VΛf(t, ω − ξ) and (VΛTaf)(t, ω) = e−iωaVΛf(t− a, ω)

for a, t, ξ and ω ∈ R and f ∈ L2(R).Periodizing the Fourier transform of an Ω-bandlimited function f gives the point-

wise almost everywhere convergent series f =∑

k∈Z(M2kΩf ) , which is also obtained

from the natural, 2Ω-periodic extension of the Fourier series of f . All the samplevalues of f are encoded in the Fourier coefficients of f . This type of periodizationbecomes problematic if bandlimited signals are sampled at a rate which is insufficientfor the recovery with the sampling theorem. Because frequency content of a signal isthen duplicated, the result of such an irreversible periodization is generally referredto as aliasing.

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3.1. Frequency periodization for the short-time Fourier transform. Ifthe short-time Fourier transform is similarly periodized in the frequency variable,then we can recover the sample values of a bandlimited f with an integral transform.Unlike the periodization of the Fourier transform, this procedure does not introducethe unwanted irreversible effects of aliasing because it allows perfect recovery of un-dersampled bandlimited signals. This is true even for any function with an integrableFourier transform, which we show next.

Definition 3.1. For f ∈ L2(R) and α ∈ R, we define the quasi-periodizedshort-time Fourier transform QαVΛf by

QαVΛf(t, ω) =∑m∈Z

e−2miαt(fTtg) (ω − 2mΛ) =∑m∈Z

e−2miαtVΛf(t, ω − 2mΛ)

whenever (t, ω) ∈ R2 is chosen so that the above series converges absolutely.The quasi-periodization introduced here is the same as in the Zak transform [20,

Chapter 8]. The motivation for inserting the complex exponential in the sum is thatwith the choice α = Λ, the function f can be obtained by a simple integral transform,as shown in the next proposition.

Proposition 3.2. If f is a continuous function in L2(R) with an integrableFourier transform, Λ > 0, and α ∈ R, then QαVΛf is continuous and bounded.Moreover, if α = Λ, then

f(t) =1

2π

∫ Λ

−Λ

QΛVΛ(t, ω)eitωdω .

Proof. By definition of QαVΛf and the fact that f ∈ L1(R), for fixed t the func-tion ω �→ QαVΛf(t, ω) is in L1([−Λ,Λ]). This follows from

∑m∈Z

e−2miαtV f(t, ω −2mΛ) =

∑m∈Z

e−2miαtT2mΛ(fTtg) (ω) and from T2mΛ(fTtg) = f ∗ T2mΛ(Ttg) to-gether with the Gaussian decay of g. This implies that all the partial sums of theseries

∑m e−2miαtT2mΛ(Ttg) are dominated by a constant, independent of t. Thus,

Holder’s inequality and dominated convergence ensure that QαVΛf(t, ω) defines abounded and continuous function on R2.

In the special case α = Λ, the definition of QΛVΛf implies

QΛVΛf(t, ω)eitω =

∑m∈Z

VΛf(t, ω − 2mΛ)eit(ω−2mΛ), (t, ω) ∈ R2 .

Consequently, we can eliminate the summation by changing the domain of integration,∫ Λ

−Λ

QΛVΛf(t, ω − 2mΛ)eit(ω−2mΛ)dω =

∫ ∞

−∞VΛf(t, ω)e

itωdω .

Now the reconstruction formula in Proposition 2.2 for functions with an integrableFourier transform gives the claimed identity.

In order to recover regularly spaced sample values of f it is also possible to usethe alternative periodization Q0VΛf .

Corollary 3.3. If f is a continuous function in L2(R) with an integrable Fouriertransform and tn = nπ/Λ, then

f(tn) =1

2π

∫ Λ

−Λ

Q0VΛf(tn, ω)eitωdω .

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Proof. By definition, QΛVΛf(tn, ω) = Q0VΛf(tn, ω).With the help of the sampling theorem we immediately obtain the following con-

sequence for Ω-bandlimited functions if we choose Λ = Ω.Corollary 3.4. If f ∈ L2(R) is continuous and Ω-bandlimited, then

f(t) =1

2π

∞∑n=−∞

sin(Ωt− nπ)

Ωt− nπ

∫ Ω

−Ω

Q0VΩf(nπ/Ω, ω)eitωdω .

Henceforth, we will use the scaling Λ = Ω for the Gaussian window in the short-time Fourier transform of Ω-bandlimited functions. This scaling is primarily moti-vated by the sampling theorem, but is also natural on intuitive grounds because thenthe width of the window is inverse proportional to the cut-off frequency Ω.

3.2. Time-frequency periodization of the short-time Fourier transform.Next, we investigate whether we can use a suitable time-frequency representation torecover an Ω-bandlimited function with finitely many nonzero sample values.

Definition 3.5. Let sl(t) = sin(Ω(t− tl))/(Ω(t − tl)), tl = lπ/Ω, for t ∈ R andl ∈ Z. We define Hk,Ω to be the k-dimensional subspace of Ω-bandlimited functionsof the form

f =k−1∑l=0

clsl

with nonzero sample values cl = f(tl) ∈ C, l ∈ {0, 1, . . . , k − 1}.For the purpose of reconstructing f ∈ Hk,Ω, we wish to use a time-frequency

quasi-periodization of its short-time Fourier transform.Definition 3.6. For f ∈ L2(R), k ∈ N, Ω > 0, and α ∈ R, we denote

Pk,αVΩf(t, ω) =∑

n,m∈Z

(−1)(k+1)ne−2miαt−iknπω/ΩVΩp(t− knπ/Ω, ω − 2mΩ)

with the preperiodized function p = (f + (−1)k+1Tkπ/Ωf)/2 whenever the series ex-pression on the right-hand side is absolutely convergent.

The preperiodization step in this definition serves to improve the summability ofthe series expression defining Pk,αVΩf , so that it can be applied to all f ∈ Hk,Ω. It

ensures that when f is bounded on R, then the frequency quasi-periodization doesnot introduce discontinuities because p extends continuously to limω→±Ω p(ω) = (1+

(−1)k+1eikπ)f(±Ω)/2 = 0.We prepare the sampling theorem for Hk,Ω with an alternative expression for the

time-frequency quasi-periodization of the short-time Fourier transform.Lemma 3.7. Let Ω > 0, k ∈ N, ωm = 2mΩ/k for m ∈ Z and Ω′ = Ω/k if k

is even, otherwise Ω′ = 0. If f ∈ L2(R) is Ω-bandlimited and f is continuous andbounded on (−Ω,Ω), then

Pk,0VΩf(t, ω) =

√2

k√πe−i(ω−Ω′)t

∑m,l

e−(ωm−ω+Ω′)2/2Ω2

eiωmtf(ωm +Ω′ − 2lΩ) .

Proof. For any function g for which the Poisson sum formula is valid [20], we have∑n

e−itnk(ω−Ω′)g(s− tnk) =Ω

kπe−i(ω−Ω′)s

∑m

e2imsΩ/k g(2mΩ/k − ω +Ω′) ,

where tnk = nkπ/Ω for n ∈ Z.

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In particular, for g(s) = e−Ω2s2/2, this implies∑n

e−inkπ(ω−Ω′)/Ωg(s− tnk) =

√2

k√πe−i(ω−Ω′)s

∑m

e2imsΩ/ke−(2mΩ/k−ω+Ω′)2/2Ω2

.

Replacing s by t− s and inserting this identity into the short-time Fourier trans-form for the preperiodized function p gives∑

n

e−inkπ(ω−Ω′)/ΩVΩp(t− tnk, ω)

=

√2

k√πe−ik(ω−Ω′)t

∑m

e2imΩt/ke−(2mΩ/k−ω+Ω′)2/2Ω2

p(2mΩ/k +Ω′) .

Together with the frequency periodization, we have by the intertwining relation-ship with modulations∑

n,l

e−inkπ(ω+Ω′)/ΩVΩp(t− nkπ/Ω, ω − 2lΩ)

=

√2

k√πe−i(ω−Ω′)t

∑m,l

e−(ωm−ω+Ω′)2/2Ω2

eiωmtp(ωm +Ω′ − 2lΩ).

Finally, we eliminate the preperiodized function p in favor of f ,

p(Ω′ + ωm − 2lΩ) = (1 + (−1)k+1eikπ(ωm+Ω′)/Ω)f(Ω′ + ωm − 2lΩ)/2

= f(Ω′ + ωm − 2lΩ) .

Theorem 3.8. Let Ω > 0, k ∈ N, and tn = nπ/Ω for n ∈ Z. If f ∈ Hk,Ω, then

f(tn) =1

2π

∫ Ω

−Ω

Pk,0VΩf(tn, ω)eitnωdω, n ∈ {0, 1, . . . , k − 1} .

Proof. Integrating over the integral [−Ω,Ω] gives by the preceding lemma∫ Ω

−Ω

Pk,0VΩf(tn, ω)eitnωdω

=

√2

k√πeiΩ

′tn∑m,l

∫ Ω

−Ω

e−(ωm−ω+Ω′)2/2Ω2

eiωmtf(ωm +Ω′ − 2lΩ) .

Now shifting the summation index ωm = 2mΩ/k by 2lΩ in the sum and integrating,we have ∫ Ω

−Ω

Pk,0VΩf(tn, ω)eitnωdω

=

√2

k√πeiΩ

′tn∑m,l

∫ Ω

−Ω

e−(ωm+2lΩ−ω+Ω′)2/2Ω2

eiωmtf(ωm +Ω′)

=

√2

k√πeiΩ

′tn∑m

∫ ∞

−∞e−(ωm−ω+Ω′)2/2Ω2

eiωmtn f(ωm +Ω′)dω

=2Ω

k

∑m

ei(Ω′+ωm)tn f(ωm +Ω′) .D

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139

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136.

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With the orthogonality of complex exponentials of the form en : ω �→ einπ(Ω′+ω)

on the interval [−Ω,Ω] as well as on the set {wm +Ω′ : m ∈ Z} ∩ [−Ω,Ω], we convertthe Riemann sum to an integral which provides the sample value of the Ω-bandlimitedfunction

1

k

∑m

ei(Ω′+ωm)tn f(ωm +Ω′) =

1

2Ω

∫ Ω

−Ω

eiωtn f(ω)dω =π

Ωf(tn) .

Put together, this establishes the desired identity.In conjunction with the usual sampling theorem, we recover f ∈ Hk,Ω from its

quasi-periodized short-time Fourier transform.Corollary 3.9. If f ∈ Hk,Ω, then

f(t) =1

2π

k−1∑n=0

sin(Ωt− nπ)

Ωt− nπ

∫ Ω

−Ω

Pk,0VΩf(tn, ω)eitnωdω , t ∈ R.

4. Signal recovery from the roots of the quasi-periodized short-timeFourier transform. Next, we derive a reconstruction formula for f in terms of theroots of Pk,0VΩf . To this end, we relate Pk,0VΩf with a quasi-periodized analyticfunction PkAΩf and use some complex analysis.

Definition 4.1. Let k ∈ N, Ω > 0, and Ω′ = Ω/k if k is even, otherwise Ω′ = 0.If f ∈ Hk,Ω, we define

PkAΩf(Ωt+ iω/Ω) = e−itωeω2/2Ω2

Pk,0VΩf(t,−ω +Ω′) .

Lemma 4.2. If f ∈ Hk,Ω, then PkAΩf is analytic and

PkAΩf(x− iy + kπ) = PkAΩf(x− iy)

and

PkAΩf(x− iy + 2i) = e−2i(x−iy)+2f(x− iy) .

Proof. To see the analyticity, we compare the function PkAΩf with AΩp, wherep = (f + (−1)k+1Tkπ/Ωf)/2. As a first step, we use Definition 3.6 and the relation

VΩf(t, ω) = e−itωe−ω2/(2Ω2)AΩf(Ωt− iω/Ω) to obtain

Pk,0V f(t,−ω +Ω′) =∑m,n

(−1)(k+1)neit(ω+Ωm−Ω′)e−(ω+Ωm−Ω′)2/2Ω2

AΩp(zn,m),

where we have abbreviated Ωm = 2mΩ, tn = knπ/Ω, and zn,m = Ω(t − tn) + i(ω +Ωm − Ω′)/Ω. Next, we combine this identity with Definition 4.1 and regroup terms,

PkAΩf(Ωt+ i

ω

Ω

)= e−itωeω

2/2Ω2

Pk,0VΩf(t,−ω +Ω′)

= eω2/2Ω2 ∑

m,n

(−1)(k+1)neit(Ωm−Ω′)e−(ω+Ωm−Ω′)2/2Ω2

AΩp(zn,m)

=∑m,n

(−1)(k+1)ne−iΩ′(t+iω/Ω2)e2m(itΩ−ω/Ω)e(Ωm−Ω′)2/2Ω2

AΩp(zn,m) .

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Reinserting the expressions for tn, Ωm, and zn,m, we identify PkAΩf as the quasi-periodization

PkAΩf (Ωt+ iω/Ω) =∑

n,m∈Z

(−1)(k+1)ne−iΩ′(t+i(ω−Ω′)/Ω2)e2m(itΩ−(ω−Ω′)/Ω−m)

×AΩp(Ωt− knπ + i(ω − Ω′)/Ω− 2mi) .

The convergence properties of Pk,0VΩf used in the proofs of Lemma 3.7 and of PkAΩ

are identical. Because of the bandlimitedness of h and of the Gaussian decay of thewindow function, the quasi-periodization is uniformly convergent on compact sets inC. This implies that PkAΩf is analytic.

The quasi-periodized short-time Fourier transform of f ∈ Hk,Ω obeys

Pk,0VΩf(t+ kπ/Ω, ω) = (−1)k+1e−ikπω/ΩPk,0VΩf(t, ω)

and

Pk,0VΩf(t, ω + 2Ω) = Pk,0VΩf(t, ω) .

Consequently,

PkAΩf(x− iy + kπ) = ei(x+kπ)yey2/2Pk,0VΩf((x+ kπ)/Ω,Ωy +Ω′))

= eixyey2/2Pk,0VΩf(x/Ω,Ωy +Ω′) = PkAΩf(x− iy)

and

PkAΩf(x− iy + 2i) = eix(y−2)e(y−2)2/2Pk,0VΩf(x/Ω,Ωy − Ω′))

= e−2i(x−iy)eixyey2/2+2Pk,0VΩf(x/Ω,Ωy +Ω′))

= e−2i(x−iy)+2PkAΩf(x− iy)

for all x, y ∈ R.We know that the range of PkAΩ is a k-dimensional vector space. It turns out

that the analyticity and the quasi-periodicity conditions characterize the range space.Definition 4.3. We define Fk, k ∈ N, to be the vector space of analytic functions

on C which observe the quasi-periodicity conditions for each z ∈ C,1. periodicity f(z + kπ) = f(z) and2. quasi-periodicity f(z + 2i) = e2e−2izf(z).

We also equip this space with the inner product

〈f, g〉 = Ω

2π3/2

∫[0,kπ]×[0,2]

f(z)g(z)e−z22dz1dz2 .

Examining the quasi-periodicity properties of PkAΩf for f ∈ Hk,Ω, we observethat the range of PkAΩ is contained in Fk. But in fact the range space has dimensionk because PkAΩ is invertible, and the dimension of the range coincides with thedimension of Fk. This fact is generally known to researchers of theta functions; see,e.g., [4].

Theorem 4.4 (Bellman [4]). Let k ∈ N; then the dimension of Fk is k.Proof. Since f is analytic and periodic, it restricts to a smooth, kπ-periodic

function on R. The Fourier coefficients of this function are

cn =1

kπ

∫ kπ

0

f(x, 0)e−2inx/kdx .

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Integrating the analytic function e−2inz/kf counterclockwise along the boundaryof the rectangle {(x, y) : 0 ≤ x ≤ kπ, 0 ≤ y ≤ 2} gives zero, and by the periodicitycondition (4.3) the contributions of the contour parallel to the imaginary axis cancel,which leaves ∫ kπ

0

f(x)e−2inx/kdx =

∫ kπ

0

f(x+ 2i)e−2in(x+2i)/kdx .

Together with the quasi-periodicity condition (4.3), this yields∫ kπ

0

f(x)e−2inx/kdx = e2e4n/k∫ kπ

0

e−2i(n+k)x/kf(x)dx .

This provides a recursion relation for the Fourier coefficients,

cn = e2e4n/kcn+k .

Consequently, after specifying {c0, c1, . . . , ck−1}, all the other coefficients are deter-mined.

Moreover, by iteration of the recursion relation one can verify that the Fourier co-efficients decay exponentially as n → ±∞. This means every choice of {c0, c1, . . . , ck−1}leads to a Fourier series

f(z) =

∞∑n=−∞

cne2inz/s,

which is uniformly convergent on compact sets in the complex domain. The limit ofthe series is an analytic function with the desired periodicity and quasi-periodicityproperties.

We conclude that there is a vector-space isomorphism between Fk and Ck, whichmeans that Fk is a k-dimensional vector space.

Corollary 4.5. The map PkAΩ from Hk,Ω to Fk is a vector-space isomor-phism.

One of the peculiarities of the space Fk is that the functions in it have a fixednumber of roots in the fundamental domain T = {x+iy ∈ C : 0 ≤ x < kπ, 0 ≤ y < 2}.This will be verified with the evaluation of Cauchy integrals.

When computing the contour integral∮Γm(z)dz of a meromorphic function m

on C with respect to a closed, piecewise smooth contour Γ, we follow the conventionthat poles on the contour contribute with the principal value, e.g.,∮

Γ

1

z − 1dz = πi

for Γ with the parametrization Γ(s) = eis, s ∈ [0, 2π].Proposition 4.6. If f is a function in the space Fk , then

1

2πi

∮Γ

f ′(z)f(z)

dz = k ,

where Γ is the contour which traverses the edges of T = {z = x + iy ∈ C : 0 ≤ x ≤kπ, 0 ≤ y ≤ 2} in counterclockwise orientation. This means Γ is a closed contourconsisting of line segments between the points 0, kπ, kπ + 2i, 2i, and 0.

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Proof. Using the periodicity and quasi-periodicity conditions, we see

f ′(z + kπ)

f(z + kπ)=

f ′(z)f(z)

and

f ′(z + 2i)

f(z + 2i)= −2i+

f ′(z)f(z)

.

This implies

1

2πi

∮Γ

f ′(z)f(z)

dz =1

2πi

∫ kπ

0

(f ′(x)f(x)

− f ′(x+ 2i)

f(x+ 2i)

)dx = k .

Corollary 4.7. By Cauchy’s residue theorem, the preceding result implies thatthe number of roots of f ∈ Fk enclosed by Γ is k. This follows from our convention,by which each root on Γ contributes half, and because such roots appear in pairs dueto the periodicity and quasi-periodicity of f .

Proposition 4.8. If f is a function in the space Fk without roots on the contourΓ which traverses the edges of {z = x + iy ∈ C : 0 ≤ x ≤ kπ, 0 ≤ y ≤ 2} incounterclockwise orientation, then

1

2πi

∮Γ

zf ′(z)f(z)

dz =π

2k2 + ik + (kπM + 2iN) ,

with some integers N,M ∈ Z.Proof. Using the periodicity and quasi-periodicity again, we have

(z + kπ)f ′(z + kπ)

f(z + kπ)= z

f ′(z)f(z)

+ kπf ′(z)f(z)

and

(z + 2i)f ′(z + 2i)

f(z + 2i)= −2iz + 4 + z

f ′(z)f(z)

+ 2if ′(z)f(z)

,

which gives

1

2πi

∮Γ

zf ′(z)f(z)

dz =1

2πi

∫ kπ

0

(2iz − 4− 2i

f ′(z)f(z)

)dz +

1

2πi

∫ 2i

0

kπf ′(z)f(z)

dz .

Due to the periodicity and quasi-periodicity and the multivalued nature of the complexlogarithm, we know that

1

2πi

∫ kπ

0

f ′(z)f(z)

dz ∈ Z

and

1

2πi

∫ 2i

0

f ′(z)f(z)

dz ∈ 1

πi+ Z .

This implies

1

2πi

∮Γ

zf ′(z)f(z)

dz =k2π

2+ 2ik − 2iN +

1

ik + πkM,

with integers N and M .

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By the usual residue calculus, this Cauchy integral provides us with some infor-mation about the sum of the roots.

Corollary 4.9. If f ∈ Fk, and f has roots {z1, z2, . . . , zk} (repetitions allowed),none on the boundary of the contour Γ which traverses the edges of {z = x+ iy ∈ C :0 ≤ x ≤ kπ, 0 ≤ y ≤ 2}, then

k∑j=1

zj =π

2k2 + ik + (kπM + 2iN) .

The roots give rise to a factorization formula for any function in Fk. The deriva-tion of this fact is analogous to the classical treatment of the Jacobi theta functions[40].

Theorem 4.10. Let h(z) =∏∞

n=1(1 + e−4n+2 cos(2z/k) + e−8n+4). If f ∈ Fk

has roots at the positions {z1, z2, . . . , zk} (repetitions allowed), then

f(z) = Ck∏

j=1

h(z + z0 − z′j)

with z0 = kπ/2 + i, z′j = zj for j ≤ k − 1, z′k = z0 −∑k−1

j=1 zj, and some constantC ∈ C.

Proof. We first show that h has periodicity properties which resemble what weneed. The function h has only one root at z0 = kπ/2+ i. This follows from Cauchy’sintegral formulas

1 =1

2πi

∮Γ

h′(z)h(z)

dz

and

z0 =1

2πi

∮Γ

zh′(z)h(z)

dz ∈ D ,

which determines N and M because z0 is in the rectangle D = {z = x + iy ∈ C, 0 ≤x ≤ kπ, 0 ≤ y ≤ 2}.

We note that h(z + kπ) = h(z) and that from factoring

h(z) =( ∞∏

n=1

(1 + e−πτ(2n−1)/ke2iz/k))( ∞∏

n=1

(1 + e−πτ(2n−1)/ke−2iz/k))

we get

h(z + 2i) = h(z)(1 + e−2/ke2iz/k)−1(1 + e2/ke−2iz/k) = e2/ke−2iz/kh(z) .

Inserted into the contour integral, this gives z0 = kπ/2 + i.We now form the function

g(z) =

k∏j=1

h(z + z0 − z′j),

where z0 is as above and z′j = zj for j ≤ k − 1 and z′k = z0 −∑k−1

j=1 zj. From the

identity for∑k

j=1 zj in the preceding theorem, we know that zk−z′k ∈ kπZ+2iZ, and

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thus the roots in T are identical when passing from zj to z′j . From h(z+z0−z′j+kπ) =

h(z+ z0 − z′j) and h(z+ z0 − z′j +2i) = e2/ke−2i(z−z′j+z0)/kh(z− z′j + z0) we conclude

that

g(z + 2i) = e2e−2izg(z),

which means g ∈ Fk .Dividing f by g gives a doubly periodic analytic function with no zero or pole in

the entire complex plane. Therefore, by Liouville’s theorem this quotient must be aconstant.

Theorem 4.11. If f ∈ Hk,Ω, then Pk,0VΩf has k roots (multiplicity counted)at points {(xj , yj)}kj=1 in D = {z = x + iy : 0 ≤ x ≤ kπ/Ω, 0 ≤ y ≤ 2Ω}, and if

zj = Ωxj − iyj/Ω for j ≤ k − 1, z0 = kπ/2 + i, and zk = z0 −∑k−1

j=1 zj, then thereexists a constant C ∈ C such that

f(t) = C∞∑

n=−∞

sin(Ω(t− tn))

Ω(t− tn)

∫ Ω

−Ω

e−ω2/2Ω2k∏

j=1

h(Ωtn − i(ω − Ω′)/Ω− zj + z0)dω

for all t ∈ R, where h(z) =∏∞

n=1(1 + e−4n+2 cos(2z/k) + e−8n+4).Proof. By comparing Pk,0VΩf and PkAΩf , we convert the expression for PkAΩf

in terms of its roots into the corresponding one for the short-time Fourier transform,

Pk,0VΩf(x, y) = Ce−ixye−y2/2k∏

j=1

h(Ωx− i(y − Ω′)/Ω− zj + z0) .

Consequently, the sample values of f are given by

f(tn) = C

∫ Ω

−Ω

e−ω2/2Ω2k∏

j=1

h(Ωtn − i(ω − Ω′)/Ω− zj + z0)dω , n ∈ {1, 2, . . . , k} .

Now applying the sampling theorem gives the desired identity.

5. Recovery of sparse signals from a subset of their roots. Next, weinvestigate whether it is possible to recover a sparse signal from the location of a fewroots. To prepare this, we discuss norm identities and reproducing kernels.

5.1. Norm identities and reproducing kernels for the short-time Fouriertransform and for its quasi-periodization. The short-time Fourier transformmaps into L2(R2) and becomes an isometry when the range space is equipped witha suitable norm (see, e.g., [20]). To keep the exposition self-contained, we show thiswell-known fact.

Proposition 5.1. Let f ∈ L2(R); then its norm relates to the L2-norm of theshort-time Fourier transform VΩf by

‖f‖ =( Ω

2π3/2

∫R2

|VΩf(t, ω)|2dtdω)1/2

.

Similarly, we have the norm identity for AΩf ,

‖f‖ =( Ω

2π3/2

∫C

|AΩf(z)|2e−z22dz1dz2

)1/2.

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Proof. By definition, abbreviating the window function g(t) = e−Ω2t2/2 and usingPlancherel’s theorem results in∫

R2

|VΩf(t, ω)|2dtdω =

∫R2

|(fTtg) (ω)|2dtdω = 2π

∫R2

|f(s)(Ttg)(s)|2dsdt .

Next, carrying out the integration with respect to t gives the factor∫R|Ttg(s)|2dt =∫

Re−Ω2(s−t)2dt =

√π/Ω. The result for AΩf follows from inserting VΩf(t, ω) =

e−itωe−ω2/(2Ω2)AΩf(Ωt− iω/Ω) and by a change of variables.This implies that the range of VΩ is a closed subspace of L2(R2).From the norm identity, we see that the suitably normalized Hilbert adjoint

Ω2π3/2V

∗Ω is a left inverse of V , which provides another reconstruction formula.

A similar norm identity holds for functions in Hk,Ω.Proposition 5.2. If f ∈ Hk,Ω, then

‖f‖ =

(Ω

2π3/2

∫[0,kπ/Ω]×[−Ω,Ω]

|Pk,0VΩf(t, ω)|2dtdω)1/2

and

‖f‖ =

(Ω

2π3/2

∫[0,kπ]×[−1,1]

|PkAΩf(z)|2e−z22dz1dz2

)1/2

.

Proof. Using the notation and the identity for Pk,0VΩf from Lemma 3.7,∫[0,kπ/Ω]×[−Ω,Ω]

|Pk,0VΩf(t, ω)|2dtdω

=2

kΩ

∑m,l

∫ Ω

−Ω

e−(ωm−ω+Ω′)2/Ω2 |f(ωm +Ω′ − 2lΩ)|2dω

=2

kΩ

∑m

∫ ∞

−∞e−(ωm−ω+Ω′)2/Ω2 |f(ωm +Ω′)|2dω

=2√π

k

∑m

|f(ωm +Ω′)|2 .

Next, when f restricted to (−Ω,Ω) is a trigonometric polynomial of the form

f =∑k−1

m=0 cme−imπω/Ω then, by the Plancherel identities for the discrete and thecontinuous Fourier transform,

1

k

∑m

|f(ωm +Ω′)|2 =1

2Ω

∫ Ω

−Ω

|f(ω)|2dω =π

Ω‖f‖2 .

For the short-time Fourier transform this implies∫[0,kπ/Ω]×[−Ω,Ω]

|Pk,0VΩf(t, ω)|2dtdω =2π3/2

Ω‖f‖2 .

Inserting the relation between Pk,0VΩf and PkAΩf gives the corresponding result forthe analytic function.

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Theorem 5.3. The map PkAΩ : Hk,Ω → Fk is unitary.Proof. The fact that Hk,Ω and Fk are both k-dimensional reduces the proof to

showing that PkAΩ is an isometry. This was established with the preceding proposi-tion.

In order to establish recovery from roots, we associate these locations with pointevaluation functionals.

Definition 5.4. Let z ∈ C; then Λz denotes the point evaluation functionalcorresponding to z, Λzf = f(z) for all f ∈ Fk .

Proposition 5.5. For each z ∈ C, the corresponding point evaluation functionalis continuous on the space Fk, which is equipped with the norm induced by the innerproduct. Moreover, there is a vector ηz such that

〈f, ηz〉 = f(z)

for all f ∈ Fk.Proof. From the mean value property of analytic functions, we have for any fixed

r > 0 that

f(w) =1

πr2

∫B(w,r)

f(z)dz1dz2,

where the integral is over a disk B(w, r) of radius r centered at w. We choose r < 1so that when B(w, r) is shifted by kπZ or 2iZ, the shifted copies form a disjoint unionW = ∪m,n∈ZB(w + kπm+ 2in, r).

Next, Jensen’s inequality gives

|f(w)| ≤(

1

πr2

∫B(w,r)

|f(z)|2dz1dz2)1/2

,

and using the periodicity relations for f , there is a function q which is bounded awayfrom zero on compact sets such that

1

πr2

∫B(w,r)

|f(z)|2dz1dz2 ≤∫W∩T

q(z)|f(z)|2e−z22dz2dz2 .

We thus conclude that there is C > 0 with

|f(w)| ≤ C‖f‖Fk.

By the Riesz representation theorem [39], we conclude that for each point evaluationfunctional Λz, there is an associated vector ηz with the desired property.

Definition 5.6. We denote the vector associated with Λz by ηz, so 〈f, ηz〉 =f(z).

By definition, if f ∈ Fk has a root at z, then f and ηz are orthogonal.Proposition 5.7. Given f ∈ Fk with k distinct roots {z1, z2, . . . , zk}, and

the associated vectors {ηz1 , ηz2 , . . . , ηzk}, then any subset S of at most k − 1 of thesevectors is linearly independent.

Proof. Suppose we have chosen a subset of k − 1 vectors, say S = {ηz1 , ηz2 , . . . ,ηzk−1

}. We need to show that none of them is a linear combination of the other vectors.Let us pick z1, for example. If ηz1 were a linear combination of the other vectors inS, then any function in Fk vanishing at the remaining k − 2 points z2, z3, . . . , zk−1

would also vanish at z1. However, we can construct a function which vanishes at these

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k− 2 points but not at z1. To this end, we take a function f vanishing at the distinctpoints z1, z2, . . . , zk. By the factorization result,

f(z) = C

k∏j=1

h(z + z0 − z′j) ,

with z′j = zj for j ≤ k − 1 and z′k = z0 −∑k−1

j=1 zj. We can now perturb the location

of z′1 and of z′k in such a way that z0 = 1k

∑kj=1 z

′j remains unchanged, for example,

z′′1 = z′1 + ε and z′′k = z′k − ε with some ε ∈ C . For sufficiently small ε, the points{z′′1 , z′2, z′3, . . . , z′k−1, z

′k} are distinct. Then the resulting function

fε(z) = Ch(z + z0 − z′1 − ε)h(z + z0 − z′k + ε)k−1∏j=2

h(z + z0 − z′j)

vanishes at z2, z3, . . . , zk−1 but not at z1. The fact that no ηzj is a linear combinationof the other vectors in S follows similarly.

Having shown that all sets of size k−1 are linearly independent, the independenceof smaller subsets follows.

The above result on linear independence is possible because of the analyticitystemming from the use of a Gaussian window. For other window functions it is stillan open problem [22]. The role of linear independence is central to guaranteeing sparserecovery. The first results of this type, based on frequency measurements, relied onthe special structure of Vandermonde matrices [19]. We define an analogous propertyfor the kernel functions in our transform space.

Definition 5.8. Let {e1, e2, . . . , ek} denote the canonical basis which gives thesample values for Fk , so

g(tn) = 〈PkAΩg, en〉for n ∈ {1, 2, . . . , k}. We say that a family of vectors {ηz1 , ηz2 , . . . , ηzs} has the sparserecovery property if for any subset J of {1, 2, . . . , k} of size |J | ≥ s, and any linearcombination

g =

s∑j=1

cjηzj ,

the orthogonality relations 〈g, ej〉 = 0 for all j ∈ J imply that g = 0.Theorem 5.9. Let v ∈ Fk be nonzero and let the roots {z1, z2, . . . , zk} of v be such

that {ηzj}sj=1 has the sparse recovery property for s ≤ k − 2. If v ∈ span{ej : j ∈ J},where J ⊂ {1, 2, . . . , k} and |J | = s+ 1, then there is no other vector w which is notcollinear with v, is orthogonal to all {ηzj}sj=1, and has support size s+1 or smaller.

Proof. Let W be the span of any s basis vectors {ej1 , ej2 , . . . , ejs} and v ∈W with v ⊥ ηzj for all 1 ≤ j ≤ s. From the linear independence shown in thepreceding proposition, dim span{ηzj}sj=1 = s. By the sparse recovery property, W⊥∩span{ηzj}sj=1 = {0}. This means dim(W⊥ + span{ηzj}sj=1) = n − s + s = n. Thus,

if v ⊥ W⊥ and v ⊥ ηzj for all 1 ≤ j ≤ s, then v = 0. This implies that there cannotbe any nonzero v with support size of at most s.

Now, assume v has support of size dimW = s+ 1 and v ⊥ ηzj for all 1 ≤ j ≤ s.Then by the sparse recovery property dim(W⊥ + span{ηzj}sj=1) = n − s − 1 + s =

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n − 1, so there is a one-dimensional subspace which is orthogonal to W⊥ and tospan{ηzj}sj=1.

If there were two noncollinear vectors v and u perpendicular to {ηzj}sj=1, eachwith support size s + 1, then, using the preceding argument of dimension counting,they cannot have the same support. Moreover, we could project u onto the orthogonalcomplement of v and then the projection would have support outside of that of v. Sowithout loss of generality we can assume that there is a vector u ⊥ v and u ⊥ ηzj forall j.

Again dimension counting, we conclude that u ∈ W⊥. Otherwise u would projectto a nonzero vector in W and then v would be orthogonal to that projection, to W⊥,and to {ηzj}sj=1, which forces v = 0. Consequently, v + εu has for sufficiently small

ε support strictly containing the support of v. Next, span{v + εu} ∩ W⊥ = {0},v + εu ⊥ ηzj , and thus dim(W⊥ + span{ηzj}sj=1 + span{v + εu}) = n, but then{v, v + εu} cannot be linearly independent.

The preceding theorem is central to what follows. Informally, it means that“dimension counting” gives the right answer: Knowing s roots of a function reducesthe dimension of the subspace in which it can be found from k to k−s. Moreover, theintersection of this subspace with an (s+1)-dimensional coordinate subspace (spannedby s + 1 vectors of an orthonormal basis) has the generically expected dimension,1 = (k − s) + (s + 1) − k. Finally, there can only be one such coordinate subspaceintersecting nontrivially with the subspace determined by the roots. This means an(s + 1)-sparse function can be determined, up to an overall multiplicative constant,from s roots.

5.2. Sparse recovery for random signals. Next, we show that a genericdistribution of zeros has the sparse recovery property.

Theorem 5.10. Assume that when choosing s roots {z1, z2, . . . , zs}, s ≤ k − 1,of a random signal f ∈ Fk , their distribution is absolutely continuous with respect tothe Lebesgue measure on Ts. Then the sparse recovery property holds with probabilityone.

Proof. For the choices of {z1, z2, . . . , zs} which do not have the sparse recoveryproperty, there exists a nontrivial linear combination such that

∑sj=1 cj〈ηzj , el〉 = 0

for all l ∈ J , with some index set J ⊂ {1, 2, . . . , k} of size |J | = s. In other words,the sparse recovery property is violated if the matrix (Zj,l(z))j≤s,l∈J with entriesZj,l(z) = 〈ηzj , el〉 does not have full rank for some J , |J | ≤ s. By a union bound,we only need to show that this occurs with probability zero for a given set J of size|J | = s.

To this end, we use the analyticity in the problem. We know that the entries ofZ(z) are actually el(zj), given by anti-analytic functions, and that there are nontrivialsolutions if and only if detZ(z) = 0. However, the determinant is a polynomial in theentries of Z, and thus detZ(z) is analytic in all zj.

We first establish that detZ as a function of {zj}sj=1 is not identically zero. With-out loss of generality, we can assume that J = {1, 2, . . . , s}. The Leibniz rule expressesthe determinant as the linear combination detZ(z) =

∑π(−1)σ(π)

∏l∈J eπl(zl)

summed over all permutations of the index set J , with σ(π) ∈ {0, 1} the parityof each permutation π. Because of the orthogonality of {el} and the norm identity,cross terms in the corresponding expression for | detZ(z)|2 do not contribute whenintegrated over Ts with respect to the natural measure, which results in

Ωs

2sπ3s/2

∫Ts

| detZ(x+ iy)|2∏l∈J

e−y2l dxldyl = s! .

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This shows that detZ is not identically zero.Finally, we establish that detZ is nonzero with probability one. This follows from

the Weierstrass preparation theorem [23], or from a weaker version we describe below.Fixing all but the last variable makes Z(z) analytic in zs, and thus because analyticfunctions have isolated zeros unless they are identically zero, the determinant canonly vanish on the disjoint countable union of submanifolds of codimension one in Cs.This set has Lebesgue measure zero, and thus any distribution which is absolutelycontinuous with respect to the Lebesgue measure gives probability zero.

5.3. Sparse recovery for deterministic signals. At the cost of an additionalrandomization, deterministic signals can be recovered as well. To this end, we applya random unitary to the unknown vector before selecting a number of roots for itsrecovery. With the additional randomization, the recovery problem is equivalent tothat of a random vector which is selected with respect to the unitarily invariantprobability measure on the unit sphere in the Hilbert space Fk. More precisely, werecover the one-dimensional subspace generated by this vector, which is identified asa point in the projective space PFk. The following lemma is similar to the version ofthe Lefschetz embedding theorem presented by Debarre [15, Theorem 6.8].

Lemma 5.11. The map u : Tk−1 → PFk which associates (z1, z2, . . . , zk−1) with

the complex line Cg in Fk generated by g(w) = h(w +∑k−1

j=1 zj)∏k−1

j=1 h(w+ z0 − zj),with h as above, is an isomorphism.

Proof. We define zk = z0 −∑k−1

j=1 zj and first study the map φ : Ck−1 → PFk,

which is the natural extension of u to Ck−1. If the map φ is not smooth, then thetangent map becomes singular for some point. This implies that there exists a vectorλ ∈ Ck with

∑kj=1 λj = 0 such that

k∑j=1

λjh′(z + z0 − zj)

h(z + z0 − zj)= 1

for all but finitely many z in any bounded subset of C. We integrate both sides toobtain

k∑j=1

λj lnh(z + z0 − zj) = z + c

with some constant c ∈ C. However, h, and thus the left-hand side, remains un-changed when we proceed from z to z + kπ along a path in a simply connecteddomain which does not contain singular points. This contradicts the identity withthe linear expression on the right-hand side. Thus no such λ exists and φ is smooth.Finally, restricting φ gives the injective map u on the quotient space Tk−1. Sinceit is also surjective, u is an isomorphism between the complex manifolds Tk−1 andPFk.

Theorem 5.12. Let U be a random unitary operator on Hk,Ω, chosen uniformlyat random with respect to the Haar measure, and let v be a vector in Hk,Ω whichis (s + 1)-sparse, s ≤ k − 2, in an orthonormal basis {e1, e2, . . . , ek}. Then withprobability one, any randomly selected s roots of PkAΩUv determine Cv.

Proof. By assumption, PkAΩUv is a random vector, uniformly distributed on theunit sphere S of the Hilbert space Fk. From the isomorphism between the projectivespace and the vector of roots, any randomly chosen vector of s ≤ k − 1 roots is thencontinuously distributed in Ts. Thus, Theorem 5.10 guarantees that PkAΩUv can be

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recovered with probability one up to an unknown multiplicative constant and Cv issubsequently obtained after reversing PkAΩ and U .

In most sampling scenarios, the function one would like to recover is unknown. Ifthere is only partial information about its transform, then it is not possible to applya random unitary operator to it. The reason for including the preceding result isthat there are maps such as chirp filters (the unitaries generated by the Laplacian)which could be applied to the signal before its acquisition. Experimentally, thesechirp filters, together with random shifts in time, seem to randomize the pattern ofthe roots in a way similar to the effect of random unitary operators acting on thesignal. There is a large literature on the mixing property of such unitary transforms[25, 26, 8, 27, 21].

5.4. Experimental results on recovery for noisy signals. In this section,we explore the possibility of recovering a sparse signal which has been corrupted bynoise. In contrast to some of the literature on compressive sensing, we assume thatthe noise resides in the signal and not in the measurements.

5.4.1. Removing additive noise. The analyticity properties of functions inFk imply that the nodes of the quasi-periodized short-time Fourier transform arecontinuously perturbed by additive noise.

Proposition 5.13. Consider a sequence {fj}∞j=1 in Hk,Ω defined by adding

noise to a signal, fj =∑k−1

l=0 (cl + ε(j)l )sl, with sample values {cl + ε

(j)l } and sl as in

Definition 3.5 and assume that the noise vanishes in the limit, ε(j) → 0. Then theroots of Pk,0VΩfj converge to those of Pk,0VΩg, g = limj fj.

Proof. The relationship between Pk,0VΩfj and PkAΩfj implies that it is sufficientto prove that the roots of PkAΩfj converge to those of PkAΩg. The Cauchy–Schwarzinequality, together with the Riesz representation theorem presented in Proposi-tion 5.5, implies that

|PkAΩ(fj − g)(z)| = |〈PkAΩ(fj − g), ηz〉| ≤ ‖fj − g‖‖ηz‖

and by the continuity of the map z �→ ηz , ‖fj − g‖ → 0 makes PkAΩ(fj − g)(z)converge uniformly for z ∈ T as j → ∞. Next, Cauchy’s formula and uniformconvergence imply that the number of roots enclosed in any simple closed contourremains constant for all sufficiently large j. This means that if g(z) = 0, then f (j)

has a root in a given neighborhood of z for all sufficiently large j.We examine the sensitivity to additive noise experimentally. Our experiments

use a space of signals which have five nonzero sample values in a sequence of 50consecutive sample points. The location of these sample values is chosen uniformly atrandom, and the sample values are independent and identically Gaussian distributed.

The noise added to the signal is also independent and identically Gaussian dis-tributed, chosen at a strength of 20% of the signal. In Figure 5.1, we compare thequasi-periodized short-time Fourier transforms of the noiseless and noisy signals byplotting the negative logarithm of their magnitudes. We include the locations of 20randomly selected roots for the transform of the noisy signal. The reason why 20instead of 4 roots have been chosen for the noisy signal is to compensate for the effectof the noise. These roots were found by a simple coordinate descent method withrandom initialization.

Figure 5.2 compares the noiseless signal with its recovery based on the roots of thenoisy signal. For the purpose of establishing numerical evidence of recovery, we use aniterative algorithm. First the support of the signal is estimated from the location of

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0 5 10 15 20 25 30 35 40 45 50−0.2

0

0.2

0.4

Time/sample

Am

plit

ude

(real part

)F

requency

5 10 15 20 25 30 35 40 45 50

3.14

1.57

0

−1.57

−3.14

0 5 10 15 20 25 30 35 40 45 50−0.2

0

0.2

0.4

Time/sample

Am

plit

ude

(real part

)F

requency

5 10 15 20 25 30 35 40 45 50

3.14

1.57

0.0

−1.57

−3.14

Fig. 5.1. Comparison of quasi-periodized short-time Fourier transforms of the noiseless (left)and noisy (right) signals by plotting the negative logarithm of their magnitudes. Each transform has50 roots. We have marked the 20 roots used for the recovery of the noisy signal by +. Below eachtime-frequency plot is a graph of the time series given by the sample values for the real part of thecorresponding noiseless (left) or noisy signal (right).

0 5 10 15 20 25 30 35 40 45 50−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Time/sample

Am

plitu

de(r

eal p

art)

Fig. 5.2. Comparison of the time series of sample values for the noiseless signal (solid line),with 20% noise added (dashed line) and the recovered signal (marked with “×”) based on the 20randomly selected roots in Figure 5.1.

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Recovery algorithm.Input: Set of roots {z1, z2, . . . , zs} of f =

∑j∈J cjej +

∑kj=1 εjej ∈ Hk,Ω, with unknown

support set J ⊂ {1, 2, . . . , k} of size |J | = s+ 1 in the canonical basis {ej}kj=1 and noise vector

ε =∑k

j=1 εjej .Output: Signal estimate g.Step 1 Support estimation

(a) Take the Gram matrix G = (〈ηzl , ηzj 〉)j,l of the kernel functions {ηzl}sl=1 asso-ciated with the roots. Compute the orthogonal projection matrix P which mapsonto the orthogonal complement of the range of G, P = P ∗P , and PG = GP = 0,(I − P )G = G(I − P ) = G.

(b) Among all matrices in the convex set

M = {M ∈ Ck×k : M = M∗, trM = 1, 0 ≤ M ≤ P} ,find a minimizer M∗ = argmin ‖M‖1 for the Frobenius 1-norm

‖M‖1 =k∑

j,l=1

|Mj,l| .

The set M is the set of matrices satisfying the trace normalization trM = 1 andthe operator inequalities 0 ≤ M ≤ P .

(c) Find an eigenvector belonging to the maximal eigenvalue of M∗. Estimate thesupport of the signal by the indices belonging to the s+ 1 largest coefficients.

Step 2 RecoveryFind a unit-norm vector g with the chosen support which minimizes

∑sj=1 |g(zj)|2.

Fig. 5.3. Algorithm for the recovery of an (s + 1)-sparse signal based on knowledge of s roots{z1, z2, . . . , zs} of the noise-corrupted signal.

the roots. We begin with the subspace of functions vanishing at the known locationsand associate a projection matrix P with it. Then we divide this matrix by its rank tonormalize the trace. We then perform a gradient descent to minimize the Frobenius-1norm among the convex set of matrices {M : 0 ≤ M ≤ P, trM = 1}. The reasonfor these operator inequalities is that the eigenvectors in the range of these matriceshave short-time Fourier transforms vanishing at the prescribed points. We minimizethe matrix norm to enforce sparseness and minimal rank of the resulting optimizer.After the gradient descent stabilizes, the support is estimated to be the significantcomponents of the eigenvector belonging to the largest eigenvalue. The estimate forthe vector is then obtained by selecting a nonzero vector with the prescribed supportthat has a nearly vanishing short-time Fourier transform at the prescribed roots. Thisoverdetermined system is solved by minimizing the quadratic form corresponding tothe point evaluations at the roots, subject to a fixed norm and to the desired support.We summarize the steps of the algorithm in Figure 5.3.

To compare the result with the initial vector, the estimate in our plot is multipliedby a constant so that both sum to the same value.

Finally, we illustrate the sensitivity of the recovery procedure to the noise level inFigure 5.4 by plotting how the error norm of the recovery depends on the norm of thenoise, which ranges from 0% to 50% for a unit-norm initial vector. By comparing theerror norm for the estimate with the initial, noiseless signal, we conclude when thealgorithm is successful in recovering the sparse signal from the noise-corrupted one.We define the recovery to be successful if the estimate actually denoises, meaning thesparse output of the algorithm is closer to the noiseless signal than the noisy input.For noise which has up to 10% signal strength, the algorithm recovers a sparse signal

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0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Noise norm

Rec

onst

ruct

ion

erro

r no

rm

Fig. 5.4. Comparison of the noise level to the reconstruction error. The dotted plot is theoutcome of the recovery based on 20 roots for random, unit-norm 5-sparse signals in dimensionk = 50, which have been corrupted by varying amounts of noise, ranging from a norm of 0% to 50%.The solid line indicates the initial noise level to show when the recovery removes noise (dot belowline), and when the recovery error is significantly above the noise level, for example, the outcomeswith reconstruction error

√2, which are due to incorrect support estimation.

and removes noise in 37 out of 49 cases. If the noise level goes up to 20%, then thealgorithm is successful in 59 out of 99 cases. The plot indicates that there is a nonzerofailure probability (recovery error significantly above the noise level) which increaseswith the noise. The inspection of the failed reconstructions shows that most of themresult in shifted versions of the input signal, caused by a failure to identify the supportcorrectly. Among the 99 recovery attempts for signals with a noise level of up to 20%,the support was identified correctly in 79 cases. In 18 cases, the estimated supportwas incorrectly shifted by one sample, and in 2 cases the estimated support did nothave any simple relation with that of the noiseless signal. If the shifted support isdisjoint with that of the input, then the reconstructed signal is orthogonal, whichgives an error norm of

√2, as is visible in the plot.

5.4.2. Deconvolution. One motivation for attempting recovery from the rootsis for the purpose of deconvolution. Assuming the signal has been convolved by alow-pass filter close to the identity, then the use of the Gaussian window in the short-time Fourier transform implies that filtering only slightly displaces the roots of abandlimited signal; see [7]. We briefly summarize the relevant results here.

We define an approximate identity {Hλ}λ>0 by H1 ∈ L1(R),∫RH1dt = 1, and

Hλ(x) = λH1(λx).To retain the convergence properties of the quasi-periodization of the short-time

Fourier transform after convolution with Hλ, we restrict the signal space to the so-called Feichtinger algebra.

Definition 5.14. The Feichtinger algebra S0(R) consists of all the square-integ-rable functions on R which have an integrable short-time Fourier transform,

S0(R) =

{f ∈ L2(R) : ‖VΩf‖1 ≡

∫R2

|VΩf(t, w)|dtdω < ∞}.

Theorem 5.15 (see [7]). Let {Hλ}λ≥1 be an approximate identity of scaledconvolutions as above, and let g ∈ S0(R). As λ → ∞, then the roots of Pk,0VΩHλgconverge to those of Pk,0VΩg.

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We have used this strategy to recover a sparse signal which has been corrupted byrandom “echos” consisting of the signal convolved with white noise. The performanceis comparable to the results on additive noise presented here, so we refrain fromincluding details which can be found elsewhere [7].

6. Conclusion. We conclude by summarizing the results. We have seen thatbandlimited signals with a finite number of nonzero sample values can indeed berecovered from the roots of their suitably quasi-periodized short-time Fourier trans-form. Essential for this is the fact that knowing s roots reduces the dimension ofthe subspace in which the unknown signal resides by s. Proving this fact requiresthe verification of the conjecture by Heil, Ramanathan, and Topiwala [22] in our set-ting, which is accomplished with the relation between the quasi-periodized short-timeFourier transform and a space of doubly quasi-periodic analytic functions. As a con-sequence, recovering a signal in the k-dimensional complex Hilbert space Hk,Ω up toan overall multiplicative constant requires knowledge of k − 1 roots in the transformdomain. Intuitive dimension counting also gives the right answer for a class of ran-dom, sparse signals in Hk,Ω: Assuming s ≤ k − 1 nonvanishing sample values, thenonly s − 1 roots are required. This result is again a consequence of the analyticityproperties of the quasi-periodized short-time Fourier transform. We have presented anumerical algorithm for recovery and tested its sensitivity to noise. Here, we have fo-cused on applying this algorithm for removing additive noise, and we refer the readerto another paper for results on deconvolution [7].

Acknowledgments. We wish to express our thanks for the very careful anddetailed suggestions by the anonymous reviewers.

REFERENCES

[1] R. Balan, B. G. Bodmann, P. G. Casazza, and D. Edidin, Painless reconstruction frommagnitudes of frame coefficients, J. Fourier Anal. Appl., 15 (2009), pp. 488–501.

[2] V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform,Comm. Pure Appl. Math., 14 (1961), pp. 187–214.

[3] V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform.Part II. A family of related function spaces. Application to distribution theory, Comm.Pure Appl. Math., 20 (1967), pp. 1–101.

[4] R. Bellman, A Brief Introduction to Theta Functions, Athena Series: Selected Topics inMathematics, Holt, Rinehart and Winston, New York, 1961.

[5] J. Berent, P. L. Dragotti, and T. Blu, Sampling piecewise sinusoidal signals with finiterate of innovation methods, IEEE Trans. Signal Process., 58 (2010), pp. 613–625.

[6] B. Boashash and M. Mesbah, Signal enhancement by time-frequency peak filtering, IEEETrans. Signal Process., 52 (2004), pp. 929–937.

[7] B. G. Bodmann and C. L. Liner, Stable signal recovery from the roots of the short-timeFourier transform, in Wavelets and Sparsity XIV, Proc. SPIE 8138, SPIE, Bellingham,WA, 2011, 813817.

[8] J. Bouclet and S. De Bievre, Long time propagation and control on scarring for perturbedquantized hyperbolic toral automorphisms, Ann. Henri Poincare, 6 (2005), pp. 885–913.

[9] J. Brown, Jr., Bounds for truncation error in sampling expansions of band-limited signals,IEEE Trans. Inform. Theory, 15 (1969), pp. 440–444.

[10] E. J. Candes, Compressive sampling, in International Congress of Mathematicians, Vol. III,Eur. Math. Soc., Zurich, 2006, pp. 1433–1452.

[11] E. J. Candes, J. Romberg, and T. Tao, Robust uncertainty principles: Exact signal recon-struction from highly incomplete frequency information, IEEE Trans. Inform. Theory, 52(2006), pp. 489–509.

[12] E. J. Candes, J. K. Romberg, and T. Tao, Stable signal recovery from incomplete andinaccurate measurements, Comm. Pure Appl. Math., 59 (2006), pp. 1207–1223.

[13] Z. Cvetkovic, I. Daubechies, and B. F. Logan, Interpolation of bandlimited functions from

Dow

nloa

ded

10/0

1/12

to 1

39.1

84.3

0.13

6. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p



quantized irregular samples, in Proceedings of the Data Compression Conference (DCC’02),IEEE Computer Society, Washington, DC, 2002, pp. 412–421.

[14] Z. Cvetkovic, I. Daubechies, and B. F. Logan, Single-bit oversampled A/D conversion withexponential accuracy in the bit rate, IEEE Trans. Inform. Theory, 53 (2007), pp. 3979–3989.

[15] O. Debarre, Complex Tori and Abelian Varieties, SMF/AMS Texts Monogr. 11, AMS, Prov-idence, RI, 2005; translated from the 1999 French edition by P. Mazaud.

[16] D. L. Donoho, De-noising by soft-thresholding, IEEE Trans. Inform. Theory, 41 (1995),pp. 613–627.

[17] D. L. Donoho, Compressed sensing, IEEE Trans. Inform. Theory, 52 (2006), pp. 1289–1306.[18] D. L. Donoho and I. M. Johnstone, Ideal spatial adaptation by wavelet shrinkage, Biometrika,

81 (1994), pp. 425–455.[19] D. L. Donoho and P. B. Stark, Uncertainty principles and signal recovery, SIAM J. Appl.

Math., 49 (1989), pp. 906–931.[20] K. Grochenig, An uncertainty principle related to the Poisson summation formula, Studia

Math., 121 (1996), pp. 87–104.[21] S. Gurevich and R. Hadani, Proof of the Kurlberg-Rudnick rate conjecture, Ann. of Math.

(2), 174 (2011), pp. 1–54.[22] C. Heil, J. Ramanathan, and P. Topiwala, Linear independence of time-frequency trans-

lates, Proc. Amer. Math. Soc., 124 (1996), pp. 2787–2795.[23] L. Hormander, An Introduction to Complex Analysis in Several Variables, 3rd ed., North-

Holland Math. Library 7, North-Holland, Amsterdam, 1990.[24] P. Ishwar, A. Kumar, and K. Ramchandran, Distributed sampling for dense senor networks:

A “bit-conservation principle,” in Proceedings of the Information Processing in SensorNetworks: Second International Workshop (IPSN 2003), Palo Alto, CA, 2003, L. J. Guibasand F. Zhao, eds., Lecture Notes in Comput. Sci., Springer, New York, 2003, pp. 17–31.

[25] S. Klimek, A. Lesniewski, N. Maitra, and R. Rubin, Ergodic properties of quantized toralautomorphisms, J. Math. Phys., 38 (1997), pp. 67–83.

[26] P. Kurlberg and Z. Rudnick, Hecke theory and equidistribution for the quantization of linearmaps of the torus, Duke Math. J., 103 (2000), pp. 47–77.

[27] P. Kurlberg and Z. Rudnick, On the distribution of matrix elements for the quantum catmap, Ann. of Math. (2), 161 (2005), pp. 489–507.

[28] S. Levy and P. K. Fullagar, Reconstruction of a sparse spike train from a portion of its spec-trum and application to high-resolution deconvolution, Geophysics, 46 (1981), pp. 1235–1243.

[29] C. L. Liner, Elements of 3D seismology, 2nd ed., PennWell, Tulsa, OK, 2004.[30] B. F. Logan, Information in the zero crossings of band-pass signals, Bell System Tech. J., 56

(1977), pp. 487–510.[31] M. Mahmoudi and G. Sapiro, Fast image and video denoising via nonlocal means of similar

neighborhoods, IEEE Signal Process. Lett., 12 (2005), pp. 839–842.[32] S. Mallat, Zero-crossing of a wavelet transform, IEEE Trans. Inform. Theory, 37 (1991),

pp. 1019–1033.[33] A. Papoulis, Error analysis in sampling theory, Proc. IEEE, 54 (1966), pp. 947–955.[34] C. I. Puryear and J. P. Castagna, Layer-thickness determination and stratigraphic interpre-

tation using spectral inversion: Theory and application, Geophysics, 73 (2008), pp. R37–R48.

[35] E. A. Robinson, Multichannel z-transforms and minimum-delay, Geophysics, 31 (1966),pp. 482–500.

[36] J. G. Rubino and D. Velis, Thin-bed prestack spectral inversion, Geophysics, 74 (2009),pp. R49–R57.

[37] J.-L. Starck, E. J. Candes, and D. L. Donoho, The curvelet transform for image denoising,IEEE Trans. Image Process., 11 (2002), pp. 670–684.

[38] M. Vetterli, P. Marziliano, and T. Blu, Sampling signals with finite rate of innovation,IEEE Trans. Signal Process., 50 (2002), pp. 1417–1428.

[39] J. Weidmann, Linear Operators in Hilbert Spaces, Grad. Texts in Math. 68, Springer, NewYork, 1980; translated from the German by J. Szucs.

[40] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis. An Introduction to theGeneral Theory of Infinite Processes and of Analytic Functions: With an Account of thePrincipal Transcendental Functions, 4th ed., reprinted, Cambridge University Press, NewYork, 1962.

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yrig

ht; s

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ttp://

ww

w.s

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.org

/jour

nals

/ojs

a.ph

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Documents

Spikes, Roots, and Aliasing: Recovering Bandlimited Signals from Roots of the Short-Time Fourier Transform