Sampling of Stochastic Operators

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 60, NO. 4, APRIL 2014 2359

Sampling of Stochastic OperatorsGötz E. Pfander and Pavel Zheltov

Abstract— We develop sampling methodology aimed at identi-fying stochastic operators that satisfy a support size restriction onthe autocorrelation of the operators stochastic spreading function.The data that we use to reconstruct the operator (or, in some casesonly the autocorrelation of the spreading function) is based onthe response of the unknown operator to a known, deterministictest signal.

Index Terms— Stochastic, operator sampling, autocorrelation,spreading function, scattering function, delta train, Gaborframes, Haar property, time-frequency analysis.

I. INTRODUCTION

IN WIRELESS and wired communication, in radar detec-tion, and in signal processing it is usually assumed that

a signal is passed through a filter, whose parameters haveto be determined from the output. Commonly, such systemsare modeled with a time-variant linear operator acting ona space of signals. For narrow-band signals, we can modelthe effects of Doppler shifts and multi-path propagation asthe sum of “few” time-frequency shifts that are applied to thesent signal. In general, the channel consists of a continuum oftime-frequency scatterers: the channel is formally representedby an operator with a superposition integral

(H f )(x) =∫∫

η(t, ν) MνTt f (x) dt dν, (1)

where Tt is a time-shift by t , that is, Tt f (x) = f (x −t), t ∈ R,Mν is a frequency shift or modulation given by Mν f (x) =e2π iνx f (x), ν ∈ R. We define the Fourier transform of afunction f (x) to be

F [ f ](ξ) = Fx→ξ f (ξ) = f̂ (ξ) =∫

f (x) e−2π ixξ dx .

It follows that

M̂ν f (ξ) = f̂ (ξ − ν) = Tν f̂ (ξ).

The function η(t, ν) is called the (Doppler-delay) spreadingfunction of H .

To identify the operator means to determine the spreadingfunction η of H from the response H f (x) of the operatorto a given sounding signal f (x). The not necessarily rec-tangular support of the spreading function is known as the

Manuscript received April 13, 2012; revised April 23, 2013; acceptedNovember 2, 2013. Date of publication January 20, 2014; date of currentversion March 13, 2014. G. E. Pfander and P. Zheltov were supported by theGermany Science Foundation under Grant 50292 DFG PF-4.

The authors are with the School of Engineering and Science, JacobsUniversity Bremen, Bremen 28759, Germany (e-mail: [email protected]; [email protected]).

Communicated by A. M. Tulino, Associate Editor for Communications.Color versions of one or more of the figures in this paper are available

online at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TIT.2014.2301444

occupancy pattern, and its area as spread of the operator H .The fundamental restriction for the spread to be less thanone has been shown to be necessary and sufficient for theidentifiability of channels [1]–[4]. This extends results onclasses of underspread operators that are defined as those withrectangular occupancy pattern of area less than one.

This extension of the class of underspread channels tooperators with spread less than one is particularly of interestin the field of sonar communication [5] and in the multipleinput-single output channel settings. Acoustic channels possesslarger spreads than radar and wireless channels. This is dueto the speed of sound being magnitudes lower than that ofelectromagnetic waves, resulting in time delays up to severalseconds and Doppler spreads in the tens of Hertz for high-frequency channels [6]. Another type of channels with largevalues for the area of the occupancy pattern are multipleinput – single output (MIMO) channels; they combine severaldeterministic spreading functions into one channel, thus cov-ering a larger region of the time-frequency plane [7].

In recent work, the identifiability results [1], [2], [4],and [8], have been recast within the framework of operatorsampling [3], [9]. For example, Pfander and Walnut establishconcrete reconstruction formulas for deterministic operatorsthat satisfy the spread constraints mentioned above, and whichresemble the Whittaker-Shannon interpolation formula [9].In this paper, we develop operator sampling in the stochasticsetting and give analogous reconstruction formulas.

Taking into account the random nature of real-world com-munication environments, we model such channels with sto-chastic time-variant operators [1], [2], [10], [11]. In thissetting, the spreading function η(t, ν) of the operator H in (1)is a random process† indexed by (t, ν) that is to be recoveredfrom the output process H f (x) indexed by x . For the purposesof this paper, it would be enough to think of η(t, ν; ω) asa (t, ν)-indexed family of random variables on a commonprobability space �, or as a random process with instancesfrom L2(R2).

In the sibling paper [12] we develop and use the theory ofstochastic modulation spaces to rigorously define and proveidentification results for operators with spreading functionsbelonging to a class of generalized random processes—including delta functions and white noise. The norm inequal-ities that are proven there are essential to justify use of deltatrains as sounding signals. However, in this paper, we are goingto ignore these subtleties and treat distributions on par with

†Here, and in remainder of the paper, random functions and operators aredenoted by boldface characters, x is a complex conjugate of x , x∗ conjugatetranspose. E stands for expectation, and μ for both 2D area and 4D volume.

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2360 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 60, NO. 4, APRIL 2014

TABLE I

DEVELOPMENT OF FUNCTION AND OPERATOR SAMPLING. OPERATOR SAMPLING IS BASED ON PIONEERING WORK

ON THE IDENTIFICATION OF SLOWLY TIME-VARYING CHANNELS BY KAILATH [1] AND BELLO [11]

Lebesgue integrable functions, for ease of exposition only. Thefine points of the underlying mathematical analysis will bementioned in a few side remarks.

A. Operator Sampling Theory in the Historical Perspective

The progress of operator identification theory largely fol-lows the evolution of the related theory of function sampling.The two major directions of generalization are the introductionof stochasticity (stationary and non-stationary) and the removalof the requirement that the “bandlimitation” is rectangular. Forconvenience, we summarize this development in Table I.

For example, based on the classical Shannon-Nyquist sam-pling theorem, a corresponding result for bandlimited station-ary stochastic processes was proven in great generality byLloyd [15]. We cite it here in the form given in a classicbook by Papoulis.

Theorem 1. [18, p. 378] If a stationary process x(t)is bandlimited, that is, if its power spectral densityS(ξ) � Fτ→ξE{x(t)x(t + τ )} is integrable and supp S(ξ) ⊂[−�/2,�/2], then we can recover x(t) in the mean-squaresense from the samples taken at rate � = T−1. In fact,

x(t) = l. i. m.N→∞

N∑n=−N

x(nT)sin π�(t − nT)

π�(t − nT).

The requirements of stationarity and the bandlimitation ofthe spectrum to the symmetric interval were later relaxed byLee, Cambanis and Masry, and Kluvánek [14]–[16] and [19].

In his 1963 groundbreaking paper [1], Kailath realized thatfor a deterministic time-variant channel to be identifiable, it isnecessary and sufficient that the product �T of the maximumtime delay T and maximum Doppler spread � is not greaterthan one. Since, channels were called underspread whenever�T < 1, and overspread if �T > 1 [20]. The insight ofKailath has been generalized and formalized by Kozek andPfander [8].

Following in Kailath footsteps, the seminal paper of Bellolays the groundwork for channel sampling and characterizationtools and vocabulary [11]. In the sequel [2] Bello furtherargues that it is not the product �T that matters for iden-tification of a deterministic time-variant channel, but ratherthe spread, or the area of what he calls an occupancy pattern,

that is, the not necessarily rectangular support of the spreadingfunction supp η. In particular, Bello’s assertion has been putinto a rigorous mathematical framework and was proven usingnovel tools from Gabor analysis by Pfander and Walnut [4].Also, the ideas developed in [11] were used to estimate thecapacity of the channels with non-rectangular spread [21].

A brief comment of Kailath [1] suggests sufficiency of�T ≤ 1 for the identification of stationary stochastic channelsas well as deterministic, and Bello treats this question as a sidematter, more interested in developing the estimator for η(t, ν)when the output has been contaminated by additive noise.

As with the development of function sampling, a simplerstationary model for operator identification has seen mostresearch. The channel has the property of wide-sense sta-tionarity with uncorrelated scattering (WSSUS), that is, theautocorrelation function of η(t, ν) has the form

Rη(t, ν; t ′, ν′) = E{η(t, ν) η(t ′, ν′)

}= δ(t − t ′) δ(ν − ν′) Cη(t, ν).

In other words, taps at different delays are uncorrelated andstationary. The function Cη(t, ν) is known as the scatteringfunction of H . It completely characterizes the second-orderstatistics of η and represents the power spectral density of thetransfer function of the channel. This means that the scatteringfunction represents the expected behavior of the operator. Twocommon types of methods to identify the scattering functionare deconvolution and direct measurement methods [22]–[25].In [26] we apply the methodology developed here and in[12] to study WSSUS channels in depth. Theorem 16 belowguarantees identifiability of a WSSUS channel whenever itsscattering function is merely compactly supported. In [26] wegive reconstruction results for this specific case, and developand simulate an estimator for the scattering function.

In this paper, we address a more general problem of sto-chastic spreading function reconstruction and stochastic oper-ator sampling and identification for not necessarily WSSUSchannels.

B. Overview of the Paper

Deterministic identification results in [4] and [8], allow forthe recovery of the deterministic spreading function whenever

PFANDER AND ZHELTOV: SAMPLING OF STOCHASTIC OPERATORS 2361

its support has area less than one. We discuss this in detail inSection II-A. It is easy to see that in the case of a stochasticspreading function, such deterministic reconstruction formulasare still applicable to each random variate, allowing for therecovery of an instance spreading function η(t, ν; ω) from theresponse H(ω) f (x), where ω is an element of the samplespace �. However, on its own, each instance will provide littleinformation about the average behavior of the operator H .Secondly, it is possible that the 4-dimensional volume of thesupport of the 4D autocorrelation function Rη(t, ν; t ′, ν′) isless than one, while some instances η(t, ν; ω) have 2D areagreater than one.

The contributions of this paper follow. In Section II-Cwe consider the case when the 4-dimensional autocorrelationfunction Rη(t, ν; t ′, ν′) of the operators’ spreading functionis supported on a 4D region U = supp Rη(t, ν; t ′, ν′) thatcan be expressed as a tensor product of some 2D region Swith itself. In this scenario, we prove that it is possible—andgive an explicit reconstruction formula (10)—to recover thestochastic spreading function of the channel in the mean-square sense from the response H f of a channel to a periodicweighted delta train, provided that the set S occupies a regionof area less than one. This case will include the specialcase deterministic operators as their autocorrelation functionssatisfy Rη(t, ν; t ′, ν′) = η(t, ν) η(t ′, ν′).

In Section II-D the case of an arbitrary support regionU = supp Rη(t, ν; t ′, ν′) is considered. With some abuseof nomenclature, we will also say that we can (stochasti-cally) identify a stochastic operator if we can recover the4-dimensional autocorrelation function Rη(t, ν; t ′, ν′) (and notthe stochastic spreading function, as in the previous case) fromthe autocorrelation function of the response H f of the channelto a periodically weighted delta train (and not the stochasticresponse itself). In Theorem 12 we prove that in this sense wecan identify the operator H provided that the support patternof the autocorrelation function is permissible.

Analogous to the deterministic criterion μ(S) < 1, therequirement μ(U) < 1 is also necessary for stochasticidentifiability of a stochastic operator, as we show in ansibling paper [12]. In this paper, we demonstrate patterns thatcorrespond to regions of 4D volume less than one but arenonetheless unidentifiable by our methods.

Permissibility of a pattern is a geometric property linkingthe operator sampling theory to finite Gabor frame theory.After giving preliminary remarks on Gabor frames in finitedimensions in Section I-C below, in Section III we show thatthe support patterns of the autocorrelation functions of theoperators are in one-to-one correspondence with the columnsubsets of special Gabor frames. We provide a (partial) clas-sification of the autocorrelation patterns and analyze severalphenomena that emerge for Gabor frames of higher dimensionSections III-A and III-B.

C. Finite-Dimensional Gabor Frames

A deterministic operator is a particular case of the stochasticoperator with a degenerate probability distribution, so theresults for stochastic operator identification must necessarily

be compatible with the deterministic ones. In the theoryof deterministic operator identification, the existing operatoridentification proofs pivot on the so-called Haar property ofGabor frames. A finite-dimensional Gabor frame in C

L isdefined as

G � {MnTkc}L−1k,n=0, (2)

where the finite-dimensional translation operators Tk andmodulation operators Mn operating on a vector c ∈ C

L aregiven by

Tkc[p] = c[p − k] and Mnc[p] � e2π inp/L c[p]. (3)

A frame has the Haar property whenever its elements arein general linear position, that is, any subset of L elementsis linearly independent. A finite-dimensional Gabor frame Ggenerated by window c as defined in (2), has the Haar propertyfor almost every choice of window c, given that the ambientdimension L is prime [27].

Application of our methods to the stochastic case spawnsa more peculiar Gabor frame on ZL × ZL , the properties ofwhich differ in the key respect of linear independence of itssubsets. In particular, such a frame has small subsets that arelinearly dependent for any choice of window c, even when theparameter L is prime.

II. OPERATOR SAMPLING

A. Deterministic Operators

Equivalently to (1), any operator H acting on one-variablesignals can be represented with its symbols,

1) its time-varying impulse response h(x, t) =∫η(t, ν) e−2π iν(x−t) dν, then

(H f )(x) =∫

h(x, t) f (x − t) dt,

2) its kernel κ(x, t) = h(x, x − t) = ∫η(x − t, ν)e−2π iνt dν,

then(H f )(x) =

∫κ(x, t) f (t) dt,

3) and its Kohn-Nirenberg symbol σ(x, ξ) = Ft→ξ h(x, t),then

(H f )(x) =∫

σ(x, ξ) f̂ (ξ) e2π ixξ dξ.

All these symbols of H can be transformed into each otherusing partial Fourier transforms and area-preserving shears,such as I2 f (x, t) � f (x + t, x) [28]. In particular, thespreading function and the Kohn-Nirenberg symbol are relatedthrough

σ(x, ξ) � Fsη(t, ν),

where the symplectic Fourier transform is given by

(Fsη) (x, ξ) =∫∫

η(t, ν) e−2π i(νx−ξ t) dt dν.

In the following, it will sometimes be advantageous topresent the results with h, κ , or σ instead of η and wewill not hesitate to do so. However, we formulate our resultsprimarily with η due to its particularly simple relationship tothe short-time Fourier transform. On the Schwartz dual space


S ′ of tempered distributions, we define the short-time Fouriertransform to be

Vϕ f (t, ν) � 〈 f, MνTt ϕ〉for any ϕ in the Schwartz space S. Then for all f, ϕ ∈ S wehave the useful equality

〈H f, ϕ〉 = 〈η, V f ϕ〉.The inner product 〈·, ·〉 is taken to be conjugate linear in thesecond variable.

We will say that H belongs to an operator Paley-Wienerspace OPW(S) if the support of the spreading functionη(t, ν) = Fsσ(x, ξ) is contained in a compact subset S ofthe time-frequency plane,

OPW(S) �{

H : L2(R) → L2(R), such that

σ ∈ L2(R2) with suppFsσ ⊆ S, compact.}

Colloquially, we refer to H as “bandlimited” to S.The connection of the operator sampling theory to the more

established function sampling is best observed on the follow-ing theorem for Hilbert-Schmidt operators with the spreadingfunction supported on a rectangle in the time-frequency plane.An operator H : L2(Rn) → L2(Rn) is Hilbert-Schmidt, H ∈HS(Rn), whenever its spreading function η(t, ν) ∈ L2(R2n).

Theorem 2. [3, Theorem 1.2] For H ∈ HS(R) such thatsupp η(t, ν) ⊆ [0, T) × [−�/2,�/2] and �T ≤ 1 we have

‖H∑k∈Z

δkT‖L2(R) = T‖η‖L2(R2),

and H can be reconstructed by

κ(x + t, x) =∑n∈Z

(HIII)(t + nT)sin πT(x − n)

πT(x − n),

with convergence in L2(R2). Here and in the following, wedenote δa(t) � δ(t − a) and III(t) = ∑

k∈ZδkT(t).

Note that if H is a multiplication operator with a bandlimitedmultiplier, Theorem 2 reduces to the classic Shannon samplingtheorem.

Remark. In order to accommodate delta functions and othergeneralized functions as identifiers we require that η(t, ν) isa mapping from the dual M∞(R2) of the Feichtinger algebraM1(R2) into the Hilbert space of zero-mean random variablesRV(�). Here, the modulation space M1(Rn) is defined viathe finiteness of the norm ‖ f ‖M1(Rn) � ‖Vg f ‖L1(R2n) < ∞,where L1(Rn) is the space of Lebesgue integrable functions,and M∞(Rn) � (M1(Rn))′ is its continuous dual [28].

Functional analytic arguments of [3], [8], [12], [29] showthat whenever η(t, ν) is compactly supported (which is aphysically reasonable assumption taken here) it is possibleto extend the domain of H from L2 to the whole of M∞and further to the Wiener amalgam space W (FL∞(R2d ), 1).Therefore, η(t, ν) is well-defined as a linear functional on thecorresponding predual space W (FL1(R2d), ∞) that includescertain tempered distributions, in particular, weighted deltatrains IIIc = ∑

k∈Zck δkT as input for H .

Below, Theorem 5 gives a deep generalization of Theo-rem 2. It provides guarantees for recovery of those operatorswhose spreading function is supported on a set of arbitraryshape — not just a rectangle — as long as the area of thesupport is less than one. Additionally, it shows that it ispossible to replace sin x/x with smooth functions with betterdecay through the use of partitions of unity generated bycontinuous functions. For a, ε > 0 we say r(t) generates an(a, ε)-partition of unity {r(t + ak)}k∈Z whenever

r(t) = 0 if t ∈ (−ε, a + ε) and∑k∈Z

r(t + ak) ≡ 1.

Analogously, in R2 we say χ̃�(t, ν) generates an (a, b, ε)-

partition of unity if χ̃�(t, ν) = 0 outside of an ε-neighborhoodof � = [0, a) × [0, b) for some ε > 0 and

∑k, ∈Z

χ̃�(t +ak, ν + b ) ≡ 1.

Definition 3. The set S is (a, b,�)-rectified if it can becovered by L = 1

ab translations � � {(k j , n j )}L−1j=0 of the

rectangle � � [0, a) × [0, b) along the lattice aZ × bZ:

S ⊂L−1⋃j=0

� + (ak j , bn j ).

Lemma 4. For any compact set S of measure μ(S) < 1contained within [0, T) × [−�/2,�/2] there exist a, b > 0and � such that S is (a, b,�)-rectified, L = 1

ab is prime,a < 1

T and b < 1� .

Proof. This is a standard result from a theory of Jordandomains. A proof can be found in [30]. Any compact set withJordan content less than one can be covered with L rectanglesof cumulative area L · 1

L = 1, and it is easy to see that we donot lose generality requiring a, b to be small enough to satisfya < 1/T, b < 1/� and L prime. �

Theorem 5. [4, Theorem 3.1], [9] Let S ⊂ [0, T) ×[−�/2,�/2] ⊂ R

2 be a compact set with measure μ(S) < 1,such that S is (a, b,�)-rectified as in Definition 3 (with �Tnot necessarily smaller than 1). Then there exists a vectorc ∈ C

L, and the test signal

IIIc =∑

n

cn mod L δn/L ,

such that for any H ∈ OPW(S)

h(x + t, t) = aLL−1∑j=0

∑q∈Z

a j,q(H IIIc)(x − a(k j + q))

×r(x − ak j ) ϕ(t + a(k j + q)) e2π ibn j t

with convergence in L2(R2). Here, the coefficients a j,k areuniquely determined by the choice of {cn}, and r(t), ϕ(t) areany functions such that r(t − ak) and ϕ̂(γ − bn) are (a, ε)-(respectively, (b, ε)-) partitions of unity in time (respectively,frequency) domains, with ε > 0 dependent on S.

Since we can always rectify a compact S with measureμ(S) < 1, Theorem 5 holds for a wider class of regions,namely, of all regions S whose so-called Jordan outer contentis less than one [4].


B. Stochastic Operator Paley-Wiener Spaces

Let H be a stochastic operator with integral representation

H f =∫∫

η(t, ν) MνTt f dt dν,

and stochastic spreading function η(t, ν) a zero-mean randomprocess such that η(t, ν; ω) ∈ L2(R2) for all ω ∈ �. Wedenote the space of all such stochastic processes by StL2(R2).The autocorrelation of the spreading function is given by

Rη(t, ν; t ′, ν′) � E{η(t, ν) η(t ′, ν′)

}.

Definition 6. We say that H is a stochastic Paley-Wiener operator bandlimited to U , whenever the support ofRη(t, ν; t ′, ν′) is contained within a closed set U ⊆ R

4, thatis,

StOPW(U) �{

H : L2(R) → StL2(R) such that

supp Rη(t, ν; t ′, ν′) ⊆ U with U closed}.

In this section, we always assume U already rectified, asdone in the deterministic case.

The symmetries of the autocorrelation function require asymmetric rectification, defined below. However, it will notcause any confusion that whenever we speak of a rectified4D region, we always mean a symmetrically rectified one.

Definition 7. The set U is (a, b,�)-symmetrically rec-tified if it can be covered by the translations � �{λ j � (k j , n j , k ′

j , n′j )}L2−1

j=0of the prototype parallelepiped

�2 � [0, a)×[0, b)×[0, a)×[0, b) along the lattice (aZ×bZ)2

U ⊂L2−1⋃j=0

�2 + (ak j , bn j , ak ′j , bn′

j )

such that the 4D volume of �2 is small: ab = 1L , L is prime,

and the right-hand side is a symmetric set.It is easy to see that the above requirements imply that

the volume of U satisfies μ(U) ≤ 1, and conversely, it canbe shown with little work that any symmetric compact setwith Jordan content less than one can be rectified with asufficiently large L [30]. This restriction on the area is crucialfor operator identification. Before giving results for generaloperators whose autocorrelation functions are supported onarbitrary sets U ⊂ R

4, we look at the sets of a special kind,U = S × S. A general case will be considered in Section II-Dbelow.

C. Sampling of Stochastic Operators Supported on S × S

Under the special circumstance that U can be represented asa product S×S of some set S in R

2, or rather, if we assume Uto be (a, b,�)-symmetrically rectified, � = � × � for some� ⊂ Z × Z such that |�| = L, the operator—via its spreadingfunction—can be reconstructed directly from the output of theweighted delta train input with an explicit formula similar tothe one presented by Pfander and Walnut [31].

We define the non-normalized Zak transform Z : L2(R) →L2 ([0, aL) × [0, b)) as

Z f (x, ν) �∑n∈Z

f (x − anL) e2π iaLnν.

We say that a series of random processes∑

n∈Zxn(t) con-

verges to x(t) in mean-square sense

x(t)m.s.=

∑n∈Z

xn(t) ⇔ limn→∞ E

{|x(t) −N∑

n=−N

xn(t)|2} = 0.

Theorem 8. Let H ∈ StOPW(S×S) such that the compactset S ⊂ R

2 has measure μ(S) < 1. Then there exist L prime,a, b > 0 with ab = 1

L , a complex vector c ∈ CL and a

sequence {α j p}Lj,p=1 depending only on c such that we can

reconstruct H from its response to an L-periodic c-weighteddelta train IIIc = ∑

k∈Z

ck mod L δak via

η(t, ν)m.s.= aL

L−1∑j=0

L−1∑p=0

α j p e−2π ia(ν−bn j )(p+k j )χ̃�(t, ν)

× (ZaL H IIIc) (t + a(p − k j ), ν − bn j ),

where the translations χ̃�(t+ak, ν+bn) generate an (a, b, ε)-partition of unity in the time-frequency domain.This means that it is always possible to find c suchthat any two operators from StOPW(S × S) can bedistinguished from their response to the pilot signalIIIc = ∑

k∈Z

ck mod L δak .

Proof: Let a, b > 0, L – prime with ab = 1L be fixed

numbers to be chosen later. This choice will depend on thesupport of η(t, ν). Let c ∈ C

L ; indices of c should always beunderstood modulo L.

H IIIc(x) =∫∫

η(t, γ ) e2π iγ x IIIc(x − t) dt dγ

=∫∫

η(t, γ ) e2π iγ x∑k∈Z

ckδ(x − t − ak) dt dγ

=∫ ∑

k∈Z

ck η(x − ak, γ ) e2π iγ x dγ

=∑k∈Z

ck+p

∫η(x − a(k + p), γ ) e2π iγ x dγ,

with all the equalities holding ω-surely, and the last equalitybeing true for arbitrary p ∈ Z. Applying Zak transform toboth sides, we get

Z(H IIIc)(x, ν)

=∑n∈Z

H IIIc(x − anL) e2π ianLν

=∑

k,n∈Z

ck+p

∫η(x−a(k+p+nL),γ)e2π i(anLν+γ (x−anL))dγ ,

where we substitute k = k + nL. With the help of Poissonsummation formula

∑e2π inaLx = b

∑δ(aLx − n), we


continue

Z(H IIIc)(x, ν)

=∑k∈Z

ck+p

∫η(x−a(k+p),γ ) e2π iγ x

∑n∈Z

e2π ianL(ν−γ ) dγ

=∑k∈Z

ck+p

∫η(x−a(k+p),γ ) e2π iγ x b

∑n∈Z

δ(ν−γ−bn) dγ

= b∑

k,n∈Z

cp−kη(x − a(p − k), ν + bn) e2π i(ν+bn)x .

where in the last line we carry out integration in γ and replacen = −n, k = −k. Substitute t = x − ap and observe in theexponent (ν + bn)x = ν(t + ap) + bnt + n

L p. For brevity, wedefine Z p(t, ν) as

Z p(t, ν) � b−1 e−2π iν(t+ap) ZH IIIc(t + ap, ν)

=∑

k,n∈Z

cp−k e2π ipn/L η(t + ak, ν + bn) e2π ibnt︸︷︷︸ηk,n (t,ν)

. (4)

For all t, ν ∈ � and all ω ∈ � we then have a mixing matrixequation

Z(t, ν) =∑

k,n∈Z

MnTkc ηk,n(t, ν), (5)

where Z(t, ν) = [Z p(t, ν)]L−1p=0 is a vector-valued function on

(t, ν) ∈ �.By Lemma 4, the set S can be (a, b, �)-rectified, that is,

there exists a collection of indices � = {(k j , n j )}L−1j=0 such that

ηk,n(t, ν) ≡ 0 on (t, ν) ∈ � for any (k, n) ∈ �. Therefore, byLemma 9 below, the infinite sums in (5) can be trimmed tofinite sums to obtain

Z(t, ν) =∑

k,n∈Z

MnTkc ηk,n(t, ν)

m.s.=∑

(k,n)∈�

MnTkc ηk,n(t, ν)

= G∣∣�

η�(t, ν), (6)

where G is the L × L2 Gabor matrix of all time-frequencyshifts of c ∈ C

L given by (2), G∣∣�

is the submatrix of Gcorresponding to columns indexed by � mod L, and

η�(t, ν) =[ηk j ,n j (t, ν)

](k j ,n j )∈�

is a column vector of nonzero patches of η(t, ν) defined by (4).Without loss of generality, no two indices in � correspond

to the same column of G, that is, (k, n) = (k ′, n′) mod L ∈ �for all (k, n) = (k ′, n′) ∈ �. Such collisions can always beavoided by choosing a different rectification (a′, b′, �′) suchthat the entire support set S is within [0, a′L ′) × [0, b′L ′),which can always be achieved using Lemma 4. By assumption,|�| ≤ L, therefore, G

∣∣�

is invertible for some complex vectorc ∈ C

L .† Denote

A = (G∣∣�

)−1 = [α j p]L−1j,p=0

†In fact, for prime L the set of c ∈ CL such that every L × L submatrix

of G is invertible is a dense open subset of CL [27]. Furthermore, all entries

of c can be additionally chosen to have absolute value one.

the inverse of G∣∣�

. The solution to (6) can now be found,giving

ηk j ,n j (t, ν)m.s.= (

G∣∣�

)−1 Z(t, ν)

= aL e−2π iνtL−1∑p=0

α j p e−2π iνapZH III(t + ap, ν).

We can now combine the patches ηk,n(t, ν) into the wholespreading function η(t, ν). In the mean-square sense we have

η(t, ν)m.s.=

L−1∑j=0

η(t, ν)χ̃�(t − ak j , ν − bn j )

m.s.=L−1∑j=0

ηk j ,n j (t − ak j , ν − bn j ) e−2π ibn j (t−ak j )

×χ̃�(t − ak j , ν − bn j )

m.s.= aLL−1∑j=0

L−1∑p=0

α j p e−2π i(ν(t+a(p−k j ))−bn j ap)

×χ̃�(t, ν) · ZH IIIc(t + a(p − k j ), ν − bn j ).

Lemma 9. Let η(t, ν) be such that supp Rη(t, ν; t ′, ν′) ⊂S × S with S ⊂ R

2 that is (a, b, �)-rectified. Then for all(t, ν) ∈ [0, a) × [0, b),

L−1∑k=0

∑l,n∈Z

ak,n,l(t, ν) η(t − a(l L + k), ν + bn)

m.s.=∑γ∈�

aγ (t, ν) ηγ (t, ν),

where ηγ (t, ν) � η(t − a(l L + k), ν + bn).Proof: Denote Q = Z/LZ× LZ×Z the set of all indices

on the left hand side. Consider

E

{|∑γ∈Q

aγ ηγ −∑γ∈�

aγ ηγ |2}

=∑

γ1,γ2∈Q\�aγ1aγ2 E

{ηγ1ηγ2

}.

The rectification of Rη guarantees that

E{ηγ1(t1, ν1) ηγ2(t2, ν2)

} = 0

whenever either γ1 or γ2 are not in �. We conclude∑γ∈Q

aγ ηγ (t, ν)m.s.=

∑γ∈�

aγ ηγ (t, ν).

Remark. Since the autocorrelation functions of WSSUSoperators have the special form

Rη(t, ν; t ′, ν′) = δ(t − t ′) δ(ν − ν′) Cη(t, ν),

Theorem 8 is applicable to WSSUS operators whenever thearea of the support of the scattering function (also known asthe spread of the operator) satisfies μ(supp Cη(t, ν)) < 1.However, it is intuitively clear that it is excessive to covera 4D diagonal (a set of measure zero in R

4) with a single4D parallelepiped of volume one. Indeed, covering it with astring of small parallelepipeds (as in Fig. 2) is enough foridentification and allows recovery of Cη for supp Cη bounded,of arbitrary size. This comes as a corollary from Theorem 13and Theorem 16.


D. Stochastic Operators With Non-Tensored Support

In this section we proceed to the most general case ofstochastic operators. Consider an operator H with a spread-ing function η(t, ν) such that the autocorrelation functionRη(t, ν; t ′, ν′) is supported on some arbitrary bounded set Uin R

4. We will see that in general it is no longer possible toreconstruct η(t, ν) itself. For instance, this happens becausewith nonzero probability some instances η(t, ν; ω) may havespread larger than one, violating the necessary condition ofthe deterministic Theorem 5. Nevertheless, one may hope torecover Rη(t, ν; t ′, ν′) from the autocorrelation of the receivedinformation R f (t, t ′).

To prove Theorem 12 below, we will need to solve equationsof the form Y = G XG∗ with both X, Y positive semi-definite.† To this end, we need a standard technique from linearalgebra. Let vectorization vec : C

n×n → Cn2

be the linearisomorphism between the space of matrices C

n×n and thespace of column vectors C

n2given by stacking of the columns

(vec A)i = Ai mod n,�i/n�, i = 0, . . . , n2 − 1.

The following identity relating vectorization and theKronecker product of matrices is well known [32], [33]:

vec(AX Bt ) = (A ⊗ B) vec X. (7)

For an arbitrary matrix A, the set �A = {(λ, λ′) : Aλ,λ′ = 0}is called the support set of A. From the properties of positivesemi-definite matrices it is easy to see that

(λ, λ′) ∈ � implies {(λ′, λ), (λ, λ), (λ′, λ′)} ⊆ �. (8)

With some abuse of terminology, we call sets that may appearas support sets of positive semi-definite matrices, and hencesatisfy (8), positive semi-definite patterns or, for short, spdpatterns.

Lemma 10. Let � be a fixed finite spd pattern, � ⊆{(0, 0), . . . , (n − 1, n − 1)}, and let G ∈ C

n×m. The followingare equivalent:

(i) For each positive semi-definite Y ∈ Cm×m, there exists a

unique X ∈ Cn×n such that a) X is positive semi-definite,

b) supp X = �, and c) Y = G XG∗.(ii) If a Hermitian matrix N ∈ C

n×n with supp N ⊆ � solvesthe homogeneous equation 0 = G NG∗, then N = 0.

(iii) The matrix G ⊗ G∣∣�

has a left inverse (is full rank).Proof: (i) ⇒ (ii) By contraposition, let there be 0 =

N ∈ Cn×n such that G NG∗ = 0; let E be the diagonal

matrix whose λth diagonal entry is one if (λ, λ) ∈ � andzero else. Then there exists a positive real number C (forexample, Gershgorin circle theorem guarantees C = ‖N‖1 =sup1≤i≤m

∑mj=1|Nij | < ∞ will be enough) such that both

C E + N and C E are positive semi-definite, and G(C E +N)G∗ = G(C E)G∗ + G NG∗ = G(C E)G∗, thus violatingthe uniqueness of X in (i), as both supp(C E + N) andsupp N ⊆ �.

(ii) ⇒ (i) Suppose, that there exist X1 = X2, both positivesemi-definite, supported on �, such that G X1G∗ = G X2G∗.

†An hermitian matrix X ∈ Cn×n is positive semi-definite if for any a ∈

Cn , a∗Xa ≥ 0.

Then N = X1−X2 = 0 is necessarily Hermitian, supp N ⊆ �,and G NG∗ = 0, a contradiction.

(iii) ⇒ (ii) Let an Hermitian matrix N be supported on �and be a solution to 0 = G NG∗. Applying vectorization toboth sides of 0 = G NG∗, we get by (7)

vec 0 = (G ⊗ G) vec N, (9)

If G ⊗ G∣∣�

is invertible, (9) implies N = 0.(ii) ⇒ (iii) Let A ∈ C

m×m be an arbitrary square matrixsupported on � such that G ⊗ G

∣∣�

vec A = 0, that is,G AG∗ = 0. The Cartesian decomposition of A is given byA = H1 + i H2 with both H1, H2 Hermitian, defined by

H1 = 1

2(A + A∗), H2 = 1

2i(A − A∗).

It is easy to see that both supp H1, supp H2 ⊆ �, andG H1G∗ = G H2G∗ = 0, since G A∗G∗ = (G AG∗)∗ = 0.Therefore, by (ii), H1 = H2 = 0, and A = 0. It follows thatvec A = 0. Since vec A is arbitrary, it follows that the columnsof G ⊗ G

∣∣�

are linearly independent, that is, G ⊗ G∣∣�

is leftinvertible.

Definition 11. We would say that a spd support set � is apermissible pattern if some c ∈ C

L generates a Gabor frameG = [MnTkc]L−1

k,n=0 such that the equivalent conditions ofLemma 10 are satisfied. If an spd pattern is not permissible,it will be called defective.

This designation reflects the emergence of these patternsas those which permit the sampling of operators with deltatrains.

Theorem 12. Let H ∈ StOPW(U) such that U is(a, b,�)-rectified for some a, b > 0, |�| = L2, and ab =1/L. If some c ∈ C

L generates G = [MnTkc]L−1k,n=0 such

that the submatrix (G ⊗ G)|� is (left) invertible, then wecan reconstruct the autocorrelation of the spreading func-tion E

{η η}

from the autocorrelation R f = E{

f f}

of theresponse f = H IIIc of H to the L-periodic c-weighted deltatrain IIIc = ∑

k∈Z

ck mod L δak with the reconstruction formula

Rη(t, ν; t ′, ν′) = b−2L2−1∑j=0

L−1∑p,p′=0

α j,p,p′ e2π iab(n j p−n′j p′)

×∑

m,m′∈Z

e−2π ia(ν(k j +mL+p)−ν ′(k′j +m′ L+p′))

×R f (t − a(k j + mL − p), t ′

−a(k ′j + m′L − p′). (10)

Proof: Again, as in the deterministic case, we start withthe support already rectified in the sense of Definition 7.We proceed in the same manner as in the direct productcase until the mixing formula (5), which guarantees that forp = 0, . . . , L − 1 for all (t, ν) within the base rectangle� = [0, a) × [0, b)

Z(t, ν) =∑

k,n∈Z

MnTkc ηk,n(t, ν),


Taking autocorrelation on both sides, we get for all p, p′ =0, . . . , L − 1, and (t, ν), (t ′, ν′) ∈ �,

E{Z p(t, ν) Z p′(t ′, ν′)

}= E

{∑k,n∈Z

(MnTkc)pηk,n(t, ν)∑

k′,n′∈Z

(Mn′Tk′c)p′ηk′,n′(t ′, ν′)}

=∑

k,n,k′,n′∈Z

(MnTkc)p (Mn′Tk′c)p′E{ηk,n(t, ν) ηk′,n′(t ′, ν′)

}.

Let supp Rη(t, ν; t ′, ν′) be covered by L2 4D parallelepipedsindexed by � = {(k, n, k ′, n′)}, that is,

Rη(t, ν; t ′, ν′)=

∑(k,n,k′,n′)∈�

E{ηk,n(t − ak, ν − bn) ηk′,n′(t ′ − ak ′, ν′ − bn′)

}.

Without loss of generality, all (k, n) and (k ′, n′) are distinctmodulo L.† For brevity, let T j be the translation

T j f (t, ν; t ′, ν′) = f (t − ak j , ν − bn j , t ′ − ak ′j , ν

′ − bn′j ).

Let us denote Yp,p′ = E{Z p Z∗

p′}

and Xk,n,k′,n′ =E{ηk,n η∗

k′,n′}, where ∗ stands for conjugate transpose. As

covariance matrices of zero-mean random vectors, bothXk,n,k′,n′ and Yp,p′ are positive semi-definite, and conversely,every positive semi-definite matrix is a covariance matrix forsome random vector [18]. With this notation, we write for each(t, ν), (t ′, ν′) ∈ � a deterministic matrix equation

Y = G XG∗, (11)

where G = [MnTkc] is a Gabor matrix as in (2).To have a a unique positive semi-definite solution X of the

underdetermined system of equations (11) with an arbitrarypositive semi-definite Y , by Lemma 10 it is necessary andsufficient that G ⊗ G

∣∣�

is left invertible. Let A denote its leftinverse,

A � [α(k,n),(p,p′)]L−1k,n,p,p′=0 such that A(G ⊗ G

∣∣�) = Id.

Such A may exist only if |�| ≤ L2.The autocorrelation of the channel spreading function can

be recovered from X by observing (starting with the definition(4) of ηk,n),

Rη(t, ν; t ′, ν′)= E

{η(t, ν) η(t ′, ν′)

}

=L2−1∑j=0

T j X (t, ν; t ′, ν′)k j ,n j ,k′j ,n′

j e−2π ia(νk j −ν ′k′j )

=L2−1∑j=0

(A−1 vecT j Y (t, ν; t ′, ν′)k j ,n j ,k′j ,n′

j e−2π ia(νk j −ν ′k′j )

=L2−1∑j=0

L−1∑p,p′=0

α j,p,p′T j Yp,p′(t, ν; t ′, ν′) e−2π ia(νk j −ν ′k′j ),

(12)

†As in the deterministic case, the parameters a, b, L can always be chosenin such a way that supp Rη(t, ν; t ′, ν′) ⊆ [0, aL)×[0, bL) up to a translation.

Fig. 1. Different types of support sets of autocorrelation functions. (a) Tensorsquare. (b) Arbitrary.

where we construct Y from the autocorrelation of the receivedsignal f (t) = H IIIc(t) via

Yp,p′(t, ν; t ′, ν′)= E

{Z p(t, ν) Z∗p′(t ′, ν′)

}= b−2 e−2π ia(νp−ν ′ p′)

E{Z f (t + ap, ν)Z f (t ′ + ap′, ν′)

}= b−2 e−2π ia(νp−ν ′ p′)∑

m,m′∈Z

e−2π ib−1(mν−m′ν ′)

× E{

f (t + a(p − mL)) f (t ′ + a(p′ − m′L))}.

Translating and simplifying, we get

Yp,p′(t − ak j , ν − bn j ; t ′ − ak ′j , ν

′ − bn′j )

= b−2 e2π iab(n j p−n′j p′)∑

m,m′∈Z

e−2π ia((p+mL)ν−(p′+m′ L)ν ′)

× R f (t − a(k j + mL − p), t ′ − a(k ′j + m′L − p).

Upon substitution of the above back into (12), we get thedesired reconstruction formula (10). This completes the proofof Theorem 12.

If the support of Rη can be factored into a tensor square ofsome set of area less than one on the time-frequency plane,then Theorem 12 guarantees reconstruction of Rη from theautocorrelation R f of the received measurement H IIIc, whichis a weaker result than Theorem 8. To wit, assume that Rη issupported on a product set S × S such that S × S is (a, b,�)-rectified, and that the area of S is less than one. It followsthat � itself can be represented as a direct product of someindex set, say, � = � × � such that S is (a, b, �)-rectified.For some vector c the submatrix of its time-frequency shiftsG� is left invertible [4], [27]. Then (G ⊗ G)|� = G� ⊗ G� isalso left invertible, and Rη(t, ν; t ′, ν′) can be recovered fromR f (t, t ′).

The strength of Theorem 12 is that it allows to exploitinformation about more complicated dependencies betweenscatterers manifested in the geometry of the support of theautocorrelation function Rη(t, ν; t ′, ν′).

For example, consider the important case of the WSSUSoperators, already mentioned in the Section II-C after Theo-rem 8. Let Hbig be some WSSUS operator whose scatteringfunction Cη(t, ν) is supported on a 2D rectangle Sbig �[0, 1)×[0, L). This rectangle Sbig of area greater than one can


Fig. 2. Different types of support sets of autocorrelation functions, WSSUScase. (a) WSSUS. (b) WSSUS, rectified.

always be rectified with a collection �big of L2 translationsof a box [0, 1) × [0, 1

L ). From the WSSUS assumption weknow that Rη(t, ν; t ′, ν′) = 0 whenever (t, ν) and (t ′, ν′) arein different boxes, therefore, the 4D set supp Rη(t, ν; t ′, ν′) is(1, 1

L ,�big)-rectifiable, as shown in Fig. 2(b). It follows thatin the sense of Theorem 12, Hbig is identifiable whenever thediagonal set �big is a permissible pattern. Since, according toTheorem 16, this is always the case with any such diagonalset, we obtain the following corollary of Theorem 12.

Theorem 13. Let H ∈ StOPW(S × S) a WSSUS operatorwith S = supp C(t, ν) a compact set of arbitrary finitemeasure μ(S) < ∞. Then there exist L prime, L > μ(S),a, b > 0 with ab = 1/L, a complex vector c ∈ C

L anda sequence {α j p}L

j,p=1 depending only on c such that wecan reconstruct the scattering function C(t, ν) of H from theautocorrelation of its response to an L-periodic c-weighteddelta train IIIc = ∑

k∈Z

ck mod Lδak using (13).

Proof: For an arbitrary ε, it is always possible to finda, b > 0, 1

ab >√

μ(S) + ε such that the support of thescattering function S is covered by N translations of therectangle [0, a) × [0, b) (as usual, indexed by �), such thatμ(S) ≤ Nab ≤ μ(S) + ε. Without loss of generality, all(k, n) ∈ � are distinct modulo L = 1

ab . We cover the4D diagonal of the set S × S with N 4D parallelepipeds ofvolume 1/(ab)2, so the total volume of the cover is N

L2 ≤ 1. ByTheorem 12, there exists c ∈ C

L such that we can reconstructthe autocorrelation of the spreading function from the responsef = H IIIc using a version of (10) that takes advantage of thestationarity of H and periodicity of IIIc:

C(t, ν)

= b−2L2−1∑j=0

L−1∑p,p′=0

∑m∈Z

α j,p,p′ e2π ia[(p−p′)(ν+bn j )−νmL]

×R f (t − a(k j − p), t − a(k j + m′L − p′). (13)

III. PERMISSIBLE PATTERNS

The preceding discussion shows that it is ultimatelyimportant to study which spd patterns � are permissible, thatis, for which sets � there exists c ∈ C

L such that G ⊗ G∣∣�

Fig. 3. Tensor rank-1 autocorrelation a patterns for L = 2. Any rank-1 tensorpattern is permissible by (16).

is full rank. Consider the spreading function η(t, ν) of Hgiven by

η(t, ν) =L−1∑k=0

L−1∑n=0

ηk,n(t − ka, ν − nb).

with ηk,n(t, ν) ∈ L2(R2) supported on [0, a) × [0, b). Dueto the specifics of the underlying time-frequency analysis, itwill be convenient to index the matrices with the finite indexset J given by

J ={(0, 0), (0, 1), . . . , (0, L−1), (1, 0), . . . , (L−1, L−1)}.(14)

We call the autocorrelation pattern of η the indicator matrixACP ∈ {0, 1}J×J

ACP(λ, λ′) =

⎧⎪⎨⎪⎩

1, if for some (t, ν), (t ′, ν′) ∈ �,

E{ηλ(t, ν) ηλ′(t ′, ν′)

} = 0

0, otherwise,

where λ = (k, n), λ′ = (k ′, n′), λ, λ′ ∈ J . Clearly, ACP mustbe symmetric, moreover, just as in (8),

ACP(λ, λ′) = 1 implies

ACP(λ′, λ) = ACP(λ, λ) = ACP(λ′, λ′) = 1, (15)

and conversely, for any matrix A ∈ CJ×J that satisfies

(15), there exists an operator with the spreading functionηA(t, ν; t ′, ν′) whose autocorrelation pattern is A. We visualizeautocorrelation patterns with diagrams such as Fig. 3, whereshaded boxes correspond to nonzero correlation betweenpatches.

Lemma 10 indicates that the problem at hand is linked to theHaar property of Gabor systems. We explore this connection


Fig. 4. Other autocorrelation patterns for L = 2.

in more detail in Section III-C. The Haar property of a finite-dimensional Gabor frame reads that any L columns of ageneric Gabor matrix G are in general linear position. Bythe result from [27], this holds for almost every G with Lprime. Therefore, any deterministic spreading function sup-ported on L boxes of area 1/L on a time-frequency planecan be identified. In other words, any set � ⊂ J such that|�| = L indexes a subset of columns of G that is linearlyindependent for almost every seed vector c. This contrastswith the stochastic setting, where we will see that there existplenty of spd patterns that correspond to submatrices of G ⊗Gwhich are rank-deficient for every choice of c.

Generally, G ⊗ G can be viewed as a Gabor system on thenon-cyclic group ZL × ZL ∼= J , generated by the windowc ⊗ c,

G ⊗ G ={

π(k, n, k ′, n′)(c ⊗ c) :(k, n), (k ′, n′) ∈ J , π(k, n, k ′, n′) = MnTkc ⊗ Mn′

Tk′c}.

The Haar property of G ⊗ G would require all subsets ofG ⊗ G of order L2 to be linearly independent, which isnot the case for any prime number L ≥ 2. However, thepositive-definiteness of the autocorrelation function demandsthe autocorrelation pattern to satisfy property (15). This pre-cludes certain subsets of G ⊗ G from being tested for lineardependence. Below we show that this restriction implies thatfor L = 2 every spd pattern is permissible; not so for L ≥ 3.

A. Case L = 2

On Figs. 3 and 4 we show all possible spd patterns forL = 2. In addition, the last pattern on Fig. 4(b), corresponds toa linear dependent set of elements of G ⊗ G . However, due tolack of symmetry, such a set cannot appear as a support set forthe autocorrelation of the spreading function of the stochasticoperator. We mention it here to highlight the connection ofthe stochastic operator identification theory with the analysisof Gabor frames on general abelian groups, for this example,Z2 × Z2. Further properties of such frames, including thispattern, can be found in [34] and [35].

Interestingly, the only admissible patterns on L = 2 belongto one of the two types, shown on Figs. 3 and 4(a). The firsttype contains sets that can be represented as rank-1 products†

†We say that an element b ∈ S ⊗ S has tensor rank k if k is the smallestnumber with the property that there exist a1, a2, . . . , ak , b1, b2, . . . , bk ∈ Ssuch that � = a1 ⊗ b1 + · · · + ak ⊗ bk .

of sets of order L in J , that is, there exists a set � or order|�| = L such that � = � × �. Such product patterns inheritthe linear independence properties of their factors, since theHaar property of G guarantees

rank(G ⊗ G

∣∣�×�

) = (rank G�)2 = L2. (16)

The second type, all boxes on the diagonal, of maximalpossible tensor rank, is also a permissible pattern for L = 2,as we observe the determinant of the matrix

det(G ⊗ G

∣∣diag

) = 4(|c0|4 − |c1|4)(c20 c1

2 − c21 c0

2) = 0

or almost all choices of the generating atom c = [c0 c1]t .This concludes the analysis of the L = 2 case with a happy

ending:Proposition 14. Let a function Rη(t, ν; t ′, ν′) be supported

on a set of measure less than or equal to one covered with4 boxes �2 +(k j a, n j b, k ′

j a, n′j b) for some {(k j , n j , k ′

j , n′j ) ∈

J × J }3j=0, and some �2 = [0, a) × [0, b) × [0, a) × [0, b)

such that ab = 1/2 and all (k, n), (k ′, n′) distinct modulo 2.Let η(t, ν) be some function with Rη(t, ν; t ′, ν′) as its auto-correlation function, and H a stochastic channel with η itsspreading function. Then we can recover Rη(t, ν; t ′, ν′) fromthe response of H to the sounding signal

IIIc(t) = c0

∑m

δ(t − a − 2am) + c1

∑m

δ(t − 2am)

for almost all c = [c0 c1]t ∈ C2, |c0| = |c1| = 1.

B. Case L > 2

Classification of all autocorrelation patterns with L = 3already presents some challenges. For once, there are over5000 possible spd patterns

1 +(

9

7

)(7

2

)+(

9

5

)(2

(5

4

)+ 3

(5

3

))= 5796.

Here, 1 corresponds to the diagonal case, plus there are(9

7

)choices of 7 boxes on the diagonal, any two of which can becorrelated corresponding to the

(72

)factor, etc.‡

The problem of classifying all possible arrangementsof boxes becomes intractable very quickly with L.In Section III-C, we show that whenever some � is a per-missible pattern for almost all c, there is a related (via thepermutations of the Gabor elements) family of �σ that willbe permissible patterns for almost all c. Such observationssimplify the classification problem, although they do notsuffice to give a complete classification.

Secondly, alas, unlike L = 2, some spd patterns correspondto rank-deficient subsets of G ⊗ G . That is, there are sets thatsatisfy (8) but are not permissible, for example, see Fig. 5. Itturns out that there are structural reasons for patterns on Fig. 6and Fig. 5 to be permissible (respectively, defective).

With little effort it can be seen that patterns with tensorrank 2 (as on Fig. 5) can never be permissible, owing solely

‡The case of even number of boxes on the diagonal is always countedwithin the larger odd case; by symmetry considerations, every correlationbetween boxes adds two boxes on the field).


Fig. 5. According to Proposition 15, any rank-2 pattern is not a permissiblepattern.

to the properties of the tensor product space and not tothe specifics of the underlying Gabor structure. In fact, forany L, permissible patterns cannot contain two complete largesets of pairwise tensor products. (Here, such two sets are{η00, η01} and {η02, η10}, and in this context, large means that# {η00, η01} + # {η02, η10} = 2 + 2 > 3.)

Proposition 15. Let G ∈ Cn×m be an arbitrary matrix,

and G1, G2 its submatrices comprising columns of G indexed,respectively, by �1 and �2 such that �1 ∩ �2 = ∅. Denote� = (�1 × �1) ∪ (�2 × �2). If |�1| + |�2| > rank G, then theset of elements in G ⊗ G

∣∣�

is linearly dependent.Proof: Since |�1| + |�2| > rank G, it follows that there

exists [b1 b2]t ∈ ker[G1|G2] = 0, that is,

G1b1 + G2b2 = 0.

Now consider B1 = b1⊗b1, B2 = b2⊗b2. Both are Hermitian,and

G1 B1G∗1 − G2 B2G∗

2 = (G1b1)(G1b1)∗ − (G2b2)(G2b2)

∗

= (G1b1)(G1b1)∗ − (−G1b1)(−G1b1)

∗

= 0.

This means that we have found a Hermitian matrix

N =⎡⎣B1 0 0

0 B2 00 0 0

⎤⎦

supported on � such that G NG∗ = 0 and, hence, G ⊗ G∣∣�

is linearly dependent. It follows that � is not a permissiblepattern.

On the other hand, the patterns with maximal tensor rank(that is, diagonal patterns) will always be permissible. Thiscorresponds to the important case of WSSUS operators. Wegive here an elementary proof for finite dimensions. (See also[36, Theorem 3.1] for a discussion of this in the realm ofHilbert-Schmidt operators in infinite dimensions.)

Theorem 16. Denote π(k, n) � MnTk . Let G ={π(k, n)c}L−1

k,n=0 be a Gabor frame generated by c. Then foralmost all c ∈ C

L , the set G ⊗ G∣∣diag of all tensor products

of each Gabor element with itself

G ⊗ G∣∣diag � {π(k, n)c ⊗ π(k, n)c}L−1

k,n=0

is linearly independent, that is, for almost all c ∈ CL the

diagonal set�diag = {(k, n), (k, n)}L−1

k,n=0

is a permissible pattern.Proof: It is a straightforward observation that the linear

independence of the set G ⊗ G∣∣diag ∈ C

J×J is equivalentto the linear independence of the family of the correspondingfinite-dimensional Hilbert-Schmidt operators Pk,n : C

L → CL

given byPk,n x = 〈x, π(k, n)c〉π(k, n)c.

Clearly, Pk,n = (MnTk)P0,0(MnTk)∗. From the spreadingfunction representation of H : C

L → CL ,

H =N−1∑

p,q=0

η[p, q]MpTq

it is easy to see with the help of the commutation relationTkMn = e−2π ikn MnTk that for any H we have

(MnTk)H (MnTk)∗ =L−1∑

p,q=0

(M(k,n)η[p, q])MpTq ,

where M(k,n)η[p, q] � e2π i(qk−pn)η[p, q]. In particular, thespreading functions of Pk,n and P0,0 satisfy ηk,n = M(k,n)η0,0.Therefore, the linear independence of the family of operators{Pk,n}L−1

k,n=0 ⊆ HS(CL) is equivalent to the linear independence

of the family {M(k,n)η0,0} ⊆ CL2

.In finite dimensions, the short time Fourier transform of c

with respect to itself takes form

Vcc[p, q] = 〈c, MpTq c〉 =∑

r

e−2π ipr/L c[r ] c[r − q].

Here and in what follows, the summation is over ZL and allthe indices of vectors c, x ∈ C

L are taken modulo L. Wecompute

P0,0x[m] = 〈x, c〉 c[m] =∑

q

x[q] c[q] c[m]

=∑

q

x[m − q] c[m − q] c[m]

=∑

q

x[m − q]∑

r

δ[r − m] c[r − q] c[r ]

=∑

q

x[m − q]∑

r

(1

L

∑p

e2π ip(r−m)/L

)c[r − q] c[r ]

=∑

p

∑q

( 1

L

∑r

e−2π ipr/L c[r ] c[r − q]) e2π imp/L x[m − q]

=∑

p

∑q

(1

LVcc[p, q]

)MpTq x[m],

which proves that η0,0[p, q] = 1L Vcc[p, q].

We claim that for almost every c ∈ CL , Vcc has full support.

For a fixed pair p, q ∈ ZL × ZL , the set of solutions Z p,q ={c ∈ C

L : Vcc [p, q] = 0} is a manifold of zero measurein C

L . Hence, Vcc has full support almost everywhere, namelywhenever c ∈ C

L \ ⋃p,q Z p,q . The proof is complete by


Fig. 6. By Theorem 16, all diagonal patterns are permissible for almostall c.

Fig. 7. Defective autocorrelation pattern on L = 5. Total area 16/25 leadsto a 16 × 25 matrix of rank 13.

observing that whenever η0,0 has full support, the family{M(k,n)η0,0}(k,n)∈ZL×ZL is linearly independent in C

L2.

Another source of deficiency that begins to appear only withL ≥ 4 is illustrated on Fig. 7.

Proposition 17. Let the height (or, equivalently, the width)of the pattern � exceed L, or, formally, for some λ0 ∈ �,

#{λ′ : (λ0, λ′) ∈ �} > L .

Then � is not the permissible pattern. The matrix G ⊗ G∣∣�

is singular for all c ∈ CL.

Proof: Let � = {λ′ : (λ0, λ′) ∈ �}. Then |�| > L,

therefore, the set G∣∣�

is linear dependent. Hence, we can finda nontrivial vector a ∈ ker G

∣∣�

. We now have (G ⊗ G)a =G ⊗ Ga = 0. That is, we have found a nontrivial linearcombination

∑i∈� ai gi = 0, and

∑i∈�

ai gi ⊗ g1 =∑i∈�

ai (gi ⊗ g1) = 0.

The theorem immediately follows from Lemma 10 byobserving that whenever there exists a non-trivial A supportedon �, not skew-Hermitian and such that G A = 0, (here,A = [

a 0]), then a non-trivial Hermitian matrix A + A∗

satisfies G(A + A∗)G∗ = 0.The existence of defective spd sets shows that one must

be careful when trying to reconstruct the autocorrelation ofthe spreading function by sampling an operator with the delta

train. Luckily, the task of weeding out defective patterns isuncoupled from the sampling procedure. All defective patterns�bad can be discovered numerically and inexpensively bytesting a Gabor matrix generated by a random vector c. Itis easy to see that the rank deficiency of G ⊗ G

∣∣�

will (withprobability one) indicate whether � is good or bad.

Remark. Theorem 12 gives a procedure to recover E{η η}

from the autocorrelation E{

f f}

of the response f =H IIIc of H to an L-periodic c-weighted delta train IIIc =∑

k∈Zck mod L δak . The result is based on finding a permissible

rectification of the set U . If some rectification satisfies thesymmetry conditions for autocorrelations, but is not permis-sible, one can seek alternative (possibly finer) rectificationswith the hope that one of them is permissible. However,Propositions 15 and 17 indicate that for some sets U thereare no permissible rectifications, that is, Theorem 12 doesnot apply to StOPW(U) of such U . Indeed, for example,if for some (t0, ν0), we have a 2D measure of a sliceμ(E{η(t0, ν0) η(t ′, ν′)}) > 1, then every rectification will havethe defect discussed in Proposition 17, and our procedure torecover Rη(t, ν; t ′, ν′) is not applicable.

Similarly, if for two sets S1, S2 we have μ(S1)+μ(S2) > 1and (S1 × S1) ∪ (S2 × S2) ⊆ U . Consider U be(a, b,�)-symmetrically rectified, and S′

1 be the induced(a, b, �1)-rectification of S1, and S′

2 be the induced(a, b, �2)-rectification of S2 \ S′

1. (Note that �2 is not emptyunless S2 ⊂ S′

1, but then μ(S′1) > 1, and the rectification

is again defective by Theorem 17). Clearly, �1 ∩ �2 = ∅,(�1 × �1) ∪ (�2 × �2) ⊂ �, and μ(S′

1) + μ(S′2) ≥

μ(S1) + μ(S2) > 1, and Proposition 15 applies.This observation does not imply that there exist symmetric

U with 4D volume less than one and the property that notest signal g allows to recover E

{η η}

from the autocorrelationE{

Hg Hg}. In fact, recovery may be possible using a different

type of a pilot signal, for example, a non-periodic delta train.

C. Equivalence Classes of Permissible Patterns

Let SJ be the symmetry group of all permutations on thegroup J of order L2 defined in (14).

Definition 18. We call two spd patterns � and �σ on thegrid J × J homologous whenever there exists a permutationσ ∈ SJ such that

(λ, λ′) ∈ � ⇔ (σ (λ), σ (λ′)) ∈ �σ .

Two homologous patterns � and �σ are equivalent if foralmost all choices of the window c ∈ C

L generating G,

rank G ⊗ G∣∣�

= rank G ⊗ G∣∣�σ .

For example, the patterns on Fig. 8 are all homologous, withthe first two provably equivalent, according to Theorem 19below.

Numerical evidence shows that linear independence of thesubsets of the tensored Gabor frame G ⊗ G is invariant underhomology in the sense of Definition 18, for example, all thepatterns on Fig. 8 are permissible, as tested on a large numberof randomly chosen c. Also, for L = 3, 5, and 7, entire orbitsof patterns {�σ }σ∈SJ are permissible.


Fig. 8. Three homologous patterns for L = 3.

Additional indirect support to the hypothesis that any twopatterns that are homologous with respect to the permutationof the Gabor atoms have the same linear independence statusfor almost all c comes from the equivalence of all patterns forthe case L = 2 (see Fig. 3) and the universality of TheoremsTheorem 15, 16, 17 with respect to the permutations of theGabor atoms.

We will describe the permutations σ ∈ SJ such that� and �σ are provably equivalent for any � for almost allc ∈ C

L . Clearly, all such permutations form a subgroup ofSJ . It remains an open question whether this subgroup is thewhole group SJ .

For sake of notational simplicity, let us identify the atomsof a Gabor frame G generated by a window c ∈ C

L withthe elements of the torus ZL × ZL . The (k, n)th atom g =MnTkc = π(k, n) c would correspond to (k, n) ∈ ZL × ZL ,where finite-dimensional operators M and T are defined in (3),and π(k, n) = MnTk .

Proposition 19. Let A denote the group of permutationsσ ∈ SJ such that � and �σ are equivalent for all spd patterns� ⊆ J for all c ∈ C

L, except maybe for a set of measurezero.

a) translations of the torus σa(k, n) � (k + q, n + p) mod Lbelong to A for all p, q ∈ Z.

b) reflection of the torus σb(k, n) � (−n, k) mod L belongsto A.

c) rotation of the torus σc(k, n) � (k,−n) mod L belongsto A.

For an illustration of these transformations in the case ofZ4 × Z4, see Figs. 9 and 10.

Proof: Note that the linear independence of the subsetG� of elements of G is invariant under scaling the atomsby scalars pλ, complex conjugation, and any invertible lineartransformation T . Let G = [MnTkc]L−1

k,n=0. First, we constructseveral Gabor frames closely related to G.

a) It is easy to see that the Gabor frame Ga constructed fromthe atom ca = ϕa(c) � MpTq c is a column permutationof G up to constant multiples of columns by C = e2π ipk .

Fig. 9. Actions of the time-frequency shift (a) and conjugation (b) aretranslation and horizontal reflection, respectively.

Fig. 10. Action of the Fourier transform is 90◦ rotation.

Indeed,

π(k, n) ca = MnTk(MpTqc)

= Mn e2π ipkMpTkTqc

= e2π ipkMn+pTk+q c

= Caπ(k + q, n + p)c.

b) Similarly, constructing the Gabor frame Gb with the Gaborwindow being the Fourier transform of the atom cb =ϕb(c) � Fc produces a Fourier image of the original Gabormatrix. This holds since the Fourier transform commuteswith time-frequency shifts in the following sense:

π(k, n)cb = MnTkFc

= F(T−nMkc)

= F( e2π iknMkT−nc)

= Cb Fπ(−n, k)c.

The Fourier transform is a unitary operation which actson the Gabor matrix from the left, hence it does notchange the linear independence of the columns, scalingof columns by the constant Cb notwithstanding. In otherwords, Gb is a frame that is unitarily equivalent to aparticular permutation of G, namely, it corresponds torotating the torus, see Fig. 10 [37], [38].Curiously, the vertical flip operation Ic j � c− j also createsan equivalent pattern. This can be understood by observingI = F2. This corresponds to the reflection of both rowsand columns of the torus ZL × ZL .


c) Another operation that preserves linear independence isconjugation. Let the Gabor frame Gc be generated by thewindow cc = ϕc(c) � c. Then Gc contains all the atoms ofG in the order that corresponds to the horizontal reflectionof the torus group on itself. Note that taking conjugationdoes not destroy the linear independence of the vectors.

Now let � be any spd pattern. From the above constructionsof the Gabor frames Gα, α = a, b, c, it is evident that for allgenerating atoms cα � ϕα(c) ∈ C

L , rank Gα ⊗ Gα

∣∣�σα =

rank G ⊗ G∣∣�

. For any α = a, b, c the transformationϕα : c �→ cα is a bijective mapping C L → C

L , therefore,if � is permissible for some c, then �σ is also permissible(for some other c), and moreover, if � is permissible for allc ∈ C

L except a zero set Z , then �σ is also permissiblefor all c ∈ C

L except a possibly different zero set Zσ . Andconversely, if � is defective, then �σ is also defective. Sincethere are finitely many permutations σ , the union of all badsets

⋃σ∈SJ Zσ has measure zero, and it follows that � and

�σ are equivalent.

IV. CONCLUSION

We show that a large class of stochastic operators canbe reconstructed from the stochastic process resulting fromthe application of the operator to an appropriately designeddeterministic test signal. Our results are based on combiningrecently developed sampling results for deterministic bandlimited operators [3], [4], [8] and the sampling theory ofstochastic processes [16], [18]. The classes of stochasticoperators considered here are characterized by supportconstraints on the autocorrelation of the stochastic spreadingfunction of the operator. Our results do not necessitatethe frequently used WSSUS condition, nor the so-calledunderspread condition on operators. Moreover, we show thatin some cases when the operator cannot be fully determinedfrom its action on a test signal, still, the second orderstatistics of the spreading function, that is, its autocorrelationcan be determined from the operator output. While in thedeterministic case the possibility of operator reconstructiononly depends on the support size of the spreading function,we show that in the stochastic case geometric considerationsplay a fundamental role for reconstruction.

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Götz E. Pfander (M’05) received the Ph.D. degree in Mathematics fromthe University of Maryland, College Park, in 1999. He is a Professor ofMathematics at Jacobs University Bremen, Germany since 2002. His interestsinclude harmonic analysis, wavelet and Gabor theory, and signal processing.

Pavel Zheltov received the B.S. degree in Applied mathematics and computerscience from the Moscow State University, Russia, in 2004 and the Ph.D.in Mathematics from the University of South Carolina, in 2010. He iscurrently a Postdoctoral Fellow with the BUTLER group at Jacobs UniversityBremen, Germany. His research interests include Gabor analysis, wirelesscommunications, signal processing, and learning theory.

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