AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 131459 10.1155/2014/131459 131459 Research Article Localization Operators and an Uncertainty Principle for the Discrete Short Time Fourier Transform Fernández Carmen Galbis Antonio Martínez Josep Peris Alfredo Departamento de Análisis Matemático Universidad de Valencia Burjasot 46100, Valencia Spain uv.es 2014 1322014 2014 09 12 2013 02 01 2014 13 2 2014 2014 Copyright © 2014 Carmen Fernández et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Localization operators in the discrete setting are used to obtain information on a signal f from the knowledge on the support of its short time Fourier transform. In particular, the extremal functions of the uncertainty principle for the discrete short time Fourier transform are characterized and their connection with functions that generate a time-frequency basis is studied.

1. Introduction

It is a well-known fact that a nontrivial function and its Fourier transform cannot be simultaneously well localized, and many variants of this vague statement are collected under the term uncertainty principle. We refer to  and the references therein for different versions. For functions defined on finite abelian groups, localization is generally expressed in terms of the cardinality of the support of the function. For instance, Donoho and Stark  proved the following: given a finite sequence (xi)i=1d-1 with discrete Fourier transform (x^i)i=1d-1 let Nt and Nw be the cardinals of their supports, then (1)Nt·Nwd. Meshulam  obtained a generalization for nonabelian groups. Moreover, those x for which one has the equality were characterized (see [2, 4, 5]). When d is a prime number the result was improved by Tao , who showed that the sum of the number of nonzero entries in a finite sequence and the number of nonzero entries in its Fourier transform are strictly larger than d. An extension to abelian groups of finite order is given in .

Results of this type for joint time-frequency representations in the continuous case are obtained in  among others, and for functions defined on finite abelian groups in , where it is proved that the cardinality of the support of any short time Fourier transform of a nontrivial function defined on a finite abelian group is bounded below by the order of the group. Ghobber and Jaming obtained in  an uncertainty principle for the representation of a vector in two bases, which permitted in particular to get a quantitative version of the above-mentioned result by Krahmer et al. . In Theorem 1 we present a different proof of the quantitative result in  improving the constant there obtained.

Our aim is to obtain information on a signal f from the knowledge on the support of its short time Fourier transform. To this end we introduce the localization operators in the discrete setting, which provide a filtered version of the original signal f. The importance of localization operators in this context is due to the fact that, for a given subset Λ of d×d,f=(fi)i=0d-1 is a fixed point of the localization operator LΛ,g with window g=(gi)i=0d-1 whenever Vgf is supported in Λ. In Theorem 3 we characterize those finite sequences f and g such that the cardinal of the support of Vgf is minimum; that is, we characterize the extremal functions of the uncertainty principle for the discrete short time Fourier transform.

Additional relevant information about the functions can be derived from the support of the short time Fourier transform. For instance, it is easy to see that when f and g are periodic, the short time Fourier transform Vgf is supported in a certain subgroup. We will see in Proposition 4 that a sort of converse also holds. Here again localization operators are used in the proof. In a recent paper, Gilbert and Rzeszotnik  calculated the norm of the Fourier transform from the Lp space on a finite abelian group to the Lq space on the dual group and they studied the points at which the norm is attained. In connection with this problem they consider the functions on the group such that the set of all their translations coincides (up to some constants) with the set of all their modulations and produce an orthonormal basis. In the case of cyclic groups, these functions are easily characterized (up to constants) by the support of Vff (Proposition 6). In particular, for groups of prime order these are essentially the extremal functions of the uncertainty principle for the discrete short time Fourier transform.

2. The Results

Let d denote the cyclic group d=/d in which addition is performed modulo d. If f:d is any complex-valued function on d, we define the Fourier transform f^:d by the formula (2)f^(k)=1d=0d-1f()e-2πik/d. From now on, we identify d with d and we endow it with the Euclidean norm. The translation operator Tj,jd, is the unitary operator on d given by (Tjf)()=f(-j). Similarly, the modulation operator Mk,kd, is the unitary operator defined by (3)(Mkf)()=e2πik/df(). We have Mkf^=Tkf^. The short-time Fourier transform of fd with respect to the window gd is given by () (4)(Vgf)(j,k)=1d=0d-1f()g(-j)¯e-2πik/d,j,kd.

That is, (5)(Vgf)(j,k)=(f·Tjg¯)^(k)=1df,MkTjg. It is well-known that (6)(j,k)d2|(Vgf)(j,k)|2=f2g2. This identity means that {MkTjg:(j,k)d2} is a tight frame for d whenever gd{0}.

If A is a set we denote by |A| its cardinal. For fA we represent by f0 the cardinal of the set suppf={aA:f(a)0}. Clearly, ·0 is not a norm.

The first result we present was given in  as a quantitative version of a result of Krahmer et al.  which states that Vgf0d for every f,gd{0}. Its proof was an adaptation of a method that was originally developed in [11, 12] in the continuous setting. We give a different proof improving the estimate.

Theorem 1 (Ghobber and Jaming [<xref ref-type="bibr" rid="B15">15</xref>]).

Let S{0,1,,d-1}2 be a subset with cardinal |S|<d. Then (7)fg1(1-(|S|/d))((j,k)S|(Vgf)(j,k)|2)1/2.

Proof.

From the definition we get (8)|(Vgf)(j,k)|1dfg. Hence (9)f2g2=(j,k)S|(Vgf)(j,k)|2+(j,k)S|(Vgf)(j,k)|2(j,k)S|(Vgf)(j,k)|2+1df2g2|S|, from where the conclusion follows.

Our next aim is to investigate under which conditions the support of Vgf has the smallest possible cardinal. The proof of the result that follows is based on an analysis of the discrete localization operators. These operators where introduced by Daubechies  in the continuous case in order to localize a signal both in time and frequency.

It is well known that f can be recovered from Vgf as (10)f()=g-21d(j,k)d2(Vgf)(j,k)g(-j)e2πik/d.

Definition 2.

Let gd and Λd2 be given such that g=1. Then, the localization operator LΛ,g:dd is defined by (11)(LΛ,g  f)()=1d(j,k)Λ(Vgf)(j,k)g(-j)e2πik/d.LΛ,gf is a filtered version of f, as only those values of Vgf corresponding to entries in Λ are considered. Clearly LΛ,gf=f whenever Vgf is supported in Λ. Let us denote by d1 the vector space d endowed with the 1-norm and by {en}n=1d the canonical basis.

The main difficulty in the proof of the following theorem is that, a priori, f and g are arbitrary finite sequences of complex numbers. That is, we cannot assume that, after normalizing, f()=e2πi(u()/d) and g()=e2πi(v()/d) for some d-valued u and v.

Theorem 3.

Let one assume that f0+g0>d. Then Vgf0=d if, and only if, f(0)0 and there are a,λ,c such that ad=(-1)d-1, λd=cd=1, and (12)f()=ac(-1)/2f(0),g()=(λa)c(-1)/2g(0).

Proof.

For every jd the function f·Tjg¯ is different from zero; hence there is kd such that (Vgf)(j,k)0. Consequently, the hypothesis Vgf0=d means that the support of Vgf is a set (13)Λ={(j,kj):jd}. Without loss of generality we can assume that g2=1. We now consider the discrete localization operator LΛ,g:d1d1, (14)(LΛ,gh)()=1d(j,k)Λ(Vgh)(j,k)g(-j)e2πik/d. It is well known that (15)LΛ,g:d1d1=maxn=0d-1|(LΛ,gen)()|=1dmaxn=0d-1|j=0d-1g(n-j)¯g(-j)e-2πi(kj(n-)/d)|1. Since LΛ,gf=f then LΛ,g:d1d11 and we finally obtain (16)maxn1d=0d-1|g,gn|=1, where (17)g(j)=g(-j)e2πikj/d. This implies that there is 0d such that (18)|g,g0|=1d. According to Cauchy-Schwarz inequality, for every d there is μ() such that |μ()|=1 and g=μ()·g0. That is, (19)g(-j)g(0-j)=μ()·e-2πikj(-0)/d,d, which implies (20)g(-j)g(n-j)=μ()μ(n)·e-2πikj(-n)/d,n,j. We now check that μ¯·f is constant. In fact, for every nd, (21)(Vgf)(j,k)=1d=0d-1f()g(-j)¯e-2πik/d=g(n-j)¯μ(n)¯e-2πikjn/d1d×=0d-1f()μ()¯e2πi(kj-k)/d=g(n-j)¯μ(n)¯e-2πikjn/d  μ¯·f^(k-kj). Since the support of Vgf coincides with Λ we conclude that the Fourier transform of μ¯·f vanishes at any coordinate kd{0}. Hence μ¯·f is constant and we conclude (22)g(-j)g(n-j)=f(n)¯f()¯·e-2πikj(-n)/d,n,j. After replacing ,n, and j by +1,n+1, and j+1 in the previous identity we obtain (23)e-2πikj+1(-n)/df(n+1)¯f(+1)¯=e-2πikj(-n)/df(n)¯f()¯. We take =n+1 and conclude that (24)e2πi(kj+1-kj)/d does not depend on j. That is, kj+1-kj is constant (modulo d) and there is p such that (25)kj+1=k0+pj(modulod). We put λ=e-2πik0/d and take n=j=0 in (22) to obtain (26)g()·f()¯=λf(0)¯g(0). In order to simplify we normalize the function f so that (27)f(0)¯g(0)=1. We take n=j in (22) to obtain (28)g(-j)=e-2πi(k0+pj)(-j)/df(j)¯f()¯g(0), that is, (29)λ-jf(-j)¯=λ-je-2πipj(-j)/df(j)¯f()¯g(0),f()=f(-j)f(j)e2πipj(-j)/dg(0)¯. Then j=1 gives (30)f()=f(-1)f(1)e2πip(-1)/dg(0)¯. Finally, we consider (31)a=f(1)g(0)¯,b=λa¯,c=e2πip/d. Then (32)f()=ac-1f(-1)d, which implies (33)f()=ac(-1)/2f(0)d. Since f()=f(+d) we conclude (34)adcd(d-1)/2=1. That is, ad=(-1)d-1. Now, using that |a|=|c|=1, (35)g()=λf()¯=(λa¯)c(-1)/21f(0)¯=(λa)c(-1)/2g(0). The necessary condition is proved. In order to show the sufficiency, let us consider a,λ,c with λd=cd=1,ad=(-1)d-1 and (36)f()=ac(-1)/2,g()=(λa)c(-1)/2. Then (37)(Vgf)(j,k)=(λa)jd=0d-1(λ¯)c(2j-j2-j)/2e-2πik/d=(λa)jdc(-j2-j)/2=0d-1rj,k, where (38)rj,k=λ¯cje2πik/d. Consequently, (39)(Vgf)(j,k)0rj,k=1. That is, using that λ¯=e2πim/d, we have that (40)(Vgf)(j,k)0k=m+jp(modulo  d).

According to the previous result, the support of Vgf is highly regular for instance when g does not vanish and Vgf0=d.

Next we prove that the periodicity of f and g is related to the fact that the short time Fourier transform Vgf is supported in a certain subgroup.

Proposition 4.

Let d=pq and f,gd{0} be given. Then, the following conditions are equivalent.

There is λ such that λp=1 and(41)f(+q)=λf(),g(+q)=λg(),d.

The support of  Vgf is contained in d×pd.

Proof.

(1) implies (2) is obvious and we only prove (2) implies (1). Without loss of generality, we can assume that g2=1. We take Λ=d×pd and consider the localization operator LΛ,g:d1d1 defined by (42)(LΛ,gh)()=1d(j,k)Λ(Vgh)(j,k)g(-j)e2πik/d. Condition (2) implies LΛ,gf=f; hence (43)LΛ,g:d1d11. That is, (44)maxn=0d-1|(LΛ,gen)()|1. On the other hand, (Vgen)(j,k)=1dg(n-j)¯e-2πi(nk/d) and (45)(LΛ,gen)()=1d(j=0d-1g(n-j)¯g(-j))×(k=0q-1e-2πi((n-)k/q)). Consequently, (46)(LΛ,gen)()=0whenevernmoduloq,(LΛ,gen)()=1pj=0d-1g(n-j)¯g(-j)whenevern=moduloq. From (44), Cauchy-Schwarz inequality, and condition g2=1 we finally conclude that, for some nd, (47)m=0p-11p|j=0d-1g(n-j)¯g(n+mq-j)|=1. Hence (48)|j=0d-1g(n-j)¯g(n+mq-j)|=1for everym=0,1,,p-1. In particular, by Cauchy-Schwarz inequality, there is λ with |λ|=1 such that (49)g(n+q-j)=λg(n-j),j=0,1,,d-1. That is, g(+q)=λg() for every =0,1,,d-1. From g()=g(+d)=λpg() we obtain λp=1. Since |Vgf(j,k)|=|Vfg(-j,-k)| we can proceed as before and obtain f(+q)=μf() for some μ with μp=1. Finally, from the fact that (50)(Vgf)(j,k)=1d(=0q-1f()g(-j)¯e-2πi(k/q))×(m=0p-1(λ¯μ)me-2πi(mk/p)) is different from zero for some kpd we conclude λ¯μ=1. That is, λ=μ and the proposition is proved.

In connection with the norm attaining points of the Fourier transform the following definition was given in .

Definition 5.

We say that f=(fi)i=0d-1d gives rise to a time-frequency basis if the translations of (1/d)f form an orthonormal basis and this basis is equal to {ck(1/d)Mkf:kd} for some constants ck𝕋.

If f generates a time-frequency basis it must be biunimodular; that is, |fi|=|f^i|=1, but the opposite does not hold. Biunimodular vectors and vectors generating time-frequency basis are easily characterized in terms of the support of Vff.

Proposition 6.

Let fd{0} be given. Then

f is biunimodular if and only if  Vff(0,k)=Vff(k,0)=δk,0,

a multiple of f generates a time-frequency basis if and only if(51)suppVff={(j,π(j)):j=0,,d-1},

where π:{0,,d-1}{0,,d-1} is a permutation.

Proof.

(1) Follows from Vff(0,k)=|f|2^(k) and Vf^f^(j,k)=Vff(k,j).

(2) Let us first assume that fd generates a time-frequency basis. Accordingly, |fi|=1 and there is a permutation π:{0,,d-1}{0,,d-1} such that π(0)=0 and (52)f(-j)=λje-2πiπ(j)/df(),|λj|=1. Then (53)(Vff)(j,k)=λj¯d=0d-1e-2πi(k-π(j))/d, which is nonzero only for k=π(j) (mod d).

To show the converse implication we first note that (Vff)(0,0)=(1/d)f2>0 implies π(0)=0. As Vff(j,k)=fTjf¯^(k) one has (54)fTjf¯^=λjeπ(j) for some λj, from where (55)f()f(-j)¯=μje2πiπ(j)/d for some μj. Since π(0)=0 then |f()|2=μ0. Therefore |f()| is constant. We now deduce (56)|μj|=|f()f(-j)|=μ0 for every j. In particular, we can write (57)μ1=μ0e-2πir1 and we obtain (58)f()f(-1)¯=μ0e-2πir1e2πiπ(1)/d,d. Proceeding by recurrence we finally obtain (59)f()=f(0)e-2πi(r1+k1(+1))/2d, and then it is easy to see that for some constants cj𝕋,Tjf=cjMjk1f. Finally, as Vff(j,0)=0 for j0 we get that {Tjf:j=1,,d-1} is an orthogonal basis, and this basis coincides up to constants with the set of all modulations.

As an application we recover [16, Theorem 4.5].

Corollary 7.

Let fd{0} be given. Then a multiple of f generates a time-frequency basis if and only if there exist a and c such that ad=(-1)d-1,c is a primitive d-root of unity, and (60)f()=ac(-1)/2f(0),d.

Proof.

Let us assume that f generates a time-frequency basis and f(0)=1. According to Theorem 3, there exist a and c such that ad=(-1)d-1,cd=1, and (61)f()=ac(-1)/2f(0),d. We put c=e-2πip/d. From the proof of the same result we have (Vff)(j,k)0 if and only if k=pj (modulo d). Hence, Proposition 6 gives that the map dd,jpj, is injective, or equivalently, p and d have no common prime divisors. That is, c is a primitive d-root of unity.

Corollary 8.

Let d be a prime number. Then Vff0=d if and only if f satisfies one of the following conditions:

f0=1.

f^0=1.

(A multiple of) f generates a time-frequency basis.

Proof.

We assume d>2. If 2f0>d we can apply Theorem 2.5 to conclude that f()=f(0)ac(-1)/2 with ad=cd=1. If c1 it is a primitive d-root of the unity and therefore f generates a time-frequency basis. In case c=1,f^0=1.

If 2f0d then, by , we have 2f^0>d; hence as before either f^ generates a time-frequency basis or the support of f^ is a singleton.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research of C. Fernández and A. Galbis was partially supported by MEC and FEDER Projects nos. MTM2010-15200 and GVA Prometeo no. II/2013/013. The research of J. Martínez was supported by MEC Project no. MTM2008-04594.

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