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Localization Operators and an Uncertainty Principle for the Discrete Short Time Fourier Transform
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.
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 with discrete Fourier transform let and be the cardinals of their supports, then Meshulam  obtained a generalization for nonabelian groups. Moreover, those for which one has the equality were characterized (see [2, 4, 5]). When 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 . 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 [8–13] 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 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 . The importance of localization operators in this context is due to the fact that, for a given subset of is a fixed point of the localization operator with window whenever is supported in . In Theorem 3 we characterize those finite sequences and such that the cardinal of the support of 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 and are periodic, the short time Fourier transform 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 space on a finite abelian group to the 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 (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 denote the cyclic group in which addition is performed modulo . If is any complex-valued function on , we define the Fourier transform by the formula From now on, we identify with and we endow it with the Euclidean norm. The translation operator , is the unitary operator on given by . Similarly, the modulation operator , is the unitary operator defined by We have . The short-time Fourier transform of with respect to the window is given by ([17–19])
That is, It is well-known that This identity means that is a tight frame for whenever .
If is a set we denote by its cardinal. For we represent by the cardinal of the set . Clearly, 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 for every . 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 ). Let be a subset with cardinal . Then
Proof. From the definition we get Hence from where the conclusion follows.
Our next aim is to investigate under which conditions the support of 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 can be recovered from as
Definition 2. Let and be given such that . Then, the localization operator is defined by is a filtered version of , as only those values of corresponding to entries in are considered. Clearly whenever is supported in . Let us denote by the vector space endowed with the -norm and by the canonical basis.
The main difficulty in the proof of the following theorem is that, a priori, and are arbitrary finite sequences of complex numbers. That is, we cannot assume that, after normalizing, and for some -valued and .
Theorem 3. Let one assume that . Then if, and only if, and there are such that , , and
Proof. For every the function is different from zero; hence there is such that . Consequently, the hypothesis means that the support of is a set Without loss of generality we can assume that . We now consider the discrete localization operator , It is well known that Since then and we finally obtain where This implies that there is such that According to Cauchy-Schwarz inequality, for every there is such that and . That is, which implies We now check that is constant. In fact, for every , Since the support of coincides with we conclude that the Fourier transform of vanishes at any coordinate . Hence is constant and we conclude After replacing , and by , and in the previous identity we obtain We take and conclude that does not depend on . That is, is constant (modulo ) and there is such that We put and take in (22) to obtain In order to simplify we normalize the function so that We take in (22) to obtain that is, Then gives Finally, we consider Then which implies Since we conclude That is, . Now, using that , The necessary condition is proved. In order to show the sufficiency, let us consider with and Then where Consequently, That is, using that , we have that
According to the previous result, the support of is highly regular for instance when does not vanish and .
Next we prove that the periodicity of and is related to the fact that the short time Fourier transform is supported in a certain subgroup.
Proposition 4. Let and be given. Then, the following conditions are equivalent.(1)There is such that and(2)The support of is contained in .
Proof. (1) implies (2) is obvious and we only prove (2) implies (1). Without loss of generality, we can assume that . We take and consider the localization operator defined by Condition (2) implies ; hence That is, On the other hand, and Consequently, From (44), Cauchy-Schwarz inequality, and condition we finally conclude that, for some , Hence In particular, by Cauchy-Schwarz inequality, there is with such that That is, for every . From we obtain . Since we can proceed as before and obtain for some with . Finally, from the fact that is different from zero for some we conclude . That is, 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 gives rise to a time-frequency basis if the translations of form an orthonormal basis and this basis is equal to for some constants .
If generates a time-frequency basis it must be biunimodular; that is, , but the opposite does not hold. Biunimodular vectors and vectors generating time-frequency basis are easily characterized in terms of the support of .
Proposition 6. Let be given. Then (1) is biunimodular if and only if , (2)a multiple of generates a time-frequency basis if and only ifwhere is a permutation.
Proof. (1) Follows from and .
(2) Let us first assume that generates a time-frequency basis. Accordingly, and there is a permutation such that and Then which is nonzero only for (mod ).
To show the converse implication we first note that implies . As one has for some , from where for some . Since then . Therefore is constant. We now deduce for every . In particular, we can write and we obtain Proceeding by recurrence we finally obtain and then it is easy to see that for some constants . Finally, as for we get that 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 be given. Then a multiple of generates a time-frequency basis if and only if there exist and such that is a primitive -root of unity, and
Proof. Let us assume that generates a time-frequency basis and . According to Theorem 3, there exist and such that , and We put . From the proof of the same result we have if and only if (modulo ). Hence, Proposition 6 gives that the map , is injective, or equivalently, and have no common prime divisors. That is, is a primitive -root of unity.
Corollary 8. Let be a prime number. Then if and only if satisfies one of the following conditions:(1). (2). (3)(A multiple of) generates a time-frequency basis.
Proof. We assume . If we can apply Theorem 2.5 to conclude that with . If it is a primitive -root of the unity and therefore generates a time-frequency basis. In case .
If then, by , we have ; hence as before either generates a time-frequency basis or the support of is a singleton.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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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