On Discrete Time Wilson Systems

Kohli, Teena; Panwar, Suman; Kaushik, S. K.

doi:https://doi.org/10.1155/2020/8426897

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Wavelets and Wavelet-Based Numerical Methods

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Research Article | Open Access

Volume 2020 | Article ID 8426897 | https://doi.org/10.1155/2020/8426897

On Discrete Time Wilson Systems

Teena Kohli,¹Suman Panwar,²and S. K. Kaushik³

Academic Editor: Yongqiang Fu

Received13 Aug 2020

Revised04 Nov 2020

Accepted13 Nov 2020

Published07 Dec 2020

Abstract

In this paper, we define the discrete time Wilson frame (DTW frame) for and discuss some properties of discrete time Wilson frames. Also, we give an interplay between DTW frames and discrete time Gabor frames. Furthermore, a necessary and a sufficient condition for the DTW frame in terms of Zak transform are given. Moreover, the frame operator for the DTW frame is obtained. Finally, we discuss dual pair of frames for discrete time Wilson systems and give a sufficient condition for their existence.

1. Introduction

The idea of frame as a redundant peer of a basis was originated in 1952 by Duffin and Schaeffer [1]. It came to limelight only with the historic paper of Daubechies et al. [2]. A sequence of vectors is termed as a frame (or Hilbert frame) for a separable Hilbert space if there exist constants such that

The positive numbers and are termed as lower and upper frame bounds of the frame, respectively. The bounds may not be unique. If , then is called an -tight frame, and if , then is said to be a Parseval frame. The inequality in (1) is recognized as the frame inequality of the frame .

A sequence of vectors is called a Riesz basis if is complete and there are positive constants and such that

Gabor frame for (which is a Riesz basis) has bad localization properties in either time or frequency. Thus, a system to replace Gabor systems which do not have bad localization properties in time and frequency was required. Wilson [3, 4] suggested a system of functions which are localized around the positive and negative frequency of the same order. The idea of Wilson was used by Daubechies et al. [5] to construct orthonormal “Wilson bases” which consist of functions given bywith a smooth well-localized window function . For such bases, the disadvantage described in the Balian–Low theorem is completely removed. Independent of the work of Daubechies et al. [5], orthonormal local trigonometric bases consisting of the functions , where , were introduced by Malvar [6], where window functions are assumed to be compactly supported, and only two immediately neighbouring windows are allowed to have overlapping support. Some generalizations of Malvar bases were studied in [7, 8]. To obtain more freedom for the choice of window functions, biorthogonal bases were investigated in [9]. A drawback of Malvar’s construction is the restriction on the support of the window functions. Therefore, it was preferred to consider Wilson bases of Daubechies et al. [5].

Feichtinger et al. [10] proved that Wilson bases of exponential decay are not unconditional bases for all modulation spaces on including the classical Bessel potential spaces and the Schwartz spaces. Also, Wilson bases are not unconditional bases for the ordinary spaces for , as shown in [10]. Approximation properties of Wilson bases are studied by Bittner [11], and Wilson bases for general time-frequency lattices are studied by Kutyniok and Strohmer [12]. Generalizations of Wilson bases to nonrectangular lattices are discussed by Sullivan et al. [3], with motivation from wireless communication and cosines modulated filter banks. Wojdyllo [13] studied modified Wilson bases and discussed Wilson system for triple redundancy in [14]. Discrete time Wilson frames with general lattices are studied by Lian et al. [15]. Motivated by the fact that one has different trigonometric functions for odd and even indices, Bittner [11, 16] considered Wilson bases introduced by Daubechies et.al [5] with nonsymmetrical window functions for odd and even indices. This generalized system of Bittner was later studied extensively by Kaushik and Panwar [17–19] and Jarrah and Panwar [20].

In this article, we consider the system defined by Bittner [16] to define the discrete time Wilson frame (DTWF) and give examples for its existence. Some observations related to properties of discrete time Wilson frames are given. Also, a relationship between DTW frames and the discrete time Gabor frames is discussed. Furthermore, a necessary and a sufficient condition for the DTW frame in terms of Zak transform are obtained and the frame operator for the DTW frame is constructed. Finally, dual pair of frames for discrete time Wilson systems is defined and a sufficient condition for its existence is given.

The discrete time Wilson (DTW) system associated with is defined aswhere , and .

The DTW system given by (4) can be rewritten for any as

Remark 1. For , the DTW system has the formwhere , and .

2. Outline of the Paper

In this article, we define discrete time Wilson frames (DTW frames) and discuss various properties of DTW frames (see Observations (I) to (VIII)). An interplay between DTW frames and discrete time Gabor frames has been given in Theorem 1. Also, a necessary and a sufficient condition for the DTW frame in terms of Zak transform are given in Theorem 3 and 4, respectively. The construction of the frame operator for the DTW frame is discussed in Theorem 5. Finally, we discuss dual pair of frames for discrete time Wilson systems and give a sufficient condition for its existence. Various examples are given to illustrate the discussion.

3. Discrete Time Wilson Frames

In this section, we define the discrete time Wilson frame based on the Wilson system considered by Bittner [11, 16], explore their existence through examples, and investigate various properties including its relationship with discrete time Gabor systems. We begin with the following definition.

Definition 1. The discrete time Wilson system:where is as defined in (4) and is called a discrete time Wilson frame (DTWF) if there exist constants satisfyingThe constants and are called lower and upper frame bounds for the DTWF . The supremum of all lower frame bounds and the infimum of all upper frame bounds are called optimal lower and optimal upper frames bounds, respectively.
In case the system satisfy only the right-hand side of inequality (8), then the system is called a discrete time Wilson Bessel sequence for .
In order to show the existence of discrete time Wilson Bessel sequences which are not DTWF for , we give the following examples.

Example 1. (i)Let . Then, Therefore, we obtain Hence, is a discrete time Bessel sequence for with Bessel bound . However, it is a DTW frame if and only if .(ii)Let Then, we have Hence, is a discrete time Bessel sequence for with Bessel bound . Furthermore, it is not a frame as it does not satisfy the lower frame condition for . Moreover, note that(1)If , then is a DTW frame with frame bounds and .(2)If , then is a DTW frame with frame bounds and .Next, we give examples of Wilson systems which are discrete time Wilson frames for .

Example 2. (i)Let Then, using the fact that we have Therefore, is a discrete time Wilson frame for .(ii)Let Note that Therefore,Hence, is a DTW frame for .
In view of the above discussion, we have the following observations in relation to DTW frames. (I)Let and let be the translation operator on , where and . Then, Indeed, it follows from the fact that (II) Let be a DTW system for . Then, Indeed, one can compute that In view of Observations (I) and (II), we obtain (III). (III) Let be a DTW system for . Then, Using Observations (II) and (III), we have (IV). (IV) For all , (V) If for the DTW system , then for all , and we obtain (VI) Let and be two DTG Bessel sequences with Bessel bounds and , respectively. Then, the system is a DTW Bessel sequence with Bessel bound . Indeed, using observation (III) and the hypothesis, we have

Remark 2. The converse of observation (VI) may not be true even if additionally we assume that the system is a frame for .

Example 3. Let , , and . Then, Thus, the systems and are not DTG frames for . Now, using observation (III), we obtain Hence, is a DTW frame for with frame bounds 8 and 16. (VII) Let be such that Then, the system is a DTW Bessel sequence for with Bessel bound . (VIII) If are functions having bounded support, then the DTW system is a DTW Bessel sequence for .Indeed, one may perceive that, since the functions and have bounded support, and as defined in observation (VII) are finite, and hence the DTW system is a DTW Bessel sequence for .
Now, we prove a result related to DTW systems for the particular case when .

Lemma 1. For , we have

Proof. Using observation (V), we obtainHence, we computeIn the following result, we give an interplay between the DTW frame and DTG frame for .

Theorem 1. The Wilson system is a DTW frame for if and only if is a DTG frame for .

Proof. Let and be the positive constants such thatThen, using Lemma 1, it is easy to conclude that is a DTG frame for with frame bounds and .
Conversely, let be a DTG frame for . Then, there exist positive constants and such thatAgain, by utilizing Lemma 1, we deduce that is a DTW frame with frame bounds and .
Now, we define discrete time tight Wilson frame for and investigate their relationship with discrete time Gabor frame for .

Definition 2. The discrete time Wilson system given by (4) is called a discrete time tight Wilson frame (DTTWF) if there exists a constant such thatIf , then the frame is called Discrete Time Parseval frame.
Next, we state two results whose proofs can be worked out using Lemma 1.

Proposition 1. Let be a DTTW frame for with frame bound . Then, is a DTG frame with frame bounds and .

Proposition 2. Let be a DTTG frame for with frame bound . Then, is a DTW frame for with frame bounds and .

4. Discrete Zak Transform and Discrete Time Wilson Frames

Various properties of the Zak transform (continuous version) were studied by Janssen [21,22] and the discrete version is discussed by Heil [23] who gave the following definition of discrete Zak transform.

Definition 3 (see [23]). The discrete Zak transform of a sequence is given bywhere is a fixed parameter and is the dual group of .
Next, we state a result related to Zak transform proved by Heil [23].

Theorem 2 (see [23]). Given a fixed and . If , then the system is a frame for with frame bounds and .
Now, we give a necessary condition for DTW frame in terms of the discrete Zak transform.

Theorem 3. Let . If is a DTW frame for with frame bounds and , then

Proof. Since is a DTW frame for with frame bounds and , using Theorem 1, the system is a DTG frame for with frame bounds and .
Hence, the result follows using Theorem 2.
Towards, the converse of Theorem 3, we have the following result.

Theorem 4. Let . If there exists and such that the following inequality holdsthen is a DTW frame for with frame bounds and .

Proof. It can be worked out using Theorem 1 and Theorem 2.

Remark 3. For , the system is not a frame for .

5. Discrete Time Wilson Frame Operator

The frame operator for a frame is constructed by the composition of two important operators, namely, the analysis operator and the synthesis operator. The frame operator is positive, bounded, invertible, and self-adjoint. It ensures the existence of a canonical dual frame of a given frame, i.e., if is a frame and is the frame operator, then is a frame called the canonical dual of the frame . It is known that the canonical tight frame leads to a perfect reconstruction when used for both analysis and synthesis. Keeping this in mind, we make an attempt to construct the frame operator for the discrete time Wilson frame. We begin with the following definition.

Definition 4. Let be a discrete time Bessel sequence for . Then, DTWF operator is defined asIn the following result, we construct the frame operator for the discrete time Wilson frame with the help of the frame operators of the two associated discrete time Gabor Bessel sequences.

Theorem 5. For , let and be DTG Bessel sequences with frame operators and , respectively. Then, the frame operator for DTW system is given by , where

Proof. By hypothesis, we have and . Since and are DTG Bessel sequences, we obtainAlso, using observation (VI), we deduce that the systems and are DTG Bessel sequences and the system is a DTW Bessel sequence with their frame operators denoted by , , and , respectively. Then, for all , we obtainwhereNow, for all , we computeThus, Similarly, it can be proved that .
Hence, we conclude that

6. Dual Pair of Frames for Discrete Time Wilson Systems

In this section, we study dual pair of frames and obtain a sufficient condition for the existence of a dual pair of discrete time Wilson systems. First, we state the definition of a dual pair of frames discussed by Christensen [24, 25].

Definition 5 (see [25]). Let be a Hilbert space and let , , , and be Bessel sequences. Then, and are dual pair of frames ifIn the following result, we give a sufficient condition for the existence of a dual pair of discrete time Wilson systems.

Theorem 6. Let and let and be two DTW Bessel sequences for . Then, there exist DTW Bessel sequences and such that and are dual pair of frames for .

Proof. Let and be the preframe operators for the DTW Bessel sequences and , respectively. Then, and are given bywhere . Then,Also, the operator is bounded on . Furthermore, . Let and be a pair of DTW dual frames for . Then, using Proposition 2.1 of [25], we computeAlso, using Lemma 6.3.2 of [24], we deduce that and form a dual pair of frames for if is a DTW Bessel sequence. Now, observe that is a DTW system given by . Since is a DTW Bessel sequence and is a bounded operator, is a DTW Bessel sequence.
Finally, we prove a result related to compact support of functions generating DTW Bessel sequences.

Theorem 7. Let , and let and be two DTW Bessel sequences for . If the functions , and are compactly supported, then the functions , and are also compactly supported.

Proof. Suppose that and be such that , and be compactly supported. Then, in view of the proof of Theorem 6, one can conclude that the functions , and are compactly supported if and are compactly supported.
By assumption, , and are all compactly supported. Therefore, there exists an such thatSince is compactly supported. Similarly, is compactly supported.

7. Conclusion

Gabor frame for (which is a Riesz basis) has bad localization properties in either time or frequency. Wilson [3, 4] suggested a system of functions which are localized around the positive and negative frequency of the same order. Based on the Wilson systems, Wilson frames for were introduced and studied in [17–20]. In this article, discrete time Wilson frames (DTWF) are defined and their relationship with discrete time Gabor frames is investigated. Also, frame operator for the DTWF has been constructed. Finally, keeping duality in mind, dual pair of frames for the discrete time Wilson systems have been studied.

Data Availability

No data were used to support the findings of the study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright

Copyright © 2020 Teena Kohli 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.

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