Mathematical Problems in EngineeringVolume 2011 (2011), Article ID 303460, 9 pagesdoi:10.1155/2011/303460

Research Article

A Note on Operator Sampling and Fractional Fourier Transform

Qingyue Zhang

Department of Mathematics and LPMC, Nankai University, Tianjin 300071, China

Received 28 June 2011; Accepted 12 July 2011

Academic Editor: Blas M. Vinagre

Copyright © 2011 Qingyue Zhang. 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.

Abstract.

This paper presents that the kernel of the fractional Fourier transform (FRFT) satisfies the operator version of Kramer's Lemma (Hong and Pfander, 2010), which gives a new applicability of Kramer's Lemma. Moreover, we give a new sampling formulae for reconstructing the operators which are bandlimited in the FRFT sense.

1. Introduction and Notations

Sampling theory for operators motivated by the operator identification problem in communications engineering has been developed during the last few years [1–4]. In [4], Hong and Pfander gave an operator version of Kramer's Lemma (see [4, Theorem 25]). But they did not give any explicit kernel satisfying the hypotheses in [4, Theorem 25] other than the Fourier kernel. In this paper, we present that the kernel of the fractional Fourier transform satisfies the hypotheses in [4, Theorem 25]. Therefore, we give a new applicability of Kramer's method.

The FRFT—a generalization of the Fourier transform (FT)—has received much attention in recent years due to its numerous applications, including signal processing, quantum physics, communications, and optics [5–7]. Hong and Pfander studied the sampling theorem on the operators which are bandlimited in the FT sense (see [4]). In this paper, we generalize their results to bandlimited operators in the FRFT sense.

For , its FRFT is defined by where , and the transform kernel is given by where is Dirac distribution function over , , and . The inverse FRFT is the FRFT at angle , given by where the bar denotes the complex conjugation. Whenever , (1.2) reduces to the FT. Through this paper, we assume that .

In FRFT domain, the function space with bandwidth is defined by For the sake of simplicity, when , is written as .

In the following, we use the notation if there exist positive constants and such that for all objects in the set .

Let be a Hilbert space and be a sequence in . The set is said to be a frame [8, 9] for if

Let with . is a set sampling for if

2. The Properties of the Kernel of FRFT

In this section, we consider under what conditions is a frame for for every . The following theorem gives a necessary and sufficient condition for to be a frame for for every .

Theorem 2.1. For any and . is a frame for if and only if is a frame for .

Remark 2.2. By Theorem 2.1, when taking appropriate , is a frame for each . Therefore, we give a kernel satisfying the hypotheses in [4, Theorem 25], which gives a new applicability of Kramer's Lemma.

To prove Theorem 2.1, we need to introduce the following results.

Lemma 2.3. is a set of sampling for if and only if is a frame for for every .

Proof. Suppose that is a set of sampling for . Then, for any , there exists such that . Since we have Therewith, is a frame for for any .On the other hand, suppose that is a frame for for any . Specifically, is also a frame for . Then, by (2.1), This completes our proof.

The following proposition gives a necessary and sufficient condition about is set of sampling for .

Proposition 2.4 (see [10, Lemma 3.5]). is set of sampling for if and only if is a frame for .

Lemma 2.5. is set of sampling for if and only if is a frame for .

Proof. Since where denotes the FT operator, we have By Proposition 2.4, this completes the proof.

Proof of Theorem 2.1. By Lemmas 2.3 and 2.5, we immediately get the claim.

3. Sampling of Operators Related to FRFT

In this section, a new sampling formulae for operator is proposed. First we introduce some definitions and notations about sampling of operator.

The class of Hilbert-Schmidt operators consists of bounded linear operators on which can be represented as integral operators of the form with kernel . Let . We call the time-varying impulse response of . If , then the operator norm of is defined by .

The Feichtinger algebra is defined by where is the short-time Fourier transform of with respect to the Gaussian . An operator class is identifiable if all extend to a domain containing a so-called identifier and The operator class permits operator sampling if one can choose in (3.3) with discrete support in in the distributional sense. In that case, is called sampling set for .

For , let

The following theorem states that a bandlimited operator in FRFT sense permits operator sampling.

Theorem 3.1. For and , choose with on and with and r = 1 on . Then permits operator sampling as and operator reconstruction is possible by means of the -convergent series

Before we give the proof of Theorem 3.1, the following two propositions are needed.

Proposition 3.2 (see [4, Theorem 8]). For and , choose with on and with and r = 1 on . Then permits operator sampling as and operator reconstruction is possible by means of the -convergent series

Proposition 3.3 (see [11, Lemma 1]). Assume a signal . Let Then .

Proof of Theorem 3.1. Due to (2.4), we have By the proof of Proposition 3.2, Applying Proposition 3.3, we obtain where Using Proposition 3.2 again, Therewith,

Next, we give an important multichannel operator sampling theorem, namely, derivative operator sampling. Checking the proof of [4, Theorem 32], we have following lemma.

Lemma 3.4. Let and denotes the th derivative of in the distributional sense. Then, and operator reconstruction is possible by means of the -convergent series where is a Riesz basis for for each fixed , and .

For the operators which are bandlimited in FRFT sense. We have the following theorem.

Theorem 3.5. Let and denotes the th derivative of in the distributional sense. Then, and operator reconstruction is possible by means of the -convergent series where is a Riesz basis for for each fixed and .

Proof. Due to (2.4), we have By Proposition 3.3, we obtain where . Put By Lemma 3.4, we have where By the proof of Lemma 3.4, (3.18) holds. Moreover, by (3.21) we obtain

Remark 3.6. After careful development of pertinent tools, one can formulate extensions of results in this paper to the linear canonical transform case (see [11–13]). We have presented the FRFT case because of its simplicity and applicability.

4. Conclusion

Kramer's Lemma is very important in the proofs of a number of sampling theorems. In [4, Theorem 25], Hong and Pfander proved an operator sampling version of Kramer's Lemma. But they did not give any explicit kernel satisfying the hypotheses in [4, Theorem 25] other than the Fourier kernel. In this paper, we find that the kernel of the fractional Fourier transform satisfies the hypotheses in [4, Theorem 25]. This observation gives a new applicability of Kramer's method. Moreover, we give a new sampling formulae for reconstructing the operators which are bandlimited in the FRFT sense. This is an extension of some results in [4].

References

W. Kozek and G. E. Pfander, “Identification of operators with bandlimited symbols,” SIAM Journal on Mathematical Analysis, vol. 37, no. 3, pp. 867–888, 2005.

G. E. Pfander and D. F. Walnut, “Operator identification and Feichtinger's algebra,” Sampling Theory in Signal and Image Processing, vol. 5, no. 2, pp. 183–200, 2006.

G. E. Pfander, “Sampling of operators,” http://arxiv.org/abs/1010.6165

Y. M. Hong and G. E. Pfander, “Irregular and multi-channel sampling of operators,” Applied and Computational Harmonic Analysis, vol. 29, no. 2, pp. 214–231, 2010.

X.-G. Xia, “On bandlimited signals with fractional Fourier transform,” IEEE Signal Processing Letters, vol. 3, no. 3, pp. 72–74, 1996.

T. Erseghe, “Unified fractional Fourier transform and sampling theorem,” IEEE Transactions on Signal Processing, vol. 47, no. 12, pp. 3419–3423, 1999.

J. Shi, Y. Chi, and N. Zhang, “Multichannel sampling and reconstruction of bandlimited signals in fractional Fourier domain,” IEEE Signal Processing Letters, vol. 17, no. 11, pp. 909–912, 2010.

O. Christensen, An Introduction to Frames and Riesz Bases, Birkhäauser, Boston, Mass, USA, 2003.

I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, Pa, USA, 1992.

K. Seip, “On the connection between exponential bases and certain related sequences in ,” Journal of Functional Analysis, vol. 130, no. 1, pp. 131–160, 1995.

B. -Z. Li, R. Tao, and Y. Wang, “New sampling formulae related to linear canonical transform,” Signal Processing, vol. 87, no. 5, pp. 983–990, 2007.

J. J. Healy and J. T. Sheridan, “Sampling and discretization of the linear canonical transform,” Signal Processing, vol. 89, no. 4, pp. 641–648, 2009.

A. Stern, “Sampling of linear canonical transformed signals,” Signal Processing, vol. 86, no. 7, pp. 1421–1425, 2006.