Sampling rate conversion for linear canonical transform

doi:10.1016/j.sigpro.2008.06.008

Signal Processing

Volume 88, Issue 11, November 2008, Pages 2825-2832

https://doi.org/10.1016/j.sigpro.2008.06.008 Get rights and content

Abstract

The linear canonical transform (LCT) has been shown to be a powerful tool for optics and signal processing. This paper investigates the sampling rate conversion problem in the LCT domain. Firstly, the discrete-time LCT is introduced and the formulas of interpolation and decimation in the LCT domain are derived. Then, based on the sampling theorem expansion in the LCT domain, the formulas of sampling rate conversion by real factors for the LCT in time domain are proposed. The spectral analysis of sampling rate conversion by real factors in the LCT domain is also illustrated. The sampling rate conversion theories in the Fourier domain and the fractional Fourier domain are shown to be special cases of the achieved results. The simulations verify the effectiveness of the obtained results.

Introduction

The linear canonical transform (LCT) [1], [2], [3], [4], [5] is an integral transform with three free parameters. It was introduced in the 1970s and many transforms such as the Fourier transform (FT), the fractional Fourier transform (FrFT), the Fresnel transform and scaling operations are special cases of the LCT [4], [5], [6]. Recently, along with applications of the FrFT in the signal processing community, the role of the LCT for signal processing has also been noticed. It has found many applications in radar system analysis, filter design, phase retrieval, pattern recognition, and many other areas [3], [4], [5], [6], [7], [8].

The sampling process is central in almost any domain and it explains how to sample continuous signals without aliasing. The sampling theorem expansions for the LCT have been derived in [9], [10], [11], which provides the link between the continuous signals and the discrete signals, and can be used to reconstruct the original signal from their samples satisfying the Nyquist rate of that domain. Moreover, with the development of digital signal processing, computational amount and storage load have gradually increased. To reduce the computational load as well as saving the storage space, different sampling rates and the conversion between them are typically required in many applications such as communications, image processing, digital audio and multimedia. The sampling rates conversion theory is the basis of multirate signal processing and it helps to convert the sampling rate of a discrete-time signal to obtain another discrete-time version without significantly destroying the signal components of interest. Since the LCT has shown to be a powerful signal processing tool, it is necessary to study the sampling rates conversion theory in the LCT domain. The conventional sampling rate conversion by rational factors and irrational factors in the FT domain has been studied in [12], [13], [14], [15], [16], [17], [18], respectively. Recently, the sampling rate conversion by rational factors associated with the FrFT has been analyzed by Tao [19] and Meng [20]. However, the two papers focused on the spectral analysis of interpolation and decimation in the FrFT domain and did not give closed form formulas of interpolation and decimation in time domain. In addition, their methods could not deal with the sampling rate conversion problem by irrational factors. To the best of our knowledge, the sampling rate conversion associated with the LCT has never been presented before. It is therefore worthwhile to study the sampling theory in the LCT domain, which can not only generalize the sampling theories of the FT [12], [13], [14], [15], [16], [17], [18] and the FrFT [19], [20], but also provide a theoretical support for reducing the sampling rate and advance applications such as filter banks theorem for the LCT.

To overcome the sampling rate conversion problem in the LCT domain, we need to define discrete-time LCT (DTLCT) to analyze the spectrum of sampled signal in the LCT domain. In this paper, we firstly give the definition of DTLCT by introducing digital frequency in that domain, which is a generalization of the classical DTFT. Then the formulas for interpolation and decimation associated with the DTLCT are derived using the results in the FT domain. Moreover, based on the sampling theorem expansion in the LCT domain, the formulas of sampling rate conversion by real factors for the LCT in time domain are deduced. The LCT domain analysis of sampling rate conversion by real factors is also given. The sampling rate conversion theories in the FT domain and the FrFT domain are special cases of the achieved results. The simulations verify the effectiveness of the obtained results.

The rest of this paper is organized as follows. In Section 2, the definition of the LCT and the sampling theorem expansions in the LCT domain are reviewed, then the definition of DTLCT is given. In Section 3, the formulas of interpolation and decimation in the LCT domain are derived. Moreover, the formulas of the sampling rate conversion by real factors for the LCT in time domain are deduced and the LCT domain analysis of sampling rate conversion by real factors is also given. In Section 4, some examples are given to verify the achieved results. Finally, we make conclusions in Section 5.

Section snippets

The LCT

The LCT of a signal x(t) with parameter (a, b, c, d) is defined as [1], [3] $X_{(a, b, c, d)} (u) = L_{(a, b, c, d)} (x (t)) (u) = {\begin{matrix} \sqrt{\frac{1}{j 2 π b}} e^{j (d / 2 b) u^{2}} \int_{- \infty}^{+ \infty} x (t) e^{j (a / 2 b) t^{2}} e^{- j (1 / b) ut} d t, & b \neq 0, \\ \sqrt{d} e^{j (cd / 2) u^{2}} x (d u), & b = 0, \end{matrix}$ where ad−bc=1. In general, parameter a, b, c, d are real numbers and we only consider the case of b≠0, since the LCT is just a chirp multiplication operation if b=0. When (a, b, c, d)=(cos α, sin α, −sin α, cos α), the LCT reduces to the FrFT, i.e., $L_{(\cos α, \sin α, - \sin α, \cos α)} (x (t)) (u) = \sqrt{e^{- j α}} F_{α} (x (t)) (u),$ where F_α(x(t))(u) is

Sampling rate conversion in the LCT domain

In this section, we assume that the original discrete-time signal x(n) is sampled with sampling period Δt_x from continuous signal x(t) band-limited in the LCT domain, i.e., X₍_a_, _b_, _c_, _d₎(u)=0 for |u|>u_c, where Δt_x⩽π|b|/u_c.

Simulations

Some examples are given in this section to verify the achieved results. There are some literatures discussing numerical algorithms for the LCT such as [21], [22], [23], [24]. Here, we use the DAFT algorithm proposed in [21]. In the following, we assume that the parameter of the LCT (a, b, c, d)=(1, 5, 1/5, 2). The original discrete signal x(n) is sampled from continuous signal x(t) band-limited in the LCT domain with cutoff frequency u_c=9.3779. The sampling period Δt_x=1 and the sampling

Conclusions

In this paper, we discuss the sampling rate conversion problem for the LCT. At first, we define DTLCT by introducing digital frequency in that domain; then the formulas for interpolation and decimation in the LCT domain are derived in a simple way based on the results in the Fourier domain. Furthermore, using the sampling theorem expansion in the LCT domain, the formulas for sampling rate conversion by real factors for the LCT in time domain are deduced, which can remove unwanted images and

Acknowledgments

The authors would like to thank the anonymous reviewers for their valuable comments and suggestion that improved the clarity and quality of this manuscript. We would also like to thank Dr. Bingzhao Li of Beijing Institute of Technology for many fruitful discussions and the proof reading of the paper.

Sponsored in part by the National Nature Science Foundation of China (nos. 60232010 and 60572094) and the National Science Foundation of China for Distinguished Young Scholars (no. 60625104).

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View full text

Sampling rate conversion for linear canonical transform

Abstract

Introduction

Section snippets

The LCT

Sampling rate conversion in the LCT domain

Simulations

Conclusions

Acknowledgments

Opt. Commun.

Opt. Commun.

Signal Processing

Signal Processing

Linear canonical transformations and their unitary representations

J. Math. Phys.

ABCD matrix formalism of fractional Fourier optics

Opt. Eng.

Eigenfunctions of linear canonical transform

IEEE Trans. Signal Process.

Theory and Applications of the Fractional Fourier Transform

The Fractional Fourier Transform with Applications in Optics and Signal Processing

The fractional Fourier transform and time–frequency representations

IEEE Trans. Signal Process.

Convolution theorems for the linear canonical transform and their applications

Science in China Ser. F—Inform. Sci.

Multirate digital filters, filter banks, polyphase networks, and application: a tutorial

Proc. IEEE