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# The analysis of decimation and interpolation in the linear canonical transform domain

- Shuiqing Xu
^{1}Email author, - Yi Chai
^{1, 2}, - Youqiang Hu
^{1}, - Lei Huang
^{1}and - Li Feng
^{1}

**Received: **2 June 2016

**Accepted: **5 October 2016

**Published: **13 October 2016

## Abstract

Decimation and interpolation are the two basic building blocks in the multirate digital signal processing systems. As the linear canonical transform (LCT) has been shown to be a powerful tool for optics and signal processing, it is worthwhile and interesting to analyze the decimation and interpolation in the LCT domain. In this paper, the definition of equivalent filter in the LCT domain have been given at first. Then, by applying the definition, the direct implementation structure and polyphase networks for decimator and interpolator in the LCT domain have been proposed. Finally, the perfect reconstruction expressions for differential filters in the LCT domain have been presented as an application. The proposed theorems in this study are the bases for generalizations of the multirate signal processing in the LCT domain, which can advance the filter banks theorems in the LCT domain.

## Keywords

- Linear canonical transform
- Decimation and interpolation
- Polyphase network
- Differential filter

## Background

The linear canonical transform (LCT) (Bodenheimer et al. 1971; Sheridan 2016; Pei and Ding 2002; Xu et al. 2015b, a), which was introduced in the 1970s, is an integral transform with three free parameters. Many widely used linear transform in optical system modeling and digital signal processing, such as the Fourier transform (FT), the fractional Fourier transform (FrFT), the Fresnel transform and scaling operations are all special cases of the LCT (Xu et al. 2015b, a; Almeida 1994; Sharma et al. 2013; Zhao et al. 2014; Shi et al. 2014). Due to its extra degrees of freedom, the LCT is more flexible and has been shown to be a powerful tool for optics, filter design, signal synthesis, time-frequency analysis, phase retrieval, pattern recognition, encryption, modulation, multiplexing in communication and many other areas (Zhang 2016b; Zhao et al. 2010, 2009; Song and Zhao 2014; Zhang 2015; Sharma and Joshi 2006; Zhang 2016a; Stern 2008; Li et al. 2007). Therefore, developing the relevant theorems for LCT are of importance in optical systems and many signal processing applications.

Simultaneously, computational amount and storage load have gradually increased due to the rapid development of digital signal processing. In order to decrease the computational amount and storage load, different sampling rates and the conversion between them are typically required in many real applications, such as image processing, digital audio and communications. Under these circumstances, the theory of multirate signal processing was introduced and improved in Vaidyanathan (1990). Decimation and interpolation are the two basic building blocks in the multirate digital signal processing systems. The decimator is utilized to decrease the sampling rate and interpolator to increase the sampling rate. What’s more, the FT and FrFT domain analysis of decimation and interpolation in the multirate digital signal processing have been well studied in Vaidyanathan (1990), Tao et al. (2008), Meng et al. (2007). As the LCT has recently been found many applications in optics and digital signal processing, the relevant theorems, such as convolution theorems, uncertainty principle theorems, sampling theorems, and others in the LCT domain have been well established (Zhang 2016b; Zhao et al. 2009, 2010; Song and Zhao 2014; Zhang 2015; Sharma and Joshi 2006; Zhang 2016a). However, to the best of our knowledge, the analysis of decimation and interpolation in the LCT domain has never been presented before. It is therefore theoretically interesting and practically useful to analyze the decimation and interpolation in the LCT domain.

Our objective in the paper is to study the LCT domain analysis of decimation and interpolation, which can not only generalize the relevant theories of the FT and the FRFT, but also act as the basis of multirate signal processing theorems in the LCT domain. We firstly give the definition of equivalent filter in the LCT domain, which is a generalization of the equivalent filter in the FRFT domain. Then, due to the polyphase decomposition is very fundamental to the efficient implementation of decimation and interpolation in the multirate digital signal processing systems, the polyphase networks for decimator and interpolator in the LCT domain have been proposed based on the definition of equivalent filter. Finally, as an application, the perfect reconstruction expressions for differential filters in the LCT domain have been presented.

## Methods

### The linear canonical transform

*x*(

*t*) denoted by \(L_x^A (u)\), is defined as Bodenheimer et al. (1971)

*a*,

*b*,

*c*,

*d*are real numbers satisfying \(ad-bc=1\), and the kernel \(K_A (u,t)\) is given by

### Discrete-time linear canonical transform

*w*into Eq. (4), we have

### Simplified linear canonical transform

## Results

### The LCT domain analysis of decimation and interpolation

To study the decimation and interpolation in the LCT domain, in this section, we firstly give the definition of equivalent filter in the LCT domain. Then, the direct implementation structure for decimation and interpolation in the LCT domain are derived based on the definition. Moreover, the polyphase networks for decimation and interpolation in the LCT domain are also deduced.

#### Equivalent FIR filter in the LCT domain

*f*(

*t*),

*x*(

*t*),

*h*(

*t*), respectively. From Eq. (9), it is easy to know that the convolution theorem for the DTLCT contains an extra chirp factor and hence does not easily implement in the time domain. On the other hand, by multiplying the \(e^{ - j\frac{{dw^2 }}{{2bT^2}}}\) to both sides of Eq. (9), we can obtain

###
**Definition 1**

*h*(

*nT*), then we define \(\bar{H}_A (w)\) as the equivalent filter in the LCT domain.

#### The polyphase implementation of decimation and interpolation in the LCT domain

The polyphase decomposition is very fundamental to the efficient implementation of decimation and interpolation in the multirate digital signal processing systems. It can be applied for the derivation of new sampling theorems and the recovering bandlimited signal from nonuniformly sampled versions. Theories and applications of polyphase decomposition for the decimation and interpolation in the FT and FRFT domain have been well studied in Vaidyanathan (1990), Tao et al. (2008), Meng et al. (2007). To obtain the polyphase implementation of decimation and interpolation in the LCT domain, we study the direct implementation structure for decimation and interpolation in the LCT domain at first.

*D*of the direct implementation structure in Fig. 2. Similar to the decimator case, the efficiency implementation structure for interpolator in the LCT domain also can be obtained.

From above analysis, the direct efficiency implementation structure for decimator and interpolator in the LCT domain have been derived. To obtain the polyphase networks for decimation and implementation in the LCT domain, the polyphase networks for equivalent filter have been derived in the following at first.

*H*(

*w*) is written in the form

Furthermore, the polyphase implementation structure for decimation and interpolation in LCT domain have many real applications, such as it can perform digital filtering the general case, it also can offer significant saving in computation rate and hardware in two important applications, sampling rate alternation and realization of filter banks.

## Discussion

### Application in the LCT differential filter

## Conclusions

This paper has analyzed the decimation and interpolation in the LCT domain, which can advance the theorems for multirate signal processing in the LCT domain. First, the definition of equivalent FIR filter in the LCT domain has been proposed. By using the definition, the direct implementation structure for decimation and interpolation in the LCT domain have been derived. In addition, the polyphase implementation of decimation and interpolation in the LCT domain also have been obtained. Finally, as an application, the perfect reconstruction expressions for differential filters in the LCT domain have been obtained. The future research and work will be in the direction of real applications of the multirate signal processing theories in the LCT domain.

## Declarations

### Authors' contributions

Xu participated in the design of the study and drafted the manuscript. Chai, Hu, Huang and Feng helped to construct the study design and to draft the manuscript. All authors read and approved the final manuscript.

### Acknowledgements

The authors would like to thank the editor and the anonymous referee for their valuable comments and suggestions. This work was funded by Chongqing University Postgraduates’ Innovation Project (CYB15051) and the National Natural Science Foundation of China (61374135, 61633005, 61673076).

### Competing interests

The authors declare that they have no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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