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Convolution theorems for the linear canonical sine and cosine transform and its application
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摘要:
针对奇、偶信号的去噪问题，提出了一种基于线性正则正(余)弦变换卷积定理的乘性滤波器设计方法。在现有线性正则变换域卷积理论的基础上，研究了两类线性正则正(余)弦变换卷积定理，利用所得卷积定理，通过合理选择滤波函数，设计了一类基于卷积定理的线性正则正(余)弦变换域带限信号的乘性滤波模型，并对算法的复杂度进行分析。研究表明，这种滤波模型特别适合处理奇、偶信号，并能有效降低乘积滤波的计算复杂度，提高运算效率。
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关键词:
- 线性正则变换 /
- 线性正则正(余)弦变换 /
- 卷积定理 /
- 滤波
Abstract:For the denoising problem of odd and even signals, a multiplicative filter design method based on the convolution theorem of the linear canonical sine and cosine transform is proposed. Two kinds of convolution theorems associated with the linear canonical sine and cosine transform based on the existing linear canonical transform domain convolution theory are derived. Using this two convolution theorems, two kinds of the multiplicative filtering models of the band-limited signal are designed by choosing an appropriate filter function in linear canonical sine and cosine transform domain. And the complexity of these schemes is analyzed. The results indicate that these filtering models are particularly suitable for handling odd and even signals, and can effectively improve computational efficiency by reducing computational complexity.
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Overview: In the modern optical signal processing domain, the collected signals must be denoised before the signal is analyzed and processed. The multiplicative filtering is one of the effective denoising methods in signal processing field based on the convolution theorem. The classical convolution theorem shows that the convolution of the two signals in time domain is leads to simple multiplication of their Fourier transforms in the Fourier transform domain. But Fourier transform is a holistic transformation based on the time domain or frequency domain, which is not suitable for modern optical signal processing.
As generalization of the Fourier transform and the fractional Fourier transform, Linear canonical transform has become a one of the powerful tools for modern optical signal analysis and processing, and has achieved fruitful research results in recent years. In order to further reduce computation and improve computing efficiency, convolution theory and application based on linear canonical transform has become one of the hot topic research in modern optical signal processing. Therefore, this paper will mainly focus on the research of convolution theory and application based on canonical sine transform and canonical cosine transform which have very close relations with the linear canonical transform, and have important role in signal processing, optics and other fields. Because canonical sine transform has no even eigenfunction and canonical cosine transform have no odd eigenfunction, therefore, it is much more efficient to use the canonical sine transform to deal with the odd signal and use the canonical cosine transform to deal with the even signal. Moreover, the complexity of the canonical sine transform and canonical cosine transform is one half of the complexities of the linear canonical transform, then, it is more suitable for engineering applications.
Hence, for the denoising problem of odd and even signals, a multiplicative filter design method based on the convolution theorem of the canonical sine and cosine transform is proposed. Two kinds of the convolution theorems associated with the canonical sine and cosine transform based on the existing linear canonical transform domain convolution theory are derived. Using this two convolution theorems, a kind of the multiplicative filtering model of the band-limited signal is designed by choosing an appropriate filter function in canonical sine and cosine transform domain. And the complexity of this scheme is analyzed. The results indicate that this filtering model is particularly suitable for handling odd and even signals, and can effectively improve computational efficiency by reducing computational complexity.
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图 1 线性正则正(余)弦变换的带通乘积滤波器
Figure 1. The bandpass multiplicative filter in the CST or CCT domain
图 2 线性正则正弦变换的带通乘积滤波器的时域实现
Figure 2. The method to achieve the bandpass multiplicative filters in the CST domain by convolution in the time domain
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