分数阶傅里叶变换在倒谱分析中的应用
Application of Fractional Fourier Transform in Cepstrum Analysis
摘要：
震源子波估计是地震信号处理中解决地下构造特征的关键。基于倒谱分析的同态反褶积是一种有效的子波估计方法。通常，信号傅里叶变换的对数的傅里叶变换的逆是该信号的倒谱域表示。时域中两个信号的卷积运算变成倒谱域中的加法。分数阶傅里叶变换（FRFT）是标准傅里叶变换（FT）的推广。在FRFT中，变换核是一组线性啁啾，而FT的核由复正弦曲线组成。根据分数阶，信号可以在多个域中表示。这使得FRFT在信号分析中比标准FT具有额外的自由度。在本文中，我们利用了FRFT的多域性质，并将其应用于倒谱分析。我们将这种组合称为分数倒谱（FC）。我们推导出真正的FC公式，并给出一个例子，使用小波显示传统的倒谱与不同的分数阶FRFT的多域表示。
英文摘要：
Source wavelet estimation is the key in seismic signal processing for resolving subsurface structural properties. Homomorphic deconvolution using cepstrum analysis has been an effective method for wavelet estimation for decades. In general, the inverse of the Fourier transform of the logarithm of a signal's Fourier transform is the cepstral domain representation of that signal. The convolution operation of two signals in the time domain becomes an addition in the cepstral domain. The fractional Fourier transform (FRFT) is the generalization of the standard Fourier transform (FT). In an FRFT, the transformation kernel is a set of linear chirps whereas the kernel is composed of complex sinusoids for the FT. Depending on the fractional order, signals can be represented in multiple domains. This gives FRFT an extra degree of freedom in signal analysis over the standard FT. In this paper, we have taken advantage of the multidomain nature of the FRFT and applied it to cepstral analysis. We term this combination the Fractional-Cepstrum (FC). We derive the real FC formulation, and give an example using wavelets to show the multidomain representation of the traditional cepstrum with different fractional orders of the FRFT.
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