頻率變形及其在訊號處理上的應用

Frequency Warping and Its Applications on Signal Processing

李凱婷(Kai-Ting Lee)

指導教授 ： 貝蘇章

國立臺灣大學/電機資訊學院/電信工程學研究所/碩士(2004年)

https://doi.org/10.6342/NTU.2004.00552

摘要

頻率變形在訊號處理上的應用已被廣泛地討論。在以往，我們計算離算傅立葉轉換相當於在z平面的單位圓上均勻的取樣，然而，在某些應用上，非均勻的取樣會更有效率。
頻譜分析在訊號處理上扮演著很重要的角色，在此論文中，首先介紹利用頻率變形來達到非相等頻寬之頻譜分析，我們使用全通濾波器來完成頻率變形，並藉由調整全通濾波器的參數來達成想要的變形效果。更近一步地，爲了有效重建訊號，我們以拉葛爾轉換來實現頻率變形。接著，我們將頻率變形的概念延伸離散傅立葉轉換、離散餘弦轉換以及離散小波轉換，把不均勻頻率解析度的原理應用在這些訊號處理常使用的轉換式上。
離散頻率變形傅立葉轉換主要應用在頻譜分析，將其應用在估計受雜訊破壞的弦波參數時，比使用傳統的離散傅立葉轉換估計來的更有效率，另外，我們可設計出可調的有限脈衝響應濾波器和頻率變形濾波器串。離散餘弦轉換目前已被應用在靜態影像壓縮標準影像壓縮技術上，假使將離散頻率變形餘弦轉換應用在影像壓縮上，會得到比靜態影像壓縮標準更好的效益。但此方法計算複雜度頗高，若進一步使壓縮失真比理想化，不僅可降低複雜度，也能得到更好的效益。

關鍵字

頻率變形

並列摘要

The applications of frequency warping on signal processing have been discussed extensively. Conventionally, computing the discrete Fourier transform that is equivalent to sampling of the z transform of the input sequence at equally spaced angles around the unit circle. However, in some applications, it is better to sample it at unequally spaced angles.
Spectral analysis plays an important role in the field of signal process. In this thesis, we first introduce unequal bandwidth spectral analysis, which utilizes digital frequency warping. Here we use allpass maps to achieve frequency warping. We can fulfill any desired warping by selecting the warped parameter of the allpass filter. Moreover, in order to recover the original signal efficiently, the frequency warping is implemented by Laguerre filter instead. Then, the concept of frequency warping is then extended to discrete Fourier transform (DFT), discrete cosine transform (DCT) and discrete wavelet transform (DWT), i.e., applying the idea of nonuniform frequency resolution to these common transforms.
Warped discrete Fourier transform (WDFT) is mainly applied to spectral analysis. In the application of sinusoidal parameter estimation of noise-corrupted data, using WDFT is more efficient than using DFT. In addition, we can design tunable finite impulse response (FIR) filter and warped filter bank. DCT has been used in the standard of image compression of joint photograph experts group (JPEG) at present. Provided that WDCT is used in image compression, we will obtain better performance than ordinary DCT. However, the method has a defect of high computation complexity. If we further modify the image compression algorithm in the rate-distortion sense, not only the computation load will be reduced but the performance will also be improved.

並列關鍵字

frequency warping

參考文獻

Frequency Warping

[1] Alan V. Oppenheim, Don H. Johnson, and Kenneth Steiglitz, “Computation of spectra with unequal resolution using the fast Fourier transform,” Proceedings of the IEEE, vol. 59, pp. 299-301, Feb. 1971.

[2] Alan V. Oppenheim and Don H. Johnson, “Discrete Representation of Signals,” Proceedings of the IEEE, vol. 60, pp. 681-691, June 1972.

[4] Thomas von Schroeter, “Frequency warping with arbitrary allpass maps,” IEEE Signal Processing Letter, vol. 6, pp.116-118, May 1999.

[6] G. Evangelista and S. Cavaliere, “Audio effects based on biorthogonal time-varying frequency warping,” EURASIP Journal on Applied Signal Processing, pp. 27-35, 2001:1.