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 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。
 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.
 CHAPTER 1 Introduction 1CHAPTER 2 Frequency Warping 52.1 Introduction 52.2 General Spectral Analysis 62.3 Unequal Bandwidth Spectral Analysis 82.4 The Implementation of Frequency Warping 92.4.1 Frequency warping using an allpass transformation 92.4.2 Frequency warped by Laguerre transform 122.4.3 Dynamic frequency warping 142.5 Conclusion 17CHAPTER 3 The Theory of Warped Discrete Fourier, Cosine and Wavelet Transform 193.1 Introduction 193.2 Warped Discrete Fourier Transform 203.2.1 Definition and property 203.2.2 Implementation 243.2.3 The inverse transform 283.3 Warped Discrete Cosine Transform 29 3.3.1 Definition and property 29 3.3.2 Implementation 34 3.3.3 The inverse transform 353.4 Warped Discrete Wavelet Transform 363.4.1 Theory 363.4.2 Applications 403.5 Conclusion 41CHAPTER 4 The Applications of Warped Discrete Fourier Transform 434.1 Introduction 434.2 Frequency Estimation 44 4.2.1 Single sinusoid case 44 4.2.2 Multiple sinusoids in noise 474.3 Tunable FIR Filter Design 554.4 Discrete Multitone Transmission 564.5 Warped Filterbank Design 574.6 Conclusion 59CHAPTER 5 The Applications of Warped Discrete Cosine Transfom 615.1 Introduction 615.2 Image Compression Algorithm 62 5.2.1 Image compression algorithm using WDCT 62 5.2.2 Comparison 645.3 Rate-distortion Optimization 685.4 The Performance Comparison with JPEG 705.5 Other Applications of WDCT 725.5.1 Image-adaptive watermarking 725.5.2 Speech enhancement 745.6 Conclusion 76CHAPTER 6 Conclusion and Future Work 776.1 Conclusion 776.2 Future Works 79
 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.[3] Carlo Braccini and Alan V. Oppenheim, “Unequal bandwidth spectral analysis using frequency warping,” IEEE Transaction on Acoustics, Speech, and Signal Processing, vol. ASSP-22, pp. 236-244, Aug. 1974.[4] Thomas von Schroeter, “Frequency warping with arbitrary allpass maps,” IEEE Signal Processing Letter, vol. 6, pp.116-118, May 1999.[5] G. Evangelista, “Real-time time-varying frequency warping via short-time Laguerre transform,” Proceedings of the COST G-6 Conference on Digital Audio Effects (DAFX-00), Verona, Italy, Dec. 7-9, 2000.[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.[7] M. A. Masnadi-Shirazi and N. Ahmed, “Optimum Laguerre networks for a class of discrete-time systems,” IEEE Transactions on Signal Processing, vol. 39, pp. 2104-2108, Sept. 1991.[8] T. Oliveira e Silva, “Optimality conditions for truncated Laguerre networks,” IEEE Transactions on Signal Processing, vol. 42, pp. 2528-2530, Sept. 1994.[9] Tuomas Paatero and Matti Karjalainen, “Kautz filters and generalized frequency resolution: theory and audio applications,” J. Audio Eng. Soc., vol. 51, pp. 27-44, No. 1/2, Jan./Feb. 2003.[10] G. J. Cook, Y. H. Leung, and Y. Liu, “On the design of variable fractional delay filters with Laguerre and Kautz filters,” IEEE International Conference on Acoustics, Speech, and Signal Processing, vol. 6, pp. 281-284, 6-10 April, 2003.[11] Kwang-Pyo Choi and Keun-Yooung Lee, “An efficient audio watermarking by using spectrum warping,” IEICE Trans. Fundamentals, vol. E85-A, No. 6, June 2002.[12] Y. C. Eldar and A.V. Oppenheim, “Filterbank reconstruction of bandlimited signals from nonuniform and generalized samples,” IEEE Transactions on Signal Processing, vol. 48, Oct. 2000.Warped Discrete Fourier Transform[13] A. Makur and S. K. Mitra, “Warped discrete-Fourier transform: Theory and applications,” IEEE Transactions on Circuits and Systems—Ⅰ: Fundamental Theory and Applications, vol. 48, pp.1086-1093, Sept. 2001.[14] S. Franz, S. K. Mitra, J. C. Schmidt, and G. Doblinger, “Warped discrete Fourier transform: a new concept in digital signal processing,” IEEE International Conference on Acoustics, Speech, and Signal Processing, Proceedings. vol. 2, pp. 1205-1208, 2002.[15] L. C. Palmer, “Coarse frequency estimation using the discrete Fourier transform (Corresp.),” IEEE Transactions on Information Theory, vol. 20 , Issue: 1 , pp. 104-109, Jan. 1974.[16] D. W. Tufts and R. Kumaresan, “Estimation of frequencies of multiple sinusoids: making linear prediction perform like maximum likelihood,” Proc. IEEE, vol. 70, pp. 975-989, Sept. 1982.[17] S. Franz, S. K. Mitra, and G. Doblinger, “Frequency estimation using warped discrete Fourier transform,” Signal Processing, vol. 83, pp. 1661-1671, 2003.[18] Qing Huo Liu and Nhu Nguyen, “An accurate algorithm for nonuniform fast fourier transforms (NUFFT’s),” IEEE Microwave and Guided Wave Letters, vol. 8, No.1, Jan. 1998.[19] Qing Huo Liu and Nhu Nguyen, “Nonuniform fast Fourier transform (NUFFT) algorithm and its applications,” IEEE Antennas and Propagation Society International Symposium, vol. 3 , pp. 1782-1785, 21-26 June 1998.[20] J. A. Fessler and B. P. Sutton, “Nonuniform fast Fourier transforms using min-max interpolation,” IEEE Transactions on Signal Processing, vol. 51, Issue: 2, pp. 560-574, Feb. 2003.Warped Discrete Cosine Transform[21] Nam Ik Cho and S. K. Mitra, “Warped discrete cosine transform and its application in image compression,” IEEE Transactions on Circuits and Systems for Video Technology, vol.10, pp. 1364-1373, Dec. 2000.[22] Nam Ik Cho and S. K. Mitra, “An image compression algorithm using warped discrete cosine transform,” International Conference on Image Processing, vol. 2, pp. 834- 837, 24-28 Oct. 1999[23] Il Koo Kim, Nam Ik Cho, and S. K. Mitra, “Rate-distortion optimization of the image compression algorithm based on the warped discrete cosine transform,” Signal Processing, vol. 83, pp. 1919 vol. 83, pp. 1919-1928, 2003-1928, 2003.[24] S. Wu and A. Gersho, “Rate-constrained picture-adaptive quantization for JPEG baseline coder,” Proc. IEEE Int. Conf. Acoustics, Speech, Signal Processing, vol. 5, pp. 389-392, Apr. 1993.[25] J. R. Hernandez, M. Amado, and F. Perez-Gonzalez, “DCT-domain watermarking techniques for still images: detector performance analysis and a new structure,” IEEE Transactions on Image Processing, vol. 1, pp. 55-68, Jan. 2000.[26] N. I. Cho and S. U. Lee, “Fast algorithm and implementation of 2-D discrete cosine transform,” IEEE Transactions on Circuit System Ⅰ, vol. 38, pp. 297-305, Mar. 1991.[27] D. Slawecki and W. Li, “DCT/IDCT processor design for high data rate image coding,” IEEE Transactions on System Video Technology, vol. 2, pp. 135-146, June 1992.[28] C. I. Podilchuk, and W.Zeng, “Image-adaptive watermarking using visual modals,” IEEE Journal on Selected Areas in Communications, vol.16, pp. 525-539, May 1998.[29] Han-Seung Jung, Nam-Ik Cho, and Sang-Uk Lee, “Image-adaptive watermarking based on warped discrete cosine transform,” IEEE International Symposium on Circuits and Systems, ISCAS 2002. vol. 3, pp. 26-29, 2002.[30] Ing Yann Soon, Soo Ngee Koh and Chai Kiat Yeo, “Noisy speech enhancement using discrete cosine transform,” Speech Communication, vol.24, pp. 249-257, 1998.[31] Joon-Hyuk Chang and Nam Soo Kim, “Speech enhancement using warped discrete cosine transform,” Speech Coding, IEEE Workshop Proceedings. pp. 175-177, 6-9 Oct. 2002.Warped Wavelets Transform[32] G. Evangelista and S. Cavaliere, “Discrete frequency warped wavelets: theory and applications,” IEEE Transactions on Signal Processing, vol. 46, pp. 874-885, Apr. 1998.[33] G. Evangelista and S. Cavaliere, “Frequency-warped filter banks and wavelet transforms: a discrete-time approach via Laguerre expansion,” IEEE Transactions on Signal Processing, vol. 46, pp. 2638-2650, Oct. 1998.[34] G. Evangelista, “Pitch synchronous wavelet representations of speech and music signals,” IEEE Transactions on Signal Processing, vol. 41, pp. 3313-3330, Oct. 1993.[35] P. P. Vaidyanathan, Multirate Systems and Filter Banks. Englewood Cliffs,, NJ: Prentice-Hall, 1993.[36] R. G. Baraniuk and D. L. Jones, “Warped wavelet bases: unitary equivalence and signal processing,” IEEE International Conference on Acoustics, Speech, and Signal Processing, vol. 3, pp. 320-323,Apr. 1993.[37] R. G. Baraniuk and D. L. Jones, “Unitary equivalence: A new twist on signal processing,” IEEE Transactions on Signal Processing, vol. 43, pp. 2269-2282, Oct. 1995.[38] T. Gulzow, A. Engelsberg, and U. Heute, “Comparison of a discrete wavelet transform and a nonuniform polyphase filterbank applied to spectral-subtraction speech enhancement,” Signal Processing, vol. 64, pp. 5-19, 1998.[39] C. Herley, J. Kovacevic, K. Ramchandran, and M. Vetteri, “Tilings of the time-frequency plane: Construction of arbitrary orthogonal bases and fast tiling algorithms,” IEEE Transactions on Signal Processing, vol. 41, pp. 3341-3359, Dec. 1993.[40] T. Blu, “Iterated filter banks with rational rate changes connection with discrete wavelet transform,” IEEE Transactions on Signal Processing, vol. 41, pp. 3232-3244, Dec. 1993.Other Related Literature[41] E. I. Jury and O. Chan, “Combinatorial rules for some useful transformations,” IEEE Transactions on Circuit Theory, vol. CT-20, pp. 476-480, Sept. 1973.
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