1、1Time Frequency Analysis and Wavelet Transforms 2Fourier transform (FT) Time-Domain Frequency Domain t varies from Laplace Transform Cosine Transform, Sine Transform, Z Transform. Some things make these operations not practical:(1) Only the case where t0 t t1 is interested. (2) Not all the signals a
2、re suitable for analyzing in the frequency domain.It is hard to observe the variation of spectrum with time by these operations 2jf tXfx t edt stX sx t edtI. Introduction 3Example 1: x(t) = cos(440 t) when t 0.5, x(t) = cos(660 t) when 0.5 t 1, x(t) = cos(524 t) when t 1 The Fourier transform of x(t
3、)Frequency -400-300-200-1000100200300400-0.25-0.2-0.15-0.1-0.0500.050.10.150.24(A) Finite-Supporting Fourier Transform(B) Short-Time Fourier Transform (STFT) w(t): window function 或 mask function STFT windowed Fourier transform 或 time-dependent Fourier transform Ref L. Cohen, Time-Frequency Analysis
4、, Prentice-Hall, New York, 1995.Ref A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing, London: Prentice-Hall, 3rd ed., 2010. 002tBjf ttBXfx t edt 2,jfX t fw txed5(C) Gabor Transform S. Qian and D. Chen, Joint Time-Frequency Analysis: Methods and Applications, Prentice Hall, N.J., 19
5、96. R. L. Allen and D. W. Mills, Signal Analysis: Time, Frequency, Scale, and Structure, Wiley- Interscience.Common Features for short-time Fourier transforms and Gabor transforms (1) The instantaneous frequency can be observed(2) Without Cross Term (3) Poor clarity 22()()2,tjftxGt feexd 2()()221,2t
6、tjxGteexd 6Why Time-Frequency Analysis is Important? Many digital signal processing applications are related to the spectrum or the bandwidth of a signal. If the spectrum and the bandwidth can be determined adaptive, the performance can be improved. modulation, signal identification, multiplexing, a
7、coustics, filter design, system modeling, data compression, radar system analysis signal analysis, sampling7时频时频分析的大家族分析的大家族(1) Short-time Fourier transform (STFT)(rec-STFT, Gabor, )squarespectrogramimproveS transform (2) Wigner distribution function (WDF) combineGabor-Wigner Transform improvewindow
8、ed WDFimproveCohens Class Distribution (Choi-Williams, Cone-Shape, Page, Levin, Kirkwood, Born-Jordan, )improvePseudo L-Wigner Distribution (4) Time-Variant Basis Expansion Matching Pursuit Prolate Spheroidal Wave Function (5) Hilbert-Huang Transform (3) Wavelet transform Haar and Daubechies Coiflet, MorletDirectional Wavelet TransformAsymmetric STFT
