11月03日上课资料-.pptVIP

* VII. Other Time Frequency Distributions (II) The trend of time-frequency analysis in recent years: (1) S transform and its generalization (2) Time-variant signal expansion (3) Improvement for the Hilbert-Huang transform * VII-A S Transform [Ref] R. G. Stockwell, L. Mansinha, and R. P. Lowe, “Localization of the complex spectrum: the S transform,” IEEE Trans. Signal Processing, vol. 44, no. 4, pp. 998–1001, Apr. 1996. closely related to the wavelet transform advantages and disadvantages (Gabor transform 的修正版) * S transform 和 Gabor transform 相似。 但是 Gaussian window 的寬度會隨著 f 而改變 低頻： worse time resolution, better frequency resolution 高頻： better time resolution, worse frequency resolution t-axis f-axis The result of the S transform (compared with page 83) 6 4 2 0 -2 -4 -6 * ? General form s(f) increases with f C. R. Pinnegar and L. Mansinha, “The S-transform with windows of arbitrary and varying shape,” Geophysics, vol. 68, pp. 381-385, 2003. * Fast algorithm of the S transform Q: Can we use the FFT-based method on page 99 to implement the S transform? When f is fixed, the S transform can be expressed as a convolution form: Remember: (for every fixed f) * VI-B Generalized Spectrogram [Ref] P. Boggiatto, G. De Donno, and A. Oliaro, “Two window spectrogram and  their integrals, Advances and Applications, vol. 205, pp. 251-268, 2009. Generalized spectrogram: Original spectrogram: w1(t) = w2(t) To achieve better clarity, w1(t) can be chosen as a wider window, w2(t) can be chosen as a narrower window. * x(t) = cos(2? t) when t 10, x(t) = cos(6? t) when 10 ? t 20, x(t) = cos(4? t) when t ? 20 * x(t) = cos(2? t) when t 10, x(t) = cos(6? t) when 10 ? t 20, x(t) = cos(4? t) when t ? 20 * Generalized spectrogram: Further Generalization for the spectrogram: or * 就如同 ? Fourier series: ?m(t) = exp(j2?fmt), 部分的 Time-Frequency Analysis 也是意圖要將 signal 表示成如下的型態 並且要求在 M 固定的情形下， 為最小 VI