1、II. Short-time Fourier Transform 2,jfX t fw txedShort-time Fourier transform (STFT)Alternative definition ,jX tw txed1 S. Qian and D. Chen, Section 3-1 in Joint Time-Frequency Analysis: Methods and Applications, Prentice-Hall, 1996. 2 S. H. Nawab and T. F. Quatieri, “Short time Fourier transform,” i
2、n Advanced Topics in Signal Processing, pp. 289-337, Prentice Hall, 1987. Ref.II-A Definition 52Inverse of the STFT: To recover x(t), where w(t1 t) 0. For the alternative definition, 1211,jf tx twttX tf edf 1111,2jtx twttX ted 2,jfX t fw txed ,jX tw txedSTFT53The mask function w(t) always has the pr
3、operty of (a) even: w(t) = w(t), (b) max(w(t) = w(0), w(t1) w(t2) if |t2| |t1|(c) w(t) 0 when |t| is large w(t) = (t) (triangular function) w(t) = exp(a|t|b) Max(t) = 1 t = 1 t = 1 (hyper-Laplacian function) 54Rectangular mask STFT (rec-STFT)Inverse of the rec-STFT where t B t1 t + B The simplest fo
4、rm of the STFT Other types of the STFT may require more computation time than the rec-STFT. 2,t Bjft BX t fxed 21,jf tx tX tf edfII-B Rec-STFT55II-C Properties of the Rec-STFT(1) Integration: (a) (b) when v B t v + B, = 0 otherwise 2,0when -0,0otherwise t Bjft Bt Bt BX t f dfxedf dxdxt BtBBtB 2,jf v
5、X t f edfx v56(2) Shifting property (3) Modulation property 02200,t Bjfjft BxedX tf e 0220,t Bjfjft BxeedX t ff57(4) Special inputs: (1) When x(t) = (t), when B t B, otherwise (2) When x(t) = 1 ,1X t f,0X t f2,2 sinc 2jf tX t fBB f e58(5) Linearity property If h(t) = x(t) + y(t) and H(t, f ), X(t, f
6、 ) and Y(t, f ) are their rec-STFTs, then H(t, f ) = X(t, f ) + Y(t, f ).(6) Power integration property (7) Energy sum property (Parsevals theorem) 22,t Bt BX t fdfxd ,t Bt BX t f Yt f dfxyd 22,2X t fdf dtBxd ,2X t f Yt f df dtBxyd 59All the time-frequency analysis methods has the advantage of: The
7、instantaneous frequency can be observed.All the time-frequency analysis methods has the disadvantage of: Higher complexity for computationII-D Advantage and Disadvantage Compared with the Fourier transform: 60 Compared with other types of time-frequency analysis: The rec-STFT has an advantage of the
8、 least computation time for digital implementationbut its performance is worse than other types of time-frequency analysis.61II-E STFT with Other Windows(1) Rectangle(2) Triangle(3) Hanning(4) Hamming(5) Gaussian-B B 0.50.5cos/when 0otherwiset BtBw t-B B 0.540.46cos/when 0otherwiset BtBw t 2expw tt62(6) Asymmetric windowt = 0t-axis
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