1、117V. Wigner Distribution Function Definition 1: Definition 2: Another way for computation Definition 1:Definition 2:222jfxWt fx txted*,/*,/ 2/ 2jxWtx txtedV-A Wigner Distribution Function (WDF) 222jtxWt fXfXfed*,/22jtxWtXXed*,/ where X(f) is the Fourier transform of x(t)118Main ReferenceRef S. Qian
2、 and D. Chen, Joint Time-Frequency Analysis: Methods and Applications, Chap. 5, Prentice Hall, N.J., 1996. Other ReferencesRef E. P. Wigner, “On the quantum correlation for thermodynamic equilibrium,” Phys. Rev., vol. 40, pp. 749-759, 1932. Ref T. A. C. M. Classen and W. F. G. Mecklenbrauker, “The W
3、igner distributionA tool for time-frequency signal analysis; Part I,” Philips J. Res., vol. 35, pp. 217-250, 1980. Ref F. Hlawatsch and G. F. BoudreauxBartels, “Linear and quadratic time-frequency signal representation,” IEEE Signal Processing Magazine, pp. 21-67, Apr. 1992.Ref R. L. Allen and D. W.
4、 Mills, Signal Analysis: Time, Frequency, Scale, and Structure, Wiley-Interscience, NJ, 2004.119The operators that are related to the WDF:(a) Signal auto-correlation function: (b) Spectrum auto-correlation function: (c) Ambiguity function (AF): ,/2/2xCtx txt,/2/2xSfX fXf*2,/2/2jtxAx txtedt Ax(, )FTf
5、FTt FTf IFTtSx(, f )IFTtIFTf Cx(t, )Wx(t, f )120V-B Why the WDF Has Higher Clarity?If x(t) = exp(j2 h t)2(/2)2(/2)2222(),()jh tjh tjfxjhjfjfhWt feeedeededfh Comparing: for the case of the STFT Due to signal auto-correlation function121If x(t) = (t) 22442242244jfxjfjt fjt fWt fttedttedt et et ,/122If
6、 h(t) = g(t) + s(t) cross terms*2,/2/2jfxWt fx txted *2222222,/2/2/2/2/2/2|/2/2|/2/2 /2/2/2/2 |,|,jfhjfjfgsWt fh thtedg ts tgtstedg tgts tstg tstgts tedWt fW t 2 /2/2/2/2jffg tstgts ted V-C The WDF is not a Linear Distribution123 for 9 t 1, s(t) = 0 otherwise, f (t) = s(t) + r(t) 橫軸: t-axis, 縱軸: f -
7、axis 2exp/103s tjtj t 22exp/26 exp(4) /10r tjtj tt WDF of s(t), WDF of r(t), WDF of s(t) + r(t) -10-50510-4-2024-10-50510-4-2024-10-50510-4-2024V-D Examples of the WDF 124Simulations x(t) = cos(2t) = 0.5exp(j2t) + exp(-j2t)by the WDF by the Gabor transform 0246810-505t-axis f-axis 0246810-505f-axis
8、t-axis f-axisf-axist-axist-axis1-11-1125 (5)/4)x tt : rectangular function by the WDF by the Gabor transform 0246810-5-4-3-2-101234f-axis t-axis 0246810-505f-axis t-axis f-axisf-axist-axist-axis126 0246810-5-4-3-2-101234f-axis t-axis 0246810-505f-axis t-axis 3exp( (5)6)x tj tjt by the WDF by the Gab
9、or transform f-axisf-axist-axist-axis127 2exp(5)x tt0246810-5-4-3-2-101234f-axis t-axis 0246810-505f-axis t-axis Gaussian function: 22FTtfee Gaussian functions T-F area is minimal. by the WDF by the Gabor transform f-axisf-axist-axist-axis128 , (using = /2 )Sampling: t = nt, f = mf, = pt When x(t) i
10、s not a time-limited signal, it is hard to implement. *2,/2/2jfxWt fx txted*4,2jfxWt fx txted,2()()exp4xtftttftpWnmxnpxnpjmp V-E Digital Implementation of the WDF 129Suppose that x(t) = 0 for t n2t x(t)n1tn2t()()0ttxnpxnpif n + p n1, n2 or n p n1, n2 p 的範圍的問題的範圍的問題 (當當 n 固定時固定時)n1 n + p n2 n1 n p n2
11、 n n1 n p n2 n1 n p n2 n, n n2 p n n1 max(n1 n , n n2) p min(n2 n , n n1) min(n2 n , n n1) p min(n2 n , n n1) nt130 x(t)n1tn2tntmin(n2 n , n n1) p min(n2 n , n n1) 注意：當 n n2 或 n n1 時， 將沒有 p 能滿足上面的不等式 (n2 n)t , (n n1 )t : 離兩個邊界的距離(n n1)t (n2 n )t131,2()()exp4QxtftttftpQWnmxnpxnpjmp Q = min(n2n, nn1).
12、 p Q, Q, n n1, n2, If x(t) = 0 for t n2t possible for implementationMethod 1: Direct Implementation (brute force method)T點 F點唯一的限制條件？132When and N 2Q+112tfN 2,2()()mpNQjxtftttpQWnmxnpxnpe Method 2: Using the DFTq = p+Q p = q Q2220,2()()mQmqNNQjjxtftttqWnmexnqQxnqQe 22110,2mQmqNNNjjxtftqWnmec q e 1()
13、()ttc qxnqQxnqQfor 0 q 2Q 10c q for 2Q+1 q N1Q = min(n2n, nn1). n n1, n2,3 大限制條件133假設 t = n0t, (n0+1) t, (n0+2) t, , n1t f = m0 f, (m0+1) f, (m0+2) f, , m1 fStep 1: Calculate n0, n1, m0, m1, NStep 2: n = n0Step 3: Determine QStep 4: Determine c1(q) Step 5: C1(m) = FFTc1(q)Step 6: Convert C1(m) into
14、C( nt, mf)Step 7: Set n = n+1 and return to Step 3 until n = n1. 134,2()()exp4QxtftttftpQWnmxnpxnpjmp 222222 (),2()()tftftfQjmjpjp mxtftttpQWnmexnpxnpee Step 1 Step 2 Step 3 221,()()tfjpttx n pxnpxnpe 21,n Qp n QXn mxp c mp 22tfjmc me 222,2,tfjmtftX nmeXn m Method 3: Using the Chirp Z Transform 135V
15、-F Properties of the WDF(1) Projection property (2) Energy preservation property (3) Recovery property x*(0) 已知 (4) Mean condition frequency and mean condition timeIf , then (5) Moment properties , 2,xx tWt f df 2,xXfWt f dt 2/2,0jf txWtf edfx tx 2,/20jf txWt fedtXfX 2jtx tx te 2jfXfXfe 2,xtx tf Wt
16、fdf 2,xfXft Wt fdt2,( )nnxt Wt f dtdftx tdt 2,( )nnxf Wt f dtdffX fdf 22,xWt f dtdfx tdtXfdf 136(6) Wx(t, f ) is real(7) Region properties If x(t) = 0 for t t2 then Wx(t, f ) = 0 for t t2 If x(t) = 0 for t t1 then Wx(t, f ) = 0 for t t1(8) Multiplication theoryIf , then (9) Convolution theoryIf , th
17、en(10) Correlation theoryIf , then y tx t h t,yxhWt fWtWt fd y tx thd,yxhWt fWfWtfd y tx thd,yxhWt fWfWtfd ( ,) = ( ,)xxW t fW t f137The STFT (including the rec-STFT, the Gabor transform) does not have real region, multiplication, convolution, and correlation properties. (11) Time-shifting property
18、If , then(12) Modulation property If , then 0y tx tt0,yxWt fWttf 0exp2y tjf t x t0,yxWt fWt ff138 Try to prove of the projection and recovery properties Why the WDF is always real? What are the advantages and disadvantages it causes?139*2,/2/2jfxWt fx txtedThe importance of region property Proof of the region properties If x(t) = 0 for t t0, x(t + /2) = 0 for (t t0)/2,Therefore, if t t0 B, B is positive.If B t2 t1144(3) t = (t1 + t2)/2(1) t = t1(2) t = t2第一項第一項第一項第二項第二項第二項第二項第二項第二項-axis-axis-axis-axis-axis-axis2t22t12t12t2002t12t202t22t1t2t1t2t1t1t2t1t2BBBB
