1、70III. Gabor Transform 2() /2,3,tjxGteexd 2()2,tjfxGt feexd Alternative definitions: 2()24,2,2tjfxGt feexd 2()()22,41,2ttjxGteexd III-A DefinitionStandard form: 22()()2,1,tjftxGt feexd normalization71Main Reference S. Qian and D. Chen, Sections 3-2 3-6 in Joint Time-Frequency Analysis: Methods and A
2、pplications, Prentice-Hall, 1996.Other References D. Gabor, “Theory of communication”, J. Inst. Elec. Eng., vol. 93, pp. 429-457, Nov. 1946. (最早提出 Gabor transform) M. J. Bastiaans, “Gabors expansion of a signal into Gaussian elementary signals,” Proc. IEEE, vol. 68, pp. 594-598, 1980. R. L. Allen an
3、d D. W. Mills, Signal Analysis: Time, Frequency, Scale, and Structure, Wiley- Interscience. S. C. Pei and J. J. Ding, “Relations between Gabor transforms and fractional Fourier transforms and their applications for signal processing,” IEEE Trans. Signal Processing, vol. 55, no. 10, pp. 4839-4850, Oc
4、t. 2007. 72(1) Integration property When k 0, When k = 0, When k = 1, (recovery property) (2) Shifting property If y(t) = x(t t0), then . (3) Modulation property If y(t) = x(t)exp(j2f0t), then 22(),jftxGt feexd 2 22(1),jkt fktxGt f edfex kt 2,0txG t f dvex 2,jt fxGt f edfx t020,jf tyxGt fGttf e0,yxG
5、t fGt ff III-B Properties of Gabor Transforms 73(4) Special inputs:(a) When x( ) = ( ), (b) When x( ) = 1, (symmetric for the time and frequency domains)(5) Linearity property If z( ) = x( ) + y( ) and Gz(t, f ), Gx(t, f ) and Gy(t, f ) are their Gabor transforms, then Gx(t, f ) = Gx(t, f ) + Gy(t,
6、f ) (6) Power integration property: 2,txGt fe22,jf tfxGt fee 221.91432222 ()2 ()1.9143,utuxuGt fdfexdexd 74(7) Energy sum property Gx(t, f ) and Gy(t, f ) are the Gabor transforms of x( ) and y( )(8) Power decayed property If x(t ) = 0 for t t0, then . i.e., for t t0. If for f f0, then for f f0. *,x
7、yGt f Gt f df dtxyd 20222 ()0,t txxG t fdfeG tfdf200222 ()0(fix , vary )(fix , vary )average of ,average of ,t txxtftfGt feGtf ( )0XfFT x t200222 ()0(fix , vary )(fix , vary )average of ,average of ,ffxxftftGt feGt f75III-C Generalization of Gabor Transforms 2()24,tjfxGt feexd larger : higher resolu
8、tion in the time domain lower resolution in the frequency domainsmaller : higher resolution in the frequency domain lower resolution in the time domain76-4-3-2-101234-4-3-2-101234Gabor transform forGaussian function exp(t2) = 0.2 f-axist-axisGabor transform forGaussian function exp(t2) = 5 -4-3-2-101234-4-3-2-101234f-axist-axis
