35Hilbert - MAE 591 RANDOM DATA C. W. Lee Hilbert...

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MAE 591 RANDOM DATA C. W. Lee Hilbert Transform 1 While the Fourier transform moves the independent variable of a signal from the time to the frequency domain or vice versa, the Hilbert transform leaves the signal in the same domain. The Hilbert transform of a time signal is another time signal and the Hilbert transform of a frequency “signal” is another frequency signal. The simplest non-mathematical way of describing the Hilbert transform of a time signal is to say that it gives all frequency components of the signal a –90 degrees phase shift , or in the time domain that it shifts each component by 1/4 wave length. This effect is similar to an integration of the signal. Notice that the shift is not a given time shift, but depends on the wavelength (or frequency) of the particular component. 1. Basic Relations: i) Symmetric properties of Fourier transform ) ( ) ( ) ( ) ( ) ( t x f X t x f X t x F F F F ii) The convolution theorem ) ( ) ( ) ( ) ( )] ( ) ( [ f B f A du u t b u a t b t a = = F F ) ( ) ( ) ( ) ( )] ( ) ( [ f B f A du u f B u A t b t a = = F iii) Sign function sgn has the Fourier transforms: 0 : 0 : 1 1 < > + = t t t f j t π 1 } {sgn = F and f t j sgn 1 = F 2. Definition: i) Convolution integral = = = t t x du u t u x t x t x 1 ) ( ) ( ) ( )] ( [ ) ( ~ H ii) 2 phase shift system 2 1 J. S. Bendat and A. G. Piersol, Random Data: Analysis and Measurement Procedures , 3rd ed., John Wiley & Sons, 2000, Chapter 13. Lecture notes, ‘The Hilbert Transform,’ B&K, English BA7082-12 2 Note that ) ( ~ f X differs from the Hilbert transform of (refer to the lack of commutation property). ) ( f X
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MAE 591 RANDOM DATA C. W. Lee ) ( ) ( ) ( ) sgn ( )} ( { 1 )} ( ~ { ) ( ~ f j b e f X f X f j t x t t x f X φ π = = = = F F F
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35Hilbert - MAE 591 RANDOM DATA C. W. Lee Hilbert...

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