Lecture11_Wavelets

Lecture11_Wavelets - Last Time Started with STFT Heisenberg...

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M. Lustig, EECS UC Berkeley EE123 Digital Signal Processing Lecture 11 1 M. Lustig, EECS UC Berkeley Last Time Started with STFT Heisenberg Boxes Continue and move to wavelets Ham -- Get me the forms! 2 Δ ! = 2 N Δ t = N Δ ! · Δ t = 2 M. Lustig, EECS UC Berkeley DFT X [ k ] = N - 1 X n =0 x [ n ] e - j 2 kn/N ! t one DFT coefficient 3 X [ r, k ] = L - 1 X m =0 x [ r R + m ] w [ m ] e - j 2 km/N Δ ! = 2 L Δ t = L M. Lustig, EECS UC Berkeley Discrete STFT optional ! t one STFT coefficient 4
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M. Lustig, EECS UC Berkeley Limitations of Discrete STFT • Need overlapping Not orthogonal • Computationally intensive O(MN log N) • Same size Heisenberg boxes 5 M. Lustig, EECS UC Berkeley From STFT to Wavelets • Basic Idea: –low-freq changes slowly - fast tracking unimportant –Fast tracking of high-freq is important in many apps. –Must adapt Heisenberg box to frequency • Back to continuous time for a bit ..... 6 σ t σ ! Sf ( u, ) = Z 1 -1 f ( t ) w ( t - u ) e - j t dt Wf ( u, s ) = Z 1 -1 f ( t ) 1 p s ( t - u s ) dt u M. Lustig, EECS UC Berkeley From STFT to Wavelets • Continuous time σ t σ ! u σ t σ ! *Morlet - Grossmann 7 Z 1 -1 | ( t ) | 2 dt = 1 Z 1 -1 ( t ) dt = 0 M. Lustig, EECS UC Berkeley From STFT to Wavelets • The function is called a mother wavelet –Must satisfy: Wf ( u, s ) = Z 1 -1 f ( t ) 1 p s ( t - u s ) dt Band-Pass unit norm 8
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w ( t - u ) e j t 1 p s ( t - u s ) s = 1 lo s = 3 M. Lustig, EECS UC Berkeley STFT and Wavelets “Atoms” STFT Atoms (with hamming window)
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