Lecture11_Wavelets

# Lecture11_Wavelets - Last Time Started with STFT Heisenberg...

This preview shows pages 1–4. Sign up to view the full content.

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 coef±cient 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 coef±cient 4

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 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: ( u, s Z 1 -1 f ( t ) 1 p s ( t - u s ) dt Band-Pass unit norm 8
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)

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 12

Lecture11_Wavelets - Last Time Started with STFT Heisenberg...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online