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 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

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 ) = 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
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 ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern