Lecture11_Wavelets

# Lustig eecs uc berkeley m lustig eecs uc berkeley 7 8

This preview shows page 1. Sign up to view the full content.

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

Unformatted text preview: ECS UC Berkeley M. Lustig, EECS UC Berkeley 7 8 STFT and Wavelets “Atoms” Examples of Wavelets STFT Atoms • Mexican Hat Wavelet Atoms (with hamming window) w (t u) e 1 tu p( ) s s j ⌦t (t) = (1 s=1 ⌦hi u • Haar u 1 1 ( t) = : 0 s=3 ⌦lo u 8 < u t2 ) e t 2 /2 0t< 1 2 1 t<1 2 otherwise M. Lustig, EECS UC Berkeley M. Lustig, EECS UC Berkeley 9 10 Example: Wavelet of Chirp Wavelets VS STFT s s f u u M. Lustig, EECS UC Berkeley 11 t M. Lustig, EECS UC Berkeley 12 Example 2: “Bumpy” Signal Wavelets Transform • Can be written as linear ﬁltering W f (u, s) SombreroWavelet = = log(s) 1 p s Z 1 1 f ( t) ⇤ t f ( t) ⇤ s ( t) u ( u) ( s s )dt 1 t =p ( ) ss • Wavelet coefﬁcients are a result of bandpass ﬁltering u M. Lustig, EECS UC Berkeley M. Lustig, EECS UC Berkeley 13 14 Wavelet Transform Orthonormal Haar • Many different constructions for different signals Same scale non-overlapping – Haar good for piece-wise constant signals – Battle-Lemarie’ : Spline polynomials • Can construct Orthogonal wavelets Orthogonal between scales – For example: dyadic Haar is orthonormal 1 t 2i n ( ) i,n (t) = p i 2i 2 i = [1, 2, 3, · · · ] M. Lustig, EECS UC Berkeley M. Lustig, EECS UC Berkeley 15 16 Scaling function Scaling function 1 t 2i n ( ) i,n (t) = p 2i 2i i=m+3 i=m+2 i=m+1 1 t 2i n ( ) i,n (t) = p 2i 2i i=m i=m+1 ⌦ • Problem: i=m ⌦ • Problem: – Every stretch only covers half remaining recall, for chirp: bandwidth – Every stretch only covers half remaining bandwidth – Need Inﬁnite functions – Need Inﬁnite functions • Solution: – Plug low-pass spectrum with a scaling...
View Full Document

## This document was uploaded on 03/29/2014.

Ask a homework question - tutors are online