Lustig eecs uc berkeley m lustig eecs uc berkeley 7 8

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

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