EE123 Digital Signal Processing

Transcription

1 Discree Transforms (Finie) EE3 Digial Signal Processing Lecure 9 DFT is only one ou of a LARGE class of ransforms Used for: Analysis Comression Denoising Deecion Recogniion Aroximaion (Sarse) Sarse reresenaion has been one of he hoes research oics in he las 5 years in s Examle of secral analysis Time Deenden Fourier Transform Secrum of a bird chiring Ineresing,... bu... Does no ell he whole sory No emoral informaion x[n] To ge emoral informaion, use ar of he signal around every ime oin X[n, ) = X m= x[n + m]w[m]e jm n *Also called Shor-ime Fourier Transform (STFT) 6 Secrum of a bird chir Hz x 4 3 Maing from D D, n discree, w con. Simly slide a window and comue DTFT 4

2 Time Deenden Fourier Transform To ge emoral informaion, use ar of he signal around every ime oin X[n, ) = X m= x[n + m]w[m]e jm *Also called Shor-ime Fourier Transform (STFT) Secrogram Frequency, Hz Frequency, Hz Frequency, Hz Time, s 5 6 Discree Time Deenden FT X r [k] = LX m= L - Window lengh R - Jum of samles N - DFT lengh x[rr + m]w[m]e j km/n Tradeoff beween ime and frequency resoluion Discree Transforms (Finie) Today: Sar wih DFT Frequency only Shor-ime DFT Time-Frequency Waveles More flexible/beer Time-frequency Waveles Sarsiy Comression denoising aroximaion 7 8

3 Heisenberg Boxes Time-Frequency uncerainy rincile h:// DFT X[k] = NX n= x[n]e j kn/n = N = N = one DFT coefficien 9 Discree STFT X[r, k] = = L = L LX m= oional x[rr + m]w[m]e j km/n one STFT coefficien STFT Reconsrucion x[rr + m]w L [m] = N NX k= X[n, k]e j km/n For non-overlaing windows, R=L : x[n] = Wha is he roblem? x[n rl] w L [n rl] rl ale n ale (r + )R

4 STFT Reconsrucion x[rr + m]w L [m] = N NX k= X[n, k]e j km/n For non-overlaing windows, R=L : x[n] = x[n rl] w L [n rl] rl ale n ale (r + )R STFT Reconsrucion For sable reconsrucion mus overla window 5% (a leas) For Hann, Barle reconsruc wih overla and add. No division For sable reconsrucion mus overla window 5% (a leas) 3 4 Alicaions Time Frequency Analysis Alicaions Time Frequency Analysis Secrogram of Demodulaed FM radio (Adele on 96.5 MHz) Secrogram of FM radio KHz 38KHz 9KHz = =sec 5 6

5 Alicaions Time Frequency Analysis Alicaions Time Frequency Analysis Secrogram of digial communicaions - Frequency Shif Keying Secrogram of Orca whale 4 Frequency, Hz 3 = =sec Time, s 7 8 Alicaions Alicaion Noise removal Denoising of Sarse secrograms 4 Recall bird chir x[n] n Frequency, Hz Secrum of a bird chir Time, s Hz x 4 Secrum is sarse can imlemen adaive filer, or jus hreshold 9

6 Limiaions of Discree STFT Need overlaing No orhogonal Comuaionally inensive O(MN log N) Same size Heisenberg boxes From STFT o Waveles Basic Idea: low-freq changes slowly - fas racking unimoran Fas racking of high-freq is imoran in many as. Mus ada Heisenberg box o frequency Back o coninuous ime for a bi... From STFT o Waveles Coninuous ime u u Sf(u, ) = Wf(u, s) = Z Z f()w( u)e j d f() ( u )d s s *Morle - Grossmann 3 From STFT o Waveles The funcion Mus saisfy: Z Z is called a moher wavele () d = ()d = Wf(u, s) = Z f() ( u )d s s uni norm Band-Pass 4

7 STFT and Waveles Aoms STFT Aoms Wavele Aoms w( u)e j s ( u ) s Examles of Waveles Mexican Ha () =( )e / hi s = u u s =3 lo u u Haar 8 < () = : ale < ale < oherwise 5 6 Examle: Mexican Ha Waveles Transform Can be wrien as linear filering SombreroWavele Wf(u, s) = s Z f() ( = f() s() (u) s u )d log(s) s = s ( s ) u 7 Wavele coefficiens are a resul of bandass filering 8

8 Wavele Transform Orhonormal Haar Many differen consrucions for differen signals Haar good for iece-wise consan signals Bale-Lemarie : Sline olynomials Same scale non-overlaing Can consruc Orhogonal waveles For examle: dyadic Haar is orhonormal i,n() = i ( i n i ) i =[,, 3, ] Orhogonal beween scales 9 3 Orhonormal Haar Scaling funcion Same scale non-overlaing i,n() = i ( i n i ) i=m+3 i=m+ i=m+ i=m Orhogonal beween scales Problem: Every srech only covers half remaining bandwidh Need Infinie funcions Soluion: Plug low-ass secrum wih a scaling funcion 3 3

9 Scaling funcion i,n() = i ( i n i ) Haar Scaling funcion 8 < ale < () = : ale < oherwise i=m+ i=m Problem: Every srech only covers half remaining bandwidh Need Infinie funcions Soluion: Plug low-ass secrum wih a scaling funcion () = ale < oherwise Back o Discree Early 8 s, heoreical work by Morle, Grossman and Meyer (mah, geohysics) Lae 8 s link o DSP by Daubechies and Malla. Discree Wavele Transform NX W [k] = n= x[n] k[n] From CWT o DWT no so rivial Mus ake care o mainain roeries n 35 36

10 Examle: Discree Haar Wavele Haar for n= Examle: Discree Haar Wavele Haar for n=8 n. scaling k= k=4 s= scaling funcion k=5 k= k= s= WL k=3 ) y Fas DWT Fas DWT Fas Pyramidal Decomosiion n. - \ I..J ' 37 k=7. l. J ' k=6, deail g[n] --[@j-{f} --c_ s= moher wavele aroximaion h[n] k= k= > is ' f ' CJ{J/) Ass kjij FAT fl:_f;(q#sj)( U ctz )/ --[@j-{f} --c_ WL., ) y. J \ I '..J l ' > ' f '-... Comlexiy is O(N), less han FFT is CJ{J/) Ass kjij FAT - fl:_f;(q#sj)( U ctz )/ i

11 Haar DWT Examle x[n] Aroximaion from 5/56 coefficiens Haar Haar DFT 4 4 Examle: Denoising Noisy Signals Examle: Denoising by Thresholding noisy Haar denoised larges 5 coefficiens 43 44

12 Noisy Wavele Denoised

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