Spring 2014, EE123 Digital Signal Processing

Transcription

1 Aoucemets EE3 Digital Sigal Processig Lecture 9 Lab part I ad II posted will post III today or tomorrow Lab-bash uesday -3pm Cory hree shorter Midterms: / i class / i class /3 (or BD) i class / or / (BD) project presetatios. Posters ad demos based o slides by J.M. Kah Aoucemets Last time: Frequecy aalysis with DF Widowig oday: Cotiue Effect of zero-paddig Start Short-time Fourier rasform Widows Properties hese are characteristic of the widow type Widow Mai-lobe Sidelobe s Sidelobe log s Rect.9 M + Bartlett. M + Ha.3 M + Hammig. 3 M + Blackma. 7 M + Most of these (Bartlett, Ha, Hammig) have a trasitio width that is twice that of the rect widow. Warig: Always check what s the defiitio of M Adapted from ACourseIDigitalSigalProcessigby Boaz Porat, Wiley, Sprig, EE3 Digital Sigal Processig

2 Widows Examples Here we cosider several examples. As before, the samplig rate is s / =/ = Hz. Rectagular Widow, L = 3 w[] Rectagular Widow, L = 3 3 Sampled, Widowed Sigal, Rectagular Widow, L = 3 W(e j ) 3 3 DF of Rectagular Widow - - Ω / (Hz) DF of Sampled, Widowed Sigal Widows Examples riagular Widow, L = 3 w[]..... riagular Widow, L = 3 3. Sampled, Widowed Sigal, riagular Widow, L = 3 W(e j ) DF of riagular Widow - - / (Hz) DF of Sampled, Widowed Sigal v[]. -. V(e j ) v[]. -. V(e j ) / (Hz) / (Hz) Sprig, EE3 Digital Sigal Processig Sprig, EE3 Digital Sigal Processig Widows Examples Widows Examples Hammig Widow, L = 3 Hammig Widow, L =. Hammig Widow, L = 3 DF of Hammig Widow. Hammig Widow, L = DF of Hammig Widow w[] v[] Sampled, Widowed Sigal, Hammig Widow, L = 3 W(e j ) V(e j ) - - / (Hz) DF of Sampled, Widowed Sigal w[] v[] Sampled, Widowed Sigal, Hammig Widow, L = W(e j ) V(e j ) / (Hz) DF of Sampled, Widowed Sigal / (Hz) / (Hz) Sprig, EE3 Digital Sigal Processig 7 Sprig, EE3 Digital Sigal M. Lustig, Processig EECS UC Berkeley

3 Optimal Widow: Kaiser Miimum mai-lobe width for a give sidelobe eergy % R sidelobes H(ej! ) d! R H(ej! ) d! Example y = si(.99)+. si(.) apple < Widow is parametrized with L ad β β determies side-lobe level L determies mai-lobe width OS Eq. 9 Example Zero-Paddig I preparatio for takig a N-poit DF, we may zero-pad the widowed block of sigal samples to a block legth N L: ( v[] apple apple L L apple apple N his zero-paddig has o e ect o the DF of v[], sice the DF is computed by summig over < <. E ect of Zero Paddig We take the N-poit DF of the zero-padded v[], to obtai the block of N spectral samples: V [k], apple k apple N Sprig, EE3 Digital Sigal Processig

4 Zero-Paddig Cosider the DF of the zero-padded v[]. Sice the zero-padded v[] is of legth N, itsdfcabewritte: V (e j! )= = he N-poit DF of v[] is give by: V [k] = = v[]w k N = N v[]e j!, <! < X v[]e j( /N)k, apple k apple N = We see that V [k] correspods to the samples of V (e j! ): V [k] =V (e j! )!=k N, apple k apple N o obtai samples at more closely spaced frequecies, we zero-pad v[] to loger block legth N. hespectrumisthe same, we just have more samples. Frequecy Aalysis with DF Note that the orderig of the DF samples is uusual. V [k] = = he DC sample of the DF is k = V [] = = v[]w v[]w k N N = N X v[] = he positive frequecies are the first N/ samples he first N/ egative frequecies are circularly shifted (( k)) N = N k so they are the last N/ samples. (Use fftshift to reorder) Sprig, EE3 Digital Sigal Processig 3 Sprig, EE3 Digital Sigal Processig Frequecy Aalysis with DF Examples: Hammig Widow, L = 3, N = 3 Frequecy Aalysis with DF Examples: Hammig Widow, L = 3, Zero-Padded to N = Sampled, Widowed Sigal, Hammig Widow, L = 3, Zero-Padded to N = 3 N-Poit DF of Sampled, Widowed, Zero-Padded Sigal Sampled, Widowed Sigal, Hammig Widow, L = 3, Zero-Padded to N = N-Poit DF of Sampled, Widowed, Zero-Padded Sigal.. Zero-Padded v[] V[k] 3 k Zero-Padded v[] V[k] 3 k Spectrum of Sampled, Widowed, Zero-Padded Sigal Spectrum of Sampled, Widowed, Zero-Padded Sigal V(e j ) V[k], k = k /N V(e j ) V[k], k = k /N V[k], V(e j ) V[k], V(e j ) / (Hz) / (Hz) Sprig, EE3 Digital Sigal Processig Sprig, EE3 Digital Sigal Processig

5 idf Rect widow 7 idf Frequecy Aalysis with DF Legth of widow determies spectral resolutio ype of widow determies side-lobe amplitude. (Some widows have better tradeo betwee resolutio-sidelobe) A yo pt with a history of lower limb weakess referred for mri screeig of brai ad whole spie for cord. MRI sagittal screeig of dorsal regio shows a fait uiform liear high sigal at the ceter of the cord. he sigal abormality likely to represet: Zero-paddig approximates the DF better. Does ot itroduce ew iformatio! () Cord demyeliatio. () Syrix (spial cord disease). (3) Artifact. Aswer : Its a artifact, kow as trucatio or Gibbs artifact 9 Sprig, EE3 Digital Sigal Processig

6 Potetial Problems ad Solutios Potetial Problems ad Solutios Problem Possible Solutios. Spectral error a. Filter sigal to reduce frequecy cotet above s/ = /. from aliasig Ch. b. Icrease samplig frequecy s = /.. Isu ciet frequecy a. Icrease L resolutio. b. Use widow havig arrow mai lobe. 3. Spectral error a. Use widow havig low side lobes. from leakage b. Icrease L. Missig features a. Icrease L, due to spectral samplig. b. Icrease N by zero-paddig v[] to legth N > L. ispectrum Example Sprig, EE3 Digital Sigal Processig Discrete rasforms (Fiite) DF is oly oe out of a LARGE class of trasforms Used for: Aalysis Compressio Deoisig Detectio Recogitio Approximatio (Sparse) Sparse represetatio has bee oe of the hottest research topics i the last years i sp 3 Example of spectral aalysis Spectrum of a bird chirpig Iterestig,... but... Does ot tell the whole story No temporal iformatio! x[] Spectrum of a bird chirp 3... Hz x

7 ime Depedet Fourier rasform o get temporal iformatio, use part of the sigal aroud every time poit ime Depedet Fourier rasform o get temporal iformatio, use part of the sigal aroud every time poit X[,!) = X m= x[ + m]w[m]e j!m X[,!) = X m= x[ + m]w[m]e j!m *Also called Short-time Fourier rasform (SF) *Also called Short-time Fourier rasform (SF) Mappig from D D, discrete, w cot. Simply slide a widow ad compute DF Spectrogram 3 3 Frequecy, Hz Frequecy, Hz Frequecy, Hz ime, s 3 Discrete ime Depedet F X r [k] = LX m= L - Widow legth R - Jump of samples N - DF legth x[rr + m]w[m]e j km/n radeoff betwee time ad frequecy resolutio 7

8 Heiseberg Boxes ime-frequecy ucertaity priciple t!! t DF X[k] =! = x[]e j k/n!! = N t = N t! t = oe DF coefficiet t 9 3