EE123 Digital Signal Processing

Transcription

1 Aoucemes EE3 Digial Sigal Processig Lecure Miderm: Friday ex week Ope everyhig... bu chea shee recommeded isead Who ca o say ill pm? Opioal homework ex week Will give you miderm ad pracice quesios How s lab I goig? How do you kow his guy is isae? Las Time Frequecy Aalysis wih DFT Widowig Zero-Paddig Today: Time-Depede Fourier Trasform Heiseberg Boxes Specrum o symmeric, so ca mus be imagiary hp://xkcd.com// Discree Trasforms (Fiie) DFT is oly oe ou of a LARGE class of rasforms Used for: Aalysis Compressio Deoisig Deecio Recogiio Approximaio (Sparse) Sparse represeaio has bee oe of hoes research opics i las years i sp Example of specral aalysis Specrum of a bird chirpig Ieresig,... bu... Does o ell whole sory No emporal iformaio x[] Specrum of a bird chirp 3... Hz x

2 Time Depede Fourier Trasform To ge emporal iformaio, use par of sigal aroud every ime poi Time Depede Fourier Trasform To ge emporal iformaio, use par of sigal aroud every ime poi X[, ) = X m= x[ + m]w[m]e jm X[, ) = X m= x[ + m]w[m]e jm *Also called Shor-ime Fourier Trasform (STFT) *Also called Shor-ime Fourier Trasform (STFT) Mappig from D D, discree, w co. Simply slide a widow ad compue DTFT Discree Time Depede FT X r [k] = LX m= L - Widow legh R - Jump of samples N - DFT legh x[rr + m]w[m]e j km/n Tradeoff bewee ime ad frequecy resoluio Heiseberg Boxes Time-Frequecy uceraiy priciple hp:// DFT = N = N X[k] = = x[]e j k/n = oe DFT coefficie

3 DFT = N = N X[k] = = x[]e j k/n Discree STFT X[r, k] = = L = L LX m= opioal x[rr + m]w[m]e j km/n = Quesio: Wha is effec of zero-paddig? Aswer: Overlapped Tilig oe STFT coefficie Discree STFT X[r, k] = = L = L LX m= opioal x[rr + m]w[m]e j km/n Applicaios Time Frequecy Aalysis of Orca whale 3 Quesio: Wha is effec of R o ilig? wha effec of N? Aswer: Overlappig i ime of frequecy or boh (A) Sidelobes of Ha vs recagular widow DTFT of Recagular Widow DTFT of Hammig Widow 3 3 Wha is differece bewee specrograms? a) Widow size B<A b) Widow size B>A (B) c) Widow ype. is differe d) (A) uses overlappig widow v[]. Sampled, Widowed Sigal, Hammig Widow, L = 3 -. W(e j T ) V(e j T ) - - / (Hz) DTFT of Sampled, Widowed Sigal W(e j T ) / (Hz) / (Hz)

4 Origial Sigal Widows Examples Overlappig Ha Widows (A) Hammig Widow, L = 3 3 Hammig Widow, L = 3 DTFT of Hammig Widow. w[]. (B) 3 W(ejZ T)... Widows Examples hp://e.wikipedia.org/wiki/fm_broadcasig for more iformaio). M Samples The basebad sigal is: Z/S (Hz) Hammig LR) = cos( (f x() = (L + R) +. cos( fwidow, p ) + (L p )) +. RBDS() cos( (3fp )). Sampled, Widowed Sigal, Hammig Widow, L = 3. DTFT of Sampled, Widowed Sigal sereo pilo digial RBDS a) Widow c) Widow ype differe Task : Time-Frequecy plossize of B<A radio-frequecy specrum wih is SDR. e sigal RBDS() is a sigal cosiss of ± a cosa iervals which ecode, ad bis. This is The samples ha are obaied by SDR represe a badwidh of specrum a ceer frecrum of x(): b) Widow size B>A d) (A) uses overlappigaroud widow cy. Hece, whe demodulaig o base-bad (i.e. zero frequecy) sigal mus be imagiary sice s a o symmeric Fourier rasform. I his case, we would like o display boh sides of specrum Miki Lusig UCB. Based o 3 Course Noes by J.M Kah Sampled, Widowed Sigal, Hammig Widow, L = - Z/S (Hz) - Fall, EE3 Digial Z/S (Hz) Sigal Processig DTFT of Sampled, Widowed Sigal. ify fucio myspecrogram(x,m,fs) such ha i deecs if ipu sigal x is complex. I case, i will compue a double sided specrum ha is ceered aroud DC. For his, i would we useful e malab commads: isreal ad ffshif. v[]. V(ejZ T) moo w[] v[] W(ejZ T) V(ejZ T) Hammig Widow, L = DTFT of Hammig Widow moo sereo pilo digial RBDS Figure : Exracig segmes of a log sigal wih a Ha widow Time, s. p://e.wikipedia.org/wiki/fm_broadcasig for more iformaio). The basebad sigal is:. 3 The sigal RBDS() is a sigal cosiss of ± a cosa iervals which ecode, ad bis. This is 3 x() = (L + R) +. R) cos( (f cos( (3fp ))specrum.. Wha is cos( f differece bewee RBDS() specrograms? p ) + (L p )) +. of x(): -. frequecy while resoluio improves, emporal resoluio degrades will firs look a radio FM specrum. I US broadcas FM radio bad is.-7.mhz. Z S spli io KHz slos. This is relaively a large badwidh ad refor i is also called widebad Miki Lusig UCB. Based o Course Noes by J.M Kah Fall, EE3 Digial Sigal Processig as opposed o arrowbad FM which ca be as low as Khz. I FM radio iformaio isnow, ecoded SDR already demodulaes by carrier frequecy fc, so samples you ge are acually ofcarrier, FM of FM radio Basebad odulaig frequecy of Z 3 - y[] = y(t ) = A exp j f - / (Hz) Z T x( )d, yc () = A cos fc + f x( )d. hp://e.wikipedia.org/wiki/fm_broadcasig for more iformaio). The basebad sigal is: x() = (L + R) +. cos( fp ) + (L R) cos( (fp )) +. RBDS() cos( (3fp )). w, SDR already demodulaes by carrier frequecy fc, so samples you ge are acually moo sereo pilo digial RBDS where T is samplig x() = (Lrae. + R) +. cos( fp ) + (L R) cos( (fp )) +. RBDS() cos( (3fp ))., fc is carrier frequecy, f is frequecy deviaioz ad T x() is a ormalized basebad sigal. Thesigal RBDS()is a sigal cosiss of ± a cosa iervals which ecode, ad bis. This is moo specrum of x(): pilo sereo digial RBDS y[] = y(t ) = A exp j f x( )d, Pick your favorie radio saio (make sure re is good recepio where you live Do ry i i broadcas FM basebad sigal, x(), cosiss of moo (Lef+Righ) Chaels from 3Hz o KHz, The sigal RBDS() is a sigal cosiss of ± a cosa iervals which ecode, ad bis. This is baseme or i rooms iside Cory ha do o have widows). I ve esed good recepio from Cory hall o sigal a fsamplig modulaed Sereo (Lef - Righ) chaels aroud fp = 3KHz specrum of x(): p = KHz, ere T is rae.ampliude of FM radio h floor a.mhz ad.mhz. Acquire secods of daa a a samplig rae of, Hz. You digial RBDS, which ecodes saio iformaio, sog ame ec. a 3 fp = 7KHz. (See Compue ca use eir rl_cp or rl_sdr for ha. ad display a specrogram wih a widow of. k your favorie radio saio (make sure re is good recepio where you live Do rysamples. i i Iclude specrogram i your repor. Wha is specral ad emporal resoluio? Explai eme or i rooms iside Cory ha do o have widows). I ve esed good recepio fromwha Coryyou hallsee. Do forge o play wih di ere dyamic rages of specrogram for bes visualizaio. floor a.mhz ad.mhz. Acquire secods of daa a a samplig rae of, Hz. You use eir rl_cp or rl_sdr for ha. Compue ad display a specrogram wih a widow of of Demodulaed FM radio (Adele o. MHz) Now, SDR already demodulaes by carrier frequecy fc, so samples you ge are acually Z T mples. Iclude specrogram i your repor. Wha is specral ad emporal resoluio? I Explai may aalog radio sysems, FM demodulaio a phased-locked loop. y[] = y(t ) is = A performed exp j f x( by )d implemeig, 7KHz a you see. Do forge o play wih di ere dyamic rages of specrogram for bes visualizaio. ha racks frequecy of sigal ad exrac message x(). However, i is very easy o impleme hp://e.wikipedia.org/wiki/fm_broadcasig for more iformaio). The basebad sigal is: where T is samplig rae. a FM demodulaio whe samples are discree. Noe, ha sigal x() is proporioal o 3KHz Pick your favorie radio saio (make sure re is good recepio where you live Do ry i i isaaeous frequecy of sigal. I iside is also o derivaive We ca implebaseme or i rooms Cory proporioal ha do o have widows). I ve esed good recepioof from Coryphase hall may aalog radio sysems, FM demodulaio is performed by implemeig a phased-locked loop floor a.mhz ad.mhz. Acquire secods of daa a a samplig rae of, Hz. You me his by compuig h fiie di ereces of phase of our sigal agle(y[])-agle(y[-]). A KHz ca use eir rl_cp or rl_sdr for ha. Compue ad display a specrogram wih a widow of racks frequecy of sigal ad exrac message x(). However, i is very easy o impleme [ specrogram i your repor. Wha is produc specral ad emporal aleraive ha is robus o samples. phaseiclude wrappig is o compue y[]yresoluio? ]Explai ad compue is. wha you see. Do forge o play wih di ere dyamic rages of specrogram for bes visualizaio. FM demodulaio o SDR already = whe samples are discree. Noe, ha sigal x() is proporioal =sec Now, yourself carrier frequecy fc, so samples you phase. Covice ha demodulaes i works Iby malab i ca be simply implemeed by: ge are acually aaeous frequecy of sigal. I is also proporioal o derivaive of M. phase We ca imple Lusig, EECS UC Berkeley M. Lusig, I may aalog radio sysems, FM demodulaio is performedzbyt implemeig a phased-locked loop EECS UC Berkeley his by compuig fiie di ereces of phase of our sigal agle(y[])-agle(y[-]). >> x = A agle(y[:ed].*coj(y[:ed-])); ha racks frequecy of sigal ad exrac message x(). However, i is very easy o impleme y[] = y(t ) = A exp j f x( )d, a FM demodulaio whe samples are discree. Noe, ha sigal x() is proporioal o raive ha is robus o phase wrappig is o compue produc y[]y [ ] ad compue is isaaeous frequecy of sigal. I is also proporioal o derivaive of phase We ca implefm demodulae samplesmeyou from di ereces SDRofad display specrogram. A Noe, ha aferfm his bygo compuig fiie phase of our sigal agle(y[])-agle(y[-]). se. Covice yourself ha i works I malab i ca be simply implemeed by: aleraive ha is robus o phase wrappig is o compue produc y[]y [ ] ad compue is where T issigal samplig rae. demodulaig should realyourself adha hece half specrum be displayed. Ideify phase. be Covice i worksoly I malab i ca be simply implemeedshould by: x = agle(y[:ed].*coj(y[:ed-])); moo audio, pilo, sereo ad RBDS sigals. Noe, ha RBDS sigal may be oo weak o >> x = agle(y[:ed].*coj(y[:ed-])); Pick your favorie radio saio (make sure re is good recepio where you live Do ry i i deec or may eed beer specral resoluio. FM demodulae samples you go from SDR ad display specrogram. Noe, ha aferfm iside Cory o widows). I ve esed goodideify recepio from Cory hall sigalha shoulddo be real adhave hece oly half specrum should be displayed. demodulae samples you go from SDR ad display specrogram. Noe, ha aferfmbaseme or i roomsdemodulaig moo audio, pilo, sereo ad RBDS sigals. Noe, ha a RBDS may be oo weak o h floor a.mhz ad.mhz. Acquire secods of daa a sigal samplig rae of, Hz. You modulaig sigal should be real ad hece oly half specrum should be displayed. Ideify deec or may eed beer specral resoluio. ad display a specrogram wih a widow of o audio, pilo, sereo ad RBDS sigals. Noe, ha RBDS sigal may be oo weak oca use eir rl_cp or rl_sdr for ha. Compue samples. Iclude specrogram i your repor. Wha is specral ad emporal resoluio? Explai Subcarrier FM radio (Hidde Radio Saios) Applicaios ec or may eed beer specral resoluio. wha you see. Do forge o play wih di ere dyamic rages of specrogram for bes visualizaio. Time Frequecy Aalysis I may aalog radio sysems, FM demodulaio is performed by implemeig a phased-locked loop ha racks frequecy of sigal ad exrac message x(). However, i is very easy o impleme a FM demodulaio whe samples are discree. Noe, ha sigal x() is proporioal o isaaeous frequecy of sigal. I is also proporioal o derivaive of phase We ca impleme his by compuig fiie di ereces of phase of our sigal agle(y[])-agle(y[-]). A [Keyig digial commuicaios Frequecy Shif aleraive ha is robus oofphase wrappig is o compue- produc y[]y ] ad compue is phase. Covice yourself ha i works I malab i ca be simply implemeed by: >> x = agle(y[:ed].*coj(y[:ed-])); FM demodulae samples you go from SDR ad display specrogram. Noe, ha aferfm demodulaig sigal should be real ad hece oly half specrum should be displayed. Ideify moo audio, pilo, sereo ad RBDS sigals. Noe, ha RBDS sigal may be oo weak o deec or may eed beer specral resoluio. = =sec

5 STFT Recosrucio x[rr + m]w L [m] = N k= X[, k]e j km/n For o-overlappig widows, R=L : x[] = Wha is problem? x[ rl] w L [ rl] rl apple apple (r + )R STFT Recosrucio x[rr + m]w L [m] = N k= X[, k]e j km/n For o-overlappig widows, R=L : x[] = x[ rl] w L [ rl] rl apple apple (r + )R For sable recosrucio mus overlap widow % (a leas) STFT Recosrucio Applicaios For sable recosrucio mus overlap widow % (a leas) Noise removal For Ha, Barle recosruc wih overlap ad add. No divisio Recall bird chirp x[] Specrum of a bird chirp 3... Hz x Applicaio Deoisig of Sparse specrograms Limiaios of Discree STFT Need overlappig No orhogoal 3 Compuaioally iesive O(MN log N) Same size Heiseberg boxes Specrum is sparse ca impleme adapive filer, or jus hreshold

6 From STFT o Waveles Basic Idea: low-freq chages slowly - fas rackig uimpora Fas rackig of high-freq is impora i may apps. Mus adap Heiseberg box o frequecy Back o coiuous ime for a bi...