Computer. x c (t) x[n] y[n] y c (t) A-to-D Computer D-to-A. i A sequence of numbers i Mathematical representation:

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1 Digital Speech Processig Lecture Review of DSP Fudaetals Iput Sigal Aalog-to- Digital Coversio What is DSP? Coputer Digital-to- Aalog Coversio Output Sigal Digital ethod to represet a quatity, a pheoeo or a evet Why digital? it Sigal What is a sigal? soethig (e.g., a soud, gesture, or object) that carries iforatio a detectable physical quatity (e.g., a voltage, curret, or agetic field stregth) by which essages or iforatio ca be trasitted What are we iterested i, particularly whe the sigal is speech? Processig What id of processig do we eed ad ecouter alost everyday? Special effects? Coo Fors of Coputig ext processig hadlig of text, tables, basic arithetic ad logic operatios (i.e., calculator fuctios) Word processig Laguage processig Spreadsheet processig Presetatio processig Sigal Processig a ore geeral for of iforatio processig, icludig hadlig of speech, audio, iage, video, etc. Filterig/spectral aalysis Aalysis, recogitio, sythesis ad codig of real world sigals Detectio ad estiatio of sigals i the presece of oise or iterferece Advatages of Digital Represetatios Iput Sigal A-to-D Coverter Sigal Processor D-to-A Coverter Output Sigal Peraece ad robustess of sigal represetatios; zerodistortio reproductio ay be achievable Advaced IC techology wors well for digital systes Virtually ifiite flexibility with digital systes ulti-fuctioality ulti-iput/ulti-output Idispesable i telecouicatios which is virtually all digital at the preset tie 3 4 Digital Processig of Aalog Sigals x c (t) x[] y[] y c (t) A-to-D Coputer D-to-A A-to-D coversio: badwidth cotrol, saplig ad quatizatio Coputatioal processig: ipleeted o coputers or ASICs with fiite-precisio arithetic basic uerical processig: add, subtract, ultiply (scalig, aplificatio, atteuatio), ute, algorithic uerical processig: covolutio or liear filterig, o-liear filterig (e.g., edia filterig), differece equatios, DF, iverse filterig, AX/I, D-to-A coversio: re-quatificatio* ad filterig (or iterpolatio) for recostructio 5 Discrete-ie Sigals i A sequece of ubers i atheatical represetatio: x = { x[ ]}, < < i Sapled fro a aalog sigal, xa(), t at tie t =, x [ ] = xa ( ), < < i is called the saplig period, ad its reciprocal, FS = /, is called the saplig frequecy FS = 8 Hz = / 8 = 5μ sec FS = Hz = / = μ sec FS = 6 Hz = /6 = 6.5μ sec F = 4Hz = / 4 = 5 μ sec S 6

2 y y y y Speech Wavefor Display Varyig Saplig Rates plot( ); Fs=8 Hz Fs=6 Hz ste( ); Fs= Hz 7 8 Varyig Saplig Rates Fs=8 Hz Fs=6 Hz Fs= Hz 9 Quatizatio i x [ ] ca be quatized to oe of a fiite set of values which is the represeted digitally i bits, hece a truly digital sigal; the course aterial ostly deals with discrete-tie sigals (discrete-value oly whe oted). out Quatizatio: 7.4 rasforig a cotiuously- valued iput ito a 6.8 represetatio that assues 5. oe out of a fiite set of values 4.6 he fiite set of output values 3 i is idexed; e.g., the value has a idex of 6, or () i biary represetatio Storage or trasissio uses biary represetatio; a quatizatio table is eeded A 3-bit uifor quatizer Discrete Sigals Siewave Spectru saple 6 4 Sapled Siusoid 5si(π) quatize 6 4 Aalog siusoid, 5si(πx) Discrete siusoid roud[5si(π)] x 6 4 Quatized siusoid roud[5si(πx)] quatize -4 saple SR is a fuctio of B, the uber of bits i the quatizer x

3 Issues with Discrete Sigals what saplig rate is appropriate 6.4 Hz (telephoe badwidth), 8 Hz (exteded telephoe BW), Hz (exteded badwidth), 6 Hz (hi-fi speech) how ay quatizatio levels are ecessary at each bit rate (bits/saple) 6,, 8, => ultiately deteries the S/ ratio of the speech speech codig is cocered with aswerig this questio i a optial aer 3 aplitude he Saplig heore Sapled Hz ad 7 Hz Cosie Waves; F s = 8 Hz tie i s A badliited sigal ca be recostructed exactly fro saples tae with saplig frequecy π = Fs fax or = Ws W ax 4 Deo Exaples. 5 Hz aalog badwidth sapled at, 5,.5,.5 Hz (otice the aliasig that arises whe the saplig rate is below Hz). quatizatio to various levels,9,4,, ad bit quatizatio (otice the distortio itroduced whe the uber of bits is too low) 3. usic quatizatio 4 bit audio quatized to various levels: bit audio, bit oise (white oise) bit audio, 4 bit oise (white oise) 8 bit audio, 6 bit oise (colored) 6 bit audio, 8 bit oise (sigal correlated soewhat) Discrete-ie (D) Sigals are Sequeces x[] deotes the sequece value at tie Sources of sequeces: Saplig a cotiuous-tie sigal x[] = x c () = x c (t) t= atheatical forulas geerative syste e.g., x[] =.3 x[-] -; x[] = Ipulse Represetatio of Sequeces A sequece, a fuctio a 3 3δ[ + 3] x[ ] = x[ ] δ[ ] = a δ[ ] Value of the fuctio at Soe Useful Sequeces uit saple δ[] =, =, real expoetial x[] = α a δ[ ] a 7 δ[ 7] p[ ] = a δ[ + 3] + aδ[ ] + aδ[ ] + a δ[ 7] 3 7 uit step u[] =,, < sie wave x[] = Acos( + φ) 7 8 3

4 Variats o Discrete-ie Step Fuctio Coplex Sigal x [ ] = ( j) u [ ] sigal flips aroud 9 Coplex Sigal jθ x[ ] = ( α + jβ) u[ ] = ( re ) u[ ] r = α + β θ β α = ta ( β / ) β r θ α jθ x [ ] = r e u [ ] r is a dyig expoetial jθ e is a liear phase ter Coplex D Siusoid j x[ ] = Ae Frequecy is i radias (per saple), or just radias ot radias per secod because tie idex is diesioless oce sapled, x[] is a sequece that relates to tie oly through the saplig period Iportat property: periodic i with period π: j j( r ) Ae = Ae + π Oly uique frequecies are to π (or π to +π) Sae applies to real siusoids Periodic D Sigals A sigal is periodic with period if x[] = x[+] for all For the coplex expoetial this coditio becoes j j( + ) Ae = Ae which requires = π for soe iteger hus, ot all D siusoids are periodic! Cosequece: there are distiguishable frequecies with period e.g., = π /, =,,,- 3 Sapled Speech Wavefor x a (t) ALAB: plot ALAB: ste =.5 sec, f S =8 Hz x a (),x() x[] rap #: loss of saplig tie idex 4 4

5 Sigal Processig rasfor digital sigal ito ore desirable for x[] y[]=[x[]] [ [ ]] x[] y[] sigle iput sigle output sigle iput ultiple output, e.g., filter ba aalysis, siusoidal su aalysis, etc. 5 LI Discrete-ie Systes x[] δ[] LI Syste y[] h[] Liearity (superpositio): Τ{ ax [] + bx []}= aτ{ x []}+ bτ{ x []} ie-ivariace (shift-ivariace): x [] = x[ d ] y [] = y[ d ] LI iplies discrete covolutio: y[] = x[]h[ ] = x[] h[] = h[] x[] = 6 LI Discrete-ie Systes Exaple: Is syste y [ ] = x [ ] + x [ + ] + 3 liear? x[ ] y[ ] = x[ ] + x[ + ] + 3 x[ ] y[ ] = x[ ] + x[ + ] + 3 x[ ] + x[ ] y3[ ] = x[ ] + x[ ] + x[ + ] + x[ + ] + 3 y [ ] + y [ ] ot a liear syste! Is syste y [ ] = x [ ] + x [ + ] + 3 tie/shift ivariat? y [ ] = x [ ] + x [ + ] + 3 y [ ] = x [ ] + x [ + ] + 3 Syste is tie ivariat! Is syste y [ ] = x [ ] + x [ + ] + 3 causal? y [ ] depeds o x+ [ ], a saple i the future Syste is ot causal! 7 Covolutio Exaple 3 3 x [ ] = h [ ] = otherwise otherwise What is y [ ] for this syste? Solutio : y [ ] = x [ ]* h [ ] = h [ ] x [ ] = = ( + ) 3 = 3 = = (7 ) 4 6 = 3, 7 x[],h[] y[] = = = h[] x[] =3 =4 =5 Covolutio Exaple he ipulse respose of a LI syste is of the for: h [ ] = a u [ ] a < ad the iput to the syste is of the for: x [ ] = b u [ ] b <, b a Deterie the output of the syste usig the forula for discrete covolutio. SOLUIO: =6 =7 = y [ ] = a ub [ ] u [ ] = = b a b u[ ] = b ( a/ b) u[ ] = = ( a/ b) b a = b = u[ ] ( a/ b) b a 3 5

6 Covolutio Exaple 3 Covolutio Exaple Cosider a digital syste with iput x [ ] = for =,,,3 ad everywhere else, ad with ipulse respose h [ ] = a u [ ], a <. Deterie the respose y [ ] of this liear syste. SOLUIO: We recogize that x [ ] ca be writte as the differece betwee two step fuctios, i.e., x [ ] = u [ ] u [ 4]. Hece we ca solve for y [ ] as the differece betwee the output of the liear syste with a step iput ad the output of the liear syste with a delayed step iput. hus we solve for the respose to a uit step as: a a y[ ] = ua [ ] u [ ] = u [ ] = a y[ ] = y [ ] y [ 4] 3 Liear ie-ivariat Systes Sigal Processig Operatios easiest to uderstad easiest to aipulate powerful processig capabilities characterized copletely by their respose to uit saple, h(), via covolutio relatioship y [ ] = x [ ] h [ ] = xh [ ] [ ] = h [ ] x [ ] = y [ ] = h [ ] x [ ], where deotes discrete covolutio x[] y[]=x[] h[] h[] * = basis for liear filterig used as odels for speech productio (source covolved with syste) 33 D is a delay of -saple Ca replace D with delay eleet z 34 Equivalet LI Systes ore Coplex Filter Itercoectios x[] h[] h[] y[] h[] x[] h[] h[] y[] x[] h[] y[] x[] h[]*h[] y[] x[] h[]+h[] y[] h []*h []= h []*h [] h []+h []= h []+h [] 35 y [ ] = x [ ]* h[ ] c h [ ] = h[ ]*( h [ ] + h [ ]) + h [ ] c

7 saple uber saple uber saple uber saple uber etwor View of Filterig (FIR Filter) etwor View of Filterig (IIR Filter) D(Delay Eleet) z y [ ] = bx [ ] + bx [ ] b x [ + ] + b x [ ] y [ ] = ay [ ] + bx [ ] + bx [ ] rasfor Represetatios z-rasfor Represetatios z-trasfor: x [ ] X( z) Xz ( ) = xz [ ] = π j = x [ ] = X ( zz ) dz C ifiite power series i z, with x[] as coefficiets of ter i z direct evaluatio usig residue theore partial fractio expasio of X(z) log divisio power series expasio X(z) coverges (is fiite) oly for certai values of z: = x [ ] z < - sufficiet coditio for covergece regio of covergece: R < z < R 4 39 Exaples of Covergece Regios Exaples of Covergece Regios. x [ ] = δ[ ] -- delayed ipulse Xz z z ( ) = coverges for >, > ; z <, < ; z, =. x [ ] = u [ ] u [ ] -- box pulse - z X(z) = ( )z = -- coverges for < z < = z all fiite legth sequeces coverge i the regio < z < 3. x [ ] = au [ ] ( a< ) Xz ( ) = az = --coverges for a < z = az all ifiite duratio sequeces which are o-zero for coverge i a regio z > R aplitude aplitude aplitude 4 4. x [ ] = bu[ ] Xz ( ) = bz = --coverges for z < b = bz all ifiite duratio sequeces which are o-zero for < coverge i a regio z < R 5. x [ ] o-zero for < < ca be viewed as a cobiatio of 3 ad 4,givig a covergece regio of the for R < z < R sub-sequece for => z > R sub-sequece for < => z < R total sequece => R < z < R aplitude R R 4 7

8 Exaple z-rasfor Property If x [ ] has z-trasfor X( z) with ROC of ri < z < ro, fid the z-trasfor, Y( z), ad the regio of covergece for the sequece y [ ] = ax [ ] i ters of Xz ( ) Solutio: Xz ( ) = xz [ ] = Yz ( ) = yz [ ] = axz [ ] = = x [ ]( z/ a) = X( z/ a) = ROC: a r < z < a r i = o 43 he sequece x [ ] has z-trasfor X( z). Show that the sequece x[ ] has z-trasfor dx( z) z. dz Solutio: Xz ( ) = xz [ ] = dx( z) = x[ ] z dz = = x[ ] z z = = Zx ( [ ]) z 44 Iverse z-rasfor Partial Fractio Expasio [ ] ( ) x X zz dz π j C = where C is a closed cotour that ecircles the origi of the z-plae ad lies iside the regio of covergece R C R for X(z) ratioal, ca use a partial fractio expasio for fidig iverse trasfors Hz ( ) = bz + bz b z + az a bz + bz b = ; ( > ) ( z p )( z p )...( z p ) A A A Hz ( ) = z p z p z p Hz ( ) A A A A = ; p = z z p z p z p z p Hz ( ) A = ( z p ) i =,,..., i i z z= pi Exaple of Partial Fractios rasfor Properties z + z+ Fid the iverse z-trasfor of Hz ( ) = < z < ( z + 3z+ ) Hz ( ) z + z+ A A A = = + + z z( z+ )( z+ ) z z+ z+ z + z + z + z + A = = A = = ( z+ )( z+ ) z( z+ ) z + z+ 3 A = = zz ( + ) z= z= z= z (3/) z Hz ( ) = + < z < z+ z+ h [ ] = [ ] ( ) [ ] 3 δ u ( ) u[ ] 47 Liearity ax []+bx [] ax (z)+bx (z) Shift x[- ] z X( z) Expoetial Weightig a x[] X(a - z) Liear Weightig x[] -z dx(z)/dz ie Reversal o-causal, eed x[ x[-] -] to be causal for fiite legth sequece X(z - ) Covolutio x[] * h[] X(z) H(z) ultiplicatio of x[] w[] ( ) ( / ) π j C Sequeces doai XvWz vv dv circular covolutio i the frequecy 48 8

9 Discrete-ie Fourier rasfor Discrete- ie Fourier rasfor (DF) j z= e j = Xe ( ) = Xz ( ) j = xe [ ] j z= e ; z =,arg( z) = j π j j x [ ] = ( ) Xe e d π π evaluatio of X(z) o the uit circle i the z-plae sufficiet coditio for existece of Fourier trasfor is: 49 = x [ ] z = x [ ] <, sice z = = 5 Siple DFs DF Exaples Ipulse Delayed ipulse Step fuctio Rectagular widow Expoetial Bacward expoetial x [ ] = δ [ ], j Xe ( ) = x [ ] = δ [ ], j Xe ( ) = e j j x [ ] = u [ ], Xe ( ) = j e j j e x [ ] = u [ ] u [ ], Xe ( ) = j e j x [ ] = a u [ ], Xe ( ) =, a< j ae j x [ ] = b u[ ], Xe ( ) =, b> j be 5 x [ ] = β u [ ] β =.9 j Xe ( ) = β < j βe 5 DF Exaples x [ ] = cos( ), < < j X ( e ) = [ πδ ( + π ) + πδ ( + + π )] = j i Withi iterval π < < π, X( e ) is coprised of a pair of ipulses at ± DF Exaples

10 DF Exaples Fourier rasfor Properties.5, π/, F S/ 4, π F S/ periodicity i Xe j j( + π) ( ) = Xe ( ). 5, π, F S/, π F S.75, 3π/, 3F S/ 4, π 3F S/ period of π correspods to oce aroud uit circle i the z-plae Uits of Frequecy (Digital Doai) (rap # - loss of F S ) oralized frequecy: f,.5 (idepedet of F S ) oralized radia frequecy:, π π (idepedet of F S ) digital frequecy: f D = f / F S, F S / F S,,,, π, F S, π F S f,, f D, D 55 digital radia frequecy: D = / F S, πf S πf S 56 he DF Discrete Fourier rasfor Discrete Fourier rasfor cosider a periodic sigal with period (saples) x [ ] = x [ + ], < < x [ ] ca be represeted exactly by a discrete su of siusoids j π / X [ ] = xe [ ] = x [ ] = Xe [ ] = j π / sequece values DF coefficiets 57 exact represetatio of the discrete periodic sequece 58 DF Uit Vectors (=8) DF Exaples = = j π /8 ; e = e = j j π /8 = ; e = j j π /8 ; ( ) = e = j = e = j π /8 3; ( ) j π /8 4; = e = + j j π /8 = 6; e = j j π /8 5; ( ) = e = + j j π /8 7; ( ) 59 δ[ ] DF{ δ[ ]} 6

11 DF Exaples DF Exaples x [ ] = (.9) 3 ( = 3) 6 6 Circularly Shiftig Sequeces Issues with Periodic Sequeces Give a cotiuous tie sigal that is periodic with period ( + /) saples, is the sapled versio (saplig period ) a digital periodic sequece? the DF frequecies for a -poit sequece are =(π / ) the sigal periodicity frequecies are ' = ( π /( + /)) the digital sigal is ot periodic with period ( + /) saples (i this case it actually is periodic with period ( + ) saples) Periodic Sequeces?? DF for Fiite Legth Sequeces 65 66

12 saple uber Fiite Legth Sequeces cosider a fiite legth (but ot periodic) sequece, x[], that is zero outside the iterval Xz ( ) = xz [ ] = evaluate X(z) at equally spaced poits o the uit circle, j π/ z = e, =,,..., j π/ Xe ( ) = [ ] π / xe j, =,,..., = --loos lie DF of periodic sequece! 67 Relatio to Periodic Sequece -cosider a periodic sequece, x [ ], cosistig of a ifiite sequece of replicas of x[] x ( ) = x [ + r] r = - the Fourier coefficiets, X [ ], are the idetical to the values j π / of Xe ( ) for the fiite duratio sequece => a sequece of legth ca be exactly represeted by a DF represetatio of the for: X [ ] = [ ] π / xe j, =,,..., = x [ ] = Xe [ ] j π/, =,,..., = aplitude Wors for both fiite sequece ad for periodic sequece 68 Periodic ad Fiite Legth Sequeces x[] x[] X[] X(e j ) periodic sigal => lie spectru i frequecy fiite duratio => cotiuous spectru i frequecy 69 Saplig i Frequecy (ie Doai Aliasig) Cosider a fiite duratio sequece: j Se ( ) x [ ] for L i.e., a L poit sequece, with discrete Fourier trasfor... ( ) = L Xe j xe [ ] π = =π/ Cosider saplig the discrete Fourier trasfor by s[ ] ultiplyig it by a sigal that is defied as: j Se ( ) = δ[ π/ ] = with tie-doai represetatio - s [ ] = δ[ r] r = hus we for the spectral sequece j j j Xe ( ) = Xe ( ) Se ( ) which trasfors i the tie doai to the covolutio x [ ] = x [ ] s [ ] = x [ ] δ[ r] = x [ r] r= r= x [ ] = x [ ] + x [ ] + x [ + ] Saplig i Frequecy (ie Doai Aliasig) If the duratio of the fiite duratio sigal satisfies the relatio L, the oly the first ter i the ifiite suatio affects the iterval L ad there is o tie doai aliasig, i.e., x [ ] = x [ ] L If < L, i.e., the uber of frequecy saples is saller tha the duratio of the fiite duratio sigal, the there is tie doai aliasig ad the resultig aliased sigal (over the iterval L ) satisfies the aliasig relatio: x [ ] = x [ ] + x [ + ] + x [ ] 7 ie Doai Aliasig Exaple Cosider the fiite duratio sequece 4 x [ ] = ( + ) δ[ ] = δ[ ] + δ[ ] + 3 δ[ ] + 4 δ[ 3] + 5 δ[ 4] = he discrete Fourier trasfor of x [ ] is coputed ad sapled at frequecies aroud the uit circle. he resultig sapled Fourier trasfor is iverse trasfored bac to the tie doai. What is the resultig tie doai sigal, x [ ], (over the iterval L ) for the cases =, = 5 ad = 4. SOLUIO: For the cases = ad = 5, we have o aliasig (sice L) ad we get x [ ] = x [ ] over the iterval L. For the case = 4, the = value is aliased, givig x [] = 5 (as opposed to for x[ ]) with the reaiig values uchaged. 7

13 Periodic Sequece DF Properties Fiite Sequece Period= Legth= Sequece defied for all Sequece defied for =,,,- DF defied for all DF defied for all whe usig DF represetatio, ti all sequeces behave as if they were ifiitely periodic => DF is really the represetatio of the exteded periodic fuctio, x [ ] = x [ + r] alterative (equivalet) view is that all sequece idices ust be iterpreted odulo x [ ] = x [ + r] = x [ odulo ] = x([ ]) r = r = 73 DF Properties for Fiite Sequeces X[], the DF of the fiite sequece x[], ca be viewed as a sapled versio of the z-trasfor (or Fourier trasfor) of the fiite sequece (used to desig fiite legth filters via frequecy saplig ethod) the DF has properties very siilar to those of the z-trasfor ad the Fourier trasfor the values of X[] ca be coputed very efficietly (tie proportioal to log ) usig the set of FF ethods DF used i coputig spectral estiates, correlatio fuctios, ad i ipleetig digital filters via covolutioal ethods 74 DF Properties -poit sequeces -poit DF. Liearity ax[ ] + bx[ ] ax[ ] + bx [ ]. Shift x( e X 3. ie Reversal 4. Covolutio 5. ultiplicatio [ j π / ]) [ ] x ([ ]) X [ ] x [ ] h([ ]) X [ ] H [ ] = x [ ] w [ ] XrW [ ] ([ r]) r = 75 Key rasfor Properties y [ ] = x[ ] x[ ] Ye ( ) = X( e ) X( e ) j j j covolutio ultiplicatio y [ ] = x[ ] x[ ] Ye ( ) = X( e ) X( e ) j j j ultiplicatio circular covolutio Special Case: x[ ] = ipulse trai of period saples x [ ] = δ [ ] = j π / [ ] = δ [ ] =, =,,..., = j π / j π / = = = = X e x [ ] X [ ] e e saplig fuctio 76 Saplig Fuctio Suary of DSP-Part Part x[] X[] π 4π 6π (-)π 77 speech sigals are iheretly badliited => ust saple appropriately i tie ad aplitude LI systes of ost iterest i speech processig; ca characterize the copletely by ipulse respose, h() the z-trasfor ad Fourier trasfor represetatios eable us to efficietly i process sigals i both the tie ad frequecy doais both periodic ad tie-liited digital sigals ca be represeted i ters of their Discrete Fourier trasfors saplig i tie leads to aliasig i frequecy; saplig i frequecy leads to aliasig i tie => whe processig tie-liited sigals, ust be careful to saple i frequecy at a sufficietly high rate to avoid tie-aliasig 78 3

14 Digital Filters 79 Digital Filters digital filter is a discrete-tie liear, shift ivariat syste with iput-output relatio: y [ ] = x [ ] h [ ] = x [ ] h [ ] = Y( z) = X( z) H( z) j Hz ( ) is the syste fuctio with He ( ) as the coplex frequecy respose j j j He ( ) = Hr( e ) + jhi( e ) real, iagiary represetatio j j j jarg H( e ) agitude, phase represetatio He ( ) = He ( ) e j j j log He ( ) = log He ( ) + jarg He ( ) j j log He ( ) = Re log He ( ) j j jarg H( e ) = I log H( e ) 8 Digital Filters causal liear shift-ivariat => h[]= for < stable syste => every bouded iput produces a bouded output => a ecessary ad sufficiet coditio for stability ad for the existece of j He ( ) = h [ ] < 8 Digital Filter Ipleetatio iput ad output satisfy liear differece equatio of the for: r = r= y[ ] a y[ ] = b x[ r] evaluatig z-trasfors of both sides gives: r r = r= = r = r= r ( ) bz r Y z r = Xz ( ) az = Y( z) a z Y( z) = b z X( z) Y( z)( a z ) X( z) b z Hz ( ) = = r caoic for showig poles ad zeros 8 Digital Filters Ideal Filter Resposes H(z) is a ratioal fuctio i z O ( cr z ) r = ( dz ) = A Hz ( ) = => zeros, poles O O X X X X coverges for z > R, with R < for stability => all poles of Hz ( ) iside the uit circle for a stable, causal syste O

15 if a =, all, the FIR Systes r r = y [ ] = bx [ r] = bx [ ] + bx [ ] b x [ ] =>. h [ ] = b = otherwise. Hz ( ) = bz => ( cz ) => zeros = = 3. if h [ ] =± h [ ] (syetric, atisyetric) H(e ) = Ae ( ) e, Ae ( ) = real (syetric), iagiary (ati-syetric) j / j j j liear phase filter => o sigal dispersio because of o-liear phase => precise tie aliget of evets i sigal evet at t FIR Liear evet at t + Phase Filter fixed delay 85 FIR Filters cost of liear phase filter desigs ca theoretically approxiate ay desired respose to ay degree of accuracy requires loger filters tha o-liear phase desigs FIR filter desig ethods widow desig => aalytical, closed for ethod frequecy saplig => optiizatio ethod iiax error desig => optial ethod 86 Widow Desiged Filters LPF Exaple Usig RW Widowed ipulse respose h [ ] = h[ ] w [ ] I I the frequecy doai we get j j j He ( ) = H( e ) We ( ) I LPF Exaple Usig RW LPF Exaple Usig RW

16 Coo Widows (ie). Rectagular w [ ] = otherwise /. Bartlett w [ ] = π 4π 3. Blaca w [ ] =.4.5cos +.8cos π 4. Haig w [ ] =.54.46cos π 5. Haig w [ ] =.5.5cos I { β (( /)/( /)) } 6. Kaiser w [ ] = I β} { 9 Coo Widows (Frequecy) Widow ailobe Sidelobe Width Atteuatioti Rectagular 4 π / 3 db Bartlett 8 π / 7dB Haig 8 π / 3dB Haig 8 π / 43dB Blaca π / 58dB 9 Widow LPF Exaple Equiripple Desig Specificatios 93 = oralized edge of passbad frequecy p = oralized edge of stopbad frequecy s δ = pea ripple i passbad p δ = pea ripple i stopbad s Δ= = oralized trasitio badwidth s p 94 Optial FIR Filter Desig Equiripple i each defied bad (passbad ad stopbad for lowpass filter, high ad low stopbad ad passbad for badpass filter, etc.) Optial i sese that the cost fuctio π E= β( ) Hd ( ) H( ) d π π is iiized. Solutio via well ow iterative algorith based o the alteratio theore of Chebyshev approxiatio. 95 ALAB FIR Desig. Use fdatool to desig digital filters. Use firp to desig FIR filters B=firp(,F,A) + poit liear phase, FIR desig B=filter coefficiets (uerator polyoial) F=ideal frequecy respose bad edges (i pairs) (oralized to.) A=ideal aplitude respose values (i pairs) 3. Use freqz to covert to frequecy respose (coplex) [H,W]=freqz(B,de,F) H=coplex frequecy respose W=set of radia frequecies at which FR is evaluated ( to pi) B=uerator polyoial=set of FIR filter coefficiets de=deoiator polyoial=[] for FIR filter F=uber of frequecies at which FR is evaluated 4. Use plot to evaluate log agitude respose plot(w/pi, log(abs(h))) 96 6

17 Reez Lowpass Filter Desig Reez Badpass Filter Desig % badpass_filter_desig =iput('filter Legth i Saples:'); F=[ ]; A=[ ]; B=firp(,F,A); F=4; [H,W]=freqz(B,,F); =3 F=[.4.5 ]; A=[ ]; B=firp(,F,A) F=5; uber of frequecy poits [H,W]=freqz(B,,F); plot(w/pi,log(abs(h))); 97 figure,oriet ladscape; stitle=spritf('badpass fir desig, :%d,f: %4.f %4.f %4.f %4.f %4.f %4.f',,F); =:; subplot(),plot(,b,'r','liewidth',); axis tight,grid o,title(stitle); xlabel('ie i Saples'),ylabel('Aplitude'); leged('ipulse Respose'); subplot(),plot(w/pi,*log(abs(h)),'b','liewidth',); axis ([ -6 ]), grid o; xlabel('oralized Frequecy'),ylabel('Log agitude (db)'); leged('frequecy Respose'); 98 FIR Ipleetatio IIR Systes x [ ] x [ ] x [ ] x [ 3] x[ ] y [ ] liear phase filters ca be ipleeted with half the ultiplicatios (because of the syetry of the coefficiets) 99 y[ ] = a y[ ] + b x[ r] r = r= y [ ] depeds o y [ ], y [ ],..., y [ ] as well as x [ ], x [ ],..., x [ ] for < Hz ( ) r bz r r = A = = = dz az = = h [ ] = A( d ) u[ ] - for causal systes - partial fractio expasio h [ ] is a ifiite duratio ipulse respose (see prob.9) IIR Desig ethods Ipulse ivariat trasforatio atch the aalog ipulse respose by saplig; resultig frequecy respose is aliased versio of aalog frequecy respose Biliear trasforatio use a trasforatio to ap a aalog filter to a digital filter by warpig the aalog frequecy scale ( to ifiity) to the digital frequecy scale ( to pi); use frequecy pre-warpig to preserve critical frequecies of trasforatio (i.e., filter cutoff frequecies) IIR Filter Desig 7

18 Butterworth Desig Chebyshev ype I Desig 3 4 Chebyshev BPF Desig Chebyshev ype II Desig 5 6 Elliptic BPF Desig IIR Filters IIR filter issues: efficiet ipleetatios i ters of coputatios ca approxiate ay desired agitude respose with arbitrarily sall error o-liear phase => tie dispersio of wavefor IIR desig ethods Butterworth desigs-axially flat aplitude Bessel desigs-axially flat group delay Chebyshev desigs-equi-ripple i either passbad or stopbad Elliptic desigs-equi-ripple i both passbad ad stopbad 7 8 8

19 atlab Elliptic Filter Desig atlab Elliptic Lowpass Filter use ellip to desig elliptic filter [B,A]=ellip(,Rp,Rs,W) B=uerator polyoial + coefficiets A=deoiator polyoial + coefficiets =order of polyoial for both uerator ad deoiator Rp=axiu i-bad (passbad) approxiatio error (db) Rs=out-of-bad (stopbad) ripple (db) Wp=ed of passbad (oralized radia frequecy) use filter to geerate ipulse respose y=filter(b,a,x) y=filter ipulse respose x=filter iput (ipulse) use zplae to geerate pole-zero plot zplae(b,a) 9 [b,a]=ellip(6,.,4,.45); [h,w]=freqz(b,a,5); x=[,zeros(,5)]; y=filter(b,a,x); zplae(b,a); appropriate plottig coads; IIR Filter Ipleetatio IIR Filter Ipleetatios ==4 y[ ] = a y[ ] + b x[ r] w [ ] = aw [ ] + x [ ] y [ ] = bw [ r] r = r= = r r = A ( cr z ) r = ( dz ) = br b + b z + bz A = a z az Hz ( ) = - zeros at z= c, poles at z= d - sice a ad are real, poles ad zeros occur i coplex cojugate pairs => H(z) = A +, K = - cascade of secod order systes r Used i forat sythesis systes based o ABS ethods IIR Filter Ipleetatios Hz ( ) = c + c z K = a z az a a c c c, parallel syste Coo for for speech sythesizer ipleetatio DSP i Speech Processig filterig speech codig, post filters, pre-filters, oise reductio spectral aalysis vocodig, speech sythesis, speech recogitio, speaer recogitio, speech ehaceet ipleetatio structures speech sythesis, aalysis-sythesis systes, audio ecodig/decodig for P3 ad AAC saplig rate coversio audio, speech DA 48 Hz CD 44.6 Hz Speech 6, 8,, 6 Hz Cellular DA, GS, CDA trascodig 3 4 9

20 Saplig of Wavefors x a (t) Sapler ad Quatizer Period, x(),x() x [ ] = xa( ), < < = / 8 sec = 5 μ sec for 8Hz saplig rate = / sec = μ sec for Hz saplig rate = /6 sec = 67 μ sec for 6 Hz saplig rate = / sec = 5 μ sec for Hz saplig rate 5 he Saplig heore If a sigal x a (t) has a badliited Fourier trasfor X a (jω) such that X a (jω)= for Ω πf, the x a (t) ca be uiquely recostructed fro equally spaced saples x a (), -<<, if / F (F S F ) (A-D or C/D coverter) x a a( (t) x a () x a () = x a (t) u (), where u () is a periodic pulse trai of period, with periodic spectru of period π/ 6 Saplig heore Equatios Saplig heore Iterpretatio jωt xa() t Xa( jω ) = xa() t e dt j a jω = Ω x [ ] Xe ( ) = x( e ) jω Xe ( ) = Xa( jω+ j π / ) = o avoid aliasig eed: π / Ω >Ω π / > Ω F = / > F s case where / F, aliasig occurs < 7 8 Saplig Rates F = yquist frequecy (highest frequecy with sigificat spectral level i sigal) ust saple at at least twice the yquist frequecy to prevet aliasig (frequecy overlap) telephoe speech (3-3 Hz) => F S =64 Hz widebad speech (-7 Hz) => F S =44 Hz audio sigal (5- Hz) => F S =4 Hz A broadcast (-75 Hz) => F S =5 Hz ca always saple at rates higher tha twice the yquist frequecy (but that is wasteful of processig) 9 Recovery fro Sapled Sigal If / > F the Fourier trasfor of the sequece of saples is proportioal to the Fourier trasfor of the origial sigal i the basebad, i.e., jω π Xe ( ) = Xa ( jω), Ω < ca show that the origial sigal ca be recovered fro the sapled sigal by iterpolatio usig a ideal LPF of badwidth π /, i.e., si( π ( t ) / ) xa() t = xa( ) = π ( t )/ digital-to-aalog coverter badliited saple iterpolatio perfect at every saple poit, perfect i-betwee saples via iterpolatio

21 Deciatio ad Iterpolatio of Sapled Wavefors CD rate (44.6 Hz) to DA rate (48 Hz) edia coversio Widebad (6 Hz) to arrowbad speech rates (8Hz, 6.67 Hz) storage oversapled to correctly sapled rates--codig x[ ] = x ( ), X ( jω ) = for Ω > π F if / > F (adequate saplig) the jω π X(e ) = Xa( jω), Ω < a a Deciatio ad Iterpolatio Deciatio, = => = Iterpolatio, L= => =/ i Stadard Saplig: begi with digitized sigal: x [ ] = xa( ) Xa( jω ) =, Ω π F ( a) Fs = F X( e π ) = Xa ( jω), Ω < ( b) jω X( e ) =, πf Ω π( Fs F) jω i ca achieve perfect recovery of xa () t fro digitized saples uder these coditios Deciatio Deciatio i wat to reduce saplig rate of sapled sigal by factor of i wat to copute ew sigal x [ ] with saplig rate F ' = / ' = ( ) = F / such that x x ' d[ ] = a( ') s d with o aliasig i oe solutio is to dowsaple x [ ] = x( ) by retaiig oe out of every saples of x [ ], givig x [ ] = x[ ] d s a 3 4 Deciatio Deciatio i eed Fs' F to avoid aliasig for = ( c) i whe Fs' < F we get aliasig for = 4 ( d) 5 i DFs of x [ ] ad x[ ] related by aliasig relatioship: d j j( π)/ Xd ( e ) = X( e ) = i or equivaletly (i ters of aalog frequecy): jω' j( Ω' π )/ Xd ( e ) = X( e ) = i assuig Fs' = F, (i.e., o aliasig) we get: jω' jω'/ jω Xd ( e ) = X( e ) = X( e ) = Xa ( jω) π π = Xa ( jω), <Ω< ' ' ' 6

22 Deciatio i to deciate by factor of with o aliasig, eed to esure that the highest frequecy i x [ ] is o greater tha Fs /( ) i thus we eed to filter x [ ] usig a ideal lowpass filter with respose: < π / H ( j d e ) = π / < π i usig the appropriate lowpass filter, we ca dowsaple the reusltig lowpass-filtered sigal by a factor of without aliasig Deciatio i usig a lowpass filter gives: jω' jω π π Wd( e ) = Hd( e ) Xa( jω), <Ω< ' ' ' i if filter is used, the dow-sapled sigal, wd [ ], o loger represets the origial aalog sigal, xa (), t but istead the lowpass filtered versio of xa () t i the cobied operatios of lowpass filterig ad dowsaplig are called deciatio. 7 8 Iterpolatio i assue we have x [ ] = xa ( ), (o aliasig) ad we wish to icrease the saplig rate by the iteger factor of L i we eed to copute a ew sequece of saples of xa () t with period '' = / L, i.e., xi[ ] = xa( '') = xa( / L) i It is clear that we ca create the sigal xi[ ] = x[ / L] for =, ± L, ± L,... but we eed to fill i the uow saples by a iterpolatio process i ca readily show that what we wat is: si[ π ( '' ) / ] xi[ ] = xa( '') = xa( ) [ π ( '' )/ ] = i equivaletly with '' = / L, x[ ] = xa ( ) we get si[ π ( ) / L] xi[ ] = xa( '') = xa( ) = [ π ( )/ L] i which relates x[ ] to x[ ] directly i 9 Iterpolatio i ipleetig the previous equatio by filterig the upsapled sequece x [ / L] =, ± L, ± L,... xu[ ] = otherwise i xu[ ] has the correct saples for =, ± L, ± L,..., but it has zero-valued saples i betwee (fro the upsaplig operatio) i he Fourier trasfor of xu[ ] is siply: j j L X u ( e ) = X ( e ) jω'' jω'' L jω Xu ( e ) = X( e ) = X( e ) jω'' i hus Xu ( e ) is periodic with two periods, aely with period π / L, due to upsaplig) ad π due to beig a digital sigal 3 Iterpolatio jω ( a) Plot of X( e ) jω'' ( b) Plot of Xu ( e ) showig double periodicity for L=, '' = / () c DF of desired sigal with ( / ) X ( ) j '' a j π F Ω Ω Ω Xi ( e ) = πf <Ω π / '' i ca obtai results of () c by applyig ideal lowpass filter withgai L (to restore aplitude) ad cutoff frequecy πf = π /, givig: jl j ( / '') X( e ) < π / L Xi ( e ) = π / L π j L < π / L Hi ( e ) = π / L π Iterpolatio i Origial sigal, x[ ], at saplig period,, is first upsapled to give sigal xu[ ] with saplig period '' = / L i lowpass filter reoves iages of origial spectru givig: x[ ] = x ( '') = x ( / L) i a a 3 3

23 SR Coversio by o-iteger Factors =/L => covert rate by factor of /L eed to iterpolate by L, the deciate by (why ca t it be doe i the reverse order?) Lowpass Filter Iterpolatio LPF Deciatio LPF eed to cobie specificatios of both LPFs ad ipleet i a sigle stage of lowpass filterig ca approxiate alost ay rate coversio with appropriate values of L ad for large values of L, or, or both, ca ipleet i stages, i.e., L=4, use L=3 followed by L=3 33 Suary of DSP-Part Part II digital filterig provides a coveiet way of processig sigals i the tie ad frequecy doais ca approxiate arbitrary spectral characteristics via either IIR or FIR filters, with various levels of approxiatio ca realize digital filters with a variety of structures, icludig direct fors, serial ad parallel fors oce a digital sigal has bee obtai via appropriate saplig ethods, its saplig rate ca be chaged digitally (either up or dow) via appropriate filterig ad deciatio or iterpolatio 34 3