UNIT II : WAVEFORM CODING

Transcription

1 UNIT II : WAVEFOM CODING CONTENTS: Predictio filterig ad DPCM Delta Modulatio ADPCM & ADM ricile Liear Predictive Codig

2 Differetial Pulse Code Modulatio (DPCM) For the sigals hich does ot chage raidly from oe samle to et samle the PCM scheme is ot referred. Whe such highly correlated samles are ecoded the resultig ecoded sigal cotais redudat iformatio. By removig this redudacy before ecodig a efficiet coded sigal ca be obtaied. Oe of such scheme is the DPCM techique. By oig the ast behavior of a sigal u to a certai oit i time it is ossible to mae some iferece about the future values. The trasmitter ad receiver of the DPCM scheme is sho i the fig3. ad fig 3.3 resectively. Trasmitter: Let (t) be the sigal to be samled ad (Ts) be it s samles. I this scheme the iut to the quatizer is a sigal e(ts) = (Ts) - ^(Ts) here ^(Ts) is the redictio for uquatized samle (Ts). This redicted value is roduced by usig a redictor hose iut cosists of a quatized versios of the iut sigal (Ts). The sigal e(ts) is called the redictio error. By ecodig the quatizer outut i this method e obtai a modified versio of the PCM called differetial ulse code modulatio (DPCM). Quatizer outut v(ts) = Q[e(Ts)]= e(ts) + q(ts) ---- (3.3) here q(ts) is the quatizatio error. Predictor iut is the sum of quatizer outut ad redictor outut u(ts) = ^(Ts) + v(ts) Usig 3.3 i 3.33 u(ts) = ^(Ts) + e(ts) + q(ts) ----(3.34) u(ts) = (Ts) + q(ts) The receiver cosists of a decoder to recostruct the quatized error sigal. The quatized versio of the origial iut is recostructed from the decoder outut usig the same redictor as used i the trasmitter. I the absece of oise the ecoded sigal at the receiver iut is idetical to the ecoded sigal at the trasmitter outut. Corresodigly the receive outut is equal to u(ts) hich differs from the iut (ts) oly by the quatizig error q(ts).

3 Predictio Gai ( G): The outut sigal-to-quatizatio oise ratio of a sigal coder is defied as

4 here σ is the variace of the sigal (Ts) ad σ Q quatizatio error q(ts). The is the variace of the here σe is the variace of the redictio error e(ts) ad (SN)P is the redictio error-to-quatizatio oise ratio defied by The Predictio gai G is defied as The redictio gai is maimized by miimizig the variace of the redictio error. Hece the mai obective of the redictor desig is to miimize the variace of the redictio error The redictio gai is defied by.

5 here ρ Autocorrelatio fuctio of the message sigal POBLEM: Cosider a DPCM system hose trasmitter uses a first-order redictor otimized i the miimum mea-square sese. Calculate the redictio gai of the system for the folloig values of correlatio coefficiet for the message sigal: Solutio: (i) For ρ= 0.85 G = 3.3 I db G = 5dB (ii) For ρ = 0.95 G = 0.6 I db G = 0.dB Delta Modulatio (DM) Delta Modulatio is a secial case of DPCM. I DPCM scheme if the base bad sigal is samled at a rate much higher tha the Nyquist rate urosely to icrease the correlatio betee adacet samles of the sigal so as to ermit the use of a simle quatizig strategy for costructig the ecoded sigal Delta modulatio (DM) is recisely such as scheme. Delta Modulatio is the oe-bit (or to-level) versios of DPCM. DM rovides a staircase aroimatio to the over samled versio of a iut base bad sigal. The differece betee the iut ad the aroimatio is quatized ito oly to levels amely }δ corresodig to ositive ad egative differeces resectively Thus if the aroimatio falls belo the sigal at ay samlig eoch it is icreased by δ. Provided that the sigal does ot chage too raidly from samle to samle e fid that the stair case aroimatio remais ithi }δ of the iut sigal. The symbol δ deotes the absolute value of the to reresetatio levels of the oe-bit quatizer used i the DM. These to levels are idicated i the trasfer characteristic of Fig 3.4. The ste size Δ of the quatizer is related to δ by Δ= δ

6 Let the iut sigal be (t) ad the staircase aroimatio to it is u(t). The the basic ricile of delta modulatio may be formalized i the folloig set of relatios:

7 here Ts is the samlig eriod; e(ts) is a redictio error reresetig the differece betee the reset samle value (Ts) of the iut sigal ad the latest aroimatio to it amely ^ (Ts) = u( Ts Δ Ts) The biary quatity b (T s) is the oe-bit ord trasmitted by the DM system. The trasmitter of DM system is sho i the figure3.5. It cosists of a summer a tolevel quatizer ad a accumulator. The from the equatios of (3.43) e obtai the outut as At each samlig istat the accumulator icremets the aroimatio to the iut sigal by }δ deedig o the biary outut of the modulator.

8 I the receiver sho i fig.3.6 the stair case aroimatio u(t) is recostructed by assig the icomig sequece of ositive ad egative ulses through a accumulator i a maer similar to that used i the trasmitter. The outof bad quatizatio oise i the high frequecy staircase aveform u(t) is reected by assig it through a lo-ass filter ith a bad-idth equal to the origial sigal badidth. Delta modulatio offers to uique features:. No eed for Word Framig because of oe-bit code ord.. Simle desig for both Trasmitter ad eceiver QUANTIZATION NOISE Delta modulatio systems are subect to to tyes of quatizatio error: () sloe overload distortio ad () graular oise. If e cosider the maimum sloe of the origial iut aveform (t) it is clear that i order for the sequece of samles{u(ts)} to icrease as fast as the iut sequece of samles {(Ts)} i a regio of maimum sloe of (t) e require that the coditio i belo equatio be satisfied. aroimatio u(t) to follo a stee segmet of the iut aveform (t) ith the result that u(t) falls behid (t). This coditio is called sloe-overload ad the resultig quatizatio error is called sloe-overload distortio(oise). Sice the maimum sloe of the staircase aroimatio icreases ad decreases i u(t) ted to occur alog straight lies. For this reaso a

9 delta modulator usig a fied ste size is ofte referred to as liear delta modulatio (LDM). The graular oise occurs he the ste size local sloe characteristics of the iut ave form (t) thereby causig the staircase aroimatio u(t) to hut aroud a relatively flat segmet of the iut aveform; The graular oise is aalogous to quatizatio oise i a PCM system. The e choice of the otimum ste size that miimizes the mea-square value of the quatizig error i a liear delta modulator ill be the result of a comromise betee sloe overload distortio ad graular oise. Outut SN for Siusoidal Modulatio. Cosider the siusoidal sigal (t) = A cos(μfot) The maimum sloe of the sigal (t) is give by

10 Whe there is o sloe-overload the maimum quatizatio error }δ. Assumig that the quatizig error is uiformly distributed (hich is a reasoable aroimatio for small δ). Cosiderig the robability desity fuctio of the quatizatio error

11 Problems. Determie the outut SN i a DM system for a KHz siusoid samled at 3KHz ithout sloe overload ad folloed by a 4KHz ost recostructio filter. Solutio: Give W=4KHz f0 = KHz fs = 3KHz Usig equatio (3.54) e get (SN)0 = 3.3 or 4.9dB

12 Delta Modulatio: Problems. Cosider a Seech Sigal ith maimum frequecy of 3.4KHz ad maimum amlitude of volt. This seech sigal is alied to a delta modulator hose bit rate is set at 60bit/sec. Elai the choice of a aroriate ste size for the modulator. Solutio: Badidth of the sigal = 3.4 KHz. Maimum amlitude = volt Bit ate = 60Kbits/sec Samlig rate = 60K Samles/sec. STEP SIZE = Volts 3. Cosider a Seech Sigal ith maimum frequecy of 3.4KHz ad maimum amlitude of volt. This seech sigal is alied to a delta modulator hose bit rate is set at 0bit/sec. Elai the choice of a aroriate ste size for the modulator. Solutio: Badidth of the sigal = 3.4 KHz. Maimum amlitude = volt Bit ate = 0Kbits/sec Samlig rate = 0K Samles/sec. STEP SIZE =.068 Volts 4. Cosider a Delta modulator system desiged to oerate at 4 times the Nyquist rate for a sigal ith a 4KHz badidth. The ste size of the quatizer is 400mV. a) Fid the maimum amlitude of a KHz iut siusoid for hich the delta modulator does ot sho sloe overload. b) Fid ost-filtered outut SN Solutio: Badidth of the sigal = f0 = KHz. Nyquist ate = 8K samles/sec Samlig ate = 3K samles/sec. Ste Size = 400 mv a) For KHz siusoid Ama =.037 volts. b) Assumig LPF badidth = W= 4KHz SN = = 4.93 db

13 Adative Delta Modulatio: The erformace of a delta modulator ca be imroved sigificatly by maig the ste size of the modulator assume a time-varyig form. I articular durig a stee segmet of the iut sigal the ste size is icreased. Coversely he the iut sigal is varyig sloly the ste size is reduced. I this ay the size is adated to the level of the iut sigal. The resultig method is called adative delta modulatio (ADM). There are several tyes of ADM deedig o the tye of scheme used for adustig the ste size. I this ADM a discrete set of values is rovided for the ste size. Fig.3.7 shos the bloc diagram of the trasmitter ad receiver of a ADM System. I ractical imlemetatios of the system the ste size (T s or δ(t s ) is costraied to lie betee miimum ad maimum values. The uer limit δma cotrols the amout of sloeoverload distortio. The loer limitδmi cotrols the amout of idle chael oise. Iside these limits the adatatio rule for δ(t s) is eressed i the geeral form

14 Codig Seech at Lo Bit ates: The use of PCM at the stadard rate of 64 b/s demads a high chael badidth for its trasmissio. But chael badidth is at a remium i hich case there is a defiite eed for seech codig at lo bit rates hile maitaiig accetable fidelity or quality of reroductio. The fudametal limits o bit rate suggested by seech ercetio ad iformatio theory sho that high quality seech codig is ossible at rates cosiderably less that 64 b/s (the rate may actually be as lo as b/s).

15 For codig seech at lo bit rates a aveform coder of rescribed cofiguratio is otimized by eloitig both statistical characterizatio of seech aveforms ad roerties of hearig. The desig hilosohy has to aims i mid:. To remove redudacies from the seech sigal as far as ossible.. To assig the available bits to code the o-redudat arts of the seech sigal i a ercetually efficiet maer. To reduce the bit rate from 64 b/s (used i stadard PCM) to ad 4 b/s the algorithms for redudacy removal ad bit assigmet become icreasigly more sohisticated. There are to schemes for codig seech:. Adative Differetial Pulse code Modulatio (ADPCM) b/s. Adative Sub-bad Codig b/s. Adative Differetial Pulse Code Modulatio A digital codig scheme that uses both adative quatizatio ad adative redictio is called adative differetial ulse code modulatio (ADPCM). The term adative meas beig resosive to chagig level ad sectrum of the iut seech sigal. The variatio of erformace ith seaers ad seech material together ith variatios i sigal level iheret i the seech commuicatio rocess mae the combied use of adative quatizatio ad adative redictio ecessary to achieve best erformace. The term adative quatizatio refers to a quatizer that oerates ith a timevaryig ste size ( ) s T T s) is varied so as to match the variace σ of the iut sigal (T s). I articular e rite Δ(Ts) = Φ. σ^(ts) (3.57) here Φ Costat σ^(ts) estimate of the σ(ts) Thus the roblem of adative quatizatio is oe of estimatig σ (Ts) cotiuously. The comutatio of the estimate ^ σ (Ts) is doe by oe of to ays:. Uquatized samles of the iut sigal are used to derive forard estimates of σ (Ts)- adative quatizatio ith forard estimatio (AQF). Samles of the quatizer outut are used to derive bacard estimates of σ (Ts)- adative quatizatio ith bacard estimatio (AQB) The use of adative redictio i ADPCM is required because seech sigals are iheretly ostatioary a heomeo that maifests itself i the fact that

16 autocorrectio fuctio ad oer sectral desity of seech sigals are timevaryig fuctios of their resective variables. This imlies that the desig of redictors for such iuts should lieise be time-varyig that is adative. As ith adative quatizatio there are to schemes for erformig adative redictio:. Adative redictio ith forard estimatio (APF) i hich uquatized samles of the iut sigal are used to derive estimates of the redictor coefficiets.. Adative redictio ith bacard estimatio (APB) i hich samles of the quatizer outut ad the redictio error are used to derive estimates of the redictio error are used to derive estimates of the redictor coefficiets. () Adative Sub-bad Codig: PCM ad ADPCM are both time-domai coders i that the seech sigal is rocessed i the time-domai as a sigle full bad sigal. Adative sub-bad codig is a frequecy domai coder i hich the seech sigal is divided ito a umber of sub-bads ad each oe is ecoded searately. The coder is caable of digitizig seech at a rate of 6 b/s ith a quality comarable to that of 64 b/s PCM. To accomlish this erformace it eloits the quasi-eriodic ature of voiced seech ad a characteristic of the hearig mechaism o as oise masig. Periodicity of voiced seech maifests itself i the fact that eole sea ith a characteristic itch frequecy. This eriodicity ermits itch redictio ad therefore a further reductio i the level of the redictio error that requires quatizatio comared to differetial ulse code modulatio ithout itch redictio. The umber of bits er samle that eeds to be trasmitted is thereby greatly reduced ithout a serious degradatio i seech quality. I adative sub bad codig (ASBC) oise shaig is accomlished by adative bit assigmet. I articular the umber of bits used to ecode each sub-bad is varied dyamically ad shared ith other sub-bads such that the ecodig accuracy is alays laced here it is eeded i the frequecy domai characterizatio of the sigal. Ideed sub-bads ith little or o eergy may ot be ecoded at all. Alicatios. Hierarchy of Digital Multileers. Light ave Trasmissio Li

17 () Digital Multileers: Digital Multileers are used to combie digitized voice ad video sigals as ell as digital data ito oe data stream. The digitized voice sigals digitized facsimile ad televisio sigals ad comuter oututs are of differet rates but usig multileers it combied ito a sigle data stream. To Maor grous of Digital Multileers:. To combie relatively Lo-Seed Digital sigals used for voice-grade chaels. Modems are required for the imlemetatio of this scheme.. Oerates at higher bit rates for commuicatio carriers. Basic Problems associated ith Multileers:. Sychroizatio.. Multileed sigal should iclude Framig. 3. Multileer Should be caable hadlig Small variatios () Light Wave Trasmissio Otical fiber ave guides are very useful as trasmissio medium. They have a very lo trasmissio losses ad high badidths hich is essetial for high-seed commuicatios. Other advatages iclude small size light eight ad immuity to electromagetic iterferece. The basic otical fiber li is sho i the figure 3.. The biary data fed ito the trasmitter iut hich emits the ulses of otical oer ith each ulse beig o or off i accordace ith the iut data. The choice of the light source determies the otical sigal oer available for trasmissio.

18 The o-off light ulses roduced by the trasmitter are lauched ito the otical fiber ave guide. Durig the course of the roagatio the light ulse suffers loss or atteuatio that icreases eoetially ith the distace. At the receiver the origial iut data are regeerated by erformig three basic oeratios hich are:. Detectio the light ulses are coverted bac ito ulses of electrical curret.. Pulse Shaig ad Timig - This ivolves amlificatio filterig ad equalizatio of the electrical ulses as ell as the etractio of timig iformatio. 3. Decisio Maig: Deedig the ulse received it should be decided that the received ulse is o or off. Liear Predictive Codig: Liear redictive codig (LPC) is a method for sigal source modelig i seech sigal rocessig. It is ofte used by liguists as a format etractio tool. It has ide alicatio i other areas. LPC aalysis is usually most aroriate for modelig voels hich are eriodic ecet asalized voels. LPC is based o the source-filter model of seech sigal. Priciles. First the observatio of iut ad outut sequeces roduces a model ith a umber of oles or formats.

19 . A resultig set of coefficiets ca the describe the behaviour of a system hich is ot o yet. It is used for redictig a samle. This set of coefficiets is a all-ole model a simlified versio of the acoustic model of the seech roductio system. 3. The aalysis the estimates the values of a discrete-time sigal as a liear fuctio of revious samles. The sectral eveloe is rereseted i a comressed form usig the iformatio of the liear redictive model. Method Eveloe Calculatio The LPC method is quite close to the FFT. The eveloe is calculated from a umber of formats or oles secified by the user. Limits The formats are estimated removig their effects from the seech sigal ad estimatig the itesity ad frequecy of the remaiig buzz. The removig rocess is called iverse filterig ad the remaiig sigal is called the residue. The seech sigal source is sythesized from the buzz arameters ad the residue. The source is ra through the filter formats resultig i seech. The rocess is iterated several time is a secod ith "frames". A 30 to 50 frames rate er secod yields ad itelligible seech. The LPC erformace is limited by the method itself ad the local characteristics of the sigal. The harmoic sectrum sub-samles the sectral eveloe hich roduces a sectral aliasig. These roblems are esecially maifested i voiced ad high-itched sigals affectig the first harmoics of the sigal hich refer to the erceived seech quality ad format dyamics. A correct all-ole model for the sigal sectrum ca hardly be obtaied. The desired sectral iformatio the sectral eveloe is ot rereseted : e get too close to the origial sectra. The LPC follos the curve of the sectrum do to the residual oise level i the ga betee to harmoics or artials saced too far aart. It does ot rereset the desired sectral iformatio to be modeled sice e are iterested i fittig the sectral eveloe as close as ossible ad ot the origial sectra. The sectral

20 eveloe should be a smooth fuctio assig through the romiet eas of the sectrum yieldig a flat sequece ad ot the "valleys" formed by the harmoic eas. Liear Predictio (to reduce the samlig rate): Cosider a fiite-duratio imulse resose (FI) discrete-time filter hich cosists of three blocs :. Set of ( : redictio order) uit-delay elemets (z-). Set of multiliers ith coefficiets 3. Set of adders ( )

21 (3.6) have (3.6) e From (3.59)(3.60)ad to miimize Fid (3.6) (mea square error) erformace be Let the ide of (3.60) ˆ is redictioerror The (3.59) ) ( ˆ is the iut ) liear reditio of outut (The filter The E E E J J E e J e equatios (3.64)are called Wieer - Hof (3.64) 0 (3.63) as We may simlify ) ( atio autocorrel The ) ( 0) ]] [ [ ) is statioary rocess ith zero mea ( ( Assume J J J E T E E E E t s

22 Liear adative redictio : mi 0 mi 0 0 less tha is alays 0 (3.67) Substitutig (3.64)ito (3.63) yields ]] [ []... [] [ here (3.66) eists if as T T T T T J J r r r r r r r resetatio. of is for coveiece isa ste- size arameter ad here (3.69) at iteratio.the udate value deotes the (3.68) Defie the gradiet vector descet steeest Do iteratio usig the method of. values startig ay iitial.comute i the follosese is adative redictor The g J g

23 Figure Bloc diagram illustratig the liear adative redictio rocess Advatages Its mai advatage comes from the referece to a simlified vocal tract model ad the aalogy of a source-filter model ith the seech roductio system. It is a useful methods for ecodig seech at a lo bit rate.