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GENERAL CON CEPTSOF DIGITAL SIGNAL

PROCESSING

1-0 INTRO OUCTIO N

In recent yeara, there hns beco a rremeudous inc rcusc in the use 01digital computen und special-purpose digital ci rcui try for pcrformmgva ried signal process ing functions that wcrc orrgiua lly uchicvcd withanalog equipment. 'l'hc continucd evoluuon of relative ly inexpensrveintegrntcd circuits hns led lo a vnriety or microcom pu te rs and mi mcom ­puters thal can be uaed fol' vur-icue signu l prut.:cs~:t1ng Iunctrcns ILis now possible lo build spcciul-purposc digitnl procesaors wi t hm t hcsume size ond cost construints of systcms prevtuusty all una log 1I\

nature.Th is chaptcr will pr ovide a general discusstcu uf a few of the baste

concepts associa ted with digital signu l prccesai ng . Th c majar inl cn l ista pravide the reade r with 8 br ief ovcrview 01' thc subject before devel­oping the coucepts in detuil in later chuplcrs .

1-1 GENERAL OISCUSSION _

At the beguuung oC th is work, it is appropruuc to dracuss 8 few of thccommon terms that will be uscd and sorne of lile ussumptions that wrl lbe made. Whcrcver possible . the definitions a nd lcrm inology wil l be

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G Chup oJ /1 General Concept! of Digital S ignal Processing Seco 1-1 11 General Discus sioo 7

• (1)(e

76

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2,O 21 H .1 ,1 6T 7T aT

Ouonh ltd Somples IblI

76s•

fLJJLL1LLIt2 T 31 4T 5T 6T H aT I

OIQllol Numbtr\ (el

000 .001 ,100 110 111 100 010 001 0001 2 ' H 4T >T 6T 7T .1 I

Figure 1·2 Snmpling and digital convet sicn process.

will be diseussed later . Tbe signal is then read at in te rvals cf T seeondsby a sampler. These sa mp les must then be quantized to ene of thestandard levels. Al tho ugh thcre ore diJTerent strntegies empl oyed inthe qunntizntion procesa . onc common appron ch, which will be assumedhere, is thnt a snrnple is ussigned lo the nearest level. Thus, a sarnpleof value 4 .2 V would be qunntized to 4 V, and a sarn ple of valu é 4.6 Vwould be quant ized to 5 V.

This process for the s ig na l given is illustrated in Fig, 1·2 , (a) and(b). The pulses rep resen ti ng the signol hove bee n made very narrow toill us l rntc the Iact that other s ignnls muy be insertcd, or mu/tlple:red. inthe~ space. These pul ses may then be repr esented as bin arynumbers as illustrated in (e). In order that these numbers eould be seenon the figure. each has becn shown ayer much of the space in a givenlruerval . lnpractice , ir othe r signals are lo be inserted, the pulsesrcp rcsenting the bits of the binnry numbers cou ld be made very short.A given binary number could then be read in 8 very short interval althe bcginning of a sampling period, thu s lenving most of the tim envailable for other signnls .

The proccss by which nn analog sample is c¡uantized nnd con-

ve rted to a binary numher is called analog-to-digiuil (A I D) conuers io fl .In genera l, the dynamic rnnge of the s ignal must be compa t ible withthat of the Al D eonverter emp loyed, and the number of bits employedmust be sufficient for the requi red accurncy .

The signal can now be proce sSé"d by the type of unit app ropriatefor the application intended. ' I'his un it may be a gcue ru l-purpose corn ­puter or minicompute r, or it may be a special unit des ign ed specificallyfor this purpose. At any rate , it is composed of some combi na tion ofstandard digital ci rcuits capable of pe rforming the various arith rneticfun ct ions of addition , su btrnction , rn ullt pli c~ t i on , etc. In addition ithas logic and storage ea pability . .

At the output of the pr oeessor, the digital signal can be eonvertedto analo lorro aga m . Th lS is achieved b the rocess of dl ital-to­ana/oc IDIA) eO/l""5IO/I. n th is step, the binary nu mbe rs are [¡r st

s uccessi ve ly cORverted back lo continuous-time pul ses . Th e '~be·tween (he pu lses are Chen hlléd tn by á reconstrllct ion Id ter . 'l'hls fIlt ermay consist oCa hQJ d i ~circual, wlileh ls a speclal crrc uit designed tohold th e vnlue of a pulse between successive sample vulues . In sornecases, the ho lding ci rcuil may be design ed to cx tra poln te the oulputsignal betwcen successive points according to sorne prescribed cu rve ­fitting strntegy . In addition to a holding circuit , a basi c con t i~time 1i1ter rnay be emp loyed lo provide additional smoothing betweenpoints .

A fundamenta l question that muy ar-ise is .wheinforrna tiofi has becn lost in [h e rocess . After all • ,- ssampled only a l discrete inte rvnls O une; is there something thatmight beJDi5scd in the in te rve ning time interva ls? Furtherrnore.jn theprocese of qunnt izntion, the nctunlarnplitude is reI; laced by the nearcststandard level , whieh means tha t the"e is a possible error in amplitude.

flijIn regord to the sa mp lin uestion it will be shown in Chapt~ 3

thnt if the signa is ba n Imited ,'and if the sam Iin' rote is gr ea terthan or equn o wlce e 1tghest fl'e uene lhe si na ean theorett ­ca y e recovere rorn its discrete sam les . This corresponds to anu rum u m o wo samples per cycle al the highest frequency . In pra c­tice . thi s sampli n rat e is usu all chosen to be somewhat higher thant e ro mrnu m rate (say, three or our times l c llghest requenc ) inor er toensur.e...pract lca unp emcntntion . -or examp e, ir the highestrequency 01 the analog slgnal is 5 kHz, the theorclica l minimum ,

sampling rato is 10,000 sarnples per secc nd. and 11 practica l system Iwould employ 8 role so mc what higher. The input continuous-timesignal is often passed th rough a law -pass analog presamp{¡ng hita l o

ensure that the highest frequency is within the bounds for which tbe-signal can be rec ave red .

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" Chapo 1 11 General Concepts o{ Digital Signal Processing Se co }-} II Gen eral Discussion 5

of the values Oand 1 to represen] all DossíbIe nu rnbers. The nurnbcr oflevel s mlliaI can be represented by a num ber having II binary digits(bits) is given by

Conversely, if mis c.he number of.possjble )cycls rcQJljred ¡b e nJlmbc r

of hits rcqJlircd is fbe sma lles t intcger grcntcr Lbpn or CQm,) l o 10fTr 111........The process by which digital s igna l proccssing is achicved will be

illustrat.ed by a simplified sys Lem in whíc h the s igna l is nssu med lovary from Oto 7 volts and in which 8 possibl e level s (at 1 V increm en ta)are used for th e binary nurnbers. A block diagram is shown in Fig. l -l,and sorne wav eform s ofinterest are sh own in Fig . 1-2. Tbe signa! is firstpassed through a conLi nuous- time pr esempling ñ lter whosc Functi ou

(I .l)

o. Q'10tSt(~nol

p(OCU ~()I

m = 2"

Sompl er ondA IO Conv~rl~r

Figure 1-1 Block diagrarn o l a poss rblc di gil ~ 1 procc vsing sys!em

Pre - 50mplinQF il1~'

ArlolOQSiQnoI1,

presentoThus, re ference ca n be mud e to "a nal og sys tums ," "cont.i nu oua­time systems," "discrete-time systems," "digita l sys toms," etc.

~) A lin ear sys lem is one in whi ch the pa ra mct ers of th e s stcm arenot epen ent 00 the oature or t e level of the input exeitati on . Th isstatement 15 eqUlvalent to the staterncnt that th e principlc of su per­position applies. A linear syslem can be described by linear differcntialor dieferenee eguatj Qns A timc-inuariant linear syslpID js Qne in wh ichthe arameters do 110l val' wilh time.

:¡) A um ed system is one th a t is com ose of finite nonzero ele-, ments sa tisfying ordina ry I erential or difTerence equntion relation­

ships (as opposed lo a distributed sys tem , sa ti sfying partial differentialequation relationships). Very Iittl e refer en ee will be made in thi s tcxtto distributed systerns; the impli cation is tha l a11 systcms considercdwill be lumped unle,. otherwise noted. ¡W - >o r;<. t J, l. "'-,,;<....., o

In carrying Out various theoretical developments, it will fre­quent.ly be neeessnry lo rcfer lo systems that are eilher (a) continuous­time, linear, and time-invariant, or (b) discreto-tune. linear , a ndtime-invariant. For co~ci s cn ess , we will duaignnte (o) as a CT LT I sys­tem and (b) as a DTLTI system .~ M c''lc''o ,

The s ta nda rd forro for numeri eal processing of a diaita l s imthe binarv numher system The binary nu mber s

established in accordance with the recornmendations of the IEEEGroup on Audio and El ectroacoustics.

1) ~n analog sign a l is a fun ction that is defined ayer ª continuous _ranga o{\tlme aoa in which the amplitude mar assume a continuous~range of'values. Cornmon examples are the sinusoidal function, th e- step functi on , the output of a microphone, etc . The term "analog" ap-parently or igina ted from the field of analog eomputation, in whi ehvcltages and currents are used lo re present phy sical variables, but itha s been exte nded in usage.

2) A continuous· time signa l is a functioD tbat js defined ayer a con·tinuous [aoge oCtime, bul in which the amplitude roa either have acon mUDUS [aoge oC va ues oc a mi e nuro er ofpossible values. In thiscontext, an anal og signal could be considered as a special case of acontinuous-time signal. In practice, however, the terms "analog" and"continuous-tirne" are interchanged casua lly in usage and are oftenused to mean the sa me thing. Because of the associntion of the term"analog" with physieal analogies, pre ferenee has been established forth e term "continuous-tirne," and thi s pract iee will be followed for th emost part in this textoNevertheIess, there will be placea in which theterm "ana log" will be used for clarity, parti cularIy where it relates tothe term "digita l."

,3) Th e term guantization des cribes tbe process of representing avariable by a se t of distinct values, A uantized variable i t at maassume oDly distinct va ues,

1) - A discrete·time si gna l is a fun eti on tha t is defined onl at a ar tic-ular se t o va ues o time. This menns that the i "deot variable,tim e, is quantized . If th e amp Itu e of a dis erete·tiine signal is per­

..mlttea to assume a continuous ran e of vaIues, the fundion is sa ia toe n sam p e - ata s igna . A sarnpled-data signa eould arise from

samphng aH analog- sigual at diserete values of lime ..S ) A dig ital signal is a fundiDo jo wbjch both time and amplitude

are guantized A digital signal may always be represenled by a se·quence of nti h each number has a [¡nite number of di its .- The lerms "discrete-time" and "digita are o en mterchanged inpracti ce and are often used to mean tbe sarne thing. A great deal of thetheory underI ying discrete- time signals is applicable to pu¡.eryaigitals igna ls , so it is not always neeessary to make rigid distinctions. Thelerm "discre te-t ime" will more often be us ed in pursuing th eor eti ealdev elopm eots and the term "digita l" will more oflen be used in deserib-'ing ha rdw are or softwa re rcal iza tions.

A system ca n be describ ed by any ofthe pre eeding terms accord ingto the type of hardware or softwa re employ~ and the typ e of si gna ls

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Assum ing that a sa mple between two su ccessive quantum levels isass igned to th e nearcst quantum level , the peak quantization noise andpea k per cen tage quantiza tion noisc values are

9

(1· 6)

(I -5)s:12

.s..2V3

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17 =

S eco 1-1 11 General Discus: ion

Comparing (1-6) with (1·3) , it is see n that th e RMS noise componentis l /v'3 tim es the peak noise comp onen t.

In view of the p-eceding discussion, it appears that no informationis lost in the sampling opcration provided thaC [he sampling rate ishigh enough, and the laaü on error can b'fica ntly sma eve b choosin a suffici nt numbcr ofbits lo re rescn l.each inary number. These conce ts thcn permil us lo re resent acontinuous-ltme signa 10 terms oC a ser ies oC 15 > b' J7I rswhic mav e processed direct y wi t digital circui ta.

assumed to be uniforrnly diatr-ibuted bclween quanlum lcvcls , it ca n beshown by sta tistical analysis that the noise variance (T2 is

The root-mean-souare (RMS) (or sta nda rd dcuia tion ) valuc of this noisccomponent is

The ralher mvolved procedure 01 A1D conver SíOn, proccssing, andfinal DI A convers ion may see m lik e a lot of efforlin order to handle onesignal channel. Indeed, in many cas es such a complex procesa may nolbe economically feasib le for a s ingle s igna!..O ne of the great ad van·lages of the digital conce t is th e oss ibility of processtn' a number ofchanne s wllh lhe same arithmelic uni t. 15 process can be achi evcdby a process callea time-dlUlslM mu1tlplexillg (TD M) . It was observed

.. in lhe snmpled signnl shown in Fig . 1-2 lfinllherc was a rclotively longperiod between successive samples of the, s ignal. During this period,samples of addi tiónnl signals arc.fu.d..into thc processor. .

This concept is illuslrated in F ig. 1-3. Ea ch channel is read in asequentia l orde r , and lhc corres pond ing values are converted into 61 -.

(1·2)111

EmRx - Em',; \

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Cñ ap , 1 1/ Gen eral Concepts o{ Digi/al S ignal Processi ng

Ern llll: - Em' nq =

8

!La signa l is GQl sa mpled al a suff¡ ciently high rate, a phenome­non kh oWD as gligsing results . This concept results in a freguency'sbeing mistaken for nn entirely ditTerent freguency upon recoverrj Forexa mple , suppose a signa) with frequencies ranging from de lo 5 Hz issarn pled a l arate oí B kHz, whi ch is c1early too low to ensure recovery.If recovery is attempted, a component of the original signal at 5 kHznow appears tabe al 1 kHz, resulting in an err-onecus signa1. A com rnonexa mple of this phenomenon is ene we will cal l the "wagon wheelefTect ," pr obably noticed by the reader in western movies as th e phe­nomenon in whi ch the wheels appear to be rotatinlf backwards . Sinceeach ind ividual frame of a film is tquivalent to a discrete samplingoperation , ir the rute of spoketP';;;mg a given angl e is too large for agi ven movie Irarne rate, the wheels appear lo be tu rning either back­wards or al a ver-y slow specd. The effect of a presarny,lin1filler rernovesthe possibility thal a spurious signa} whose Ir auen v is osystem will be rnlstaken tar ane in prop er frcquency range.

With respect lo the ouantiza tlon error jt can be sern 'bat the~ error can be made as sma ll as ane chooses if the numbcr ofbits can be ¡

mad e arbi trarily ¡urge. 01 cours e,1Jiere is a practical maximum limit,-So it is necessary lo tolerate sorne error frorn this phenomenon. Even in

cont inuo us -ti rne sys tems, there ma y be noise present which wauld .r lit e 1 • e uncertainty.-"" .

P"resent in lhe di Ita! sampling process is called uantization noise.e • rna. and E rn1n represent th e máximum and minimum val ues

of the signa l , and Iet q represent the vertical distance betw een su cces­sive qu antu m levels. Using n and In as previously defined, we have

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. . . 100%Peak Percentage Quantization Noise = -- (1-4)

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OiqilolSiqno lProcessor

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Figu re 1-3 Mulnplexcd dig ita l ptoccssmg svstcrn .

~----4( Syec 1 I(1·3 )Peak Quantization Noise = ~

In many cases , the variance of the quantization noise 15 moreimportant tha n the maximum value. Th e variance is directly propor­tional to the a verage power associa ted with the noise . If the signal is

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10 Chup. 1 11 Geuera l Conceot s o( Dil!il(,JI Signal Processing Se co ]·2 11 T y p4!s o{ Preeess íñg 11

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nnry numbers in the same seguence;;\. These numbers enter into the- process jnv yoil nad , nfte r suilablc processing. apoear a l [he output in~QProprinte arder . This composjte digi tal signnl must firsL

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2

íog und Imut"si gnnls jn lb " :iVs1em are In continuous-time formoActu-all there are rollay systems in which cne or hoth s r n d di ' ta l

formoIn 9UC case onmay ool he required. th us s implifyi ng the syslem. For exnmple, assume

- trint a number af continuous-time teIemetry signals is lo be processedby a digitnl unit, but the oulput data is to be kepl in digital form forscientific dala reduct.ion Dad computntion . In this cnse,lhe A /D unit alth c input is required, bu t no convers ian is nceded 01 the output.

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Digital

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DiQi lol WOfdsper Second 114298

DiqdolCAJtpul

0 0 67<6

1-2 lYP ES O F PR OCESSING - 0411802

Figur e 1·4 Af\J lo~ and digital filler s having simil ar ch a rac te nstic s.

arca of a ¡¡enlion in eontinuous·time si nol roe g...j'he develop­ment in 1965 of t e ooley-Tokey nlg orithrns 1'01' ru pid com pu tn tio n cfthe approximate spect ru rn paved the way for new and vuried applica­tions of spectral nnalysis . With this approach, the specu-um uf signalscontaining many thousands of sample points ca n be uchieved in arnatter of milliseconds . In fuct, il has become qu ite fea sible to fi\l ersig nals by FPr transformation , numer icnl alteration of the spectrum,and inverso FFT computation .

There are many vnr-ied scientifl c disciplines that utilize spe ctralanalysis in one form or nnother and in which th e FFT ha s open ed newpotential applications . Among these are comrn"nica tierHHUgng1 ana

'y·gis, solution of bounda r e.-problems in heat an d elect r icity statís-,tica ana YSIS, oceanographic wave anal s · and vibtrona. iere are aval a e spec la__ 'ITnrocessors whi ch muy..hc....used.for.real.bme proces sing in man a) licution ddjLjon, lnflny COffi...:

puters ave" su routines available in their librarics .

Adde. Mull iptk o1ioo Deloy(by ccosrooü (T secooos)

. ,

Much of this textbook will be devoted to lhe _developmen t andapplicntjons oC lwo ;m portnnl tools for modero digital signa! proces~

in : di ' ¡tal {il ters aed as i FOllrier transforms (FFTs) .A digita 1 te r is a computuuona process In W l e the sequence

of inRUl.JUtiiili4id lA caDvcrt "d iulo a 'iQq'lence 01 º"f puLñÍJmhp rs rep,:.°resent ing the alleration oC thc data in sorne prescribed manner. A

- rom man cxample 18 the process of filte ring out a certa in range oi ¡re·que ncies in 8 signa l whi lc rejcct ing all other frequencies, wh ich is aneof the foremost classicnl npproaches to ana lag filter design o In lheclussicni conlinuous-lime ~Illle.....lhis fi¡le rins is uchioved by U suil!\!lliL

Clíñicc pf jnduclors, cnpuciLors, aed resjs!o rs arrnDgcd lo prov ide tbereguired lransmiss ion chnructeris.tics Howevcr. in UHLj igi1aLcase­this can be nchievcd comolclclv bv the nrfV'pC:Q nr (I i (y '

mulltpllcalton by constants, and del~'l'o present an exarnplo which the render is nol expected to under­

s ta nd at this point, but which is shown for molivation, ccnsider the- circu its in Fig. 1-4. A certain low-pass analog filter having a 3 dB cutofTfrequen cy of 50 Hz is shown in (a ). A digital filler having approxi­malely the sorne frequeney response from de to five times cutolT(250 k l lz} is shown in (b). The vnrious units in the fille r correspond loadditi on, multiplica tion, and delay. as indicated on the figu re .

The second rnethod thnt we wil l consider is tha f h fost Fouritransform (EPI') cg~pk-1~hQ... periTO) op olysi s em plo.y...1ne Fou ricr t~uns(ormsund _serJes hu ve long represen led an importanl

ccee 8 t> El