Signals and Systems II Part I: Signals and their representations

Transcription

1 Signals and Sysems II Par I: Signals and heir represenaions DIGIAL SOCK & PHOODISC Jerey O. Coleman his six-par series is a minicourse, ocused on sysem conceps, ha is aimed a he gap beween signals and sysems and he usual irs digial signal processing course. he goal o his series is a principles-based capabiliy o design signal-processing archiecures o conver signals beween various orms. Signals are coninuous-ime or discree-ime, real or complex, and baseband or passband. Operaions used on hose signals include analog ilering, digial ilering, inerpolaion, decimaion, and requency shis. he approach is rigorous bu largely graphical wih lile explici mahemaics. Digial Objec Ideniier 1.119/MPO JANUARY/FEBRUARY 21 Design examples ocus on sysems or sampling, reconsrucion, requency conversion, linear modulaion and demodulaion, and heir various combinaions. Among ideas no menioned are dierence equaions, z-ransorms, discree-ime Fourier ransorms, and DFs. hese, he nus and bols o DSP, can be sudied laer, aer he suden knows wha o build wih hem. his maerial has been successully augh o undergraduaes beore radiional discree-ime signals and sysems. his irs aricle in he series is abou moivaion, philosophy, and noaion. Sysems beore DSP? Why? he ashion has or many years been o consider discree-ime signals and sysems and digial signal /1/$ IEEE Auhorized licensed use limied o: Oulu Universiy. Downloaded on January 19, 21 a 3:5 rom IEEE Xplore. Resricions apply. 41

2 his series amouns boh o concise eaching noes or an undergraduae presenaion by a signals-and-sysems insrucor and o a sel-sudy mini-ex. processing in isolaion rom he coninuous-ime world o he exen possible. Bu is his realisic, or an ariice? Insrucors rouinely explain algorihms wih example signals represening speech, images, or radar or communicaion waveorms, all undamenally coninuous-ime, analog quaniies. Bu having seen he DSP world relaed o he larger coninuousime world only hrough a quick our o elemenary sampling and reconsrucion leaves many engineers a lile dizzy whenever hey are orced o approach he discree-/coninuous-ime boundary and consider how he signals and ransorms in one world relae o hose in he oher. his leads o weaker sysem design across he proession, as well as o much unnecessary anxiey, and suggess ha perhaps i is ime o rehink he undamenal philosophical approach o he subjec o DSP. In he beginning, signal processing was abou ransorming inormaionbearing signals rom available orms o needed orms compaible wih various media like circuis, anennas, ransmission lines, daa links, sorage sysems, microphones, and speakers hrough which inormaion mus be ransmied, sored, or reproduced. oday his is he elemenary aspec o signal processing, and is sudy ocuses on linear sysems operaing on deerminisic signals, which properly should include, in each o coninuous ime and discree ime, signals ha are lowpass, bandpass, real, and complex in basic naure, each compaible wih various media. ransormaions o such signals canno be discussed oday in meaningul generaliy wih discree-ime signals always considered separaely. Bu which opics in radiional discree-ime signals and sysems are acually prerequisie o working wih he range o possible simple linear-sysem srucures or changing he orms o signals? here are surprisingly ew. radiional opics in undergraduae discree-ime signals and sysems and DSP include compuaional convoluion, ranser uncions, dierence equaions and associaed iler srucures, and perhaps he plug-n-play design o ilers o simple speciicaions. Bu all o hese are acually abou realizaion or implemenaion o various processing seps, no abou he seps hemselves and wha hey should aim o do. Along he educaional way, new ransorms enlarge he Fourier menagerie bu o a purpose less han readily apparen o many sudens, who consanly ask, Why? Wha can his be used or? Indeed, wihou some sysem conex, he sudy o such implemenaion maers and he ransorms associaed wih hem is unmoivaed. Implemening various seps is ine, bu how does one decide wha seps are appropriae? Where should here be samplers, ilers, decimaors, and such? All his suggess a reordering o he convenional opic sequence. Perhaps a general sudy o linear sysems ha change deerminisic signals rom one orm o anoher should precede hese implemenaion-relaed speciics o DSP sysems. o do his requires, however, an approach o hinking and alking abou discree-ime signals and sysems ha does no require wha we usually, bu apparenly incorrecly, hink o as he major subopics o he subjec. he purpose o his series hen is o presen such an approach o he concepion o sysems wih embedded DSP, an approach boh easily augh o under graduaes who have ye o sudy discree-ime signals and sysems and easily masered by engineers whose primary specialies are elsewhere. his approach has been used successully in such conexs several imes, primarily wih seniors in a core course in nominally analog communicaions. he hear o he approach is he separaion o he idea o a signal rom he idea o is realizaion in a piece o hardware or soware. his separaion allows one o keep signals and signal processing locked in a sricly coninuous-ime world in which discree ime akes a dieren meaning, reerring o signals comprising rains o impulses. In his world, here is nohing paricularly peculiar or unnaural abou hybrid analog/dsp sysems, mulirae sysems, or complex signals. Sysem conceps are worked ou in he requency domain using manipulaion o simple Fourier skeches. he only prerequisie o his approach o preliminary sysem design is amiliariy wih he basic conceps o coninuous-ime signals and sysems. When such an approach is used in he classroom, wha is he curricular ae o radiional discree-ime signals and sysems and DSP? hese subjecs parallel or ollow aerward, repackaged only slighly. he impulse rains ha here make up discree-ime signals comprise impulses wih areas ha, when sysemaically normalized, become he amiliar sample sequences ha realize discreeime signals in compuaional DSP sysems. A radiional DSP course wih minor relabeling hen becomes he sudy o compuaional-dsp sysems ha operae on such sequences o realize he signal-processing operaions explored in his series o aricles. he remainder o his series amouns boh o concise eaching noes or an undergraduae presenaion by a signals-and-sysems insrucor and o a sel-sudy mini-ex or an engineer wih some experience, or a graduae suden, or or a moivaed undergraduae. his developmen is almos enirely graphical, wih explici mahemaics required only o esablish wo simple Fourier-ransorm properies no ypically encounered in reamens o coninuous-ime signals and sysems. he irs, immediaely below, esablishes he naure o discree-ime signals in he requency domain. he oher will come much laer, in he discussion o complex signals. Discree-ime signals Le us begin by agreeing ha a discree-ime (d..) signal x12 is one ha is nonzero only on some discree and uniormly spaced se o imes ha, le us say, includes, a condiion oddly bu convenienly saed or signal x1 2 as x12 5 x12e j2p /. (1) A 5 k, an arbirary ineger muliple o, he exponenial becomes e j2pk/ or uniy, so x12 can have any value and he equaliy will hold, bu a oher imes only x12 5 will do. Fourier ransorming (1) above o X1 2 5 X1 2 1/2 shows ha 1/, he signal s impulse rae, is a period 42 IEEE POENIALS Auhorized licensed use limied o: Oulu Universiy. Downloaded on January 19, 21 a 3:5 rom IEEE Xplore. Resricions apply.

3 o X1 2, so discree-ime signals are paired wih periodic ransorms. Since X1 2 is periodic, we can express i using a Fourier series as X1 2 5 à n52`a 2n e j2pn, where he independen variable is and he undamenal inerval is, wih hese replacing he amiliar and. he Fourier coeicien or ime n is named a n o make change o index n S n lead convenienly o X1 2 5 à n52`a n e 2j2pn, which in urn has inverse Fourier ransorm x12 5 à n52`a n d1 2 n2. From he laer we see ha discreeime signals are jus uniormly spaced impulse rains. Many will orge hese mahemaics, he mos challenging o his enire series o aricles, bu all should remember ha las senence! his series violaes wo long-esablished erminological convenions. Firs, here d.. signals are uncions o a realvalued ime variable bu are nonzero only or o he orm n, where n is an ineger. Convenionally however, d.. signals are wrien as a uncion o ha ineger n, he discree ime variable. We will deal urher wih his issue laer. Second, since we have only he one ime variable and since ha ime variable is coninuous, we can very well alk abou coninuous ime signals as disinc rom discree-ime signals as is cusomary. Insead we will use analog in an enirely common bu enirely incorrec way o reer o signals ha are no d.. and conain no d.. componen. O course i is a coninuous range raher han a coninuous domain ha makes a signal x12 properly ermed analog, bu his misuse is so widespread as o be a de aco sandard, and every elecrical engineer needs o be amiliar wih boh usages. Signals versus realizaions For us, signals are jus uncions o ime generalized o permi impulses, hence hey are absracions exising in our minds, on paper, or in our analyses. In implemenaions o sysems hese signals have various represenaions or realizaions. Analog signals are ypically realized as volages or currens, bu any physical quaniy conrollable over a coninuum o values will do. Our signal ampliudes are always dimensionless, bu dimensionless signal x12 migh be realized, or example, as volage V re x12 as in Fig. 1, wih reerence volage V re chosen or implemenaion convenience. Is choice is a hardware-design issue. Because signal ampliudes are dimensionless, signal impulse areas have ime dimensions and an impulse area is naurally realized in normalized orm as dimensionless number area/ re, ha is, area a re is naurally realized as dimensionless number a. A discree-ime signal is hen realized compuaionally in hardware or soware as a sequence o such numbers, called samples, occurring a some sample rae 1/. For he uniormly spaced impulses o a d.. signal we can simply choose consan re as he sample spacing, as in Fig. 1. Is he sample rae o he realizaion unique? Does he upper-le signal o Fig. 2 comprise simply he widely spaced impulses ha are visible? Or does i acually consis o narrowly spaced impulses wih many o heir areas zero? he second and ourh lines o he igure show wo o many possible realizaions as sequences. Noice ha halving he sandard re, sample spacing, doubles he scale o he samples by changing heir normalizaion. he ambiguiy is resolved below by adoping a sandard noaion or a d.. signal ha explicily indicaes he sample rae o be used in is realizaion. All d.. signals were assumed above o be realized compuaionally, making scale 2 2 upsample by wo signal wih obvious realizaion one alernae realizaion realizaions are scaled wih dimensioned consans. V re x() a signals have dimensionless ampliudes. (a ) x() Fig. 1 Signals are mahemaical eniies, bu heir realizaions, here a volage and a numerical sequence, are more physical. he processing o d.. signals equivalen o digial signal processing or DSP. Bu one migh jus as well scale dimensionless signal ampliude by some reerence curren I re o make a signal impulse o area a correspond o a curren impulse wih area equal o some quaniy ai re o elecric charge, wih I re chosen or implemenaion convenience. Such charge packes can be manipulaed elecronically wih swiched-capacior sysems, and mos o he developmen o ollow applies o sysems o ha ype as well as o DSP-based sysems. Noaion or signals Our sandard signal noaion is a schemaic skech in he requency domain ha mirrors he properies o a signal, showing i perhaps o be impulsive or bandlimied or symmeric. he example d.. signal on he op line o inerpolaion Fig. 2 Inerpolaion leaves he signal unchanged bu ransorms is realizaion o a higher sample rae. JANUARY/FEBRUARY Auhorized licensed use limied o: Oulu Universiy. Downloaded on January 19, 21 a 3:5 rom IEEE Xplore. Resricions apply.

4 Early DSP exs assumed ha a single sample rae was used in realizing all d.. signals in a sysem, bu here no such assumpion is made, because in modern sysems mulirae sysems are he norm. Fig. 2 is shown in boh ime and requency domains. In he specral skech, ellipses on he righ indicae he specral periodiciy ha lags a signal as d.. he only eaure o he noaion here ha is likely o seem unamiliar o mos DSP engineers is he riangular ick mark, which indicaes he sample rae o be used in he corresponding realizaion. his sample-rae ick has nohing o do wih he signal isel and so can be omied in he rare cases in which he realizaion is no o ineres. he period skeched or a signal s ransorm implies is impulse rae, and mos oen i will mach he ick-marked sample rae. Noe ha realizing impulse area a as dimensionless sample a is equivalen o scaling ha area by he ick-marked sample rae 1/ in moving rom he signal o is realizaion. In his realizaion V re a (a ) av re V re 1 a ( ) signal uni area series he scaling and he normalizaion o he samples are reerred o more or less inerchangeably. Early DSP exs assumed ha a single sample rae was used in realizing all d.. signals in a sysem, bu here no such assumpion is made, because in modern sysems mulirae sysems are he norm. For he same reason, requencies will never be normalized o he sample rae bu will always be acual and hones physical requencies. D/A conversion For he presen purposes o enabling a basic level o absrac sysem design, digial-o-analog (D/A) conversion reers o ilering a d.. inpu wih any impulse response ha is no d.. he op hree quarers o Fig. 3 shows such a conversion. A sandard conversion uses an noaion or operaion Fig. 3 D/A conversion and sampling, arranged in a loop. From op: discree-ime signal, D/A conversion, conversion oupu, and sampling (oupu a op). impulse response, as shown here, ha is a uni-area recangle cenered abou he ime origin wih widh equal o he sample inerval. Uni area implies uniy dc gain and so makes he associaed sinc 12 requency response easy o skech. Even i we did no recognize his iler s requency response as a sinc, we could sill see ha inpu sinusoids ha i only complee cycles ino impulse-response widh mus inegrae o zero in he convoluion and hereore ha his requency response mus have nulls a he sample rae and is nonzero muliples. he noncausaliy o he recangular impulse response shown o course makes i unrealizable, bu ha is o no more ineres or signiicance han he propagaion delays generally omied rom idealized models o oher circuis and compuaional sysems. Jus as we can model propagaion delay explicily when is eecs are imporan, here we can explicily model an addiional halsample delay on hose rare occasions when is eecs would be o ineres. We will rea oher ilers similarly, advancing ime wih noncausal ilers when doing so harmlessly simpliies he mahemaics. Noaion or operaions In our noaion or signal-processing operaions ha ac on signals, he inpu is o be combined in he requency domain wih he uncion skeched, using he operaor a is le. riangular icks above and below he axis mark he inpu and oupu realizaion raes (and normalizaions) respecively. he sinc skech in Fig. 3 is hereore lagged as a D/A conversion by he presence o only an inpu ick. here is no oupu ick because he oupu is no a d.. signal. A horough noaion would also indicae he reerence volage or curren by which he realizaion oupu is scaled, bu here we simpliy and consisenly omi he dimension-carrying realizaion scale consans o analog signals. In he Fig. 3 case, he ime and volage reerence levels could be absorbed ino a realizaion uni-sample response wih ampliude equal o he volage reerence. Sampling Muliplying d.. signals would o course be as nonsensical as muliplying wo co-locaed impulses anywhere bu in convoluion-like inegrals. Bu wo waveorms are easily muliplied when one is d.. and he oher is coninuous 44 IEEE POENIALS Auhorized licensed use limied o: Oulu Universiy. Downloaded on January 19, 21 a 3:5 rom IEEE Xplore. Resricions apply.

5 a he impulse imes o he irs. Such producs appear rouinely in hree conexs in DSP. he irs is he sampling operaion, an example o which appears in he boom line o Fig. 3. he ohers, decimaion and sinusoidal modulaion, are discussed laer. In Fig. 3 an inpu signal, he hird-line square wave, is muliplied by a ourhline sampling waveorm ha comprises impulses a some rae 1/ and o uniorm area. In he requency domain his is convoluion * wih he Fourier ransorm o he sampling waveorm, which consiss o uni-area specral impulses a muliples o he impulse rae. he laer ransorm is easily derived using wo Fourier properies: irs, ha periodiciy in eiher domain corresponds o uniormly spaced impulses in he oher and, second, ha he imedomain average equals he area o he dc impulse. I ollows ha he sampling operaion s oupu specrum comprises periodically shied copies o he inpu specrum. O course physical A/D converers also quanize signals and hereore inroduce approximaion eecs, bu hese are beyond he scope o his discussion. he oupu ick here ormally denoes he rae and normalizaion o he oupu samples o he analog-o-digial (A/D) converer ha realizes he sampling operaion. ha ick can also be reasonably viewed as applying o he sampling waveorm viewed as a signal. Signal reconsrucion he irs wo lines o Fig. 3 show a signal-reconsrucion operaion, realized as D/A conversion, ha undoes he sampling jus described. he squarewave signal shown happens o make he sampling and reconsrucion operaions ino exac inverses. While sampling and reconsrucion is oen depiced or resriced classes o signals, his squarewave resricion is obviously oo severe o be useul. A more ypical example appears in Fig. 4 using, as is quie common, a requency-domain descripion only. here a bandlimied real lowpass signal, perhaps music in a recording sudio, is convered o d.. orm, perhaps or sorage on or ransmission hrough some digial medium, and back again. he sysem is described by a specral skech ha saes algebraic relaionships in he requency domain. Read he lines rom op o boom: irs * second 5 hird and (hird 3 ourh) 3 ih 5 sixh. Fig. 4 Sampling and reconsrucion o he signal sampled. Upsampling and inerpolaion he hird line o Fig. 2 represens a processing sep ha increases he realizaion sample rae by some ineger acor wihou aecing he signal isel. Normalizaion in he realizaion is also aeced, wih he ne eec being simply o scale up he incoming samples by he ick-requency raio. Insering zero samples ino he sample sequence in he realizaion o increase he sample rae by an ineger acor M is upsampling by M, denoed cm. his series o aricles srains erminological convenion by using inerpolaion o reer o he ick-mark-shiing null signal operaion on he righ, which operaion corresponds boh o upsampling and renormalizaion scaling in he realizaion. Digial ilering A digial iler has a d.. impulse re - sponse and hereore a periodic requency Fig. 5 A digial iler applies a periodic requency response o a discree-ime inpu. response, and i operaes on a d.. inpu signal. In his series he erm is resriced o he common case depiced in Fig. 5 in which inpu and oupu sample raes are idenical and boh equal o he requency-response period. his makes compuing he convoluion in he realizaion relaively sraighorward. In pracice, however, digial ilering in his sense and a preceding or ollowing operaion are oen realized joinly or he sake o eicien implemenaion. Par II will coninue wih discussions o oversampling in D/A conversion and he basics o decimaion, complex signals, and Nyquis signaling. Read more abou i he maerial in he irs hree chapers o he exbook below is ypical o coverage o d.. maers in an inroducory course in signals and sysems. he remainder o ha ex s maerial would ypically be covered in a classic DSP course a he advanced undergraduae or irs-year graduae level. A.V. Oppenheim and R. Shaer, Digial Signal Processing. Englewood Clis, NJ: Prenice Hall, Figures are posed or insrucional use on he auhor s Web sie, available a hp://alum.mi.edu/www/jec. Abou he auhor Jerey O. Coleman (Jerey.Coleman@nrl.navy.mil) joined he Radar Division o he Naval Research Laboraory (NRL) in Washingon, D.C. in He le NRL in 1985 or graduae sudies, a sin wih he Boeing Company, and a aculy posiion a Michigan echnological Universiy. He reurned o NRL in His 1975/1979/1991 SBEE/MSEE/ Ph.D. degrees are rom he Massachuses Insiue o echnology, Johns Hopkins Universiy, and he Universiy o Washingon respecively, and his research is on heory and design mehods in DSP. He is a Senior Member o he IEEE. Erraa In he November/December 29 issue o IEEE Poenials, he aricle by Mudahir Funsho Akorede should have read, Guidelines or wriing an undergraduae engineering projec repor. We regre he error. JANUARY/FEBRUARY Auhorized licensed use limied o: Oulu Universiy. Downloaded on January 19, 21 a 3:5 rom IEEE Xplore. Resricions apply.