Discrete Time Signals and Fourier series

Transcription

1 Discrt Tim Signals and Fourir sris In prvious two chaptrs w discussd th Fourir sris for continuous-tim signals. W showd that th sris is in fact an altrnat rprsntation of th signal. This rprsntation can b don in a trigonomtric form with sin and cosin functions or with complx xponntials. Both forms ar quivalnts. Fourir analysis allows us to rprsnt th signal as a wightd sum of harmonic signals. Th wights of harmonic can b thought of as th spctrum of th signal. In prvious two chaptrs, our discussion was limitd to continuous tim signal. In this sction w will discuss Fourir sris for discrt signals. Proprtis of discrt signals Many of th signals w dal with ar sampld analog signals, such as voic, music, and mdical/biological signals. This is don by instantanous sampling of th undrlying signal and rcording th masurd data. A ky qustion facing th nginr is how fast to sampl? Sampling of signals Suppos w hav an analog signal and w wish to crat a discrt vrsion of it by sampling it. In Fig., w show an analog signal sampld at two diffrnt rats. It is obvious just by looking that th sampling rat chosn in Fig. (a), that th rat is not quick nough to captur all th ups and downs of th signal. Som high and low points hav bn missd. But th rat in Fig.(b) looks lik it might b too fast as it is capturing far mor sampls than w probably nd. So clarly thr is an optimum sampling rat which capturs nough information without ovrdoing it such that th undrlying analog signal can b dscribd corrctly. Discrt Tim Fourir Sris - Charan Langton Pag

2 x[n] (S Matlab Program ) Figur Continuous and a discrt signal This is whr w invok th famous Sampling Thorm by Shannon. Th thorm says: For any analog signal containing among its frquncy contnts a maximum frquncy of f max, th undrlying signal can b rprsntd faithfully by N qually spacd sampls, providd th sampling rat is at last two tims f max sampls pr scond. So for any signal, a maximum Sampling Priod that will still allow th signal to b rconstructd from its sampls is spcifid as: T s sconds 2 f (.) max Sampling frquncy is spcifid by th invrs of th sampling priod. F s sampls/scond (.2) T s Th maximum frquncy of an analog signal that can b rprsntd unambiguously by a discrt signal with a sampling priod of Ts sconds is givn by: Discrt Tim Fourir Sris - Charan Langton Pag 2

3 x[n] x[n] x(t) f or max Hz 2T s (.3) max 2 fmax radians / sc Ts Now w will show a proprty of discrt signals that is most prplxing and causs a grat dal of confusion. In Figur 2(a), w show a continuous signal, x(t). W sampl this signal at 8 sampls pr scond in Fig. 2(b) and thn with 2 sampls pr scond in Fig. 2(c) for a total of 48 sampls. (a) Tim, t (b) Sampl, n (c) Sampl, n Figur 2 (a) A continuous signal, (b) sampld at 8 sampls pr scond and (c) sampld at 2 sampls pr scond. (S Matlab Program 2) Discrt Tim Fourir Sris - Charan Langton Pag 3

4 Th signal is givn by th following xprssion. It contains frquncis, 2, 3, and 4 Hz and no othrs. x( t).25sin(2 t).7cos(4 t).5cos(6 t).5sin(8 t) Th highst frquncy in this signal is 4 Hz. Fig. 3(a) shows th Fourir sris cofficints of this continuous signal. Th cofficints span from -4 to +4 Hz and ar symmtrical about th zro frquncy. W can s that th cofficints ar.25,.35,.25,.75, for frquncis, 2, 3 and 4 Hz rspctivly. As w know, Fourir sris cofficints masur th contnt of ach frquncy and hnc th computd cofficints occur only at frquncis actually prsnt in th signal, which ar, 2, 3, and 4 Hz. This signal contains no othr frquncis so all th othr cofficints ar zro. Th spctrum is shown in Fig. 3(a). No confusion hr. Now convrt this signal to a discrt signal by sampling it at 2 sampls pr scond. This rat is slightly highr than th minimum sampling rat of 2 tims f max or 8 Hz. Th discrt vrsion is givn by th xprssion: x( t).25sin(2 k /2).7cos(4 k /2).5cos(6 k /2).5sin(8 k /2) Hr k is th sampl indx, or a point at which th signal is bing discrtizd. Now without actually going ovr th procss of how this was don, w show th spctrum of this discrt-tim signal. In Fig. 3(a), w s th spctrum of th continuous signal. It is limitd to 4 Hz as would b xpctd. But w gt an odd thing whn w comput th Fourir sris cofficints of th sampld discrt signals. As shown in Fig. 3(b), instad of finding zro s outsid of th actual bandwidth, w gt ghosts-lik copis of th sam spctrum cntrd at intgr multipl of th sampling frquncy. Contnt appars at frquncis that ar not prsnt in th signal. Discrt Tim Fourir Sris - Charan Langton Pag 4

5 Figur 3 Rplicating spctrums as a consqunc of sampling Th spctrum around zro frquncy (th cntr part in Fig. 3(a)) rpats at th sampling frquncy of th sampld signal. Evry 2 Hz, thr is a copy of th spctrum. Ths rplications occur vry 2, 24, ndlssly. In Fig. 3(c) w show th sam signal sampld at 8 Hz, and now th spctrum rpats at 8, 6,. If w sampld th continuous signal at a rat lss than 8 Hz, which is th minimum rquird by th sampling thorm, th spctrum would bgin to ovrlap and that is a problm which w will discuss in Chaptr 6. This ovrlapping is calld aliasing, rsulting from a sampling rat that is lss than twic th maximum frquncy in th signal or also calld th Nyquist rat. This rplicating of th spctrum is a consqunc of discrt sampling. W crtainly do not s it in th spctrum of a continuous-tim signal. In this sction, w ar going to discuss th Fourir sris rprsntation of discrt signals, calculation of th sris cofficints and w ar going to talk about why this spctrum rplication happns. Spcifying a discrt signal If a continuous signal is rfrrd to as x(t) thn a discrt sampld signal is writtn as x( kts ), k,, 2, WhrT s is th sampling priod, or th tim btwn any two succssiv sampls. Th indx k is calld th sampl numbr. Th quantity kt s is a masur of tim. Discrt Tim Fourir Sris - Charan Langton Pag 5

6 Figur 4 Discrt signal sampls To crat a discrt signal from a continuous signal, w tak two stps, first th continuous signal is multiplid by an impuls train of th sampling priod. But sinc mathmatically this is still a continuous signal, w thn multiply th sampld signal onc again by an impuls train, point by point. Th sum of all of thos points is th discrt signal. x( kt ) x( t) ( t kt ) s k x[ k] x( kt ) ( t kt ) k Th trm x( kt s) is considrd continuous whil th trm x[k] is its discrt quivalnt. It s clar what is happning hr, th dlta function is combing th signal to crat a discrt signal. (W us th squar brackts [ ] to dnot a discrt squnc and us rgular brackts ( ) for a continuous signal.) Lt s tak a sin wav and plot its continuous and discrt vrsions. s s s (.4) x( t) sin(2 f t), f. W rplac continuous tim t with kts and comput a fw valus of th discrt signal as follows. Ths ar plottd in Fig. 5b. x[ ] sin[2 ( 5)] x[ 9] sin[2 ( 9 5)] sin[ 3.6 ].95 x[ 8] sin[2 ( 8 5)] sin[ 3.2 ].588 x[ 7] sin[2 ( 6 5)] sin[ 2.4 ].95 x[ 6] sin[2 ( 5 5)] sin[ 2 ] x[ 5] sin[2 ( 4 5)] sin[.6 ].95 Discrt Tim Fourir Sris - Charan Langton Pag 6

7 x[n] x[n] x(t) (a) Tim, t sconds (b) Sampl, n (c) - -2pi -8pi/5-4pi/5 4pi/5 8pi/5 2pi Radians Figur 5 Sampling of a continuous signal (Matlab Program 3) Discrt signal rprsntation Thr ar two ways to spcify a sampld signal. On is by sampl numbrs. In Figur 5, w show two priods of th signal. Th signal covrs two cycls in 2 sconds. Each cycl is sampld with fiv sampls, so w hav a total of tn sampls in Fig. 5(b). This is th discrt rprsntation of signal x[k] in trms of sampls. Th rat of sampling is 5 sampls pr scond or 5 sampls pr cycl. This is a common way of showing a discrt signal particularly if th signal is not priodic. As w will s, thr ar advantags in spcifying th signal by its phas. In polar form, a priodic signal is said to covr 2 radians in on cycl. W can rplac th sampl with its phas valu for an altrnat way of dscribing a discrt signal. Thr ar fiv sampls ovr ach 2 or quivalntly a discrt angular frquncy of 2 /5 radians. Each sampl movs th signal furthr in phas by 2 / 5 radians from th prvious sampl, with two cycls or sampls covring 4 radians as in Fig. 5(c). Paramtrs of a discrt signal To crat a discrt signal w sampl this signal with sampl tim of s T. Th sampl tim as w said should b small nough to captur all important information. Th Discrt Tim Fourir Sris - Charan Langton Pag 7

8 Sampling thorm tlls us that it should b no largr than / 2 f max. Th sampling priod T s is indpndnt of th fundamntal priod T of th continuous signal and can b any numbr smallr than th Nyquist thrshold. T th fundamntal priod of th continuous signal T th sampling priod of th discrt signal s If w know th signal frquncy f and th sampling frquncy, w can also writ th signal this way rplacing T s with / F s. 2 f x[ k] sin k Fs Lt s giv th trm insid th parnthsis a spcial nam, calling it Digital frquncy. 2 f radians cycls / sc ond radians (.5) Fs sampl / sc ond sampl Th units of this frquncy ar givn as radians pr sampl and not as radians pr scond. So it is not rally a frquncy, but w call it that for lack of a bttr nam. You can also think of it as phas advanc. If w hav a signal of frquncy Hz and w sampl it with a sampling frquncy F s = 3 Hz, thn its digital frquncy is qual to 23. What dos this numbr man? It mans that ach sampl movs th signal by this many radians. If a cycl contains 2 radians, and ach sampl covrs 23 radians, thn it will tak 3 sampls to complt a cycl. Th digital frquncy in this cas is: 2 2 / 3 3 K, th priod can b sn as a ratio of th sampling frquncy to th fundamntal frquncy or th maximum frquncy: K F s 3 3 f W want this ratio to b biggr than 2 and usually much biggr than that. Th fundamntal priod K is an intgr and rprsnts numbr of sampls aftr which th Discrt Tim Fourir Sris - Charan Langton Pag 8

9 signal starts rpating again. If thr ark sampls in on priod, thn K tims th digital frquncy must qual 2. (Not th units of this frquncy ar radians pr sampl.) W dfin th priod of th signal in sampls as: 2 K 2 or K (.6) Th fundamntal frquncy is givn by: 2 (.7) K Th priod of th digital frquncy is always 2 bcaus that condition is part of its dfinition. Th smallr th digital frquncy, mor sampls ar ndd to complt on cycl. Having K in th dnominator says xactly th sam thing. If th undrlying signal is priodic, is th sampld discrt signal also priodic? Not ncssarily. It will b if and only if N is an intgr. T N mk m (.8) T s Th fundamntal priod of th continuous signal is an intgr multipl of th ratio btwn th fundamntal tim and th sampling tim. This is also sam as saying F s N mk m f (.9) For xampl a signal with fundamntal frquncy of 5 which is sampld at a rat of 2, will hav a fundamntal priod of 4 and as such this sampling rat would rsult in a discrt signal that is priodic. Discrt signal priodicity Th dfinition of a priodic signal for a discrt signal is th sam as for th continuous cas, which is x[ k] x[ k K ] (.) Th discrt valus of th signal rpat vry N sampls. It may sm lik a trivial sin n. xrcis, but w ar now going to look at th priodicity of a discrt sinusoid Discrt Tim Fourir Sris - Charan Langton Pag 9

10 Th digital frquncy hr is a gnral frquncy trm. W us th subscript whnvr w ar talking about th fundamntal frquncy and just whn w ar talking about any othr discrt harmonic frquncy. How can w show that a discrt signal sin n xprssion in Eq. (.8) to writ: is priodic? Using th abov k k K sin k K sin sin ( ) Th quation holds undr only on condition. That is if K k (.) 2 That s bcaus at vn multipls of, th valu of sin is zro. All intgr valus of k satisfy th condition. With discrt signals, thr is natural ambiguity about what frquncy a st of discrt sampls rprsnt. In Figur 6 w s two diffrnt continuous signals, on a cosin of frquncy 2 Hz and th othr of frquncy Hz, both of vry diffrnt frquncis yt thy map to th sam discrt sampls. In fact th sam sampls would fit many othr harmonics! This crats an ambiguity in discrt signal procssing that w don t hav in continuous signals. Whn w look at a discrt signal, w truly don t know what frquncy it is supposd to rprsnts. An infinity of signals fit th sam points. So intuitivly, rplication of th spctrum is saying xactly that. Sinc th algorithm (DFT, FFT tc.) dos not know which frquncy is th actual on in th signal, it just rpats th sam spctrum at all possibl harmonics. Basically saying, I don t know which on is corrct, you pick! W do of cours hav som ida of about th targt signal frquncis. Th sampling frquncy is carfully slctd to captur th undrlying signal. W ar usually not intrstd in frquncis outsid of a crtain rang. Sinc ach cycl contains xactly th sam information, w can just limit th analysis to on cycl, ignoring all th othrs rplications as truly imaginary! For this rason, th Fourir analysis to dtrmin th cofficints of a discrt signal can b limitd to just on cycl. This is why w lik to rprsnt th signal by its phas whr a 2 rang maks mor sns than daling with numbrd sampls. Any on priod contains all availabl information about th priodic signal. Discrt Tim Fourir Sris - Charan Langton Pag

11 x[n] Figur 6 A discrt signal can rprsnt any numbr of continuous signals. (Matlab Program 4) Why ar w bothring with th priodicity of a discrt signal? Arn t all sinusoids priodic by dfinition? Ys, that is tru for th continuous cas but not always tru for th discrt sinusoids. Exampl 3- What is th digital frquncy of this signal? What is its priod? 2 x[ k] cos k 3 3 Th digital frquncy of this signal is 2. Its priod K is qual to 3. Th valus of 3 x[k] rpat aftr vry thr sampls as can b sn in Fig 7. K / Sampl, n Figur 7 Discrt signal of xampl 3- Discrt Tim Fourir Sris - Charan Langton Pag

12 (Matlab Program 5) Exampl 3-2 What is th priod of this discrt signal? Is it priodic? 3k f[ k] sin 4 4 Th digital frquncy of this signal is qual to =3 / 4. Th priod of th signal is: 3 m m4 K 8, m2 Th fundamntal priod of th signal quals 8, bcaus that is th minimum numbr of sampls ndd to achiv an intgr multipl of 2. So th signals rpats aftr vry 6 radians. Exampl 3-3 Is this discrt signal priodic? Figur 8 Discrt signal of Exampl 3-2 f [ n] sin.5n Th fundamntal frquncy of this signal is.5. 2 K k 2 k Can w call this th priod of th signal? Ys, but it is not a rational numbr (a ratio of intgrs), hnc it can nvr rsult in a rpating signal. Th continuous signal as shown hr is of cours priodic but w don t s any priodicity in th discrt sampls. Discrt Tim Fourir Sris - Charan Langton Pag 2

13 x[n] Sampl, n Figur 9 Non-priodic discrt signal of xampl 3-3. (Matlab Program 6) Basis functions for Discrt-tim Fourir sris Th continuous-tim Fourir sris (CTFS) is writtn in trms of complx xponntials. Bcaus thy ar harmonic and hnc orthogonal to ach othr, ths complx xponntials form a basis st. Th sris cofficints can b sn as th projction of th signal on to ths basis functions. W ar now going to dvlop a Fourir sris rprsntation for discrt tim signals using as basis functions th discrt tim complx xponntial. Lt s xamin th discrt-tim xponntial and s how its priodicity is affctd by taking it into th discrt ralm. Th discrt complx xponntial is writtn by rplacing t with k. W can writ this in trms of th digital frquncy as: jt jk continuous signal discrt signal (.2) Not that units of both t and k ar radians. In continuous cas, a harmonic is an intgr multipl of th frquncy. In th discrt cas, th harmonic rlationship is basd on phas: 2 k (.3) Th digital frquncy is in radians pr sampl, so clarly th nxt harmonic frquncy is obtaind by incrmnting by 2, so that and 2k ar harmonics for all intgr k. This is sam as saying that and k ar harmonically rlatd for all k. Evry tim incrass by2, w gt a nw complx xponntial givn by: Discrt Tim Fourir Sris - Charan Langton Pag 3

14 j( 2 ) k jn j2 k jk (.4) Th scond trm, j 2 k is qual to bcaus: j2 k k j k cos(2 k) cos(2 ) sin(2 ) j( 2 ) k This is quit an intrsting rsult. Th xponntial is xactly th sam as th j k xponntial. Although in th continuous cas, ach and vry harmonic diffrnt, all harmonics of a discrt signal ar xactly th sam. To som xtnt this taks th thundr out of doing Fourir sris analysis on a discrt signal. Harmonics do not sm to form a usful basis st. Exampl 3-4 Show harmonics of th xponntial.25 sconds. 2 j t 3 if it is bing sampld with sampling priod of W can writ th xponntial in discrt form by rplacing t with kt k 4. s y[ k] 2 j k 2 Lt s plot this signals along with its nxt two harmonics, which ar j k j 2 k and W plot only th ral part in Fig. Why is thr only on plot? Simply bcaus th thr signals ar idntical. Discrt Tim Fourir Sris - Charan Langton Pag 4

15 x[n] Sinusoidal Squnc Sampl n (Matlab Program 7) Figur Thr discrt harmonic signals What this tlls us is that for discrt signal th traditional concpt of harmonic frquncis dos not lad to anything maningful. All harmonics ar th sam. But thn how can w do Fourir sris analysis on a discrt signal if thr ar no orthogonal basis signals to rprsnt th analysis signal? Thr is, as it turns out, availabl a st of orthogonal basis st that w can us. So far w only lookd at harmonics that diffr by phas of 2. Although thy ar harmonic in a mathmatical sns, ths ar prtty much uslss in a practical sns. Instad of looking vry 2 for a harmonic, w nd to look lswhr. W will now rval a scrt: w find hiddn insid th to 2 rang, thr xists an anothr orthogonal basis st. Lt s s what happns as digital frquncy is varid just within th to 2 rang. 2 j k 6 Tak th signal x[ k]. Its digital frquncy is qual 2 and its priod K is qual 6 to 6. W now know that th signals of digital frquncis 2 / 6 and 4 / 6 ar xactly th sam, but what about in btwn? W will incras th frquncy of this signal in 6 stps, ach tim incrasing it by 2 / 6 so that aftr 6 stps, th total incras will b 2. W can start with zro frquncy, as it maks no diffrnc whr you start. 2 ( n ) / 6 2 ( n ) / 6 2 / 6 2 ( n 2) / 6 4 / ( n 5) / 6 / 6 5 Discrt Tim Fourir Sris - Charan Langton Pag 5

16 Th variabl n stps from to K lttr n. Indx k rmains th indx of th sampl.. Thr ar N harmonics, and w indx thm with In Figur, w hav plottd th analog signal and th discrt vrsion of th sam signal. Th discrt frquncy appars to incras (mor oscillations) at first but thn aftr 3 stps (half of th priod, N) it starts to back down again. Th discrt signals for frquncis 2 /3 and th 4 /3 appar idntical. Raching th nxt harmonic at 2, th discrt signal is back to whr it startd. Furthr incrass will rpat th sam cycl. So hr w hav a significant diffrnc in how discrt and continuous signals of sam frquncis bhav. Th analog signals ar harmonic along th frquncy axis whnvr k k. Discrt signals ar harmonic in btwn ths valus whn spcifid in trms of phas. How do w know ths signals ar harmonic? Th plot of th analog signal at ths sampls in Fig. tlls us that thy ar harmonic. Of cours, if w can do th orthogonality tst, and w find that thy ar indd harmonic to ach othr. K * 2 n Ths 6 signals, which w rfr to by K, ar an orthogonal basis st that can b usd to rprsnt a discrt signals. Th wightd sum of ths K spcial signals is th discrt Fourir sris rprsntation of th signal. Unlik th continuous signal, hr th maningful rang is limitd to a finit numbr of harmonics only. Figur Discrt signals in btwn harmonic frquncis (Matlab Program 8) Discrt Tim Fourir Sris - Charan Langton Pag 6

17 Discrt Tim Fourir Sris - DTFS W not ky idas about discrt signals.. W do not know th undrlying analog signal nor its frquncy. All w know is that if th sampling frquncy is Fs, thn w can from a discrt signal unambiguously xtract signals of only of frquncy half as much. 2. A discrt signal is dfind by its digital frquncy. Th units of digital frquncy ar in radians pr sampl. W can think of it as bing dfind ovr a circl from to 2 (or to ). 3. A discrt signal of frquncy is xactly th sam as all its harmonics whn 2 k for all k. 4. Thr ar only N distinct discrt tim complx xponntial signals that ar harmonically rlatd for any givn priod N. K is th smallst such numbr and calld th fundamntal priod. Th Discrt Tim Fourir sris (DTFS) is th discrt rprsntation of a priodic signal by a linar wightd combination of N complx xponntials. Ths orthogonal xponntials xist within just on cycl. Not thr ar only N of ths, not an infinit numbr as in continuous tim Fourir sris rprsntation. Th frquncy dcomposition hr is discrt just as it is for th Continuous Tim Fourir sris (CTFS). Th cofficints dscrib th contnt of ach basis function in th signal. On can also think of th DTFS cofficints (DTFSC) as a corrlation of th complx xponntial with th targt signal. Ths cofficints form a discrt signal hnc th spctrum of a discrttim priodic signal is also discrt. If 2 5, w can writ th discrt harmonic complx xponntials as: j n k, 2 j k j k j k j k 2 j k n 2 3 ( K ) n n n n,,,,, Th indx n is usd to indicat th harmonics. Th indx k is th tim sampl. Thr sm to b many diffrnt ways popl writ th xponnt of th complx xponntial. I lik to kp th fundamntal frquncy and its indx togthr, and k th tim indx at th nd. Many sourcs also us n and k in opposit sns from th way I hav usd thm. Som us n for tim indx and k for harmonic indx. Th indxs can b confusing bcaus both th numbr of harmonics (rfrrd to by indx n) and th numbrs of sampls Discrt Tim Fourir Sris - Charan Langton Pag 7

18 (rfrrd to by k) ar sam and qual to th numbr K. Th discrt-tim rprsntation of th signal is writtn as th wightd sum of ths. x[ k] Th complx cofficints, C, C,, C ar givn as ( K ) K j n k Cn (.5) k K Cn x k K j n k [ ] (.6) n Not that w ar summing ovr just on priod. So (.8) says that a discrt Fourir sris is a dcomposition of a singl priod of th signal into a fundamntal digital frquncy, and K - harmonics of that frquncy. W said that all discrt harmonics ar th sam, and now w show that th DTFS cofficints of th nth harmonic ar xactly th sam as th cofficint for a harmonic that is an intgr multipl of mk sampls away so that: C C (.7) n ( n mk ) Hr m is an intgr. Th nth cofficint is qual to N C x[ n] n k j n k Th ( n mk ) cofficint is givn by C ( n mk) N k x[ k] j( n mk) k N k x[ k] jn k jmk k Th scond part 2 jmk k jm k is qual to. (Bcaus th valu of th complx xponntial at intgr multipls of 2 is.) So w hav: Discrt Tim Fourir Sris - Charan Langton Pag 8

19 C ( n mk ) N k x[ k] jn k C n So indd th cofficints rpat for a discrt-tim priodic signal. In practical sns, this mans w can limit th computation to just K harmonics. In arlir sction, w statd that th discrt Fourir analysis rsults in rplicating spctrums. This is a vry diffrnt situation from th cas of continuous signals, which do not hav such bhavior. Discrt signals do this bcaus as w allow n to vary ovr all valus of digital frquncy, which ar rpating vry K sampls, w can no longr tll th harmonics apart so w ar computing th sam K numbrs ovr and ovr again. Exampl 3-5 Find th discrt tim Fourir sris cofficints of this signal. 2 x[ k] sin k Th fundamntal priod of this signal is, K as w can s Figur 2 Signal of xampl 3-5 Now writ th Eulr quivalnt xprssion for this signal, w gt 2 2 j k j k x[ k] 2j 2j Discrt Tim Fourir Sris - Charan Langton Pag 9

20 From this xprssion, w can comput th DTFSC from C to C by stting n, from to 9 9. W gt th following rsult. Not that bcaus th analysis signal has only two frquncis, corrsponding to indx n =, which is th zro frquncy and n =, which corrsponds to th fundamntal frquncy th cofficints for rmaining harmonics ar zro. W can writ th cofficints as K Cn x[ k] K 9 n n x[ k] 2 j n k j n k x[k]= [.,.5878,.95,.95,.5878,.,.422,.489,.489,.422] 9 9 j o k C C x[ k] x[ k] n k k In computing th nxt cofficint, w comput th valu of th complx xponntial for n = and thn for ach valu of k, w us th corrsponding x[k] and th valu of complx xponntial. Th summation will giv us ths valus. C C 9 j k x[ k] k 2j 9 j k x[ k] 2j k Of cours, w can s th cofficints dirctly in th complx xponntial form of th signal. Th rst of th cofficints from C to C ar zro. Howvr, th cofficints 2 9 rpat aftr C so that C C. 9 ( 9 k) Exampl 3-6 Comput th DTFSC of this discrt signal. 2 2 x[ k].5.25cos k.6sin k 5 4 Th priod, K of th cosin is 5 and th priod, K of sin is 4. Priod of th whol signal is 2 bcaus it is th last common multipl of 4 and 5. This signal rpats aftr Discrt Tim Fourir Sris - Charan Langton Pag 2

21 X(f) x[n] vry 2 sampls. In tim domain, w hav highlightd th priodic sction of th 2 sampls. Th fundamntal frquncy of this signal is a Sampl n b Harmonic k Figur 3- Signal of xampl 3-6 K Cn x[ k] K n 9 x[ k] 2 n x[ k] x[] 2 j n k j n k j(2 /5) Th Fourir cofficints rpat with a priod of 2. Each complx xponntial will vary in digital frquncy by /. Basd on this knowldg, whn w look at th abov Discrt Tim Fourir Sris - Charan Langton Pag 2

22 xpansion, w can s that 2 / 4 xponntial falls at n = 5. 5 / 2 / 4 and xponntial 2 / 5 falls at n = 4 4 / 2 / 5. Also not that th Fourir sris is a brakdown of th signal in sinusoids. But hr our targt signal is convnintly alrady in sinusoids. So all w hav to do to find th cofficints it to just writ it out in th Eulr formulation and thn pick out th cofficints by inspction. W did svral xampls of this procss in Chaptr 2. W can writ this signal as [ ].5.25 k x k k.3 j k k From hr, w s that th zro-frquncy harmonic has a cofficint of.5. Th frquncy 2 / 5 has cofficints of.25 and so forth. C C n C.5.25 C C.3 j C.3 j 5 5 Working with sins and cosins is almost trivial, bcaus w alrady know what is in th signal by looking at th quation. In xampl 3-7 w will look at a signal whr th cofficint computation using closd form quations is not so simpl. Exampl 3-7 Comput th DTFS of this priodic discrt signal. Th signal rpats with priod 4 and has two impulss of amplitud 2 and. Th fundamntal frquncy of this signal is Figur 4 Signal of xampl 3-7 Discrt Tim Fourir Sris - Charan Langton Pag 22

23 W writ th xprssion for th DTFSC from Eq.(.8) C x[ k ] n 3 k j n k 2 To solv this summation in closd form is th hard part. In narly all such problms w nd to know sris summations or th quation has to b solvd numrically. In this cas w do know th rlationship. W first xprss th complx xponntial in its Eulr form. W know valus of th complx xponntial for argumnt 4 ar and rspctivly for th cosin and sin. W writ it in a concis way as: j nk 2 cos j sin ( j ) 2 2 kn nk Now substitut this into th DTFSC quation and calculat th cofficints, knowing thr ar only n = 4 harmonics in th signal bcaus th numbr of harmonics ar qual to th fundamntal priod of th signal. 3 kn C x[ k]( j ) n 4 k Sinc w ar intrstd in th harmonics, start with n =, and thn multiply ach ( ) xk [ ] with ( j) kn to gt th following rsults. Th go to n = and rpat th procss. 3 C 2 for n, k,,2, j C (2 j) for n, k,,2, C 2 for n 2, k,,2, j C 2 j for n 2, k,,2, W can stup th DTFSC quation in matrix form by stting th basic xponntial to a constant and thn writing it in trms of two variabls, th indx n and k. j W jn k nk W Discrt Tim Fourir Sris - Charan Langton Pag 23

24 Now w writ W W W W 2 3 C x W W W W [ k ] n W W W W 2 3 K W W W W n k Hr th ach column rprsnts th harmonic indx n and ach row th tim indx, k. It taks 6 xponntiations, 6 multiplications and four summations to solv this quation. W will com back to this matrix mthodology again whn w talk about DFT and FFT in Chaptr 5. W usd Matlab to comput th cofficints. Hr is what w gt. Sam as th closd form solution. j j j j j Matlab Program Exampl 3-8 Find th discrt-tim Fourir sris cofficints of this signal. This signal is part of an important class of signals that ar similar to squar pulss. Thy ar vn hardr to solv using closd form solution Figur 5 Signal of xampl 3-8 with N = 3, K = 7 W split th summation in two parts. Discrt Tim Fourir Sris - Charan Langton Pag 24

25 K N jn k jn k N C n K K k N k N jn N K jn (2N ) jn This can b simplifid (using sris summation formulas) to this form. C n sin Nn / K K sin n / K (.8) This function looks lot lik a sinc function but actually is a function calld Diric. To draw th graph, w assum N = 3 and K = 7 and K =5 and 25. Not as th signal sprads, th componnts gt mor numrous. W will com to this proprty in th nxt sction whn w talk about apriodic signals and Fourir transform. Matlab program Figur 6 Cofficints of th priodic pulss, (a) with N= 3, K = 7, (b) N =3, K = 5, (c) N = 3, K = 2 Discrt Tim Fourir Sris - Charan Langton Pag 25

26 Summary. A discrt signal can b cratd by sampling a continuous signal with an impuls train of dsird sampling frquncy. 2. Th sampling frquncy should b gratr th two tims th highst frquncy in th signal of intrst. 3. Th fundamntal priod of a discrt signal, givn by K must b an intgr for th signal to b priodic. 4. Th fundamntal discrt frquncy of th signal, givn by is qual to 2 /K. 5. Th priod of a digital frquncy is an intgr multipl of 2. Harmonic discrt frquncis vary by intgr multipl of 2, such that and 2 k ar harmonic and idntical. 6. Bcaus discrt harmonic frquncis ar idntical, w cannot us thm to rprsnt a discrt signal. 7. Instad w divid th rang from to 2 by K and us ths digital frquncis as th basis st. 8. Hnc thr ar only N = K harmonics availabl to rprsnt a discrt signal. Th Fourir analysis is limitd to ths N harmonics. 9. Byond th 2 rang of harmonic frquncis, th discrt-tim Fourir sris cofficints, (DTFSC) rpat.. In contract, th continuous-tim signal cofficints ar apriodic and do not rpat.. Somtims w can solv th coffcints using closd from solutions but in a majority of th cass, matrix mthods ar usd to find th cofficints of a signal. 2. Matrix mthod is asy to stup but is computationally intnsiv. Charan Langton Copyright 22, All Rights rsrvd langtonc@comcast.nt Discrt Tim Fourir Sris - Charan Langton Pag 26

27 %Program Chaptr 3 Program f=; Fs = 32; Ts = /Fs; t = : Ts: 2; clf; figur() % havy sampl xt=cos(2*pi*t) *sin(3*pi*t+.5)+.5*cos(4*pi*t)-.3*cos(5*pi*t+.25); ylabl('x[n]'); xlabl('sampl'); hold on plot(t/ts,xt,'-. r' ); n = : 2*Fs; xn=cos(2*pi*n*ts) *sin(3*pi*n*Ts+.5)+.5*cos(4*pi*n*Ts)-.3*cos(5*pi*n*Ts+.25); stm(n, xn); hold off figur(2) % light sampl plot(t/ts,xt,'-. r' ); hold on n = : 2*4; Ts = /4; xn=cos(2*pi*n*ts) *sin(3*pi*n*Ts+.5)+.5*cos(4*pi*n*Ts)-.3*cos(5*pi*n*Ts+.25); stm(n*8, xn); ylabl('x(t)'); xlabl('sampl'); hold off %Chaptr 3 - Program 2 t = :.: 6; x =.25*sin(2*pi**t)+.7*cos(2*pi*2*t)-.5*cos(2*pi*3*t)+.5*sin(2*pi*4*t); clf; figur(); plot(t, x) titl('(a)') ylabl('x(t)') xlabl('tim, t') figur(2); n = : 47; fs = 8; xn8 =.25*sin(2*pi**n/fs)+.7*cos(2*pi*2*n/fs)-.5*cos(2*pi*3*n/fs)+.5*sin(2*pi*4*n/fs); plot(t*8,x, '--r') ylabl('x[n]') xlabl('sampl, n') titl('(b)') hold on stm(n, xn8, '.') Discrt Tim Fourir Sris - Charan Langton Pag 27

28 axis([ ]); hold off figur(3) n2 = : fs2*6-; fs2 = 2; xn2 =.25*sin(2*pi**n2/fs2)+.7*cos(2*pi*2*n2/fs2)-.5*cos(2*pi*3*n2/fs2)+.5*sin(2*pi*4*n2/fs2); plot(t*2, x, '--r') ylabl('x[n]') xlabl('sampl, n') titl('(c)') hold on stm(n2, xn2, '.') axis([ ]); hold off figur(4); n = : 47; clf; xnd = (/48)*fft(xn8); xnd2 = abs(fftshift(xnd)); plot(n, xnd2) %Chaptr 3 - Program 3 t = -.5:.:.5; y = cos(4*pi*t); clf; subplot(3,,) plot(t, y) grid; titl('(a) '); xlabl('tim, t sconds'); ylabl('x(t)'); axis; subplot(3,,2) n = -5: 5; y2 = cos(4*pi*n*.2); stm(n, y2) grid; titl('(b) '); xlabl('sampl, n'); ylabl('x[n]'); subplot(3,,3) %n = -5: 5; n2 = -2*pi: 2*pi/5: 2*pi; y2 = cos(2*n2); stm(n2, y2) axis([-2*pi 2*pi - ]); grid; Discrt Tim Fourir Sris - Charan Langton Pag 28

29 titl('(c) '); xlabl('radians'); ylabl('x[n]'); axis([-2*pi 2*pi -..]) % Dfin x-ticks and thir labls.. st(gca,'xtick',-2*pi: pi/5: 2*pi) st(gca,'xticklabl',{'-2pi', '', '-8pi/5', '', '', '', '-4pi/5', '', '', '', '', '', '', '', '4pi/5', '', '', '', '8pi/5', '', '2pi'}) %Chaptr 3 - Program 4 f=2; Fs = 6; t = :.: ; n = : Fs*t; n2 = : Fs xt=cos(2*f*pi*t); y = cos(2*f*pi*n2/fs) xt2= cos(2*5*pi*f*t); figur() plot(t*fs, xt, t*fs, xt2, 'r') hold on stm(n2, y, 'filld') %Chaptr 3 - Program 5 f=; Fs = 3; t = :.: 4; n = : Fs*t; n2 = : Fs*4 xt=cos(2*f*pi*t); y = cos(2*f*pi*n2/fs) figur() grid; plot(t*fs, xt, 'r') xlabl('sampl, n') ylabl('x[n]') hold on stm(n2, y, 'filld') %Chaptr 3 - Program 6 f=.5/pi; Fs = ; t = :.: 3; n2 = : Fs*3 Discrt Tim Fourir Sris - Charan Langton Pag 29

30 xt=cos(2*f*pi*t); y = cos(2*f*pi*n2/fs) figur() grid plot(t*fs, xt, 'r') xlabl('sampl, n') ylabl('x[n]') hold on stm(n2, y, 'filld') %Chaptr 3 - Program 7 n = :4; w = 2*pi/2; phas = ; A =.; HShift = 2; %chang this (vn numbrs only) to s ffct of shift x = A*cos((w+(HShift*pi))*n - phas); clf; stm(n,x, 'filld'); % Plot th gnratd squnc axis([ ]); grid; titl('sinusoidal Squnc'); xlabl('sampl n'); ylabl('x[n]'); axis; %Chaptr 3 - Program 8 n=-2:2; N=9; w=2*pi/n; axis([ ]); k=; Phin=xp(j*w*k*n); subplot(3,4,); stm(n,ral(phin),'markr','.');xlabl('') t= -: /8: ; plot(t, cos(w*k*t)) axis([ ]); k=; hold on Phin=xp(j*w*k*n); subplot(3,4,2);stm(n,ral(phin),'markr','.');xlabl('2') t= -: /8: ; plot(t, cos(w*k*t)) axis([ ]); hold off k=2; Phi2n=xp(j*w*k*n); subplot(3,4,3);stm(n,ral(phi2n),'markr','.');xlabl('3') axis([ ]); Discrt Tim Fourir Sris - Charan Langton Pag 3

31 k=3; Phi3n=xp(j*w*k*n); subplot(3,4,4);stm(n,ral(phi3n),'markr','.');xlabl('4') axis([ ]); k=4; Phi4n=xp(j*w*k*n); subplot(3,4,5);stm(n,ral(phi4n),'markr','.');xlabl('5') axis([ ]); k=5; Phi5n=xp(j*w*k*n); subplot(3,4,6);stm(n,ral(phi5n),'markr','.');xlabl('6') axis([ ]); k=6; Phi6n=xp(j*w*k*n); subplot(3,4,7);stm(n,ral(phi6n),'markr','.');xlabl('7') axis([ ]); k=7; Phi7n=xp(j*w*k*n); subplot(3,4,8);stm(n,ral(phi7n),'markr','.');xlabl('8') axis([ ]); k=8; Phi8n=xp(j*w*k*n); subplot(3,4,9);stm(n,ral(phi8n),'markr','.');xlabl('9') axis([ ]); k=9; Phi9n=xp(j*w*k*n); subplot(3,4,);stm(n,ral(phi9n),'markr','.');xlabl('') axis([ ]); k=; Phin=xp(j*w*k*n); subplot(3,4,);stm(n,ral(phin),'markr','.');xlabl('') axis([ ]); k=; Phin=xp(j*w*k*n); subplot(3,4,2);stm(n,ral(phin),'markr','.');xlabl('2') axis([ ]); % Chaptr 3 - Program 9 nmin = -; nmax = 9; ND = abs(nmin)+nmax+; n = nmin: nmax; x = *cos(2*pi*n/5) -.6*sin(2*pi*n/4); clf subplot(2,,) stm(n,x, '.'); pt = sum(x.^2)*/2 x titl('a') ylabl('x[n]') xlabl('sampl n') xnd = (/ND)*dft(x, ND); subplot(2,,2) Discrt Tim Fourir Sris - Charan Langton Pag 3

32 xnd2 = fftshift(xnd); stm(n, abs(xnd2)) ylabl('x(f)') xlabl('harmonic k') pf = sum(abs((xnd2.^2))) Titl('b') %Chaptr 3 - Program %Figur 2 a = [ ] d = [ ] xom = [ a a a a a a a d a a a a a a a ] n = : lngth(xom)-; N = 256; figur() stm(n, xom) titl('(d)') figur(2) X = fft(xom, N); plot(abs(fftshift(x))) w = 6*pi * (:(N-)) / N; w2 = fftshift(w); plot(w2) w3 = unwrap(w2-2*pi); plot(w3) plot(w3, abs(fftshift(x))) xlabl('radians') plot(w3/pi, abs(fftshift(x))) xlabl('radians / \pi') %Chaptr 3, Problm om = 2*pi/4; W = xp(-j*om) for n = :4 for k = : 4 m(n,k) = W^((n-)*(k-)) nd nd m x = [ 2 ]; (/4)*x*m % Chaptr 3 - Program 2 N = 5; K = ; n = -2:2; coff = (/K)*diric(n*2*pi/K, N); stm(n, coff, '.') Discrt Tim Fourir Sris - Charan Langton Pag 32

33 Discrt Tim Fourir Sris - Charan Langton Pag 33

34 Discrt Tim Signals and Fourir sris... Proprtis of discrt signals... Sampling of signals... Spcifying a discrt signal... 5 Discrt signal rprsntation... 7 Paramtrs of a discrt signal... 7 Discrt signal priodicity... 9 Basis functions for Discrt-tim Fourir sris... 3 Discrt Tim Fourir Sris - DTFS... 7 Figur Continuous and a discrt signal... 2 Figur 2 (a) A continuous signal, (b) sampld at 8 sampls pr scond and (c) sampld at 2 sampls pr scond Figur 3 Rplicating spctrums as a consqunc of sampling... 5 Figur 4 Discrt signal sampls... 6 Figur 5 Sampling of a continuous signal... 7 Figur 6 A discrt signal can rprsnt any numbr of continuous signals.... Figur 7 Discrt signal of xampl Figur 8 Non-priodic discrt signal of xampl Figur 9 Thr discrt harmonic signals... 5 Figur Discrt signals in btwn harmonic frquncis... 6 Figur Signal of xampl Figur 2- Signal of xampl Figur 3 Signal of xampl Figur 4 Signal of xampl 3-8 with N = 3, K = Figur 5 Cofficints of th priodic pulss, (a) with N= 3, K = 7, (b) N =3, K = 5, (c) N = 3, K = Discrt Tim Fourir Sris - Charan Langton Pag 34