Discrete-time Fourier transform (DTFT) of aperiodic and periodic signals

Transcription

1 5 Discrete-time Fourier trasform (DTFT) of aperiodic ad periodic sigals We started with Fourier series which ca represet a periodic sigal usig siusoids. Fourier Trasform, a extesio of the Fourier series was developed specifically for aperiodic sigals. I chapter 4 we discussed the Fourier trasform as applied to cotiuous-time sigals. Now we examie the applicatio of Fourier trasform to discrete-time sigals. We already discussed the discrete-time Fourier series (DTFS) as applied to periodic sigals i Chapter 3. Here the same ideas are applied to aperiodic sigals to obtai the Fourier trasform. Whether periodic or o-periodic, discrete-time sigals are the mai-stay of sigal processig. Sigals are collected ad processed via samplig, or by devices which are iheretly discrete. Despite the fact that sampled sigals loo lie their aalog parets, there are some major coceptual differeces betwee discrete ad cotiuous sigals. However, the fudametal cocepts of Fourier trasform discussed i Chapter 4 for cotiuous-time sigals apply equally well to discretetime sigals with oly mior differeces i scalig. Chapter 5 - Discrete-time Fourier trasform (DTFT) of aperiodic ad periodic sigals - C. Lagto Page

2 Figure 5. A aperiodic discrete-time sigal ca be cosidered periodic if period is assumed to be ifiitely log. Let s do the same thought experimet we did for cotiuous sigals. Give a piece of a discrete ad ostesibly aperiodic sigal such as i Fig. 5., we coceptually exted its period. The sigal xis [ ] just 5 samples, but we preted that the sigal is periodic with period N. But the we say that this period ca be very log, maybe eve ifiitely log. So if we exted the period of this sigal to, we basically get bac the origial sigal x [ ] which is ow surrouded by a sea of zeros. Lim x [ ] x[ ] (5.) N N As we icrease N, i limit the result is the startig sigal, but it ca ow be cosidered a periodic sigal, although oly i a mathematical sese. We ca t see ay of the periods. They are too far apart. Ad ow sice the sigal is periodic, we ca use the discrete-time Fourier series (DTFS) to write its frequecy represetatio i terms of complex coefficiets as N C Lim xn [ ] e N N j (5.) Discrete-time Fourier Trasform (DTFT) Recall that i Chapter 3 we defied the fudametal digital frequecy of a discrete periodic sigal as, with N N as the period of the sigal i samples. As N goes to Fourier Trasform of aperiodic ad periodic sigals - C. Lagto Page

3 ifiity, from this defiitio, the fudametal frequecy goes to zeros as well. Hece the harmoics which are defied as iteger multiple of the fudametal frequecy, such as also lose meaig. We ca also thi of the fudametal frequecy as the resolutio of the frequecy respose, so if this umber is zero, the the frequecy becomes cotiuous ad, the harmoic idetifier drops out etirely. Also the sigal itself from Eq. (5.) ca be writte as a periodic sigal x[], droppig all the limits etc.. Just as we discussed i Chapter 4, the result of extedig the period to ifiity leads to a frequecy respose which is cotiuous i frequecy, eve though the sigal itself is discrete i this case. Now we defie a ew trasform called the Discrete-time Fourier Trasform of a aperiodic sigal as j DTFT X( ) x[ ] e (5.3) Here x [ ] is a aperiodic discrete-time sigal. I Chapter 4 we defied the cotiuous-time Fourier trasform as give by j t CTFT X( ) x( t) e dt (5.4) Notice the similarity betwee these two trasforms. The CTFT X ( ), of the cotiuous-time sigal x(t) is also cotiuous i frequecy. The DTFT or X( ) is cotiuous i frequecy for the same reaso that the CTFT is cotiuous: due to the extesio of the period to. Both X( ) ad X( )are show with roud bracets for this reaso. The CTFT frequecy is termed whereas the digital frequecy i DTFT is give this symbol,. The digital frequecy as we leared i Chapter 3 is uique oly over a sigle rage. The DTFT, same as the CTFT, is a way of expressig the sigal x[] usig harmoic expoetials. The spectrum i each case are the relative magitudes of these harmoics. The cotiuous ad ifiitely log fuctios are great i textboos but impractical i real life. So a cotiuous time spectrum is actually ot a desirable result. What we wat is a discrete respose, which is far more practical. Discrete data is much easier to store, maage ad maipulate. As egieers worig with umbers, we wat a spectrum that is discrete ad oe which we ca compute i a discrete maer usig computers. However we are ot quite there yet. Chapter 5 - Discrete-time Fourier trasform (DTFT) of aperiodic ad periodic sigals - C. Lagto Page 3

4 Startig from the defiitio of DTFT i Eq. (5.3), we ow derive the time domai fuctio, x [ ] that resulted i this X( ) or the DTFT. We refer to this as taig the iverse DTFT. This is writte i short-had as idtft. To tae the idtft, j we multiply both sides of Eq. (5.3) by e ad the itegrate both sides over. I this case m is the dummy variable. Commeted [Ma]: X( ) x[ ] e j X( ) e d m j j m j x[ ] e e d (5.5) Now we cotiue to maipulate the RHS. (5.6) m m [ ] j m j x[ m] e e d j (m ) x m e d The last row is still complicated looig. But we ote that the uderlied part i j (m ) the last row is summatio of the complex expoetials (CE) e ad is i fact equal to shifted delta fuctio, (m ). Now ormally i most boos, this would be left as a exercise for the studet, but we tae you o a detour to j (m ) examie this poit. However, istead of showig that itegratio of e is j ( ) equivalet to a shifted impulse, we will show oly that itegratio of e is equal to a sigle impulse. From there we as you to extrapolate the result. I fact a sic fuctio i discrete time loos just lie a delta fuctio ad is i fact equivalet to a delta fuctio. A Sic detour Fourier Trasform of aperiodic ad periodic sigals - C. Lagto Page 4

5 e d e j j j e j j e j j j e e cos( ) j si( ) cos( ) j si( ) j j si( ) si c ( ) for all iteger (5.7) The sic fuctio is for iteger values of, except at =, whe its value is as we see i Fig. 5.. Hece the sic fuctio ca be equated to a delta fuctio for the discrete case. We geeralize this fidig of the sic fuctio to the followig shifted case. j ( ) e d ( ) (5.8) The secod expressio says that the summatio of time shifted CEs gives us a time shifted delta fuctio i discrete-time. Figure 5. - (a) The discrete sic, (b) loger versio of sic still loos lie a delta fuctio, (c) the cotiuous versio of the sic ad the discrete values, ad (d) the magitude of the sic fuctio. F5_ sic Fig. 5.(a) shows the discrete sic fuctio. For every value of, a iteger other tha, the sic fuctio is equal to. It loos suspiciously lie a delta fuctio Chapter 5 - Discrete-time Fourier trasform (DTFT) of aperiodic ad periodic sigals - C. Lagto Page 5

6 here. As we icrease, we see i (b) it eeps looig lie a delta fuctio. This equivalece to a delta fuctio is i fact ot a fuctio of the legth. I (c) we plot the cotiuous values of the sic fuctio alog with the discrete values. For values of, ot itegers, we ow get o-zero values which shows us what the fuctio loos lie i cotiuous-time. The delta fuctio loo of the sic fuctio is a form of discrete deceptio. The discrete versio is picig up oly the values at certai poits which are all zero. Noe others are computed or ca we see them. So for all practical purposes, a discrete versio of the sic is a delta fuctio. I (d), we see the same cotiuous fuctios as i (c) but with its absolute value, as it is typically show i boos. Also ote that the sic fuctio is crossig the zero axis at iteger values of. So we ca write the x axis i terms as show i (d). Bac to the DTFT Substitutig Eq. (5.8) ito Eq. (5.6), we get j m X( ) e d x[] ( m ) From the siftig property of the delta fuctio, the right had side becomes x [m], ad sice this ot a fuctio of, we ed here. Hece the iverse DTFT of Eq. (5.3) is the time domai fuctio x [ ]. idtft j x[ ] X( ) e d (5.9) DTFT X x[ ] e j The forward trasform or the DTFT is deoted by symbol X. However, you will fid other ways of deotig the DTFT. Oppeheimer boo refers to it as j X e, whereas both Miral boo ad the Lathi ad Gree boos refers to it by X. These otatios are basically covetio ad ot that importat. I speaig, most all of these forms are referred to as simply as the Fourier Trasform or eve the more geeric spectrum. Ad eve more egregiously some people call this a FFT, which it may or may ot be. Sice most sigals we deal with i practice are discrete, the time qualifier is dropped ad we ca just call these as the Fourier trasform. However, i this boo we will cotiue to refer to each type of trasform by its full formal ame, CTFT, DTFT, DFT etc. Fourier Trasform of aperiodic ad periodic sigals - C. Lagto Page 6

7 Magitude ad phase spectrums The DTFT X is geerally a complex fuctio. We ca show it i two ways, just as all the other Fourier represetatios, either as its real ad imagiary compoets which are the coefficiets of the cosie ad sie harmoics or we ca show them by magitude ad phase. Both methods are ofte employed as eeded. X X( ) e Magitude j X Phase For real sigals, the magitude is symmetrical (or eve) ad the phase is odd. DTFT is cotiuous ad periodic with period of So first we say that the period of a sigal is assumed to be ifiitely log ad ow we are sayig that the DTFT is periodic. How ca that be? Well what we mea by that is the spectrum i.e. X( )repeats with. This tal of a frequecy that is measured i ' s ca be very cofusig. But we must accept the fact that the DTFT is defied i terms of the digital frequecy ad ot Hz. The sigal cosists of discrete values ad hece ca oly be represeted by N harmoics, the legth of the sigal ad othig else. I order to mae the aalysis idepedet of real time, i.e. the time betwee the samples, DTFT is istead defied i terms of radial movemet. This, if you trust us, maes the math, easier (ha!). Ulie cotiuous frequecy, discrete frequecy,, with uits of radias per sample, lacs a time dimesio. It is periodic oly i a agular sese. It is uique for values i oly oe rage. We should ot thi of it as umber of samples or, or time it taes to cover radias because that is just ot part of its defiitio. The rage of the digital frequecy ad the spectrum computed thereof is called the pricipal alias as we oted i Chapter 3. Because of this coditio, the coefficiets for harmoic frequecies outside to are just copies. Hece there is o eed to compute X( ) outside the rage. Aythig beyod that just repeats the same values from the rage, or i fact from ay such rage. We ca igore all these replicated spectrums as they are idetical to the pricipal alias. We write this property as X( ) X( m ) for all i rage of,, m a iteger (5.) This comes from the simple observatio that Chapter 5 - Discrete-time Fourier trasform (DTFT) of aperiodic ad periodic sigals - C. Lagto Page 7

8 X( m ) x[ ] e j( m) x e e X j j m [ ] ( ) Every, X ( ) is idetical to the oe before. This property simplifies the computatio as we eed oly itegrate over a rage of the digital frequecy. Sice the area uder a periodic sigal for oe period does ot chage o matter where you start the itegratio, we ca geeralize the DTFT equatio over ay rage. We ca for example write the equatio for the idtft i the secod maer, with itegratio rage writte as just, ad both are valid. j x[ ] X( ) e d j X( ) e d Comparig CTFT with DTFT Let s examie how CTFT compares to the DTFT for a aperiodic sigal. Figure Comparig CTFT with DTFT (a) aperiodic CT sigal, (b) its CTFT is cotiuous, (c) a sampled discrete sigal (d) is same as (b) but repeats with. Both CTFT ad DTFT have a very similar costruct. Assume that the CTFT of the sigal show i Fig. 5.3(a) is as show i (b). We see a sigle spectral mass Fourier Trasform of aperiodic ad periodic sigals - C. Lagto Page 8

9 aroud zero frequecy with a cotiuous frequecy resolutio. Now tae the same sigal i discrete form (c) ad it has a DTFT that is cotiuous just as the CTFT, but this oe repeats with radias. This is the same result we showed for discrete-time Fourier series. So ote that DTFS ad DTFT are very similar. We will ow show some examples of the DTFT. I these examples, we compute oly the pricipal alias which is the DTFT aroud the zero frequecy, from to. However, we must ot lose sight of the fact that the DTFT spectrum copies go o forever o each side of the pricipal alias, as we see coceptually i Fig. 5.3(d). The DTFT has the same behavior as the CTFT i most cases whe the sigal is badlimited to. The CTFT properties show i Chapter 4, Table I are equally valid i coceptual sese for discrete sigals. These properties ca be used to compute the DTFT for may sigals, startig with the owledge of the DTFT of some of the basic sigals. We ca i most cases tae a CTFT equatio, chage the cotiuous frequecy to digital frequecy otatio ad the chage cotiuous time t to discrete time otatio ad get a valid expressio for the DTFT. However, what we get this way is oly the pricipal alias because CTFT does ot repeat. We must recogize that DTFT repeats forever. The DTFT is a bridge topic to get us to the Discrete Fourier trasform (DFT), a widely employed ad a very useful algorithm. DFT is discrete i both time ad frequecy domai ad ca be calculated easily by software such as Matlab. The Fast Fourier Trasform (FFT) was developed to mae computatio of the DFT quic ad efficiet. It is just a computatio algorithm ad ot a uique type of Fourier trasform. The DTFTs for most sigals other tha a few simple oes you see i text boos are hard to compute, requirig oe to pull out itegral tables. Nor are they commoly used i real-life egieerig. So why bother with DTFT if the subject is so theoretical? The mai reaso is that util we uderstad DTFT, we caot fully appreciate the DFT. Ehe leared as a stad-aloe topic, the DFT maes sese oly i a procedural sese but oe lacs deeper uderstadig of where it is comig from. DTFT from CTFT DTFT ca be obtaied directly from a CTFT. Let s compute the DTFT of a sigal whose CTFT we ow. Chapter 5 - Discrete-time Fourier trasform (DTFT) of aperiodic ad periodic sigals - C. Lagto Page 9

10 x( t), X( ) ( ) x[ ], X( )? This is a trivial case. By maig the appropriate chages, we get X( ) ( ) for This is the pricipal alias oly. The complete DTFT repeats, just as we have repeated ourselves edlessly, so we exted the above expressio to X( ) ( ) for all For each, we get a impulse at frequecy, so this says that the spectrum of a costat discrete sigal is ever repeatig impulses at iteger multiples of. DTFT of a delayed impulse The delayed impulse x[ ] [ ], is a very importat sigal. Nearly all discrete sigals ca be decomposed as a summatio of this sigal. Agai we ca tae the CTFT of a delayed impulse ad chage the terms to their discrete equivalets but istead we will do the math for this case usig the DTFT equatio. We compute the DTFT of a delayed uit-impulse fuctio, x[ ] [ ] usig the DTFT Eq. (5.6). X[ ] [ ] e j The product of fuctios, [ j ] ad e is o-zero oly at poit, so we simplify the RHS as X[ ] e j j Sice e is ot a fuctio of, we ca igore the summatio ad the DTFT is simply equal to x[ ] ( ) X[ ] e j (5.) The magitude of this trasform is equal to Fourier Trasform of aperiodic ad periodic sigals - C. Lagto Page

11 X e j j [ ] cos( ) si( ) So matter what the shift, the magitude remais the same. The phase however is X si( ) ( ) ata cos( ) Ad will chage with the shift. This result is exactly the same as if we had applied the time-shift property to a zero-shift delta fuctio. The time shift property states that the Fourier trasform of a shifted sigal is equal to the Fourier trasform of the u-shifted sigal times a CE of frequecy, which is exactly what we have here. If the shift is equal to, the we get x[ ] ( ) X [ ] e j (5.) The trasform of the u-shifted delta sigal is of course as we see i Figure 5.(b) ad we ca see that the DTFT of this sigal is a purely cotiuous fuctio of. If =, we get x[ ] ( ) X e j j [ ] cos( ) si( ) I figure 5.4, we see the effect of the delay o the trasform of the delayed fuctio, with o chage i magitude but the phase chage by 4, with phase delay per sample delay. We see a total of 4 phase travel over the rage i (f). Chapter 5 - Discrete-time Fourier trasform (DTFT) of aperiodic ad periodic sigals - C. Lagto Page

12 Figure Comparig X ( ) of a ushifted ad shifted impulse. F5_54Impulse Liear superpositio of impulses We ow compute DTFT of a discrete sigal that combies several shifted impulse fuctios. x[ ] [ ] [ ] 4 [ ] We treat each oe of these delta fuctio idividually by applyig the liearity pricipal. X( ) [ ] e [ ] e 4 [ ] e e 4e j j j j j cos( ) j si( ) cos( ) si( ) cos( ) cos( ) jsi( ) jsi( ) real Img Note that sice the digital frequecy, has uits of radias, we do ot have a time variable to go alog with it. Fourier Trasform of aperiodic ad periodic sigals - C. Lagto Page

13 F5_55DTFT of sampleseq Figure 5.5 DTFT of (a) the discrete aperiodic sigal, (b) Pricipal alias, (c) its phase ad (d) the real repeatig DTFT. We ote that the DTFT is cotiuous ad repeats with. The spectrum show covers 3 periods, the rest are all there, outside the bouderies of the plot. We do t show them but they are ideed there. Let s tae a loo at aother way of computig the DTFT of the shifted impulses of this sigal. x[ ] [ ] 3 [ ] 5 [ 4]. Here istead of doig the math, we will apply the time-shift property to each of these delta fuctios. The DTFT of a delayed delta fuctio is a CE of frequecy, j e as per result from DTFT of a delayed impulse. From that, we write the DTFT of this composite fuctio easily, as the summatio of the idividual terms. Each of the delta fuctios correspods i frequecy domai to a frequecy of the delay as i Eq. (5.). Oce agai, ote that the uits of the digital frequecy are radias as the x-axis of the spectrum is usually give i terms of. Chapter 5 - Discrete-time Fourier trasform (DTFT) of aperiodic ad periodic sigals - C. Lagto Page 3

14 x[ ] [ ] 3 [ ] 5 [ 4] X[ ] e 3e 5e j j j4 Figure (a) The discrete aperiodic sigal, (b) the repeatig DTFT, with copies show. F5-Ex3 DTFT of a square pulse Square waves are very useful as a model of real sigals. What you lear from these sigals, you ca the geeralize to early all shapes. Let s start with a square pulse of width N discrete samples. However, ote that sigal legth is ot N, it is loger ad i fact is it ot ifiitely log? Hece the parameter N has othig to do with the legth of the sigal. It is just the width of the pulse itself ad ot the legth of the period that matters. Note that i this example, the square pulse is cetered at. We assume that N is odd. We defie this fuctio as x [ ] ( N ) ( N ) Elsewhere We compute the DTFT as ( N )/ j X( ) x[ ] e e ( N )/ N si si j (5.3) Fourier Trasform of aperiodic ad periodic sigals - C. Lagto Page 4

15 The result i the last row is the Drichlet Fuctio. (Peter Gustav Lejeue Drichlet, a Germa mathematicia) We ca also write the result as follows. X( ) Diric( N, ) (5.4) Now we plot the DTFT usig Eq. (5.4) for various values of N, which is the width of the square pulse i samples. The absolute value of the Drichlet fuctio is plotted vs. the true value i the RHS of Fig The legth of the sigal i the LHS is samples i each case. Ca you say what would happe to the DTFT o the RHS, if we icrease the legth of the sigal from samples to samples. Actually othig would chage, we would get exactly the same fuctio. DTFT is ot a fuctio of the total umber of samples beyod the pulse. The DTFT as we ca see i (5.4) is a fuctio of oly the umber of samples of the square pulse or N. That s because the formulatio of the DTFT already assumes that zeros o the sides go o forever. Figure 5.7 A pulse of legth N = 3, 5, 7 samples ad its spectrum F5-squarewaves Drichlet detour The Drichlet fuctio i the DTFT of a aperiodic square pulse is ofte called the periodic versio of the sic fuctio. These two are pheomeally importat fuctios i sigal processig. Chapter 5 - Discrete-time Fourier trasform (DTFT) of aperiodic ad periodic sigals - C. Lagto Page 5

16 si( ) si cx ( )= drichlet( x) si( N ) Nsi( ) Figure Sic ad the Drichlet fuctio We plot both of these fuctios i Fig The Sic fuctio i the top row is cotiuous ad is aperiodic. The Drichlet i the secod row appears similar to the Sic ad but is periodic with, for N = odd ad with a period of 4 whe N is eve. (We do t see this i Fig. 5.8, because the plot cotais absolute values so all the lobes are o the positive side.) Let s examie the Drichlet fuctio i a bit more detail. Fig. 5.9 shows the behavior of the Diric fuctio (the Matlab versio of the Drichlet) as a fuctio of digital frequecy. Recall that uits of digital frequecy are radias. We see that the umber of zero crossigs i the rage of, are equal to N. For N = 5, we see 4 zero crossigs, for N = 9, we see 8 zero crossig. The fuctio is o-zero oly at. O RHS, we see the same fuctio over a loger rage of digital frequecies plotted alog with a sampled-discrete versio. The discrete versio of this sigal is iterestig. It loos lie a impulse trai just as did the sic fuctio for a sigle impulse. Where the sic fuctio loos lie a sigle impulse whe sampled, Drichlet loos lie a impulse trai, with impulses preset every. The discrete versio is show by the dots. The Diric fuctio crosses zeros at all frequecies equal to Fourier Trasform of aperiodic ad periodic sigals - C. Lagto Page 6

17 ( m/ N ) where N is the order of the diric fuctio as i Matlab diric(f, N). Hece for N = 5, the zeros occur at 5, 4 5, 6 5, 8 5,., for N = 6, the zeros occur at 6, 4 6,, 8 6,. etc. Figure The Drichlet fuctio (a) N = 5, (b) N = 7, (c) N = 9. The x-axis is i radias. The zero crossigs occur at / N. F5_55Driric for N567 Applyig time-shift property to the DTFT of a square pulse What is the DTFT of a square pulse, whe ot cetered at? We ca thi of this as a square pulse located at zero frequecy but with a time-shift. Kowig the time-shift property is a very hady thig. The aalysis is same as i u-shifted case, except we are goig to add a time shift. We assume that the pulses are cetered at L samples from the origi. The time-shift is L uits. The DTFT ca ow be writte from the time shift property as simply the DTFT times the CE of j L frequecy per Eq. (5.) e as follows Chapter 5 - Discrete-time Fourier trasform (DTFT) of aperiodic ad periodic sigals - C. Lagto Page 7

18 j L X[ ] e X[ ] e j L N si si udelayed (5.5) I Fig. 5.8, we plot DTFT for two pulse widths, N= 3 ad, N = 5. Each with shift, L = samples. O the LHS, we see the u-shifted square pulse, o the RHS, the shifted versio. We see from Eq. (5.5) that the magitude of the DTFT did ot chage, the shift results i o chage i the magitude of the DTFT, oly the phase. We see the same thig for N = 5 i Fig. 5.9 Figure 5.8 Time shift property for N = 3 Figure Time Shift property for N = 5 For both cases ad i fact all cases, a time shift shifts the phase but the magitude stays the same. Fourier Trasform of aperiodic ad periodic sigals - C. Lagto Page 8

19 Time expasio property Let s tae this sigal which loos lie it has zeros iserted i a 3 sample square pulse, x []. We ca write this discrete sigal as x () ( 3) ( 6) There are may differet ways of computig the DTFT of such a sigal. Here we apply the time-expasio property to show a more efficiet method. The pulse of N = 3 has bee expaded by a factor of 3 by isertig these zeros. We write the time-expasio property as x( at) X a a (5.6) The DTFT of a square pulse for N = 3, whe iterpolated with two zeros shris by a factor of 3. We ca of course compute the DTFT directly as we did for the shifted impulse case. This is equal to si 3 N X ( ) (5.7) 3 si 6 We see that addig zeros betwee the samples, expads the sigal but compresses the DTFT. I Fig. 5.9, we see this effect as more zeros are added. What happes if N goes to ifiity? The oly oe delta fuctio is left, ad the DTFT will tur ito a flat lie. Figure 5. - Time expasio property Chapter 5 - Discrete-time Fourier trasform (DTFT) of aperiodic ad periodic sigals - C. Lagto Page 9

20 DTFT of a triagular-shaped pulse A triagular pulse is early as importat i sigal processig as the square pulse. It is the covolutio of two rectagular pulses, somethig which comes up ofte. We write the triagular pulse as x[ ] N N The pulse is N samples wide ad symmetrical. The DTFT is computed as X[ ] x[ ] e N j N cos( ) N si ( N ) Nsi ( ) e e N j j (5.8) The result is a fuctio that is the Drichlet fuctio squared. Figure 5. - A triagular shape pulse has a squared Diric sigal respose. Fourier Trasform of aperiodic ad periodic sigals - C. Lagto Page

21 We could have also computed the DTFT of a triagular fuctio by applyig the covolutio property. We recogize, that a triagle pulse is the result of a covolutio of two idetical rectagles. So we write the pulse as a covolutio. x[ ] rect * rect N N (5.9) The DTFT of this covolutio is the product of the DTFT of the idividual square pulses. From that we get N si X [ ] (5.) si So owig the properties ca mae the tas of computig FTs easier i may cases. Computig the DTFT of a Raised cosie pulse The ubiquitous raised cosie pulses are used to trasmit commuicatios sigals. They limit the badwidth of the basebad sigal ad are easily built i hardware. F s F s cos si T s T s p [ ] F F s s T T s s (5.) Here Fs is the samplig frequecy, is a real umber less tha oe ad is called the roll-off factor, T s is the iverse of symbol rate R s. The first part is called the raised cosie ad the secod part which is the sic fuctio is called the cascaded sic applied to the raised cosie pulse. If alpha =, we get a ideal rectagular shape, ad if alpha =, we get a pure raised cosie shape. These parameters set the basebad badwidth of the sigal as BW R ( ) (5.) s To compute the DTFT of this pulse we will have to resort to some heavy-duty math. But o eed, as it has already bee doe for us by better mids. Here is the equatio that gives us the CTFT of the above good looig pulse. Chapter 5 - Discrete-time Fourier trasform (DTFT) of aperiodic ad periodic sigals - C. Lagto Page

22 Ts f Ts Ts T s P( f ) cos f f Ts Ts Ts f Ts (5.3) We plot the time-domai sigal ad its DTFT i Fig. 5.. It loos very similar to a sic fuctio. Although this pulse goes o forever, for practical desig, it is clipped to a certai legth. Figure 5. - (a) Time domai root-raised cosie pulse shape (b) The spectrum of the raised cosie pulse for.5,.33,.5,.5 Note that frequecy domai loos lie a low pass filter. DTFT of a Gaussia pulse The discrete-time versio of the Gaussia sigal is give by x[ ] e (5.4) To compute the DTFT of this sigal, beig good egieers we are goig to agai sip the math. Others have already doe it for us. The DTFT of this Gaussia shaped pulse, is also Gaussia i shape. You recogize why that happes; because the pulse is a expoetial ad the itegral of such a fuctio is also a expoetial. The result is beautiful ad elegat ad a very useful thig to ow. May radom sigals are Gaussia i ature. Life is Gaussia ad it is Gaussia i all its dimesios. Fourier Trasform of aperiodic ad periodic sigals - C. Lagto Page

23 X( ) e (5.5) Figure DTFT of a Gaussia pulse for =. Note that all values are well cotaied withi 3 = 6. Note that we would have to sample the Gaussia fuctio by 4 times the maximum frequecy i order to avoid sigificat aliasig. The reaso is that this fuctio has o obvious maximum frequecy ad o matter what we choose, the sigal will still cotai frequecies higher tha that umber albeit i low amplitudes. The alterate is to first filter the sigal with a ati-aliasig filter before doig the DTFT. Now the DTFT of periodic sigals All of the sigals we looed at so far i this chapter were aperiodic, pulses stadig aloe. But what about discrete sigals that are periodic? We have a trasform for these as well ad this is a yet oe more type of Fourier trasform. We call it the DTFT of periodic sigal. The DTFT of periodic sigals is our most importat type of Fourier Trasform. Not because periodic sigals are so importat but because, the DTFT for periodic sigals turs out to be discrete. This is our real goal. We wat a discrete spectrum! We stated that as our goal. The DTFT of periodic sigals, whe modified slightly for fiite legth sigals, gives the Discrete Fourier Trasform (DFT), the most used form ad for which the well-ow Fast Fourier Trasform algorithm was writte. It too us a lot of pages i this boo ad s of years of history to get to this importat poit. However, we are ot quite there yet. Let s tae a periodic but discrete-time sigal with a period of N samples ad write its discrete Fourier series equatio. Note we did ot tal about a period whe discussig DTFT of aperiodic sigals, but we Chapter 5 - Discrete-time Fourier trasform (DTFT) of aperiodic ad periodic sigals - C. Lagto Page 3

24 will ow. Period ow becomes relevat because these sigals are periodic, so they have a period! Ad wheever, we have a period, the frequecy resolutio becomes discrete. However to derive a trasform for periodic discrete sigals, we have to go bac to discrete-time Fourier series as out startig poit. The Fourier series is writte i form of Fourier series coefficiets for discrete-time sigals as follows. (See chapter 3) j x[ ] C e (5.5) N Where N is the digital frequecy of the discrete sigal ad N is the period of the sigal. The coefficiets of the harmoics are give by C N N N x[ ] e j (5.6) Sice ow we have N samples of a periodic sigal, we ca ideed compute these coefficiets. Let s tae the DTFT of Eq. (5.6). X( ) C C e N N j j e (5.7) The coefficiets are ot a fuctio of frequecy, so they are pulled out i frot. The DTFT of the uderlied part, a summatio of complex expoetials is a trai of impulses. j e m N m ( ) Substitutig this expressio ito Eq. (5.7), we get the equatio for the DTFT of a periodic sigal. X( ) C N (5.8) Fourier Trasform of aperiodic ad periodic sigals - C. Lagto Page 4

25 It may ot be obvious here but the DTFT of a periodic discrete sigal repeats the DTFS coefficiets, C at every iteger multiple of the digital frequecy. That s what the secod part, the impulse trai is doig. This formulatio is quite differet from the DTFT of a aperiodic sigal which we computed i Eq. (5.6) ad repeat here. j X( ) x[ ] e (5.9) The DTFT i Eq. (5.8) for a periodic discrete-time sigal tells us that the DTFT of a periodic sigal cosists of its DTFS coefficiets repeated every N samples. Sice N is a fiite umber, the period of the sigal, the samples are discrete ad o loger cotiuous as they are for a aperiodic case. The spectrum is ow discrete. Just what we lie! Who wats to do itegratio whe we have computers. Repeatig this importat fact agai: The DTFT of both the aperiodic ad the periodic sigal repeats. Looig at the spectrum, it appears as if the DTFT of a periodic sigal is a sampled versio of the DTFT of a aperiodic sigal. Because of this, this case has come to be called the Discrete Fourier Trasform, ow by the acroym DFT ad ofte thought of as a sampled versio of the DTFT. I Fig. 5.4 we see the compariso of the CTFT ad DTFT for a periodic sigal. The CTFT of a cotiuous-time periodic sigal is discrete but o-repeatig. The DTFT for a discrete sigal sampled at some frequecy Fs, is discrete, however, it repeats at the samplig frequecy Fs. Chapter 5 - Discrete-time Fourier trasform (DTFT) of aperiodic ad periodic sigals - C. Lagto Page 5

26 Figure Comparig CTFT ad DTFT for periodic sigals Sice the DTFT of a periodic sigal is repeatig DTFS coefficiets, here we give a table of the DTFS coefficiets for commo sigals. Kowledge of these maes computig the DTFT of periodic sigals easy. Time domai sigal x [ ] x[ ] [ ] Impulse at 3 x[ ] [ ] A shifted impulse 4 x[ ] [ mn ] m This is a impulse trai of period N. j x[] e A periodic complex expoetial, with N 5 6 Cosie, periodic 7 Sie, periodic Table I DTFS ad DTFT of commo fuctio DTFS C N N N x[ ] e j Does ot exist DTFT Does ot exist j e N mn elsewhere mn elsewhere j mn elsewhere X( ) x[ ] e N N N j j m j Now we loo at a very basic discrete-time sigal that is periodic. As we shall see, the DTFT istead of beig cotiuous is discrete. Fourier Trasform of aperiodic ad periodic sigals - C. Lagto Page 6

27 DTFT of a costat periodic fuctio x[ ] ; Figure 5.6 DTFT of a costat discret sigal What we have for x [ ] is a impulse trai of costat amplitudes give by sample umber. Hece we ca write such a sigal as x[ ] [ ] X( ) ( ) (5.3) Recall from Chapter 4 that the amplitudes of the DTFT, is ormalized value over oe cycle ad the real amplitude of the sigal. The DTFT amplitudes just as the CTFT amplitudes are accurately related oly to each other. Specifyig thigs i digital frequecy is actually quite cofusig. Istead of specifyig the pulse trai by the sample umber, we state that the time betwee each pulse is equal to T, how does that chage Eq. (5.3)? Each sample is T secods apart, which is the real-time period of this sigal. We ca covert the digital frequecy ito real frequecy by dividig by this period as show i Fig The uits become radias per secod. Chapter 5 - Discrete-time Fourier trasform (DTFT) of aperiodic ad periodic sigals - C. Lagto Page 7

28 Figure DTFT of periodic pulse trai A alterate way to thi about the frequecy-domai impulse trai is to see it as a set of ifiite umber of impulse pairs cetered at the origi. The DTFT of each pair correspods to a cosie. Hece we have a ifiite umber of cosies, each located at a iteger multiple of the fudametal frequecy, of ever icreasig frequecy. DTFT of a sie wave Here is a ultimate studet-friedly fuctio, discrete but periodic. Assume its frequecy is. x[ ] si( ) The DTFT of this fuctio is computed as follows. First we write the siusoid as a sum of expoetials, the sice we ow that the DTFT of the sum of expoetials is a impulse trai, we apply the result from Example 5.. x[ ] e e j j j j Now we write the DTFT as X( ) ( m) ( m) j m j m Clearly these are impulse trais, with frequecy ad the repeatig with. We draw them i Fig Each pair of impulses is located aroud iteger multiple of the frequecy of the siusoid. The the whole thig repeats with. Figure DTFT of a sie wave Fourier Trasform of aperiodic ad periodic sigals - C. Lagto Page 8

29 Summary of the Fourier trasform types I Fig. 5.9, we summarize four versios of the Fourier trasform, for both periodic ad aperiodic sigals with both cotiuous ad discrete time. Each has a uique behavior, however sharig may of the properties with each other. Periodicity -> Timeresolutio Cotiuoustime (CT) Discrete-time (DT) Aperiodic CTFT DTFT Cotiuous Frequecy resolutio Spectrum does ot repeat. Cotiuous Frequecy resolutio Repeatig spectrum with pi. Periodic CTFT DTFT Discrete Frequecy resolutio Spectrum does ot repeat. Discrete Frequecy resolutio Repeatig spectrum with samplig frequecy. Figure 5.9 Four versios of the Fourier trasform Why is the Fourier series ot metioed i this figure? The reaso is that Fourier trasform ca deal with all of these types of sigals, both periodic or o-periodic, as well as discrete or cotiuous-time sigals. These various versios of the Fourier trasform ca be used to accomplish the same thig as the Fourier series ad more. The series are redudat to the periodic versios of the Fourier trasform so i our educatio they have served their purpose as a startig poit. We ca put them away ow ad move o. Table II - DTFT of commo sigals Sigal, x [ ] DTFT, X [ ], X[ ] sg[ ] e j Chapter 5 - Discrete-time Fourier trasform (DTFT) of aperiodic ad periodic sigals - C. Lagto Page 9

30 u [ ] e [ ], [ ],,,... j e j a [ ] b [ ] j ae be a u[ ], a ae j e, real Square pulse of width, cetered at = Square pulse of width, cetered at = j j si si si si e j cos cos j e e j si si j e e j Fourier Trasform of aperiodic ad periodic sigals - C. Lagto Page 3

31 Summary of Chapter 5. The DTFT of a discrete-time aperiodic sigal is developed by assumig that the period of the discrete pulse is ifiitely log.. Because the period is presumed very log, the frequecy resolutio approaches zero, hece the DTFT, specified by X ( ), becomes a cotiuous fuctio of frequecy. 3. The DTFT of a aperiodic sigal is a fuctio of the digital frequecy, which is uique oly i rage. The DTFT computed aroud the frequecy is called the pricipal alias. 4. We eed to compute the DTFT oly i this rage as DTFT i all other frequecy rages are idetical to the pricipal alias. 5. The DTFT is give by X( ) x[ ] e j The frequecy is cotiuous. 6. The idtft is give by j x[ ] X( ) e d 7. The DTFT of a aperiodic discrete sigal is cotiuous ad repeatig. 8. The DTFT of a periodic discrete sigal is give by X( ) C N 9. The DTFT of a periodic discrete sigals is the discrete-time Fourier series coefficiets, C repeatig at the samplig frequecy of the sigal.. DTFT is most similar i behavior to discrete-time Fourier series, DTFS. The amplitudes however are ormalized i DTFT.. The DTFT of a discrete periodic sigal is discrete, with frequecy resolutio of / N with N equal to the samples per period.. DTFT leads us to the Discrete Fourier Trasform (DFT) which ca be used for fiite legth sigals. Fourier Trasform of aperiodic ad periodic sigals - C. Lagto Page 3

32 Copyright 5 Chara Lagto All Rights reserved Fourier Trasform of aperiodic ad periodic sigals - C. Lagto Page 3