EEL: Discrete-Time Sigals ad Systems The Discrete-Time Fourier Trasorm (DTFT) The Discrete-Time Fourier Trasorm (DTFT). Itroductio I these otes, we itroduce the discrete-time Fourier trasorm (DTFT) ad explore some o its properties.. Discrete-Time Fourier Trasorm (DTFT) A. Itroductio Previously, we have deied the cotiuous-time Fourier trasorm (CTFT) as, X () xt ()e jπt dt () where xt () is a cotiuous-time sigal, X () is the CTFT o the cotiuous-time sigal, t deotes the time variable (i secods), ad deotes the requecy variable (i Hertz). The discrete-time Fourier trasorm (DTFT) is deied by, Xe ( j ) e j where is a discrete-time sigal, Xe ( j ) is the DTFT o the discrete-time sigal, deotes the time idex, ad deotes the requecy variable. Comparig the deiitios or the CTFT ad the DTFT, we observe the ollowig diereces. For the DTFT, the itegratio over time t has bee replaced by a summatio over the time idex ; also, the requecy variable is uitless. I order to relate the requecy variable to a real requecy, we eed to kow the samplig requecy s. That is, i the discrete-time sequece represets a sampled cotiuous-time sigal, sampled at requecy s, the ad the correspodig real requecy are related as ollows: () s π ------, -------. () π s B. Aalytic example We begi our study o the DTFT by lookig at the DTFT o a simple discrete-time sigal, amely a pulse o width M + cetered at : M > M () While i geeral it is much more diicult to derive a aalytic expressio or the DTFT tha or the CTFT, or the example sigal i () it is possible to derive a closed-orm expressio. Applyig deiitio (), M Xe ( j ) e j M M Xe ( j ) e j( M) M Xe ( j ) e jm e j () () (7) - -
EEL: Discrete-Time Sigals ad Systems The Discrete-Time Fourier Trasorm (DTFT) Xe ( j ) ejm e j M ------------------------------------ ( + ) e j From (7) to (8) above, we used the ollowig iite geometric sum idetity: (8) N α α -------------------------- ( N + ), α. (9) α where, or our case, α e j ad N M. () Equatio (8) ca be urther simpliied by symmetrizig the powers o the expoetials i the umerator ad deomiator: Xe ( j ) ejm e j ( M + ) ( e j( M + ) e j( M + ) ) -------------------------------------------------------------------------------------------------------------- e j ( e j e j ) Xe ( j ) ejm e j M --------------------------------- ( + ) e j( M + ) e j( M + ) e j ----------------------------------------------------------------------- e j e j Note that, ejm e j ( M + --------------------------------- ) e j () () () so that equatio () reduces to: Xe ( j ) e j( M + ) e ----------------------------------------------------------------------- j( M + ) e j e j e j( M + ) e ----------------------------------------------------------------------- j( M + ) j --------------------------------- j e j e j () Xe ( j ) si -- ( M + ) -------------------------------------- si -- () I Figure below, we plot Xe ( j ) or a pulse with M. Note that this DTFT appears to be periodic i the requecy variable with period T π. This is ot oly the case or this example, but or the DTFT o ay discrete-time sigal. The periodic property o the DTFT is easily show by goig back to the deiitio i equatio (): Xe ( j( + π) ) e j ( + π ) e jπ e j Note that or iteger, so that: e jπ e jπ Xe ( j ) () (7) Xe ( j( + π) ) Xe ( j ). (8). Strictly speakig, equatio (9) is ot valid or ± π, {,,, }, sice α or those values o. For these values o, Xe ( j ) M +. Usig L Hopital s Rule, this is, however, precisely the limitig value o the ial expressio or Xe ( j ) i equatio (). Thereore, we do ot treat this case separately. - -
EEL: Discrete-Time Sigals ad Systems The Discrete-Time Fourier Trasorm (DTFT) M 8 Xe ( j ).. - - - - - - 8 Xe ( j ) 8 Xe j ( ) - - - - - - - Figure Hece, the DTFT o ay discrete-time sigal is periodic i wit period T π. This result is closely related to our previous discussio o samplig ad aliasig as we shall see shortly. Now, let us see how the DTFT i equatio () chages as a uctio o the pulse width parameter M. I Figure, we plot ad Xe ( j ) or M 8,,,,. Note the strog similarity o Figure below, ad Figure o page 7 o the Fourier Series to Fourier Trasorm otes. For the arrowest pulse ( M, δ[ ] ), the requecy cotet is uiormly distributed or π π, while or the wider pulses, the requecy cotet becomes more ad more cocetrated aroud. This is the same pheomeo as or the CTFT (Figure, Fourier Series to Fourier Trasorm otes).. Some additioal DTFT illustratios Below, we explore additioal examples o the DTFT. A. Fiite-legth cosie wave Here we cosider the DTFT o a discrete-time sigal, sampled rom the cotiuous-time sigal x c () t, x c () t cos( πt) (9) with samplig requecy s Hz or a total o samples, such that, x c ( s ) {,,, 8, 9}. () elsewhere Applyig deiitio (), 9 Xe ( j ) x c ( s )e j () I Figure, we plot Xe ( j ) ad Xe ( j ) or the discrete-time sequece i () as a uctio o the dimesioless requecy variable, ad as a uctio o the real requecy variable (i Hertz), where we make - -
EEL: Discrete-Time Sigals ad Systems The Discrete-Time Fourier Trasorm (DTFT) M Xe ( j )... - -. - - - M Xe ( j ).... - - - - M - - - - - - - - Xe ( j ) M 8 Xe ( j ).. - - - - - - M 8 Xe ( j ).. - - - - - Figure - -
EEL: Discrete-Time Sigals ad Systems The Discrete-Time Fourier Trasorm (DTFT). -. - Xe ( j ) - - - - - Xe ( j ) - - - - - - Xe ( j ) π s - - - Figure Xe ( j ) π s - - the substitutio i equatio (). That is, the bottom two plots i Figure correspod to the two expressios below: Xe ( j ) π s ad Xe ( j ) π () s These plots are more meaigul to us, sice they clearly shows two spikes i the DTFT cetered at ±Hz. Now, let us make a ew observatios. First, ote that whe plotted as a uctio o, oe period o the DTFT covers the requecy rage [ s, s ] [ Hz, Hz] ( s Hz ). Sice the DTFT is periodic, this same requecy spectrum will be repeated outside this requecy rage. Recall rom our discussio o samplig, this is exactly what we said would occur amely, that the requecy spectrum o the origial cotiuous-time waveorm will be repeated at osets o ± k s, k {,,, }. I Figure, or example, we plot Xe ( j ) π s or Hz < < Hz, correspodig to π < < π. At this poit, the reader might object to the above discussio o the grouds that we have previously said that the cotiuous-time spectrum o the cosie wave x c () t i equatio (9) is give by, X c () --δ ( + ) + --δ ( ) () - -
EEL: Discrete-Time Sigals ad Systems The Discrete-Time Fourier Trasorm (DTFT) Xe ( j ) π s - - Figure That is, the CTFT has two distict spikes at ±Hz, ad is zero everywhere else. The CTFT i () is, however, correct oly or a cosie wave that is ot time-limited. Note that by restrictig our samplig o the cosie wave to periods o the Hz cosie wave ( samples), we implicitly assumed that the discrete-time sigal is zero everywhere else. That is, the DTFTs i Figures ad represet the discrete-time requecy spectrum o a time-limited cosie wave. For compariso, we ca compute the CTFT or the ollowig time-limited cotiuous-time uctio, xt () cos( πt) [ ut () ut ( ) ] () by applyig deiitio (): X () xt ()e jπt dt cos( πt)e jπt dt () Usig the iverse Euler relatios or the cosie uctio: X () X () --e jπt + --e jπt e jπtdt -- e [ jπt( ) + e jπt( + ) ] dt () (7) X () -- --------------------------- e jπt( ) jπ( ) + e jπt( + ) --------------------------- jπ( + ) t t (8) X () -- e jπ( ) e --------------------------- jπ( + ) jπ( ) + --------------------------- jπ( ) --------------------------- jπ( ) + --------------------------- jπ( ) (9) X () j -- e jπ( ) e ------------------------------------ jπ( + ) π( ) + ------------------------------------ π( ) () Figure below plots the time-limited, cotiuous-time sigal i () ad its CTFT magitude spectrum X (). Note that except or a scalig dierece, the CTFT ad the DTFT appear very similar over the requecy rage [ s, s ]. Thus, the act that the DTFT spectral represetatio o the sampled, time-limited cosie wave is ot etirely localized at ±Hz (as might have bee expected) is ot a cosequece o the DTFT itsel, but rather a cosequece o the iite-legth samplig process. We reer to this pheomeo, amely, the spreadig o requecy cotet rom the idealized peaks to the etire requecy rage due to iite-legth samplig, as spectral leakage. The loger the sequece legth, the more cocetrated the requecy cotet o that discrete-time sigal will be i the eighborhood o its domiat requecies. Below, we cosider the DTFT or the discrete-time sigal, - -
EEL: Discrete-Time Sigals ad Systems The Discrete-Time Fourier Trasorm (DTFT) xt (). -. -. t - X ().. - - Figure x c ( s ) {,,, 98, 99} elsewhere () where x c () t is agai a Hz cosie wave, ad the samplig requecy is agai give by s Hz. Note that the dierece betwee the discrete-time sigals i equatios () ad () is that ow cosists o samples ( cycles o the cosie wave), istead o samples ( cycles o the cosie wave). I Figure, we plot the DTFT as a uctio o requecy or i equatio (). Comparig the plots i Figure to those i Figure ( samples), ote how much more tightly ocused the requecy cotet is about the requecies ±Hz. B. Fiite-legth sum o cosies Here we cosider the DTFT o a discrete-time sigal, sampled rom the cotiuous-time sigal x c () t, x c () t + cos( πt) + cos( πt) () with samplig requecy s Hz or a total o samples, such that, x c ( s ) {,,, 8, 9}. () elsewhere Applyig deiitio (), 9 Xe ( j ) x c ( s )e j () I Figure 7, we plot Xe ( j ) ad Xe ( j ) or the discrete-time sequece i () as a uctio o the dimesioless requecy variable, ad as a uctio o the real requecy variable (i Hertz), where we agai - 7 -
EEL: Discrete-Time Sigals ad Systems The Discrete-Time Fourier Trasorm (DTFT). -. - 8 Xe ( j ) π s - Xe ( j X )() π s - make the substitutio i equatio (). That is, the bottom two plots i Figure 7 correspod to the two expressios below: Xe ( j ) π s ad Xe ( j ) π () s Note the peaks i the requecy spectrum at Hz, ± Hz ad ± Hz, correspodig to the three terms i equatio (). I Figure 8, we plot Xe ( j ) π s or Hz < < Hz, correspodig to π < < π. Note agai that the requecy spectrum o the time-limited cotiuous-time waveorm is replicated at osets o ± k s, k {,,, } or the time-limited sampled waveorm. Note the similarity o the spectrum i Figure 8 to that o the idealized, iiite-time sampled waveorm, plotted i Figure 9 below. I the examples above, the samplig requecy s was larger tha the Nyquist requecy o max. I the ext sectio, we will look at the DTFT whe the samplig requecy is less tha the Nyquist requecy.. Aliasig ad the Discrete-Time Fourier Trasorm (DTFT) A. Itroductio - - - - I the previous sectio, we saw a example o the discrete-time Fourier Trasorm (DTFT) or iite-legth sequeces, - Figure x c ( s ) {,,, 8, 9} elsewhere () where, x c () t + cos( πt) + cos( πt) (7) ad the samplig requecy is give as s Hz, which is greater tha the Nyquist requecy, max Hz Hz. (8) - 8 -
EEL: Discrete-Time Sigals ad Systems The Discrete-Time Fourier Trasorm (DTFT) - 8 Xe ( j ) - - - - - Xe ( j ) - - - - 8 Below, we explore the DTFT or the same cotiuous-time sigal, but or two samplig requecies below the Nyquist requecy. B. Udersampled examples - - Here we look at the DTFT o the ollowig iite-legth sequece: Xe ( j ) π s - - - Figure 7 Xe ( j ) π s - - x c ( s ) {,,,, } elsewhere (9) where s Hz ad x c () t is give by equatio (7) above. That is, we sample the cotiuous-time sigal x c () t or the same legth o time, oly at a lower samplig requecy (below the Nyquist requecy o Hz) tha beore. I order to relate our preset discussio to our previous discussio o samplig ad aliasig, we plot the ollowig uctios i Figure below: () the origial sigal x c () t ; () the sampled sigal ; () the iiite-legth, cotiuous-time requecy spectrum X c () (CTFT) o the cotiuous-time sigal x c () t ; () the sampled req. spectrum X s () (CTFT) or the iiite-legth, cotiuous-time sigal x s () t, x s () t x' [ ]δ t -- s () - 9 -
EEL: Discrete-Time Sigals ad Systems The Discrete-Time Fourier Trasorm (DTFT) 8 Xe ( j ) π s - - Figure 8.. - - Figure 9: Sampled spectrum o iiite-legth sampled sum o cosies. where x' [ ] x c ( s ), ; ad () the magitude DTFT Xe ( j ) π s o the iite-legth, discretetime sigal as a uctio o requecy. Note that except or the spectral leakage (deied previously) caused by the iite-legth o, the DTFT has peaks o the same relative magitude ad at the same requecies as X s (). I Figure, we plot the same uctios as i Figure, except that ow the iite-legth sequece is give by: x c ( s ) {,,,, 7} elsewhere () where s.hz ad x c () t is agai give by equatio (7) above. Agai, ote the similarity betwee the DTFT ad X s (). C. Oversampled example Fially or compariso, we geerate the same ive plots as above or the oversampled case rom the previous sectio (Figure ). The iite-legth sequece is ow the same as last time [equatio ()] with s Hz. I all three o these examples, the DTFT or the iite-legth sampled sequeces geerates a similar requecy distributio as the CTFT or the iiite-legth sampled sequeces (represeted i the cotiuoustime domai as x s () t ), with two mai diereces: () scalig ad () spectral leakage caused by the iitelegth samplig processes i equatios (), (9) ad ().. Coclusio The Mathematica otebook dtt.b was used to geerate the examples i this set o otes. Next time, we will itroduce the discrete Fourier trasorm (DFT) ad show how it is related to the DTFT. - -
EEL: Discrete-Time Sigals ad Systems The Discrete-Time Fourier Trasorm (DTFT) s Hz x c () t - - t 8. X c (). X s ().... - - - - - - DTFT Xe ( j ) π s - - - Figure - -
EEL: Discrete-Time Sigals ad Systems The Discrete-Time Fourier Trasorm (DTFT) x c () t s.hz - - t. 7... X c (). X s ().... -7. - -.. 7. -7. - -.. 7. DTFT Xe ( j ) π s -7. - -.. 7. Figure - -
EEL: Discrete-Time Sigals ad Systems The Discrete-Time Fourier Trasorm (DTFT) s Hz x c () t - - t. X c (). X s ().... 8 - - DTFT - - Xe ( j ) π s - - Figure - -