Lecture #7: Discrete-time Signals and Sampling
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1 EEL335: Discree-Time Signals and Sysems Lecure #7: Discree-ime Signals and Sampling. Inroducion Lecure #7: Discree-ime Signals and Sampling Unlike coninuous-ime signals, discree-ime signals have defined values only a a fixed number of poins in ime; discree-ime signals ypically arise when we sample a coninuous-ime signal. In his class, we will consider discree-ime signals whose defined values occur only a regularly spaced inervals, in oher words, discree-ime signals wih a consan sampling frequency. To conver a coninuous-ime signal x c () o a discreeime signal xn [ ], we can use he following equaion: xn [ ] = x c ( nt s ) = xn ( ) () where T s denoes he sampling period, denoes he sampling frequency, and, = T s (2) Noe ha in equaion () we define he discree-ime signal xn [ ] as a funcion of he dimensionless ime index n, n {, 2,, 2,,, }. Figure plos he discree-ime sequences xn [ ] of ow coninuous-ime signals x (), and a sampling frequency of = 5 Herz (samples/second). 4 x () 4 xn [ ] = 5Hz sampling n.2 x ().2 xn [ ] = 5Hz.. sampling Figure If we wan o reconsruc a coninuous-ime signal from a discree-ime signal, knowledge of he sampling frequency is criically imporan, since wihou i, we do no know o which real ime each ime index n corresponds. For example, music recorded on a CD is sored as a sequence of numbers. By inernaional agreemen, all music sored on CDs is sampled a 44. khz; wihou his criical piece of informaion, he sored music could no be faihfully reconsruced in he coninuous-ime domain (on your sereo). Compuer music files, such as mp3 s, have heir sampling frequency encoded in he file iself. If he encoded sampling frequency in he mp3 file is incorrec for some reason, he music ha your compuer would play would sound eiher oo fas or oo slow. To demonsrae his poin, I ook a shor piece of music (from Kenny Roger s The Gambler ), sampled a 32kHz, and played he original music during lecure, assuming ha he sampling frequency was 24kHz (incorrec), 32kHz (correc) and 4kHz (incorrec). The piece of music played back wih he incorrec sampling frequency of -. n - -
2 EEL335: Discree-Time Signals and Sysems Lecure #7: Discree-ime Signals and Sampling 24kHz sounds slowed down, while i sounds sped up wih he incorrec sampling frequency of 4kHz. All hree music files are posed on he web sie in wav and mp3 formas. 2. Sampling A. How fas is enough? An exremely imporan quesion ha mus be answered in he sampling process is How fas is enough? Tha is, how fas do we have o sample a coninuous-ime signal in order o be able o reconsruc i from jus he discree-ime signal and knowledge of he sampling frequency? A ruly remarkable heorem, he Shannon Sampling Theorem, answers his quesion: Shannon Sampling Theorem: A coninuous-ime signal x c () wih frequencies no higher han f max can be reconsruced exacly from is samples xn [ ] = x c ( nt s ), if he samples are aken a a sampling frequency > 2f max, ha is, a a sampling frequency greaer han 2f max. The frequency 2f max is known as he Nyquis frequency. We have already seen ha any signal can be convered ino is frequency domain represenaion, which ells us wha frequencies are conained wihin ha signal. Now, we begin o see one reason why ha frequencydomain represenaion is so imporan. I allows us o deermine (or filer our signal) so ha we know f max ; wih ha knowledge, we can confidenly sample a a frequency greaer han he Nyquis frequency and lose no informaion in ha sampling process. As an example, consider a Hz cosine wave: x c () = cos( 2π) (3) For his signal, f max = Hz, so ha he Nyquis frequency is 2Hz. In Figure 2, we plo he sequences xn [ ] corresponding o eigh differen sampling frequencies: Hz, Hz, 5Hz, 3Hz, 2.Hz,.9Hz,.4Hz and.hz. The Sampling Theorem ells us ha we should be able o reconsruc x c () from jus he samples for all sampling frequencies greaer han 2Hz (op five plos in Figure 2). This seems surprising given he relaive sparsiy of daa for sampling frequencies 5Hz, 3Hz and 2.Hz. As you will remember, mos of he class hough ha he Hz sampling frequency would be sufficien for reconsrucing he original waveform; some of he class hough he Hz sampling frequency would be sufficien; however, very few hough ha any of he lower frequencies would be sufficien for perfec reconsrucion of he original signal. Therefore, he Sampling Theorem is no only remarkably powerful, bu also remarkably surprising. B. Ideal reconsrucion of discree-ime signal Noe ha he Sampling Theorem does no ell us specifically how we should reconsruc he coninuous-ime signal from he discree-ime samples. Clearly, if we are o reconsruc he original sine wave in Figure 2 from he samples corresponding o he smaller sampling frequencies above 2Hz, we canno simply use linear inerpolaion. Figure 3 shows, for example, wo naive reconsrucions of he cosine wave for he sampling frequency = 2. Hz. The reconsrucion on he leimply approximaes he reconsruced funcion x r () as a sequence of pulses spaced T s seconds apar. The reconsrucion on he righ linearly inerpolaes beween consecuive samples; neiher reconsrucion could, in any sense of he phrase, be considered a perfec reconsrucion of he original signal. To achieve he ideal reconsrucion, each sample should be he weighed and ime-shifed peak of he sinc funcion, sinc() sin( ) = (4) ploed in Figure 4 below. Specifically, he ideal reconsrucion is given by,. We will see laer why his is so
3 EEL335: Discree-Time Signals and Sysems Lecure #7: Discree-ime Signals and Sampling = Hz = Hz Hz = = 3Hz = 2.Hz =.9Hz =.4Hz =.Hz Figure 2: A Hz cosine wave and differen sampling frequencies x r () = xn [ ]sinc[ π ( n )]. (5) n where x r () refers o he reconsruced, coninuous-ime signal. You may wonder why he value of he sinc() funcion is a =, since boh he numeraor and denominaor of equaion (4) is zero a =. When we have such an undefined raio, we can use L Hospial s Rule (from calculus) o deermine he value a = : - 3 -
4 EEL335: Discree-Time Signals and Sysems Lecure #7: Discree-ime Signals and Sampling = 2.Hz f Figure 3: wo bad reconsrucions of cosine wave s = 2.Hz.8 sinc() lim sinc() Figure 4 d sin( ) sin( ) d cos( ) = lim = lim d = lim = lim cos( ) =. (6) () d Now, in Figure 5 below we plo his ideal reconsrucion for sampling frequencies = 5Hz, 3Hz, 2.Hz,.9Hz,.4Hz and.hz, using equaion (5) above. Noe ha for sampling frequencies greaer han 2Hz (he Nyquis frequency for he Hz signal), he reconsrucion is very good, while for sampling frequencies less han 2Hz, he reconsruced signal x r () does no approximae x c () in equaion (3) very well. Hz, while good, is no per- A his poin, he observan reader will noice ha he reconsrucion for fec. The reason for his is ha we are no acually reconsrucing, = 2. x c () = cos( 2π) [equaion (3)] (7) bu insead are reconsrucing he ime-limied coninuous-ime signal below: x c () = cos( 2π) [ u () u ( 5) ]. (8) As we will see laer, he coninuous-ime magniude frequency rerpresenaion X c () f for (8) above is acually no frequency-limied such ha f max = Hz, as illusraed in Figure 6, which plos X c () f for he imelimied signal (8). Noe ha while he frequency conen of he ime-limied cosine waveform sill has dominan peaks a ± Hz, ime limiing he cosine causes he frequency specrum o spread ou over he enire frequency specrum. Therefore, he 2.Hz sampling frequency is sricly speaking no larger han 2f max, so ha he reconsrucion in Figure 5 is no perfec. Furhermore, here is a pracical problem wih he reconsrucion in equaion (5); he value of he reconsruced signal x r () is dependen on all samples for all values of. For example, suppose we waned o - 4 -
5 EEL335: Discree-Time Signals and Sysems Lecure #7: Discree-ime Signals and Sampling = 5Hz = 3Hz = 2.Hz =.9Hz Hz = =.Hz Figure 5: Ideal reconsrucions from samples X c () f Figure 6: Time-limied frequency specrum of Hz cosine signal f - 5 -
6 EEL335: Discree-Time Signals and Sysems Lecure #7: Discree-ime Signals and Sampling know he value of he original signal for = 2 ; in he ideal reconsrucion, x r ( 2) is dependen no only on samples close o = 2, bu all samples, no maer how disan (in ime) from = 2. Therefore, as a pracical maer, he ideal reconsrucion is no realizable in real sysems. We can compensae for his fac by oversampling; ha is sampling a signal x c () a a frequency significanly larger han he Nyquis frequency. Le us again consider he example of CDs. The human ear canno hear frequencies higher han approximaely 2 khz; a sampling frequency of 44. khz is herefore more han 2f max, where f max is he maximum frequency ha we should be ineresed in for an audio sysem inended for human enjoymen. C. Conclusion Nex ime, we will coninue our invesigaion of he sampling process o undersand wha is so criical abou he Nyquis frequency. No surprisingly, our developmen will rely on he ransformaion beween he ime and frequency domain. Finally, all he figures relaed o sampling were generaed wih he Mahemaica noebook inro_sampling.nb
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