Signal Processing and Linear Systems-B.P.Lathi copy

Thus a discrete time sinusoid of any frequency no m

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ------------~__ I 47r r ----;-::;-.-::----....;....._J 27r F ig. 8 .11 sinusoid. A graphical artifice to determine the reduced frequency of a discrete-time The First Explanation R ecall t hat s ampling a c ontinuous-time s inusoid cos O t a t u nit t ime i ntervals = 1) g enerates a d isc:ete-time s inusoid cos Ok. T hus, b y s ampling a t u nit mterv~ls, we ~enerate a d Iscrete-time sinusoid o f f requency 0 ( rad/sample) from a. c ontmuou.s-tlme s.inuso.id of. frequency 0 ( rad/s). Superficially, i t a ppears t hat ~mce .a c ontm.uous-:lme ~musO!d w aveform is u nique for e ach v alue o f 0 , t he r esultmg d Iscrete-tIme sm~sO!d m ust also h ave a u nique w aveform for e ach O. R ecall ~owev~r, t hat t here IS a u nit t ime i nterval between samples. I f a continuous-tim~ sl~usO!d exe~u:es s.ev~ral cycles d uring u nit t ime ( between successive s amples), i t WIll n ot b e VISIble m ItS s amples. T he s inusoid m ay j ust as well n ot h ave e xecuted t hose cycles. A~other low frequency c ontinuous-time s inusoid could also give t he s~me ~amp~es. Flg.ure 8.12 shows how t he s amples o f t wo very different c ontinuous~Ime sm~sO!ds o f ~Ifferent frequencies g enerate i dentical d iscrete-time sinusoid. T his ~llustrat.lOn e xplams w hy two discrete-time sinusoids whose frequencies 0 a re n ommally dIfferent have t he s ame waveform. ~T n f\ - --. .' .f: t '! 0 0 J N onuniqueness o f d iscrete-time s inusoids is easy t o p rove m athematically. B ut w hy d oes i t h appen p hysically? We now give h ere two different physical explanations o f t his i ntriguing p henomenon. 2 3 j I \ Physical Explanation o f Nonuniqueness o f Discrete-Time Sinusoids 2.57r ~ VV n~ f ! 6 7/ ~ A 8 S 10 4 k_ . J V V U··. /._.. . V J F ig. 8 .12 Physical explanation of nonuniqueness of Discrete-time sinusoid waveforms. 554 8 Discrete-time Signals a nd Systems H uman Eye is a Lowpass Filter F igure 8.12 also brings o ut one interesting fact; t hat a h uman eye is a lowpass filter. Both t he continuous-time sinusoids in Fig. 8.12 have t he s ame set q f samples. Yet, w hen we see t he samples, we i nterpret them as t he samples of t he lower frequency sinusoid. T he eye does not see (or cannot reconstruct) t he wiggles of t he higher frequency sinusoid between samples because the eye is basically a lowpass filter. 8.2 Some Useful Discrete-time Signal models 555 T hus, m a signaI ej flk , t he frequency 11 = 7r + X a ppears a s frequency 7r - x. . . Therefore, as 11 increases beyond 7r, t he actual frequency decreases, until a t 11 = 27r (x = 7r), t he a ctual frequency is zero (7r - X = 0). As we increase 11 beyond 27r, t he same cycle of events repeats. For instance, 11 = 2.57r is t he same as 11 = 0.57r. [[[I[[[I[[[[[[[ I[[[[[[[[[rnntl' t 12 -8 -4 0 4 (a) 12 k_ (b) F ig. 8 .13 Another physical explanation of nonuniqueness of discrete-time sinusoid wave- forms. T he S econd Explanation Here we s hall p resent a quantitative argument using a discrete-time exponential r ather t han a d i...
View Full Document

Ask a homework question - tutors are online