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

Thus a discrete time sinusoid of any frequency no m

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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 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...
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