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

611k 8 cos o4d 8 cos o4d 8 813 t

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Unformatted text preview: screte-time sinusoid. As explained earlier, a discrete-time exponential e jm c an b e viewed as a phasor r otating counterclockwise a t a uniform angular velocity of 11 r ad/sample, as shown in Fig. 8.7a. A similar argument shows t hat the exponential e - jflk is a phasor rotating clockwise a t a uniform angular velocity of 11 r adians per sample, as depicted in Fig. 8.7b. T he a ngular velocity of b oth these r otating p hasors is 11 rad. Therefore, as t he frequency 11 increases, t he a ngular velocity also increases. This, however, is t rue only for values of 11 in t he r ange 0 t o 7r. S omething v ery i nteresting happens when t he frequency 11 increases beyond 1r. Let 11 = 7r + x w here x < 7r. Figure 8.13a shows t he phasor progressing from k = 0 t o k = 1, a nd F ig. 8.13b shows t he same phasor progressing from k = 1 t o k = 2. Because t he p hasor r otates a t a s peed of 11 = 7r + x r adians/sample, t he phasor angles a t k = 0, 1, a nd 2 are 0, 7r + x a nd 27r + 2 x = 2 x, respectively. In b oth t he figures, t he p hasor is p rogressing counterclockwise a t a velocity of (7r + x) r ad/sample. But we may also i nterpret t his motion as the phasor moving clockwise (shown in gray) a t a lower s peed of (7r - x) r ad/sample. E ither of these interpretations describes t he p hasor m otion correctly. I f t his motion could be seen by a human eye, which is a lowpass filter, it will automatically interpret t he speed as 7r - x, t he lower of t he two speeds. This is t he stroboscopic effect observed in movies, where a t certain speeds, c arriage wheels appear t o move backwards.t t A s troboscope i s a source of light t hat flashes periodically o n a n object, t hus g enerating a sampled image o f t hat o bject. W hen a s troboscope flashes o n a r otating o bject, such as a wheel, t he wheel a ppears t o r otate a t a c ertain speed. Now increase t he a ctual speed o f r otation (while maintaining t he s ame f lashing r ate). I f t he s peed is i ncreased beyond some critical value, t he wheels appear t o r otate b ackwards b ecause o f t he lowpass filtering effect described above in t he t ext. As we c ontinue t o i ncrease t he s peed further, t he b ackward r otation a ppears t o slow down continuously t o zero speed ( where t he wheels a ppear s tationary), a nd reverse t he d irection again. T his effect is o ften o bserved i n movies in scenes w ith r unning c arriages. A movie reel consists of a sequence of p hotographs s hot a t d iscrete instants, a nd is basicaily a sampled signal. F ig. 8 .14 Highest Oscillation Rate in a Discrete-Time sinusoid occurs at !1 = 7r. Highest Oscillation Rate in a D iscrete-Time Sinusoid Occurs a t 11 = 7r T his discussion shows t hat t he highest r ate of oscillation occurs for t he frequency 11 = 7r. T he r ate of oscillation increases continuously as 11 increases from o t o 7r, t hen decreases as 11 increases from 7r t o 27r. Recall t hat a frequency 7r + X a ppears as the frequency 7r - x. T he frequency 11 = 27r...
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