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

In other words we c an recover t he signal j t from

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Unformatted text preview: f (t) before sampling f (t). T his way we lose only t he c omponents beyond t he folding frequency Hz; these components now cannot reappear t o corrupt t he components with frequencies below t he folding frequency. This suppression of higher frequencies can be accomplished by a n ideal lowpass filter o f b andwidth F s/2 Hz. This filter is called t he a ntialiasing f ilter. N ote t hat t he a ntialiasing operation must be performed before the signal is sampled. '1'1- 5 Sampling 328 5.1 T he S ampling Theorem 329 F (OJ) ( b) o - 0.2 ( d) 0.2 1 -- -5 9 '(Hz)-+ (b) • •• - 0.3 F (OJ) -0.1 0 0.2 t -- Lowpa" filter ~' 1(1) :0.05 (d) I .... ...• .3 .... : -0.1 .... "'~ : , . , :~, ~/ t -- 0.2 -49lt - 10lt 101t - 20 0 ~ -5 5 401t OJ-+ F ig. 5.7 Effect of practical sampling. T he a ntialiasing filter, being a n ideal filter, is unrealizable. In practice, we use a steep cutoff filter, which leaves a sharply a ttenuated residual s pectrum beyond t he folding frequency F,/2. Practical Sampling I n proving t he sampling theorem, we assumed ideal samples obtained by multiplying a signal f (t) by a n impulse t rain which is physically nonexistent. In practice, we multiply a signal f (t) by a t rain of pulses of finite width, depicted in Fig. 5.7b. T he s ampled signal is i llustrated in Fig. 5.7c. We wonder whether it is possible to recover or reconstruct f (t) from t he s ampled signal f (t) in Fig. 5.7c. Surprisingly, t he answer is affirmative, provided t hat t he sampling rate is n ot below t he Nyquist rate. T he s ignal f (t) c an be recovered by lowpass filtering f (t) as if it were sampled by impulse t rain. T he p lausibility of this result becomes a pparent when we consider t he fact t hat r econstruction of f (t) requires t he knowledge of t he Nyquist sample values. This information is available or built in t he s ampled signal 7(t) in Fig. 5.7c because the k th s ampled pulse s trength is f (kT). To prove t he result analytically, we observe t hat t he s ampling pulse t rain PT(t) d epicted in Fig. 5.7b, being a periodic signal, c an b e expressed as a trigonometric Fourier series 00 PT(t) = Co + L Cn cos (nwst + On) n =l 27r w S=T 20 9 '(Hz)-+ F ig. 5.8 An example of practical sampling. and f (t) = f (t)PT(t) = f (t) [co + ~ Cn cos (nw,t + On)] 00 = C of(t) + L C nf(t) cos (nw.t + On) (5.11) n =l T he s ampled signal f (t) consists of C of(t), C d(t) cos (w,t + OIl, C 2f(t) cos (2wst + (2), . ... N ote t hat t he first t erm C of(t) is t he desired signal a nd all t he o ther terms are modulated signals with s pectra c entered a t ± ws, ± 2w., ±3ws, . .. , as illustrated in Fig. 5.7e. Clearly t he signal f (t) c an be recovered by lowpass filtering o f f (t), provided t hat w , > 47r B (or F . > 2 B). • E xample 5 .3 To demonstrate practical sampling, consider a signal f(t) = sinc 2 (57rt) sampled by a rectangular pulse sequence PT(t) illustrated in Fig. 5.8c. The period of PT(t) is 0.1 second, so t hat the fundamental frequency (which is the sampling frequency) is 10 Hz. Hence, w, = 207r. The Fourier series for PT(t) can be expressed as PT(t) = Co + L Cn co...
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