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Unformatted text preview: only o ne t erm ( corresponding t o k = 0) i n t he
s ummation o n t he r ighthand s ide o f Eq. (5.10b) survives. T hus f it) = sinc(211"Bt) T his signal is i llustrated i n Fig. 5.4b. O bserve t hat t his is t he o nly signal t hat h as a
b andwidth B Hz a nd t he s ample values f(O) = 1 a nd f (nT) = 0 (n # 0). No o ther s ignal
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satisfies these conditions. 5 .12 Practical Difficulties in Signal Reconstruction k = L J(kT) sinc[211"B(t  kT)] (5.10a) k J (kT) sinc (211" B t  k1T) =L (5.10b) k t Figure 5.3a shows t hat the impulse response of this filter is noncausal, and this filter is not
realizable. In practice, we make it realizable by delaying the impulse response by T / 2. This
merely delays the o utput of the filter by T / 2. I f a signal is s ampled a t t he Nyquist r ate :F. = 2 B Hz, t he s pectrum F(w) consists of repetitions of F (w) w ithout any gap between successive cycles, as depicted
in Fig. 5.5a. To recover J (t) from J (t), we need t o pass t he s ampled signal J (t)
t hrough a n i deallowpass filter, shown d otted in Fig. 5.5a. As seen in Sec. 4.5, such
~ filter is unrealizable; i t c an be closely approximated only with infinite time delay
I II t he response. In other words, we c an recover t he signal J (t) from its samples
with infinite time delay. A practical solution to this problem is t o sample t he signal
~ a r ate higher t han t he Nyquist r ate (:F. > 2B or W s > 411" B ). T he result is
F(w), consisting of repetitions of F(w) w ith a finite band gap between successive 5 Sampling 326 ro, 327 T he Sampling Theorem finite d uration or width. We c an demonstrate (see P rob. 5.110) t hat a signal cannot
be timelimited a nd b andlimited simultaneously. I f a signal is timelimited, i t c annot
be bandlimited a nd vice versa (but it can be simultaneously nontimelimited and
nonbandlimited). Clearly, all practical signals, which are necessarily timelimited,
are nonbandlimited; they have infinite bandwidth, a nd t he s pectrum F(w) consists
of overlapping cycles of F(w) r epeating every F s Hz (the sampling frequency)' as
i llustrated in Fig. 5.6. Because of infinite bandwidth in this case, t he s pectral overlap
is a c onstant feature, regardless of t he sampling rate. Because of t he overlapping
tails, F (w) no longer has complete information a bout F(w), a nd i t is no longer
possible, even theoretically, t o recover f (t) from t he s ampled signal f (t). I f t he
sampled signal is passed t hrough a n i deallowpass filter, t he o utput is n ot F(w) b ut
a version o f F(w) d istorted as a result o f two separate causes: YVfffriV (.)
2 nH 5.1 r o 1. T he loss of t he t ail of F(w) beyond (b) F ig. 5 .5 Spectra of a signal sampled at (a) the Nyquist rate (b) above the Nyquist rate.
cycles, as i llustrated in Fig. 5.5b. We c an now recover F(w) from F(w) using a
lowpass filter w ith a g radual cutoff characteristic, shown d otted in Fig. 5.5b. But
even in this case, t he filter gain must be ze...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.
 Spring '13
 Bayliss
 Signal Processing, The Land

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