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Unformatted text preview: quencies. T he i deallowpass filter (Fig. 4.27), for
example, allows all components below w = W r ad/s t o p ass w ithout d istortion a nd
s uppresses a ll c omponents above w = W . F igure 4.28 illustrates ideal highpass a nd
b andpass f ilter characteristics.
T he i deal lowpass filter in Fig. 4.27a has a linear phase of slope - td, which
results in a t ime d elay o f t d seconds for all i ts i nput c omponents of frequencies 4.5 Ideal a nd P ractical F ilters 271
h ( I) 00_ -w w o H (oo)=-oot ; ;.---d (a) (b) F ig. 4 .27 Ideallowpass filter: its frequency response and impulse response.
below W r ad/s. T herefore, if t he i nput is a signal f (t) b andlimited t o W r ad/s, t he
o utput y(t) is f (t) delayed by t d; t hat is, y(t) = f (t - td) T he s ignal f (t) is t ransmitted b y this system w ithout d istortion, b ut w ith t ime
delay t d. For t his filter \H(w)\ = r ect(2';;;) a nd LH(w) = e - jwtd , so t hat H(w) = r ect (2~) e - jwtd (4.60a) T he u nit i mpulse response h(t) o f t his filter is o btained from p air 18 ( Table 4.1)
a nd t he t ime-shifting p roperty
1 h (t) = F - [ rect W =- (2~) e - jwtd ] sinc [W (t - td)J (4.60b) 7r Recall t hat h(t) is t he s ystem response t o i mpulse i nput 6(t), which is a pplied a t t = O. F igure 4.27b shows a curious fact: t he r esponse h(t) begins even before t he
i nput is applied ( at t = 0). Clearly, the filter is non causal a nd t herefore physically
unrealizable. Similarly, one can show t hat o ther i deal filters (such as t he ideal
highpass or t he ideal b andpass filters depicted in Fig. 4.28) are also physically
For a physically realizable system, h(t) m ust b e causal; t hat is, h(t) = 0 for t < 0 I n t he frequency domain, t his c ondition is equivalent t o t he well-known P aleyWiener c riterion, which s tates t hat t he n ecessary a nd sufficient condition for t he
a mplitude response \H (w) \ t o b e realizable is 1 00 - 00 I lnIH(w)lld
1+w (4.61) I f H(w) does n ot s atisfy t his c ondition, i t is unrealizable. Note t hat if IH(w)1 = 0
over any finite b and, IlnIH(w)11 = 0 0 over t hat b and, a nd t he c ondition (4.61) is violated. If, however, H (w) = 0 a t a single frequency (or a s et o f discrete frequencies), 274 4 Continuous-Time Signal Analysis: T he Fourier Transform When they t hink o f a system, they t hink o f its impulse response h (t). T he w idth
of h (t) indicates t he t ime constant (response time); t hat is, how fast t he s ystem
is c apable of responding to a n i nput, and how much dispersion (spreading) i t will
cause. This is t he time-domain perspective. From t he frequency-domain perspective, these engineers view a system as a filter, which selectively transmits certain
frequency components and suppresses t he o thers [frequency response H(w)J. Knowing t he i nput s ignal spectrum and t he frequency response of t he system, they create
a mental image o f t he o utput signal spectrum. This concept is precisely expressed
by Y(w) = F(w)H(w).
We can analyze LTI systems by ti...
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