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Unformatted text preview: wpass pulse, t hat is i ts s pectrum is c oncentrated
a t low frequencies. Verify this behavior by considering a system whose u nit impulse
response is h (t) = r ect (~). T he i nput pulse is a triangle pulse p(t) = l l.( ~).
T he a rea u nder t his pulse is A = 0.5 X 1 0- 6 • Show t hat t he s ystem response to this
pulse is v ery n early t he s ystem response t o t he i nput A6(t).
A lowpass system time constant is o ften defined as t he w idth of its u nit impulse
response h (t) (see Sec. 2.7-2). An i nput pulse p(t) t o t his system passes practically
without d istortion, if t he w idth of p(t) is much greater t han t he s ystem t ime c onstant.
Assume p( t ) t o be a lowpass pulse, t hat is its spectrum is c oncentrated a t low frequencies. Verify this behavior by considering a system whose u nit impulse response
is h (t) = r eet (~). T he i nput pulse is a t riangle pulse p(t) = ll.(t). Show t hat t he
s ystem o utput t o t his pulse is very nearly k p(t), where k is t he s ystem g ain t o a dc
signal, t hat is, k = H(O). Show t hat t his filter is physically unrealizable by using t he t ime-domain criterion
[noncausal h(t)l a nd t he frequency-domain (Paley-Wiener) criterion. C an t his filter
b e m ade approximately realizable by choosing a sufficiently large to? Use y our own
(reasonable) criterion o f a pproximate realizability t o d etermine to.
Hint: Use p air 22 in Table 4.1.
4.5-2 Show t hat a filter with transfer function H (w) .!. Joo
7r - 00 X (w)
W- Y a nd X (w) = _.!. Joo
7r - 00 R(w)
W- Y 5 2(10 )
+ 10 l0e is unrealizable. C an t his filter b e m ade approximately realizable by choosing a sufficiently large to? Use y our own (reasonable) criterion of approximate realizability t o
d etermine to.
Hint: Show t hat t he impulse response is noncausal.
4.5-3 Determine if t he filters with t he following transfer functions are physically realizable.
I f t hey are not realizable, c an t hey be realized exactly or approximately by allowing
a finite time delay in t he response? H(w) = ( a) 1 0- 6 sinc (1O- 6 w) ( b) 1O- 4 ll. ( _ _ _) ( c) 27r 6(w)
4.6-1 Show t hat t he e nergy of a Gaussian pulse j (t) = --l.= e-~
U V21r is 2 "0.. Verify this result by deriving t he energy E j from F(w) using P arseval's
Hint: See p air 22 in Table 4.1. Use t he fact t hat
OO J - 00 4.6-2 e _x2 2 / dx = J 2'; Show t hat J OO assuming t hat h( t ) h as no impulse a t t he origin. This pair of integrals defines t he
H ilbert t ransform.
Hint: Let he(t) a nd holt) b e t he even a nd o dd c omponents of h (t). Use results in
P rob. 4.1-3. See Fig. 1.24 for t he r elationship between he(t) a nd holt). Recall t hat
s gn(t) ¢ =? 2 /jw. Use convolution property.
This problem s tates o ne of t he i mportant p roperties of causal systems: t hat t he real
a nd i maginary p arts of t he t ransfer function of a causal system are related. I f one = w2 A causal signal h (t) h as a Fourier transform H(w). I f R(w) a nd X (w) a re t he real
a nd t he i maginary p arts...
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