Consider a system with a causal impulse response h(t). Its system transfer funct
ID: 1927489 • Letter: C
Question
Consider a system with a causal impulse response h(t). Its system transfer function H(s) has the following pole-zero plot, Given that this impulse response h(t) is a causal signal, what is the region of convergence? Find h(t) (you can keep any constant multiplier in your solution). Is the system with this impulse response h(t) a BIBO stable system? We form another causal LTI system, whose system transfer function is H1(s) = (s + 2 + j)H(s). Let h1(t) be the impulse response of this new system. Is h1(t) a real signal? Justify your answer.Explanation / Answer
From given pole zero plot we can say that it has four poles at
s = -3,+1, -2±j
and hence H(s) = K/(s+3)(s-1)(s2+4s+5)
where K is any constant.
a) For h(t) to be causal signal the region of convergence will be
Re(s) > +1
because any pole doesnt lie in ROC.
b) by Partial fractioning we can write H(s) as
H(s) = c1/(s+3) + c2/(s-1) + (c3s+c4)/((s+2)2+ 1)
where c1,c2.. are constants , values of which can not be determined from the pole zero plot.
and from laplace transform table we can find it's inverse is easily
=>
h(t) = [c1e-3t + c2et + ke-2tcos(t)] u(t)
c) The presence of pole in positive real part side of the pole zero plot will make it as BIBO unstable system.
pole at s = 1 is causing this unstability.Bounded input wont produce a bounded input
d) H1(s) = (s+2+j) H(s)
= K/(s+3)(s+1)(s+2-j)
on partial fractioning we can write
H1(s) = a1/(s+3) +a2/(s+1) +a3/(s+2-j)
where a1,a2 and a3 can be imaginary
h1(t) = [a1e-3t +a2et + a3e-2t.ejt] u(t)
hence prsence of imaginary cofficient a1,a2,a3 and ejt will make h1(t) imaginary..
therefore h1(t) is not a real signal.
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