a) The impuhe respome of a LTI system is given as Comider A(t) to be the prodact of the two signas f() =nd g(t) sach that A(e) =S(1)«(e) where s)-e"M1) Using the fuy comoktion property, or H(J) = placmg-fQ, sing)+ {cosp)(? (a+ j4l}° +£L; Hiet: To obtain te same resut une abo te time shiing property kr b) Ir 1-a'+ a evalute a, B, e and 2 and express k(t) uning these vakaes. e) Let the isput to a continon-time LTI system with the tramfer instion (foquene respome) HJA) ghen in part (b) be of the bm 1.4 cost d) Fa in the blarks (nclaling A) in the table below, showing every step of you cakulatiom.

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Q.6
a) The impuke response of a LTI system is given as
h(e) = Be" cos(2+p)u(t); a>0.
Consider h(t) to be the product of the two signals f(t) and g(t) such that
h(e) = f (t)g(*)
where s) =e"ut)
g(t) = B cos(2 + 9) = B cos2, (t+)
and
Using the frequercy comvoltion property, or
F(jn)• 5(ntn,)= F(j[n+n,]) show that
(acoso-2, sing) + j(cos o)2
(a+ jn)' +2
H(jn) = B-
a+ jn
Hint: To obtain the same result use abo the time shiting property for
cos (24 +9) = cosf2,(t+.
b)
If
H(n) =
evalate a, B, o and 2, and express h(t) using these values.
Let the input to a continuous-time LTI system with the transfer finction (frequency
respose) H(Jn) given in part (b) be of the form
c)
x(1) =-+
cost+cos 3t +cos St+,
25
d)
Fill in the blarks (including k) in the table bekow, showing every step of your
cakuktions.
Transcribed Image Text:Q.6 a) The impuke response of a LTI system is given as h(e) = Be" cos(2+p)u(t); a>0. Consider h(t) to be the product of the two signals f(t) and g(t) such that h(e) = f (t)g(*) where s) =e"ut) g(t) = B cos(2 + 9) = B cos2, (t+) and Using the frequercy comvoltion property, or F(jn)• 5(ntn,)= F(j[n+n,]) show that (acoso-2, sing) + j(cos o)2 (a+ jn)' +2 H(jn) = B- a+ jn Hint: To obtain the same result use abo the time shiting property for cos (24 +9) = cosf2,(t+. b) If H(n) = evalate a, B, o and 2, and express h(t) using these values. Let the input to a continuous-time LTI system with the transfer finction (frequency respose) H(Jn) given in part (b) be of the form c) x(1) =-+ cost+cos 3t +cos St+, 25 d) Fill in the blarks (including k) in the table bekow, showing every step of your cakuktions.
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