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Question

# Consider the cascade interconnection of three causal LTI system shown below; and the overall impulse response is h(n) where h2(n)=u(n)−u(n−2) y1(n) is response of first system for the input. Then x(n)=δ(n)−δ(n−1). ∑∞n=−∞y(n)= ____0

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Solution

## The correct option is A 0 h(n)=h1(n)∗[h2(n)∗h2(n)] No.of non zero samples in h(n) = No. of samples in h1(n) + No. of samples in (h2(n)∗h2(n))−1 7 = No, of samples in h1(n)+3−1 No. of samples in h1(n)=7−2 = 5 Let , h2(n)=a,b,c,d,e h1(n)=abcde h2(n)∗h2(n)=1,21a,b,c,d,e 2a 2b 2c 2d 2e h(n)=a,b,c,d,e[a,b+2a,c+2b+a,d+2c+b,c+2d+c,2e+d,e] Compare with h(n), h(n)=[1,5,10,11,8,4,1] a = 1 b + 2a = 5 b + 2 = 5 b = 3 c + 2b + a = 10 c + 2(3) + (1) = 10 c + 6 + 1 = 10 c = 3 d + 2(3) + 3= 11 d + 9 = 11 d = 2 e + 2d + c = 8 e + 4 + 3 = 8 e = 1 ∴h1(n)=1,3,3,2,1 y1(n)=x(n)∗h1(n) x(n)1−11,3,3,2,1 y1(n)=−1,−3,−3,−2,−11,2,0,−1,−1,−1 ∑∞n=−∞y1(n)=1+2+0−1−1−1=0

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