Question
Two discrete-time linear time-invariant systems with impulse responses h1[n] = δ[n - 1] + δ[n + 1] and h2[n] = δ[n] + δ[n - 1] are connected in cascade, where δ[n] is the Kronecker delta. The impulse response of the cascaded system is
Options :
δ[n - 1] δ[n] + δ[n + 1] δ[n - 1]
δ[n - 2] + δ[n + 1]
δ[n - 2] + δ[n - 1] + δ[n] + δ[n + 1]
δ[n] δ[n - 1] + δ[n - 2] δ[n + 1]
Answer :
δ[n - 2] + δ[n - 1] + δ[n] + δ[n + 1]
Solution :
Given:
h1[n] = δ[n - 1] + δ[n + 1] h2[n] = δ[n] + δ[n - 1]
If h1[n]and h2[n] are cascaded connected then
h[n] = h1[n] * h2[n]
Where '*' denotes convolution.
h[n] = h1[n] * h2[n]
Taking z-transform both side
H[z] = H1[z] ⋅ H2[z] H[z] = (z-1 + z) ⋅ (1 + z-1) = (z-1 + z-2 + z + 1 )
Taking inverse z-transform both side
h[n] = δ[n-1] + δ[n-2] + δ[n+1] + δ[n]
∴ Impulse response of the cascaded system is δ[n - 2] + δ[n - 1] + δ[n] + δ[n + 1]
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