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

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 Correct Answer δ[n - 2] + δ[n - 1] + δ[n] + δ[n + 1]

Concept:

The z-transform of a unit impulse function or Kronecker delta δ  ↔ 1

The time-shifting affects the z-transform as:

x = z -n0 X(z)

Application:

Given:

h1 = δ + δ 

h2 = δ + δ 

If h₁and h₂ are cascaded connected then h = h₁ * h₂

Where '*' denotes convolution.

h = h1 * h2

Taking z-transform both side

H = H₁ ⋅ H₂

H = (z^-1 + z) ⋅ (1 + z^-1) = (z^-1 + z-2 + z + 1 )

Taking inverse z-transform both side

h = δ + δ + δ + δ

∴ Impulse response of the cascaded system is δ + δ + δ + δ