One of the important properties of convolution, in both continuous and discrete time, is...

One of the important properties of convolution, in both continuous and discrete time, is the associativity property. In this problem, we will check and illustrate this property.

(a) Prove the equality

[x(t) * h(t)] * g(t) = x(t) * [h(t) * g(t)] (P2.43-1)

by showing that both sides of eq. (P2.43-1) equal

(b) Consider two LTI systems with the unit sample responses h₁ [n] and h₂ [n] shown in Figure P2.43(a). These two systems are cascaded as shown in Figure P2.43(b). Let x[n] = u[n].

(i) Compute y[n] by first computing w[n] = x[n] × h ₁ [n] and then computing

(ii) Now find y[n] by first convolving h₁ [n] and h₂ [n] to obtain g [n] = h₁ [n] × h₂ [n] and then convolving x[n] with g[n] to obtain y[n] =

The answers to (i) and (ii) should be identical, illustrating the associativity property of discrete-time convolution.

(c) Consider the cascade of two LTI systems as in Figure P2.43(b), where in this case

And

and where the input is

Determine the output y[n]. (Hint: The use of the associative and commutative properties of convolution should greatly facilitate the solution.)