Final answer:
The zero-state response of the system when x[n] = u[n] is found by using the convolution sum of the unit step function and the given impulse response. This involves summing over a geometric series of terms that are determined by whether n is even or odd, due to the cosine function in the impulse response.
Step-by-step explanation:
Calculating the Zero-State Response (ZSR)
To calculate the zero-state response of the system when the input x[n] = u[n], where u[n] is the unit step function, we use the convolution sum. Since the impulse response of the system h[n] is given, the zero-state response can be found using:
y[n] = x[n] * h[n]
For x[n] = u[n], the convolution y[n] is:
y[n] = sum(u[k] × h[n-k]) from k=0 to n
Because u[n] is 1 for all n ≥0 and 0 otherwise, the sum simplifies to:
y[n] = sum(h[n-k]) from k=0 to n
Substituting the given impulse response:
y[n] = sum(√2(2)^(n-k) cos(π/2 × (n-k) + π/4)u[n-k]) from k=0 to n
Since u[n-k] is only 1 for indices n-k ≥ 0, we only sum over terms where k ≤ n. For each term in the sum, cos(π/2 × (n-k) + π/4) simplifies to √2/2 for even values of n-k and 0 for odd values, due to the properties of the cosine function and the factor of π/2.
The final expression for y[n] would be the sum of these terms which requires simplification depending on whether n is even or odd. This process involves summing a geometric series for even terms of n.