Question

The impulse response function of a discrete-time, linear, time-invariant system is h(n) = 0.6n−1u[n − 1]

a) Find its output when its input is given by x[n] = u[n + 3] − u[n − 4]

b) check if it is causal, also check for if it is bibo stable

Answer 1

To find the output of the given system with a specific input, convolve the input with the impulse response function. The system is causal and BIBO stable.

Step-by-step explanation:

To find the output of the system given the input x[n] = u[n + 3] - u[n - 4], we can convolve the input with the impulse response function h(n). The convolution is given by y[n] = x[n] * h[n].

Using the convolution sum, we can calculate the output for each value of n:

y[n] = ∑[k=-∞ to ∞] x[k] * h[n-k]

Substituting the given values for x[n] and h(n), we get:

y[n] = ∑[k=1 to ∞] (u[k + 3] - u[k - 4]) * 0.6(n - k - 1)u[n - k - 1]

For the causality check, we can see that the impulse response h(n) is causal because it is only nonzero for n ≥ 1. Therefore, the system is causal.

To check if the system is BIBO (Bounded-Input Bounded-Output) stable, we need to examine the impulse response h(n). If the sum of the absolute values of the impulse response is finite, the system is BIBO stable. In this case, since the impulse response h(n) is a geometric sequence that decays exponentially, the system is BIBO stable.