Question

Compute y[n] for the convolution of the following discrete-time signals:

x[n]={0.5, 1≤n≤8
{0,​ otherwise​
h[n]={1, 0≤n≤4
{0,​ otherwise​
Find and display the resulting y[n] for this convolution.

Answer 1

The convolution y[n] of x[n] and h[n] is calculated by summing the product of overlapping elements for each shift in the range of n from 1 to 12.

Step-by-step explanation:

The convolution of two discrete-time signals x[n] and h[n] is performed by sliding one signal past the other and calculating the area of overlap for each shift. Given x[n] is non-zero from 1 to 8 and h[n] is non-zero from 0 to 4, we can compute the convolution y[n] by considering all possible overlaps between these two signals.

Steps for computing y[n]:

1. Determine the range of n for which y[n] will be non-zero. In this case, it's from n=1 to n=12. This is because the earliest point of non-zero overlap occurs at n=1 (the first point of x[n] with h[0]) and the latest point of non-zero overlap occurs when h[4] crosses the last point of x[n] at n=8+4=12.
2. For each n in the determined range, sum the product of x[k] and h[n-k], where k ranges over all the values that produce non-zero products.
3. Display the results in a list or sequence that provides the value of y[n] for each n.