Question

Compute the CT convolution for each of the following signals x and h using the convolution integral and clearly sketch the output.

x(t) = {sin(πt),​ if 0 {0, otherwise ​

h(t)​ = {2,​ if 1 {0, otherwise ​​

Answer 1

The CT convolution of x(t) = sin(pi*t) for t >= 0 and h(t) = 2 for 0 <= t <= 1, both zero otherwise, involves integrating the product of x(t) and a shifted h(t) over the range where they overlap. The result is a piece-wise function that can be sketched by evaluating the integrals over the key intervals.

Step-by-step explanation:

CT Convolution of Signals

To compute the CT convolution of the signals x(t) = sin(πt) for t ≥ 0 and h(t) = 2 for 0 ≤ t ≤ 1, and zero otherwise, we use the convolution integral:

y(τ) = ∫ x(t) h(τ - t) dt,
which simplifies considering the non-zero intervals of both x(t) and h(t).

The convolution integral will result in a piece-wise function because the overlap between the two signals changes as the convolution kernel h(t) is shifted over signal x(t). This function will describe the output of the convolution process.

Sketching the output would involve identifying the key intervals over which the two functions overlap and then integrating the product of the functions over those intervals. Since sin(πt) is periodic and h(t) is constant over its non-zero interval, the characteristics of the convolution will reflect this behavior.