Question

Please figure out the convolution of the given two signals:
x(t) = u(t−1) and v(t) = e³ᵗu(t)

Answer 1

The convolution of the given signals can be found using the convolution integral. After substituting the signals x(t) = u(t-1) and v(t) = e^3t u(t), we can simplify the integral to find the value of the convolution at any time t.

Step-by-step explanation:

The convolution of the given signals is the integral of the product of the two signals when one is reversed and shifted. In this case, we have x(t) = u(t-1) and v(t) = e^3t u(t). To calculate the convolution, we need to reverse and shift x(t) by the time duration of v(t). This results in the convolution:

(x*v)(t) = int[0 to t] x(τ) v(t-τ) dτ

Substituting the given signals into the convolution integral, we get:

(x*v)(t) = int[0 to t] u(τ-1) e^3(t-τ)u(τ) dτ.

Simplifying this integral will give us the value of the convolution at any given time t.