Q7Signals and Systems

Question

Determine the convolution of the given signals: $x (t) = u (t - 3) - u (t - 5)$ and $h (t) = e^{- 3 t} u (t)$ Where u(t) is unit step function.

Answer 1

Given signals: $x (t) = u (t - 3) - u (t - 5)$ (A rectangular pulse from $t = 3$ to $t = 5$ ) and $h (t) = e^{- 3 t} u (t)$ .

The convolution is $y (t) = \int_{- \infty}^{\infty} x (τ) h (t - τ) d τ$ .

The condition $u (t - τ) = 1$ implies $τ \leq t$ . We evaluate the integral across three regions of $t$ .

1. For $t < 3$ : The integration bounds $[3, 5]$ and $τ \leq t$ do not overlap. Thus, $y (t) = 0$ .

2. For $3 \leq t < 5$ : The active integration region is from $3$ to $t$ .

3. For $t \geq 5$ : The active integration region covers the entire pulse from $3$ to $5$ .

Question