Question

Determine the continuous time convolution integral and choose correct option for:
\(y(t) = [u(t) - u(t - 2)] * u(t)\)

• A. \(tu(t) + (2 - t)u(t - 2)\)
• B. \((2 - t)u(t) + tu(t - 2)\)
• C. \(tu(t) - (t - 2)u(t - 2)\)
• D. \((t - 2)u(t) + tu(t - 2)\)

Hint:

Break the input signal into standard step functions and apply convolution integrals piecewise to find the output.

Answer 1

The Correct Option is A

We are convolving a rectangular pulse with a unit step function.

Step 1: Understand signals:
\( u(t) - u(t - 2) \): rectangular pulse of width 2 from \( t = 0 \) to \( t = 2 \)
\( u(t) \): unit step function

Step 2: Convolution of pulse with step:
Convolution integral:
\[ y(t) = \int_{-\infty}^{\infty} [u(\tau) - u(\tau - 2)] \cdot u(t - \tau) \, d\tau \]

This results in:
\[ y(t) = \begin{cases} 0, & t < 0 \\ t, & 0 \leq t < 2 \\ 2, & t \geq 2 \end{cases} \]

Convert to unit step form:
\[ y(t) = t \cdot u(t) + (2 - t) \cdot u(t - 2) \]