Question

Determine the system response y(t) for an LTI system, if x(t) = e⁽⁻²ᵗ⁾u(t) and h(t) = e⁽⁻³ᵗ⁾u(t).

Answer 1

The system response y(t) for an LTI system can be obtained by convolving the input signal x(t) with the impulse response h(t). In this case, the system response y(t) is given by y(t) = (1/5)e⁽⁻³t⁾(e⁽⁻²t⁺³⁾ - 1).

Step-by-step explanation:

The system response y(t) for an LTI system can be obtained by convolving the input signal x(t) with the impulse response h(t). In this case, x(t) = e⁽⁻²ᵗ⁾u(t) and h(t) = e⁽⁻³ᵗ⁾u(t).

Using the convolution integral, we can write the system response y(t) as:

y(t) = ∫[0 to t] x(tau)h(t-tau)dtau

Substituting the given expressions for x(t) and h(t), we have:

y(t) = ∫[0 to t] e⁽⁻²tau⁾e⁽⁻³(t-tau)⁾u(tau)dtau

Simplifying the integral and evaluating it gives the system response y(t) as:

y(t) = (1/5)e⁽⁻³t⁾(e⁽⁻²t⁺³⁾ - 1)