Question

Consider the continuous-time LTI system with impulse response

h(t)=e⁻³⁽ᵗ⁻¹⁾ u(t−1)
Calculate the output of this system to the input
x(t)=4,0≤t≤10;0, otherwise.
y(t)=x(t) ∗ h(t)=?

Answer 1

The output y(t) of a continuous-time LTI system is calculated by using the convolution integral between the input signal x(t) and the impulse response h(t), taking into account the limits of integration set by the non-zero durations of x(t) and h(t).

Step-by-step explanation:

Output of a Continuous-Time LTI System

To calculate the output y(t) of a continuous-time LTI system with impulse response h(t) = e⁻³⁵(t−1) u(t−1) and input x(t) = 4 for 0 ≤ t ≤ 10; 0 otherwise, we use the convolution integral.

The formula for output response y(t) = x(t) * h(t) involves integrating over all possible values of τ, where the input and shifted impulse response overlap. The system has an impulse response that starts at t = 1, thus the output will also be zero for t < 1. We integrate from τ = max(1, t-10) to τ = t to account for the non-zero duration of x(t). Due to the Heaviside step function in h(t), the integral will be zero before t = 1.

The output is calculated as:

1. Substitute x(t) and h(t) into the convolution integral.
2. Perform the integration within the bounds determined by the intersection of the durations of x(t) and h(t).