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)=? 2

Answer 1

To find the output y(t) of the continuous-time LTI system for the given input x(t), compute the convolution of x(t) with the impulse response h(t). The output is determined by integrating the product of the input with the shifted impulse response over their overlapping range.

Step-by-step explanation:

The student question relates to the output of a continuous-time linear time-invariant (LTI) system given an input signal and the system's impulse response. The output y(t) of the system is the convolution of x(t) with h(t). Since the input x(t) = 4 is a constant signal from t = 0 to t = 10 and zero elsewhere, the convolution integral simplifies as we only need to consider the region where x(t) is non-zero.

The impulse response given is h(t) = e^{-3(t-1)}u(t-1), where u(t-1) is the unit step function that starts at t = 1. To compute the output y(t), the convolution integral can be set up and solved over the range where x(t) and h(t) overlap. Therefore, y(t) = x(t) * h(t) will involve integrating the product of the input and shifted impulse response over the valid range of t.