Question

output signal y(t) and input signal x(t) of a continuous LTI sysfem are y(t)=e⁻ᵗ u(t),x(t)=u(t). Find the impulse response h(t).

Answer 1

The impulse response h(t) for the given continuous LTI system is found by differentiating the output signal y(t), which results in h(t) = -e⁻¹ u(t).

Step-by-step explanation:

To find the impulse response h(t) of a continuous Linear Time-Invariant (LTI) system given the output signal y(t) = e⁻¹ u(t) and the input signal x(t) = u(t), one must understand the relationship between the input, output, and impulse response in LTI systems. This relationship is characterized by the convolution integral in the time domain. Since the input signal x(t) is the unit step function u(t), we know the system's response to an impulse δ(t) would be the derivative of the output y(t). Taking the derivative of y(t) yields h(t) = -e⁻¹ u(t), since the derivative of the step function u(t) is the impulse function δ(t) and the derivative of e⁻¹ is -e⁻¹.