Question

A linear time-invariant system has frequency response H(f)= 1/1+j2πfand input x(t)=e⁻⁽ᵗ⁻¹⁾ u(t−1). Determine the system's output y(t)=x(t)∗h(t).

Answer 1

To determine the system's output y(t) from the given information, we need to convolve the input signal x(t) with the impulse response h(t). The impulse response can be found by taking the inverse Fourier transform of the frequency response. Using the given frequency response, we can find that the impulse response is constant and non-zero at t = 0, resulting in an output signal y(t) that is the same as the input signal x(t) for t ≥ 0.

Step-by-step explanation:

To determine the system's output y(t), we need to convolve the input signal x(t) with the impulse response h(t). The impulse response h(t) can be found by taking the inverse Fourier transform of the frequency response H(f):

h(t) = Inverse Fourier Transform [H(f)]

Using the given frequency response H(f) = 1 / (1 + j2πf), we can substitute f = 0 into H(f) to find the impulse response at t = 0:

h(t = 0) = 1 / (1 + j2π(0)) = 1.

Since the impulse response h(t) is constant, the output signal y(t) will be the same as the input signal x(t), but with an additional delay equal to the time at which the impulse response is non-zero. In this case, the impulse response h(t) is non-zero at t = 0. Therefore, y(t) = x(t) for t ≥ 0.