Problem 1 (15 pts) State whether the following statements are TRUE or FALSE. Provide a brief explanation for each part. (a) (5 pts) A system with the following impulse response function is time- invariant: h(t, au) = e^{-2t-2 au} (sin(t) cos( au) - cos(t) sin( au))u(t - au) (1) (b) (5 pts) A system with the following input-output relationship is causal: y(t) = x(t - 3) + int_{t-3}^{3t} e^{-(t-sigma)}u(t - sigma)x(sigma)dsigma (2) (c) (5 pts) Let z(t) = frac{1}{3}x(t - 2) + 4, where x(t) is a band-limited signal with maximum frequency 100 rad/s. The minimum sampling frequency (according to Nyquist theorem) to sample z(t) is omega_s = 2 imes frac{1}{3} imes 100 = frac{200}{3} rad/s.
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A system is time-invariant if its impulse response function does not depend on time. In this case, the impulse response function depends on both t and T, so the system is not time-invariant. (b) TRUE. A system is causal if the output at any time depends only on Show more…
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