1. Let's try some starter convolutions to get the swing of how to perform continuous con- volutions. For each problem find y(t)=x(t)*h(t) and report on what kind of filter h(t) is. (a) x(t) = -3e^{-t-3}u(t), h(t) = \delta(t + 3)
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PROBLEM 4 Determine and sketch the convolution of the following two signals x(t) = {t+1 0 ≤ t ≤ 1, 2-t 1 < t ≤ 2, 0 otherwise} h(t) = ͆(t+2) - 2͆(t) PROBLEM 5 Suppose that x(t) = {1 0 ≤ t ≤ 1, 0 otherwise}. h(t) = x(2t) a. Determine and sketch y(t) = x(t) * h(t) b. Determine d/dt y(t). How many discontinuities does d/dt y(t) have? PROBLEM 6 Consider a signal x(t) and the impulse response h(t) given by x(t) = u(t-2) - u(t-4) and h(t) = e^-2tu(t) a. Compute y(t) = x(t) * h(t) b. Compute g(t) = (d/dt x(t)) * h(t) c. How is g(t) related to y(t)?
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Problem 1 (20 Points). A system with impulse response h(t), input x(t) and output y(t) can be modeled using the convolution integral y(t) = h(t) * x(t) = ∫ (τ=-∞ to ∞) x(τ)h(t - τ)dτ (1) Consider the system with the input-output relationship y(t) = x(t - 1) (2) a) Determine whether this system is 1) linear, 2) causal, and 3) time-invariant b) Determine the impulse response h(t) of the system. c) Obtain the system response to the input x(t) = e^{jt}.
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Consider a Linear and Time-Invariant System (LTI) whose impulse response is given by h(t). Find the output y(t) corresponding to the input x(t) using the Convolution Integral x(t) = 2[u(t) - u(t - 3)] + [u(t - 3) - u(t - 5)] h(t) = e^{3t}[u(t) - u(t - 3)]
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