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4. (a) Consider an LTI system with an impulse response h(t) = -2\delta(t) + 5e^{-2t}u(t). Without using convolution, find the output when the input is x(t) = \cos(t). (b) Consider an LTI system with an impulse response h(t) = u(t+1) - u(t-3). Let the input be given by x(t) = e^{-(t-2)}u(t-2). Verify the convolution property of CTFT by doing the following: i. Convolve: y(t) = x(t) \ast h(t), and then find Y(j\omega). ii. Compute: X(j\omega)H(j\omega), and verify it is equal to Y(j\omega).

          4. (a) Consider an LTI system with an impulse response
h(t) = -2\delta(t) + 5e^{-2t}u(t).
Without using convolution, find the output when the input is
x(t) = \cos(t).
(b) Consider an LTI system with an impulse response
h(t) = u(t+1) - u(t-3).
Let the input be given by
x(t) = e^{-(t-2)}u(t-2).
Verify the convolution property of CTFT by doing the following:
i. Convolve: y(t) = x(t) \ast h(t), and then find Y(j\omega).
ii. Compute: X(j\omega)H(j\omega), and verify it is equal to Y(j\omega).

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4. (a) Consider an LTI system with an impulse response
h(t) = -2δ(t) + 5e^-2tu(t).
Without using convolution, find the output when the input is
x(t) = cos(t).
(b) Consider an LTI system with an impulse response
h(t) = u(t+1) - u(t-3).
Let the input be given by
x(t) = e^-(t-2)u(t-2).
Verify the convolution property of CTFT by doing the following:
i. Convolve: y(t) = x(t) ∗h(t), and then find Y(jω).
ii. Compute: X(jω)H(jω), and verify it is equal to Y(jω).

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University Physics with Modern Physics

Hugh D. Young 14th Edition

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Solved by Expert Adi S

Step 1

We know that the output of an LTI system to a sum of inputs is equal to the sum of the outputs to each individual input. Let's break down the input x(t)cos(t) into two parts: x(t) and cos(t). The output to x(t) can be found by convolving x(t) with the impulse Show more…

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(a) Consider an LTI system with an impulse response h(t) = -28t + Se^(-2t)u(t). Without using convolution, find the output when the input is x(t)cos(t). b) Consider an LTI system with an impulse response h(t) = u(t + 1) - u(t - 3). Let the input be given by 2(t) = e^-(t-2)u(t - 2). Verify the convolution property of CTFT by doing the following: Convolve: y(t) = r(t) * h(t), and then find Y(jw). ii_ Compute: X(jw)H(jw), and verify it is equal to Y(jw).

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