Question

Exercise 3. TWO WAYS TO CALCULATE LTI OUTPUTS Recall the two complementary approaches to calculating the output of an LTI system. • Directly compute the convolution $y(t) = x(t) * h(t)$. • Compute the convolution in frequency domain $y(t) = \mathcal{F}^{-1}(\mathcal{F}(x(t))\mathcal{F}(h(t)))$. Based on the impulse response $h(t)$ and the input $x(t)$, sometimes one method will be easier than the other. You need to decide when to use which. Determine when to use which method for the following situations, and find the output $y(t)$. (a) $x(t) = e^{-t}u(t)$, $h(t) = u(t) - u(t - 2)$. (b) $x(t) = e^{-t}u(t)$, $h(t) = e^{-2t}u(t)$. (c) $x(t) = \cos(3t)$, $h(t) = \text{sinc}(t)$.

          Exercise 3. TWO WAYS TO CALCULATE LTI OUTPUTS
Recall the two complementary approaches to calculating the output of an LTI system.
• Directly compute the convolution $y(t) = x(t) * h(t)$.
• Compute the convolution in frequency domain $y(t) = \mathcal{F}^{-1}(\mathcal{F}(x(t))\mathcal{F}(h(t)))$.
Based on the impulse response $h(t)$ and the input $x(t)$, sometimes one method will be easier than the other.
You need to decide when to use which. Determine when to use which method for the following situations,
and find the output $y(t)$.
(a) $x(t) = e^{-t}u(t)$, $h(t) = u(t) - u(t - 2)$.
(b) $x(t) = e^{-t}u(t)$, $h(t) = e^{-2t}u(t)$.
(c) $x(t) = \cos(3t)$, $h(t) = \text{sinc}(t)$.

Exercise 3. TWO WAYS TO CALCULATE LTI OUTPUTS
Recall the two complementary approaches to calculating the output of an LTI system.
• Directly compute the convolution y(t) = x(t) * h(t).
• Compute the convolution in frequency domain y(t) = ℱ^-1(ℱ(x(t))ℱ(h(t))).
Based on the impulse response h(t) and the input x(t), sometimes one method will be easier than the other.
You need to decide when to use which. Determine when to use which method for the following situations,
and find the output y(t).
(a) x(t) = e^-tu(t), h(t) = u(t) - u(t - 2).
(b) x(t) = e^-tu(t), h(t) = e^-2tu(t).
(c) x(t) = cos(3t), h(t) = sinc(t).

Added by Jeremy Y.

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