Homework 9 Solutions, ECE301 Spring 2011 Prof. Boutin
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Question 1
(a) The given signal has finite duration and hence it is not band-limited. (b) The given signal has finite duration and hence it is not band-limited. (c) The given signal has infinite duration and hence it is band-limited. (d) The given signal has infinite duration and hence it is band-limited.
Question 2
$ x(t)=e^{j\pi t}\frac{\sin(\pi t)}{t}=\pi e^{j\pi t}\frac{\sin(\pi t)}{\pi t} $
Using frequency shift property of FT, we get: $ \begin{align} \mathcal{X}(\omega)&=\left\{\begin{array}{ll} \pi, & \text{ for } |\omega - \pi|<\pi \\ 0, & \text{ elsewhere} \end{array} &=\left\{\begin{array}{ll} \pi, & \text{ for } -\pi<\omega-\pi<\pi \\ 0, & \text{ elsewhere} \end{array} &=\left\{\begin{array}{ll} \pi, & \text{ for } 0<\omega<2*\pi \\ 0, & \text{ elsewhere} \end{array} \end{align} $
