Homework 1 Solution, ECE438, Fall 2014, Prof. Boutin

A complex exponential

$ x(t)=e^{j2 \pi f_0 t} $

From table, $ e^{j\omega_0t} \leftrightarrow 2\pi \delta(\omega - \omega_0) $, therefore
$ \begin{align} e^{j2\pi f_0 t } \leftrightarrow &2\pi \delta(2\pi f - 2\pi f_0) \\ &=\delta(f - f_0) \end{align} $
Where the last line is by the scaling property of the delta function.

A sine

$ x(t)=sin(t) $

A cosine

$ x(t)=cos(t) $

A periodic function

$ x(t)=x(t-T) $

An impulse train

$ x(t)=\sum_{n=-\infty}^{\infty} \delta (t-nT) $

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