# HW1 Solution ECE438 Fall 2013

In this homework, you were asked to start from a table of CT Fourier transforms in terms of $\omega$ in radians and to obtain the corresponding relationships in terms of frequency f (in hertz). Below are the solutions.

To get the justification for each transform/property, click on the corresponding link.

CT Fourier Transform Pairs and Properties (frequency f in hertz per time unit) (info)
(Click title to see explanation on how to obtain the formula in terms of f in hertz)
Definition CT Fourier Transform and its Inverse
Justification CT Fourier Transform $X(f)=\mathcal{F}(x(t))=\int_{-\infty}^{\infty} x(t) e^{-i2\pi ft} dt$
Justification Inverse CT Fourier Transform $\, x(t)=\mathcal{F}^{-1}(X(f))=\int_{-\infty}^{\infty}X(f)e^{i2\pi ft} df \,$
CT Fourier Transform Pairs
x(t) $\longrightarrow$ $X(f)$
Justification CTFT of a unit impulse $\delta (t)\$ $1 \! \$
CTFT of a shifted unit impulse $\delta (t-t_0)\$ $e^{-i2\pi ft_0}$
CTFT of a complex exponential $e^{iw_0t}$ $\delta (f - \frac{\omega_0}{2\pi}) \$
$e^{-at}u(t)\$, where $a\in {\mathbb R}, a>0$ $\frac{1}{a+i2\pi f}$
$te^{-at}u(t)\$, where $a\in {\mathbb R}, a>0$ $\left( \frac{1}{a+i2\pi f}\right)^2$
CTFT of a cosine $\cos(\omega_0 t) \$ $\frac{1}{2} \left[\delta (f - \frac{\omega_0}{2\pi}) + \delta (f + \frac{\omega_0}{2\pi})\right] \$
CTFT of a sine $sin(\omega_0 t) \$ $\frac{1}{2i} \left[\delta (f - \frac{\omega_0}{2\pi}) - \delta (f + \frac{\omega_0}{2\pi})\right]$
CTFT of a rect $\left\{\begin{array}{ll}1, & \text{ if }|t|<T,\\ 0, & \text{else.}\end{array} \right. \$ $\frac{\sin \left(2\pi Tf \right)}{\pi f} \$
CTFT of a sinc $\frac{2 \sin \left( W t \right)}{\pi t } \$ $\left\{\begin{array}{ll}1, & \text{ if }|f| <\frac{W}{2\pi},\\ 0, & \text{else.}\end{array} \right. \$
CTFT of a periodic function $\sum^{\infty}_{k=-\infty} a_{k}e^{ikw_{0}t}$ $\sum^{\infty}_{k=-\infty}a_{k}\delta(f-\frac{kw_{0}}{2\pi}) \$
CTFT of an impulse train $\sum^{\infty}_{n=-\infty} \delta(t-nT) \$ $\frac{1}{T}\sum^{\infty}_{k=-\infty}\delta(f-\frac{k}{T}) \$
CT Fourier Transform Properties
x(t) $\longrightarrow$ $X(f)$
multiplication property $x(t)y(t) \$ $X(f)*Y(f) =\int_{-\infty}^{\infty} X(\theta)Y(f-\theta)d\theta$
convolution property $x(t)*y(t) \!$ $X(f)Y(f) \!$
time reversal $\ x(-t)$ $\ X(-f)$
Other CT Fourier Transform Properties
Parseval's relation $\int_{-\infty}^{\infty} |x(t)|^2 dt = \int_{-\infty}^{\infty} |X(f)|^2 df$

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