10/10,13,16,17 - Fourier Transform Properties

Properties of the Fourier Transform

Linearity

$F[ax(t)+bx(t)]$	$=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}[ax(t)+bx(t)]e^{-j2\pi ft}dt$
	$=a\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x(t)e^{-j2\pi ft}dt+b\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x(t)e^{-j2\pi ft}dt$
	$=aF[x(t)]+bF[x(t)]$

Time Invariance (Delay)

$F[x(t-t_0)]$	$=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x(t-t_0)e^{-j2\pi ft}dt$	Let $u=t-t_0$ and $du=dt$
	$=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x(u)e^{-j2\pi f(u+t_0)}du$
	$=e^{-j2\pi f{t}_0}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x(u)e^{-j2\pi fu}du$
	$=e^{-j2\pi f{t}_0}F[x(t)]$

??

$F[e^{j2\pi ft}x(t)]$	$=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}[e^{j2\pi f_{0}t}x(t)]e^{-j2\pi ft}dt$
	$=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x(t)e^{-j2\pi (f-f_0)t}dt$
	$=X(f-f_0)$

??

$F[cos(2\pi f_{0}t)\cdot x(t)]$	$=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{e^{j2\pi f_{0}t}+e^{-j2\pi f_{0}t}}{2}x(t)e^{-j2\pi ft}dt$
	$=\frac{1}{2}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x(t)[e^{-j2\pi (f-f_0)t}+e^{-j2\pi (f+f_0)t}]dt$
	$=\frac{1}{2}X(f-f_0)+\frac{1}{2}X(f+f_0)$

??

$x(t)$	$=F^{-1}[X(f)]$
$F\left[\frac{dx}{dt}\right]$	$=F\left[\frac{d}{dt}{F}^{-1}[X(f)]\right]$
	$=F[\frac{d}{dt}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}X(f)e^{j2\pi ft}df]$
	$=F[\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}j2\pi fX(f)e^{j2\pi ft}df]$
	$=F[j2\pi f{F}^{-1}[X(f)]]$
	$=j2\pi fX(f)$	Thus $\frac{dx}{dt}$ is a linear filter with transfer function $j2\pi f$