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

??

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