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

Properties of the Fourier Transform

Linearity

$F\left[a\,x(t)+b\,x(t)\right]$	$=\int _{-\infty }^{\infty }\left[a\,x(t)+b\,x(t)\right]e^{-j\,2\pi f\,t}\,dt$
	$=a\int _{-\infty }^{\infty }\,x(t)\,e^{-j\,2\pi f\,t}\,dt+b\int _{-\infty }^{\infty }\,x(t)\,e^{-j\,2\pi f\,t}\,dt$
	$=a\,F[x(t)]+b\,F[x(t)]$

Time Invariance (Delay)

$F[x(t-t_{0})]\,\!$	$=\int _{-\infty }^{\infty }x(t-t_{0})\,e^{-j\,2\pi f\,t}\,dt$	Let $u=t-t_{0}\,\!$ and $du=dt\,\!$
	$=\int _{-\infty }^{\infty }x(u)\,e^{-j\,2\pi f\,(u+t_{0})}\,du$
	$=e^{-j\,2\pi f\,t_{0}}\int _{-\infty }^{\infty }x(u)\,e^{-j\,2\pi f\,u}\,du$
	$=e^{-j\,2\pi f\,t_{0}}\,F[x(t)]$

??

$F\left[e^{j\,2\pi f\,t}x(t)\right]$	$=\int _{-\infty }^{\infty }\left[e^{j\,2\pi f_{0}\,t}x(t)\right]e^{-j\,2\pi f\,t}\,dt$
	$=\int _{-\infty }^{\infty }x(t)\,e^{-j\,2\pi (f-f_{0})\,t}\,dt$
	$=X(f-f_{0})\,\!$

??

$F[cos(2\pi f_{0}\,t)\cdot x(t)]$	$=\int _{-\infty }^{\infty }{\frac {e^{j\,2\pi f_{0}\,t}+e^{-j\,2\pi f_{0}\,t}}{2}}x(t)\,e^{-j\,2\pi f\,t}\,dt$
	$={\frac {1}{2}}\int _{-\infty }^{\infty }x(t)\left[e^{-j\,2\pi (f-f_{0})\,t}+e^{-j\,2\pi (f+f_{0})\,t}\right]\,dt$
	$={\frac {1}{2}}X(f-f_{0})+{\frac {1}{2}}X(f+f_{0})$