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

$f(n+1)$	$=(n+1)^{2}$
	$=n^{2}+2n+1$