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

Frequency Shifting

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

Double Sideband Modulation

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

Differentiation in Time

$x(t)\,\!$	$=F^{-1}\left[X(f)\right]$
$F\left[{\frac {dx}{dt}}\right]$	$=F\left[{\frac {d}{dt}}F^{-1}\left[X(f)\right]\right]$
	$=F\left[{\frac {d}{dt}}\int _{-\infty }^{\infty }X(f)\,e^{j\,2\pi f\,t}\,df\right]$
	$=F\left[\int _{-\infty }^{\infty }j\,2\pi fX(f)e^{j\,2\pi f\,t}\,df\right]$
	$=F\left[j\,2\pi f\,F^{-1}[X(f)]\right]$
	$=j\,2\pi f\,X(f)$	Thus ${\frac {dx}{dt}}$ is a linear filter with transfer function $j\,2\pi f$

The Game (frequency domain)

Input	LTI System	Output	Reason
$\delta (t)\,\!$	$\Longrightarrow$	$h(t)\,\!$	Given
$\delta (t)\,e^{-j\,2\,\pi f\,t}$	$\Longrightarrow$	$h(t)\,e^{-j\,2\,\pi f\,t}$	Proportionality
$\int _{-\infty }^{\infty }\delta (t)\,e^{-j\,2\,\pi f\,t}\,dt=F[\delta (t)]=1$	$\Longrightarrow$	$\int _{-\infty }^{\infty }h(t)\,e^{-j\,2\,\pi f\,t}\,dt=F[h(t)]=H(f)$	Superposition
$F[\delta (t-\lambda )]=1\cdot e^{j\,2\,\pi f\,\lambda }$ The notes have $e^{-j\,2\,\pi f\,\lambda }$ Is this an error?	$\Longrightarrow$	$H(f)\cdot e^{j\,2\,\pi f\,\lambda }$	Time Invariance
$x(\lambda )\cdot 1\cdot e^{j\,2\,\pi f\,\lambda }$	$\Longrightarrow$	$x(\lambda )\cdot H(f)\cdot e^{j\,2\,\pi f\,\lambda }$	Proportionality
$\int _{-\infty }^{\infty }x(\lambda )\cdot 1\cdot e^{j\,2\,\pi f\,\lambda }\,d\lambda =X^{-1??}(F)$	$\Longrightarrow$	$\int _{-\infty }^{\infty }x(\lambda )x(\lambda )\cdot H(f)\cdot e^{j\,2\,\pi f\,\lambda }\,d\lambda =X^{-1??}(F)\,H(f)$	Superposition