10/09 - Fourier Transform

$\left\langle e^{j2\pi nt/T}\mid e^{j2\pi mt/T}\right\rangle$	$=\int _{-\infty }^{\infty }e^{j2\pi nt/T}e^{-j2\pi mt/T}\,dt$
	$=\int _{-\infty }^{\infty }e^{j2\pi (n-m)t/T}\,dt$
	$=\int _{-T/2}^{T/2}e^{j2\pi (n-m)t/T}\,dt$	Assuming the function is perodic with the period T
	$=T\delta _{m,n}\,\!$

Fourier Transform

Remember from 10/02 - Fourier Series

• $\alpha _{m}={\frac {1}{T}}\int _{-T/2}^{T/2}x(t)e^{-j2\pi mt/T}\,dt$
• $x(t)=x(t+T)=\sum _{n=-\infty }^{\infty }\alpha _{m}e^{j2\pi m/T}$

If we let $T\rightarrow \infty$

${\frac {1}{T}}$	$\rightarrow df$
${\frac {n}{T}}$	$\rightarrow f$	Remember $f={\frac {2\pi n}{T}}\,\!$
$T\,\!$	$\rightarrow \infty$
$\sum _{n=-\infty }^{\infty }{\frac {1}{T}}$	$\rightarrow \int _{-\infty }^{\infty }()df$

Definitions

$F[x(t)]\,\!$	$=X(f)\,\!$	$=\int _{-\infty }^{\infty }x(t)e^{-j2\pi ft}dt$	$=\left\langle x(t)\mid e^{j2\pi ft}\right\rangle$
$F^{-1}[x(t)]\,\!$	$=x(t)\,\!$	$=\int _{-\infty }^{\infty }X(f)e^{j2\pi ft}dt$	$=\left\langle X(f)\mid e^{-j2\pi ft}\right\rangle$