10/09 - Fourier Transform: Difference between revisions

Revision as of 17:53, 17 November 2008

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

If we let $T\rightarrow \infty$

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

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

$F^{-1}[F[x(t)]]\,\!$	$=\int _{-\infty }^{\infty }\left[\int _{-\infty }^{\infty }x(\lambda )e^{-j2\pi f\lambda }d\lambda \right]e^{j2\pi ft}df$	$=\int _{-\infty }^{\infty }X(f)e^{j2\pi ft}df$	$=x(t)\,\!$
		$=\int _{-\infty }^{\infty }x(\lambda )\int _{-\infty }^{\infty }e^{j2\pi f(t-\lambda )}dfd\lambda$	$=\int _{-\infty }^{\infty }x(\lambda )\delta (t-\lambda )d\lambda$	$=x(t)\,\!$
	$=\int _{-\infty }^{\infty }\left[\int _{-\infty }^{\infty }x(\lambda )e^{-j\omega \lambda }d\lambda \right]e^{j\omega t}{\frac {1}{2\pi }}d\omega$	$=\int _{-\infty }^{\infty }x(\lambda )\left[{\frac {1}{2\pi }}\int _{-\infty }^{\infty }e^{j(t-\omega )\lambda }d\omega \right]d\lambda$	$=\int _{-\infty }^{\infty }x(\lambda )\delta (t-\omega )d\lambda$	$=x(t)\,\!$

$\int _{-\infty }^{\infty }e^{j2\pi ft}e^{-j2\pi f\lambda }df$	$=\left\langle e^{j2\pi ft}\mid e^{j2\pi ft}\right\rangle _{f}$	$=\delta (t-\lambda )\,\!$
$\int _{-\infty }^{\infty }e^{j2\pi tf}e^{-j2\pi tf_{0}}dt$	$=\left\langle e^{j2\pi tf}\mid e^{j2\pi tf_{0}}\right\rangle _{t}$	$=\delta (f-f_{0})\,\!$

@@ Line 65: / Line 65: @@
 |<math>=\int_{-\infty}^{\infty}x(\lambda) \int_{-\infty}^{\infty}e^{j2\pi f(t-\lambda)} df d\lambda</math>
 |<math>=\int_{-\infty}^{\infty}x(\lambda) \delta(t-\lambda) d\lambda</math>
+|<math>=x(t)\,\!</math>
+|-
+|
+|<math>=\int_{-\infty}^{\infty}\left [ \int_{-\infty}^{\infty} x(\lambda) e^{-j\omega\lambda}d\lambda\right ]e^{j\omega t} \frac{1}{2\pi}d\omega </math>
+|<math>=\int_{-\infty}^{\infty}x(\lambda) \left [ \frac{1}{2\pi} \int_{-\infty}^{\infty}  e^{j(t-\omega) \lambda}d\omega\right ] d\lambda</math>
+|<math>=\int_{-\infty}^{\infty}x(\lambda) \delta(t-\omega) d\lambda</math>
 |<math>=x(t)\,\!</math>
 |}