10/09 - Fourier Transform: Difference between revisions

Revision as of 14:10, 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 $f={\frac {2\pi n}{T}}\,\!$
$T\,\!$	$\rightarrow \infty$
$\sum _{n=-\infty }^{\infty }{\frac {1}{T}}$	$\rightarrow \int _{-\infty }^{\infty }()df$

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

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

@@ Line 46: / Line 46: @@
 |<math>=X(f)\,\!</math>
 |<math>=\int_{-\infty}^{\infty} x(t) e^{-j2\pi ft}dt</math>
-|<math>=\left \langle x(t) \mid e^{j2\pi ft}\right \rangle</math>
+|<math>=\left \langle x(t) \mid e^{j2\pi ft}\right \rangle_t</math>
 |-
 |<math>F^{-1}[x(t)]\,\!</math>
 |<math>=x(t)\,\!</math>
-|<math>=\int_{-\infty}^{\infty} X(f) e^{j2\pi ft}dt</math>
+|<math>=\int_{-\infty}^{\infty} X(f) e^{j2\pi ft}df</math>
-|<math>=\left \langle X(f) \mid e^{-j2\pi ft}\right \rangle</math>
+|<math>=\left \langle X(f) \mid e^{-j2\pi ft}\right \rangle_f</math>
 |}
 ==Examples==
@@ Line 57: / Line 57: @@
 |-
 |<math>F^{-1}[F[x(t)]]\,\!</math>
+|<math>=\int_{-\infty}^{\infty}\left [ \int_{-\infty}^{\infty} x(t) e^{-j2\pi ft}dt\right ]e^{j2\pi ft} df</math>
-|<math>=</math>
 |-
 |
-|<math>=n^2 + 2n + 1</math>
+|<math>=\int_{-\infty}^{\infty}X(f) e^{j2\pi ft} df</math>
+|-
+|
+|<math>=x(t)\,\!</math>
 |}