10/09 - Fourier Transform: Difference between revisions

Revision as of 12:56, 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
	$=T\delta _{m,n}\,\!$

@@ Line 3: / Line 3: @@
 |<math>\left \langle e^{j2\pi n t/T} \mid e^{j2\pi m t/T} \right \rangle</math>
 |<math>=\int_{-\infty}^{\infty}e^{j2\pi n t/T} e^{-j2\pi m t/T}\,dt</math>
+|
 |-
 |
 |<math>=\int_{-\infty}^{\infty}e^{j2\pi (n-m) t/T}\,dt</math>
+|
+|-
+|
+|<math>=\int_{-T/2}^{T/2}e^{j2\pi (n-m) t/T}\,dt</math>
+|Assuming the function is perodic with the period T
+|-
+|
+|<math>=T\delta_{m,n}\,\!</math>
+|
 |}
-*This is undefined due to the limits. The notes say that the integral = T, but no limits were defined. You would only know to do -T/2 to T/2 if you knew the function was periodic with period T.
 ==Fourier Transform==