10/09 - Fourier Transform: Difference between revisions

Revision as of 13:56, 17 November 2008

$⟨ e^{j 2 π n t / T} ∣ e^{j 2 π m t / T} ⟩$	$= \int_{- \infty}^{\infty} e^{j 2 π n t / T} e^{- j 2 π m t / T} d t$
	$= \int_{- \infty}^{\infty} e^{j 2 π (n - m) t / T} d t$
	$= \int_{- T / 2}^{T / 2} e^{j 2 π (n - m) t / T} d t$	Assuming the function is perodic with the period T
	$= T δ_{m, n}$

@@ 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==