10/09 - Fourier Transform: Difference between revisions

Revision as of 15:10, 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
	$= T δ_{m, n}$

If we let $T \to \infty$

$F [x (t)]$	$= X (f)$	$= \int_{- \infty}^{\infty} x (t) e^{- j 2 π f t} d t$	$= {⟨ x (t) ∣ e^{j 2 π f t} ⟩}_{t}$
$F^{- 1} [x (t)]$	$= x (t)$	$= \int_{- \infty}^{\infty} X (f) e^{j 2 π f t} d f$	$= {⟨ X (f) ∣ e^{- j 2 π f t} ⟩}_{f}$

$F^{- 1} [F [x (t)]]$	$= \int_{- \infty}^{\infty} [\int_{- \infty}^{\infty} x (t) e^{- j 2 π f t} d t] e^{j 2 π f t} d f$
	$= \int_{- \infty}^{\infty} X (f) e^{j 2 π f t} d f$
	$= x (t)$

@@ 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>=</math>
+|<math>=\int_{-\infty}^{\infty}\left [ \int_{-\infty}^{\infty} x(t) e^{-j2\pi ft}dt\right ]e^{j2\pi ft} df</math>
 |-
 |
-|<math>=n^2 + 2n + 1</math>
+|<math>=\int_{-\infty}^{\infty}X(f) e^{j2\pi ft} df</math>
+|-
+|
+|<math>=x(t)\,\!</math>
 |}