10/09 - Fourier Transform: Difference between revisions

Revision as of 18:09, 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 $T \to \infty$

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

$\int_{- \infty}^{\infty} e^{j 2 π f t} e^{- j 2 π f λ} d f$	$= {⟨ e^{j 2 π f t} ∣ e^{j 2 π f t} ⟩}_{f}$	$= δ (t - λ)$
$\int_{- \infty}^{\infty} e^{j 2 π t f} e^{- j 2 π t f_{0}} d t$	$= {⟨ e^{j 2 π t f} ∣ e^{j 2 π t f_{0}} ⟩}_{t}$	$= δ (f - f_{0})$

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

@@ Line 54: / Line 54: @@
 |}
 ==Examples==
+{| border="0" cellpadding="0" cellspacing="0"
+|-
+|<math>\int_{-\infty}^{\infty}e^{j2\pi ft}e^{-j2\pi f\lambda}df</math>
+|<math>=\left\langle e^{j2\pi ft}\mid e^{j2\pi ft}\right\rangle_f</math>
+|<math>=\delta(t-\lambda)\,\!</math>
+|-
+|<math>\int_{-\infty}^{\infty}e^{j2\pi tf}e^{-j2\pi tf_0}dt</math>
+|<math>=\left\langle e^{j2\pi tf}\mid e^{j2\pi tf_0}\right\rangle_t</math>
+|<math>=\delta(f-f_0)\,\!</math>
+|}
 {| border="0" cellpadding="0" cellspacing="0"
 |-
@@ Line 75: / Line 86: @@
 ===Sifting property of the delta function===
-{| border="0" cellpadding="0" cellspacing="0"
-|-
-|<math>\int_{-\infty}^{\infty}e^{j2\pi ft}e^{-j2\pi f\lambda}df</math>
-|<math>=\left\langle e^{j2\pi ft}\mid e^{j2\pi ft}\right\rangle_f</math>
-|<math>=\delta(t-\lambda)\,\!</math>
-|-
-|<math>\int_{-\infty}^{\infty}e^{j2\pi tf}e^{-j2\pi tf_0}dt</math>
-|<math>=\left\langle e^{j2\pi tf}\mid e^{j2\pi tf_0}\right\rangle_t</math>
-|<math>=\delta(f-f_0)\,\!</math>
-|}