Signals and systems/GF Fourier: Difference between revisions

Revision as of 12:07, 29 October 2006

The Fourier series is used to analyze arbitrary periodic functions by showing them as a composite of sines and cosines.

A function is considered periodic if $x (t) = x (t + T)$ for $T \neq 0$ .

The exponential form of the Fourier series is defined as $x (t) = \sum_{n = - \infty}^{\infty} α_{n} e^{j 2 π n t / T}$

$e^{j θ} = \cos θ + j \sin θ$

$\sin x = \frac{e^{j x} - e^{- j x}}{2 j}$

$\cos x = \frac{e^{j x} + e^{- j x}}{2}$

$⟨ n ∣ m ⟩ = T δ_{n, m}$

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 ==Changing Basis Functions==
-==Notes==
+==Identities==
 <math>e^{j \theta} = \cos \theta + j \sin  \theta \, </math>
+<math>\sin x = \frac{e^{jx}-e^{-jx}}{2j} \,</math>
+<math>\cos x = \frac{e^{jx}+e^{-jx}}{2} \,</math>
 <math> \left \langle n \mid m \right \rangle = T \delta_{n,m} \,</math>