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Revision as of 06:24, 5 April 2012

This Collective table of formulas is proudly sponsored
by the Nice Guys of Eta Kappa Nu.

Visit us at the HKN Lounge in EE24 for hot coffee and fresh bagels only $1 each!

                                         HKNlogo.jpg


Discrete Fourier Transform

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Discrete Fourier Transform Pairs and Properties (info)
Definition Discrete Fourier Transform and its Inverse
Let x[n] be a periodic DT signal, with period N.
N-point Discrete Fourier Transform $ X [k] = \sum_{n=0}^{N-1} x[n]e^{-j 2\pi \frac{k n}{N}} \, $
Inverse Discrete Fourier Transform $ \,x [n] = (1/N) \sum_{k=0}^{N-1} X[k] e^{j 2\pi\frac{kn}{N}} \, $
Discrete Fourier Transform Pairs (info)
$ x[n] \ \text{ (period } N) $ $ \longrightarrow $ $ X_N[k] \ \ (N \text{ point DFT)} $
$ \ \sum_{k=-\infty}^\infty \delta[n+Nk] = \left\{ \begin{array}{ll} 1, & \text{ if } n=0, \pm N, \pm 2N , \ldots\\ 0, & \text{ else.} \end{array}\right. $ $ \ 1 \text{ (period } N) $
$ \ 1 \text{ (period } N) $ $ \ N\sum_{m=-\infty}^\infty \delta[k+Nm] = \left\{ \begin{array}{ll} N, & \text{ if } n=0, \pm N, \pm 2N , \ldots\\ 0, & \text{ else.} \end{array}\right. $
$ \ e^{j2\pi k_0 n} $ $ \ N\delta[((k - k_0))_N] $
$ \ \cos(\frac{2\pi}{N}k_0n) $ $ \ \frac{N}{2}(\delta[((k - k_0))_N] + \delta[((k+k_0))_N]) $
Discrete Fourier Transform Properties
$ x[n] \ $ $ \longrightarrow $ $ X[k] \ $
Linearity $ ax[n]+by[n] \ $ $ aX[k]+bY[k] \ $
Circular Shift $ x[((n-m))_N] \ $ $ X[k]e^{(-j\frac{2 \pi}{N}km)} \ $
Duality $ X[n] \ $ $ NX[((-k))_N] \ $
Multiplication $ x[n]y[n] \ $ $ \frac{1}{N} X[k]\circledast Y[k], \ \circledast \text{ denotes the circular convolution} $
Convolution $ x(t) \circledast y(t) \ $ $ X[k]Y[k] \ $
$ \ x^*[n] $ $ \ X^*[((-k))_N] $
$ \ x^*[((-n))_N] $ $ \ X^*[k] $
$ \ \Re\{x[n]\} $ $ \ X_{ep}[k] = \frac{1}{2}\{X[((k))_N] + X^*[((-k))_N]\} $
$ \ j\Im\{x[n]\} $ $ \ X_{op}[k] = \frac{1}{2}\{X[((k))_N] - X^*[((-k))_N]\} $
$ \ x_{ep}[n] = \frac{1}{2}\{x[n] + x^*[((-n))_N]\} $ $ \ \Re\{X[k]\} $
$ \ x_{op}[n] = \frac{1}{2}\{x[n] - x^*[((-n))_N]\} $ $ \ j\Im\{X[k]\} $
Other Discrete Fourier Transform Properties
Parseval's Theorem $ \sum_{n=0}^{N-1}|x[n]|^2 = \frac{1}{N} \sum_{k=0}^{N-1}|X[k]|^2 $

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