Discrete Fourier transforms (DFT) Pairs and Properties

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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