Engineering Tables/DTFT Transform Properties

Property	Time domain $x[n] \!$	Frequency domain $X(\omega) \!$	Remarks
Linearity	$a x[n] + b y[n] \!$	$a X(e^{i \omega}) + b Y(e^{i \omega}) \!$
Shift in time	$x[n - k] \!$	$X(e^{i \omega}) e^{-i \omega k} \!$	integer k
Shift in frequency	$x[n] e^{i a n} \!$	$X(e^{i (\omega-a)}) \!$	real number a
Time reversal	$x[- n] \!$	$X(e^{-i \omega}) \!$
Time conjugation	$x[n]^* \!$	$X(e^{-i \omega})^* \!$
Time reversal & conjugation	$x[-n]^* \!$	$X(e^{i \omega})^* \!$
Derivative in frequency	$\frac{n}{i} x[n] \!$	$\frac{d X(e^{i \omega})}{d \omega} \!$
Integral in frequency	$\frac{i}{n} x[n] \!$	$\int_{-\pi}^{\omega} X(e^{i \vartheta}) d \vartheta \!$
Convolve in time	$x[n] * y[n] \!$	$X(e^{i \omega}) \cdot Y(e^{i \omega}) \!$
Multiply in time	$x[n] \cdot y[n] \!$	$\frac{1}{2 \pi} X(e^{i \omega}) * Y(e^{i \omega}) \!$
Correlation	$\rho_{xy} [n] = x[-n]^* * y[n] \!$	$R_{xy} (\omega) = X(e^{i \omega})^* \cdot Y(e^{i \omega}) \!$

Where:

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