Engineering Tables/DTFT Transform Properties

Property	Time domain $x[n]\!$	Frequency domain $X(\omega )\!$	Remarks
Linearity	$ax[n]+by[n]\!$	$aX(e^{i\omega })+bY(e^{i\omega })\!$
Shift in time	$x[n-k]\!$	$X(e^{i\omega })e^{-i\omega k}\!$	integer k
Shift in frequency	$x[n]e^{ian}\!$	$X(e^{i(\omega -a)})\!$	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	${\frac {n}{i}}x[n]\!$	${\frac {dX(e^{i\omega })}{d\omega }}\!$
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:

• $*\!$ is the convolution between two signals
• $x[n]^{*}\!$ is the complex conjugate of the function x[n]
• $\rho _{xy}[n]\!$ represents the correlation between x[n] and y[n].