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	$\frac{i}{n}x[n]$	$\underset{-\pi }{\overset{\omega }{\int }}X(e^{i\vartheta })d\vartheta$
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].