Engineering Tables/DTFT Transform Properties

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

Where:

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