# 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].