# 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].