# Engineering Tables/DTFT Transform Properties

Property Time domain
$x[n] \!$
Frequency domain
$X(\omega) \!$
Remarks
Linearity $a x[n] + b y[n] \!$ $a X(e^{i \omega}) + b Y(e^{i \omega}) \!$
Shift in time $x[n - k] \!$ $X(e^{i \omega}) e^{-i \omega k} \!$ integer k
Shift in frequency $x[n] e^{i a n} \!$ $X(e^{i (\omega-a)}) \!$ 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 $\frac{n}{i} x[n] \!$ $\frac{d X(e^{i \omega})}{d \omega} \!$
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].