View source for Engineering Tables/DTFT Transform Properties

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{| class="wikitable"
|-
! Property
! Time domain <br /><math>x[n] \!</math>
! Frequency domain <br /> <math>X(\omega) \!</math>
! Remarks
|-
| Linearity
| <math>a x[n] + b y[n] \!</math>
| <math> a X(e^{i \omega}) + b Y(e^{i \omega}) \!</math>
|
|-
| Shift in time
| <math>x[n - k] \!</math>
| <math>X(e^{i \omega}) e^{-i \omega k} \!</math>
| integer ''k''
|-
| Shift in frequency
| <math>x[n] e^{i a n} \!</math>
| <math>X(e^{i (\omega-a)}) \!</math>
| real number ''a''
|-
| Time reversal
| <math>x[- n] \!</math>
| <math>X(e^{-i \omega}) \!</math> 
|
|-
| Time conjugation
| <math>x[n]^* \!</math>
| <math>X(e^{-i \omega})^* \!</math>
|
|-
| Time reversal & conjugation
| <math>x[-n]^* \!</math>
| <math>X(e^{i \omega})^* \!</math>
|
|-
| Derivative in frequency
| <math>\frac{n}{i} x[n] \!</math>
| <math>\frac{d X(e^{i \omega})}{d \omega} \!</math>  
|
|-
| Integral in frequency
| <math>\frac{i}{n} x[n] \!</math> 
| <math>\int_{-\pi}^{\omega} X(e^{i \vartheta}) d \vartheta \!</math>
|
|-
| Convolve in time
| <math>x[n] * y[n] \!</math>
| <math>X(e^{i \omega}) \cdot Y(e^{i \omega}) \!</math>
|
|-
| Multiply in time
| <math>x[n] \cdot y[n] \!</math>
| <math>\frac{1}{2 \pi} X(e^{i \omega}) * Y(e^{i \omega}) \!</math>
|
|-
| Correlation
| <math>\rho_{xy} [n] = x[-n]^* * y[n] \!</math>
| <math>R_{xy} (\omega) = X(e^{i \omega})^* \cdot Y(e^{i \omega}) \!</math> 
|
|}

'''Where:'''

* <math>*\!</math> is the convolution between two signals
* <math>x[n]^*\!</math> is the complex conjugate of the function ''x[n]''
* <math>\rho_{xy} [n]\!</math> represents the correlation between ''x[n]'' and ''y[n]''.