Engineering Tables/DTFT Transform Properties

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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].