Linearity of the DTFT

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Linearity of the DTFT

$\displaystyle \zbox {\alpha x_1 + \beta x_2 \leftrightarrow \alpha X_1 + \beta X_2}$

$\displaystyle \hbox{\sc DTFT}(\alpha x_1 + \beta x_2) = \alpha\cdot \hbox{\sc DTFT}(x_1) + \beta \cdot\hbox{\sc DTFT}(x_2)$

where $\alpha, \beta$ are any scalars (real or complex numbers),

and

are any two discrete-time signals (real- or complex-valued functions of the integers), and

are their corresponding continuous-frequency spectra defined over the unit circle in the complex plane.

Proof: We have

$\begin{eqnarray*} \hbox{\sc DTFT}_\omega(\alpha x_1 + \beta x_2) & \isdef & \su... ...{-j\omega n}\\ &\isdef & \alpha X_1(\omega) + \beta X_2(\omega) \end{eqnarray*}$

(where $\isdef$ means ``is defined as''). One way to describe the linearity property is to observe that the Fourier transform ``commutes with mixing.''

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Spectral Audio Signal Processing
    Fourier Transforms for Continuous/Discrete Time/Frequency
       Fourier Theorems for the DTFT
          Linearity of the DTFT