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Embedded SystemsFPGAElectronics

Spectral Audio Signal Processing
    Fourier Transforms for Continuous/Discrete Time/Frequency
       Selected Continuous Fourier Theorems
          Poisson Summation Formula

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Poisson Summation Formula

As shown in §2.4.13 above, the Fourier transform of an impulse train is an impulse train with inversely proportional spacing:

$\displaystyle \psi_R(t)\;\longleftrightarrow\; {\frac{1}{R}}\cdot\psi_{\frac{1}... ...isebox{0.8em}{\rotatebox{-90}{\resizebox{1em}{1em}{\ensuremath{\exists}}}}(Rf) $

where

$\displaystyle \psi_R(t) \isdef \sum_{m=-\infty}^\infty \delta(t-mR) = \frac{1}... ...box{-90}{\resizebox{1em}{1em}{\ensuremath{\exists}}}}\left(\frac{t}{R}\right). $

Using this Fourier theorem, we can derive the continuous-time PSF using the convolution theorem for Fourier transforms:3.5

$\displaystyle \sum_m w(t-mR) = \psi_R \ast w \;\longleftrightarrow \; \Psi_R \cdot W = \frac{1}{R}\cdot\psi_{\frac{1}{R}}\cdot W $

Using linearity and the shift theorem for inverse Fourier transforms, the above relation yields

\begin{eqnarray*} \sum_m w(t-mR) &=& \frac{1}{R} \hbox{\sc IFT}_t \left[W(f)\... ...) \right]\\ [5pt] &=& \frac{1}{R} \sum_k W(f_k)e^{j 2\pi f_k t}. \end{eqnarray*}

We have therefore shown

$\displaystyle \zbox {\sum_{m=-\infty}^{\infty} w(t-mR) = \frac{1}{R} \sum_{k=-\infty}^{\infty} W(f_k)e^{j 2\pi f_k t}, \quad f_k\isdef \frac{k}{R}.} \protect$ (3.10)

Compare this result to Eq.$ \,$(8.4). The left-hand side of (2.10) can be interpreted $ s_R(t)=\hbox{\sc Alias}_R(w)$, i.e., the time-alias of $ w$ on a block of length $ R$. The function $ s_R(t)$ is periodic with period $ R$ seconds. The right-hand side of (2.10) can be interpreted as the inverse Fourier series of $ W(f)$ sampled at invervals of $ 1/R$ Hz. This sampling of $ W(\omega)$ in the frequency domain corresponds to the aliasing of $ w(t)$ in the time domain.

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Previous: Impulse Trains
Next: Sampling Theory
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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