Poisson Summation Formula

Consider the summation of N complex sinusoids having frequencies uniformly spaced around the unit circle [243]:

$\begin{eqnarray*} x(n) &\mathrel{\stackrel{\Delta}{=}}& \frac{1}{N} \sum_{k=0}^{... ... \\ \end{array} \right. \\ &=& \hbox{\sc IDFT}_n(1 \cdots 1) \end{eqnarray*}$

where $\omega_k \mathrel{\stackrel{\Delta}{=}}2\pi k / N$ .

$\begin{psfrags} % latex2html id marker 22869\psfrag{d(n)}{\LARGE $\displaystyl... ...rain created by a sum of sampled complex sinusoids.} \end{figure} \end{psfrags}$

Setting (the FFT hop size) gives

$\displaystyle \zbox {\sum_m \delta(n-mR) = \frac{1}{R} \sum_{k=0}^{R-1}e^{j\omega_kn}}$

where $\omega_k \mathrel{\stackrel{\Delta}{=}}2\pi k/R$ (harmonics of the frame rate).

Let us now consider these equivalent signals as inputs to an LTI system, with an impulse response given by , and frequency response equal to $W(\omega)$ .

$\begin{psfrags} % latex2html id marker 22896\psfrag{w(n)}{\LARGE $w(n)$\ }\... ...tems theory proof of the Poisson summation formula.} \end{figure} \end{psfrags}$

Looking across the top of Fig.8.18, for the case of input signal $\sum_m \delta(n-mR)$ we have

$\displaystyle y(n) = \sum_m w(n-mR).$

Looking across the bottom of the figure, for the case of input signal

$\displaystyle x(n) = \frac{1}{R} \sum_{k=0}^{R-1}e^{j\omega_kn},$

we have the output signal

$\displaystyle y(n) = \frac{1}{R} \sum_{k=0}^{R-1} W(\omega_k)e^{j\omega_kn}.$

This second form follows from the fact that complex sinusoids $e^{j\omega_kn}$ are eigenfunctions of linear systems--a basic result from linear systems theory [243,242].

Since the inputs were equal, the corresponding outputs must be equal too. This derives the Poisson Summation Formula (PSF):

$\displaystyle \zbox {\underbrace{\sum_m w(n-mR)}_{\hbox{\sc Alias}_R(w)} = \und... ...e}_{\frac{2\pi}{R}}(W)\right]}} \quad \omega_k \isdef \frac{2\pi k}{R} \protect$

(9.4)

Note that the PSF is the Fourier dual of the sampling theorem [247], [243, Appendix G].

The continuous-time PSF is derived in §2.4.14.

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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
    Overlap-Add (OLA) STFT Processing
       Dual of Constant Overlap-Add
          Poisson Summation Formula