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Poisson Summation Formula

Consider the summation of N complex sinusoids having frequencies uniformly spaced around the unit circle [263]:

\begin{eqnarray*}
x(n) &\mathrel{\stackrel{\Delta}{=}}& \frac{1}{N} \sum_{k=0}^{N-1}e^{j\omega_kn} =
\left\{
\begin{array}{ll}
1 & n=0 \quad (\hbox{\sc mod}\ N) \\
0 & \mbox{elsewhere} \\
\end{array} \right. \\ [5pt]
&=& \hbox{\sc IDFT}_n(1 \cdots 1)
\end{eqnarray*}

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


\begin{psfrags}
% latex2html id marker 22678\psfrag{d(n)}{\normalsize $\displaystyle\normalsize \sum_l\delta(n-lN)$\ }\psfrag{\makebox[0pt][l]{2}N}{\normalsize $-2N$}\psfrag{\makebox[0pt][l]{N}}{\normalsize $-N$}\psfrag{0}{\normalsize $0$}\psfrag{N}{\normalsize $N$}\psfrag{n}{\normalsize $n$}\psfrag{2N}{\normalsize $2N$}\psfrag{1}{\normalsize $1$}\begin{figure}[htbp]
\includegraphics[width=3.5in]{eps/delta}
\caption{Discrete-time impulse train created
by a sum of sampled complex sinusoids.}
\end{figure}
\end{psfrags}

Setting $ N=R$ (the FFT hop size) gives

$\displaystyle \zbox {\sum_m \delta(n-mR) = \frac{1}{R} \sum_{k=0}^{R-1}e^{j\omega_kn}}$ (9.26)

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 $ w(n)$ , and frequency response equal to $ W(\omega)$ .


\begin{psfrags}
% latex2html id marker 22707\psfrag{w(n)}{\normalsize $w(n)$\ }\psfrag{W(w)}{\normalsize $W(\omega)$\ }\psfrag{timesum}{\normalsize $\displaystyle\sum_l\delta(n-lR)$\ }\psfrag{freqsum}{\normalsize $\frac{1}{R}\displaystyle\sum_k e^{j\omega_kn}$\ }\psfrag{t}{\normalsize $\displaystyle\sum_l w(n-lR)$\ }\psfrag{f}{\normalsize $\frac{1}{R}\displaystyle\sum_k W(\omega_k)e^{j\omega_kn}$\ }\psfrag{time}{\normalsize Time}\psfrag{freq}{\normalsize Frequency}\begin{figure}[htbp]
\includegraphics[width=4in]{eps/poisson}
\caption{Linear systems theory proof of the Poisson summation formula.}
\end{figure}
\end{psfrags}

Looking across the top of Fig.8.16, for the case of input signal $ \sum_m \delta(n-mR)$ we have

$\displaystyle y(n) = \sum_m w(n-mR).$ (9.27)

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

$\displaystyle x(n) = \frac{1}{R} \sum_{k=0}^{R-1}e^{j\omega_kn},$ (9.28)

we have the output signal

$\displaystyle y(n) = \frac{1}{R} \sum_{k=0}^{R-1} W(\omega_k)e^{j\omega_kn}.$ (9.29)

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 [263,262].

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

$\displaystyle \zbox {\underbrace{\sum_m w(n-mR)}_{\hbox{\sc Alias}_R(w)} = \underbrace{\frac{1}{R}\sum_{k=0}^{R-1} W(\omega_k)e^{j\omega_k n}}_{\hbox{\sc DFT}_R^{-1} \left[\hbox{\sc Sample}_{\frac{2\pi}{R}}(W)\right]}} \quad \omega_k \isdef \frac{2\pi k}{R} \protect$ (9.30)

Note that the PSF is the Fourier dual of the sampling theorem [269], [263, Appendix G].

The continuous-time PSF is derived in §B.15.


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