Poisson Summation Formula - Distributional Formulation

Distributional Formulation

These equations can be interpreted in the language of distributions (Córdoba 1988; Hörmander 1983, §7.2) for a function or distribution, whose derivatives are all rapidly decreasing (see Schwartz function). Using the Dirac comb distribution and its Fourier series:

(Eq.5)

Eq.1 readily follows:

$\begin{align} \sum_{k=-\infty}^\infty \hat f(k) &= \sum_{k=-\infty}^\infty \left(\int_{-\infty}^{\infty} f(x)\ e^{-i 2\pi k x} dx \right) = \int_{-\infty}^{\infty} f(x) \underbrace{\left(\sum_{k=-\infty}^\infty e^{-i 2\pi k x}\right)}_{\sum_{n=-\infty}^\infty \delta(x-n)} dx \\ &= \sum_{n=-\infty}^\infty \left(\int_{-\infty}^{\infty} f(x)\ \delta(x-n)\ dx \right) = \sum_{n=-\infty}^\infty f(n). \end{align}$

Similarly:

$\begin{align} \sum_{k=-\infty}^{\infty} \hat s(\nu + k/T) &= \sum_{k=-\infty}^{\infty} \mathcal{F}\left \{ s(t)\cdot e^{-i 2\pi\frac{k}{T}t}\right \}\\ &= \mathcal{F} \bigg \{s(t)\underbrace{\sum_{k=-\infty}^{\infty} e^{-i 2\pi\frac{k}{T}t}}_{T \sum_{n=-\infty}^{\infty} \delta(t-nT)}\bigg \} = \mathcal{F}\left \{\sum_{n=-\infty}^{\infty} T\cdot s(nT) \cdot \delta(t-nT)\right \}\\ &= \sum_{n=-\infty}^{\infty} T\cdot s(nT) \cdot \mathcal{F}\left \{\delta(t-nT)\right \} = \sum_{n=-\infty}^{\infty} T\cdot s(nT) \cdot e^{-i 2\pi nT \nu}. \end{align}$

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