# A Sampling theorem for power signals

Any function $f \in PW_{\pi}$ (where $PW_{\pi}$ is the Paley-Wiener space), can be expanded in terms of the orthonormal basis $\{e^{i n x}\}_{n\in \mathbb Z}$ as $$\hat{f}(x)=\sum_{n\in \mathbb Z}\left\langle\hat{f},e^{i n x}\right\rangle_{L^2(-\pi,\pi)} e^{i n x}$$ where $\langle g, h\rangle_{L^2(-\pi,\pi)}=\frac{1}{2\pi}\int_{-\pi}^\pi g(x) \overline{h(x)} dx$. Taking the inverse Fourier transform in previous equation, we obtain the Whittaker-Kotelnikov-Shannon (WKS) sampling theorem, $$f(x)=\sum_{n\in \mathbb Z}f(n) \operatorname{sinc}(x-n), \ \ \ x\in \mathbb R$$ where $\operatorname{sinc}(x)$ is the normalized sinc function. My question is the following. Sampling theorem, at least in the version that I wrote above, is related to $L^2$ space and so, to scalar product: $$\langle g, h\rangle_{L^2(-\pi,\pi)}=\frac{1}{2\pi}\int_{-\pi}^\pi g(x) \overline{h(x)} dx$$ Instead, usually, for power signals, we define the inner product to be $$\left\langle g\,,\,h\right\rangle = \lim_{T\rightarrow \infty} \frac 1 {2T} \int_{-T}^T g(x)\overline{h(x)}\,dx \tag{1}$$ Is there a version of Sampling theorem related to scalar product (1)? Is it used in signal processing?

References and answers are welcome.

(I premise that I'm not an expert on the signal theory, therefore I do apologise if this question isn't much precise.)

## 1 Answer

Interesting question! The only example I could easily find is this paper.

Casinovi, G., "Sampling and Ergodic Theorems for Weakly Almost Periodic Signals," in Information Theory, IEEE Transactions on, vol.55, no.4, pp.1883-1897, April 2009 doi: 10.1109/TIT.2009.2013021