# proof of sampling theorem

## 0.1 Set-up

Let $w>0$ be the (two-sided) bandwidth. The variable $\xi$ below will denote frequency, and the variable $t$ will denote time. (Both $w$ and $\xi$ are measured in .)

Consider the space of functions:

 $\mathcal{H}^{\prime}=\{g\in\mathbf{L}^{2}(\mathbb{R})\colon g(\xi)=0\text{ for% almost all }\lvert\xi\rvert>w/2\}$

which is clearly seen to be a complex Hilbert space with the usual inner product for $\mathbf{L}^{2}(\mathbb{R})$.

Let $\mathcal{F}$ denote the Fourier transform on $\mathbf{L}^{2}(\mathbb{R})$, which is a unitary transform by Plancherel’s theorem. So,

 $\mathcal{H}=\{f\in\mathbf{L}^{2}(\mathbb{R})\colon(\mathcal{F}f)(\xi)=0\text{ % for almost all }\lvert\xi\rvert>w/2\}=\mathcal{F}^{-1}\mathcal{H}^{\prime}$

is also a Hilbert space.

## 0.2 Computation of orthonormal basis

One orthonormal basis for $\mathcal{H}^{\prime}$ consists of the usual Fourier functions on the interval $[-w/2,w/2]$, extended to be zero on $\mathbb{R}\setminus[-w/2,w/2]$:

 $\phi_{n}(\xi)=\begin{cases}\frac{1}{\sqrt{w}}e^{-2\pi in\xi/w}\,,&\lvert\xi% \rvert\leq w/2\\ 0\,,&\lvert\xi\rvert>w/2\,,\end{cases}\quad n\in\mathbb{Z}\,.$

Mapping these by $\mathcal{F}^{-1}$ produces an orthonormal basis for $\mathcal{H}$:

 $\displaystyle(\mathcal{F}^{-1}\phi_{n})(t)$ $\displaystyle=\frac{1}{\sqrt{w}}\int_{-w/2}^{w/2}e^{-2\pi in\xi/w}\,e^{2\pi i% \xi t}\,d\xi$ $\displaystyle=\frac{1}{\sqrt{w}}\int_{-w/2}^{w/2}e^{2\pi i\xi(t-n/w)}\,d\xi$ $\displaystyle=\frac{1}{\sqrt{w}}\bigl{(}w\,\operatorname{sinc}(w(t-n/w))\bigr{% )}=\sqrt{w}\,\operatorname{sinc}(wt-n)\,,$

where we have used the fact that the Fourier transform of $t\mapsto w\,\operatorname{sinc}(wt)$ (normalized sinc function) is the rectangular box function of bandwidth $w$, and vice versa.

## 0.3 Expansion by orthonormal basis

Given $f\in\mathcal{H}$, let $g=\mathcal{F}f\in\mathcal{H}^{\prime}$. We can expand $g$ in a Fourier series with respect to the $\{\phi_{n}\}$:

 $g(\xi)=\sum_{n\in\mathbb{Z}}\langle g,\phi_{n}\rangle\,\phi_{n}(\xi)\,,$

with the infinite sum converging in $\mathbf{L}^{2}(\mathbb{R})$. Taking $\mathcal{F}^{-1}$ of both sides, we obtain:

 $f(t)=(\mathcal{F}^{-1}g)(t)=\sum_{n\in\mathbb{Z}}\langle g,\phi_{n}\rangle\,(% \mathcal{F}^{-1}\phi_{n})(t)=\sqrt{w}\sum_{n\in\mathbb{Z}}\langle g,\phi_{n}% \rangle\,\operatorname{sinc}(wt-n)\,.$

Moreover,

 $\langle g,\phi_{n}\rangle=\frac{1}{\sqrt{w}}\int_{-\infty}^{\infty}g(\xi)\,e^{% 2\pi in\xi/w}\,d\xi=\frac{1}{\sqrt{w}}(\mathcal{F}^{-1}g)\Bigl{(}\frac{n}{w}% \Bigr{)}=\frac{1}{\sqrt{w}}f\Bigl{(}\frac{n}{w}\Bigr{)}\,.$

(Since $g$ is also in $\mathbf{L}^{1}$, its inverse Fourier transform $\mathcal{F}^{-1}g$ is a continuous function. Provided that we modify $f$ on a set of measure zero, we can assume that $f=\mathcal{F}^{-1}(\mathcal{F}f)=\mathcal{F}^{-1}g$ is continuous. So it is legal to talk about the pointwise values $f(n/w)$.)

## 0.4 Result

Hence, we arrive at the representation:

 $f(t)=\sum_{n\in\mathbb{Z}}f\Bigl{(}\frac{n}{w}\Bigr{)}\,\operatorname{sinc}(wt% -n)\,,$

thereby reconstructing any $f\in\mathcal{H}$ — a square-integrable band-limited function — from its samples at every time period of length $1/w$.

## 0.5 Uniform and absolute convergence of series

The infinite series for $f$ converges in $\mathbf{L}^{2}$ by construction, but in fact it also converges uniformly and absolutely. To see this, first note that by the Cauchy-Schwarz inequality,

 $\sum_{n\in\mathbb{Z}}\lvert f(\tfrac{n}{w})\rvert\,\lvert\operatorname{sinc}(% wt-n)\rvert\leq\Bigl{(}\sum_{n\in\mathbb{Z}}\lvert f(\tfrac{n}{w})\rvert^{2}% \Bigr{)}^{1/2}\Bigl{(}\sum_{n\in\mathbb{Z}}\operatorname{sinc}^{2}(wt-n)\Bigr{% )}^{1/2}\,.$

The series $\sum_{n}\lvert f(n/w)\rvert^{2}$ converges by Parseval’s theorem ($w^{-1/2}f(n/w)$ are the Fourier coefficients of $g$). Also, the series $\sum_{n}\operatorname{sinc}^{2}(wt-n)$ is uniformly bounded for all $t\in\mathbb{R}$. To prove this, it suffices to restrict to $t$ bounded inside $[0,1/w]$ as the function $t\mapsto\sum_{n}\operatorname{sinc}^{2}(wt-n)$ is $1/w$-periodic; and then it becomes an easy estimate using the fact that $\sum_{n}n^{-2}<\infty$. It follows that the series $\sum_{n}\lvert f(n/w)\rvert\lvert\operatorname{sinc}(wt-n)\rvert$ is uniformly bounded for all $t$, and its tail tends to zero uniformly in $t$.

## 0.6 Illustrations

• http://aux.planetmath.org/files/objects/8650/sample.pyPython program to produce the three figures

Title proof of sampling theorem ProofOfSamplingTheorem 2013-03-22 16:28:57 2013-03-22 16:28:57 stevecheng (10074) stevecheng (10074) 13 stevecheng (10074) Proof msc 42A38 msc 94A20 PlancherelsTheorem