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

Figure 1: Reconstructing a Bessel function (of the first kind), using a $\operatorname{sinc}$ series with $n=-10,-9\ldots,0,\ldots,10$ . Theoretically exact bandwidth is $w=1/\pi\approx 0.32$ .

Dlmf — Figure 1: Reconstructing a Bessel function (of the first kind), using a $\operatorname{sinc}$ series with $n=-10,-9\ldots,0,\ldots,10$ . Theoretically exact bandwidth is $w=1/\pi\approx 0.32$ .

Figure 2: Effect of under-sampling and over-sampling beyond the actual bandwidth of the function

Figure 3: The basis functions $\operatorname{sinc}(t)$ , $\tfrac{1}{2}\operatorname{sinc}(t\pm 1)$ , $\tfrac{1}{4}\operatorname{sinc}(t\pm 2)$

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

Title	proof of sampling theorem
Canonical name	ProofOfSamplingTheorem
Date of creation	2013-03-22 16:28:57
Last modified on	2013-03-22 16:28:57
Owner	stevecheng (10074)
Last modified by	stevecheng (10074)
Numerical id	13
Author	stevecheng (10074)
Entry type	Proof
Classification	msc 42A38
Classification	msc 94A20
Related topic	PlancherelsTheorem