proof of sampling theorem

0.1 Set-up

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

Consider the space of functions:

$${\mathscr{H}}^{\prime }=\{g\in {𝐋}^2(ℝ):g(\xi )=0\text{ for almost all }|\xi |>w/2\}$$

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

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

$$\mathscr{H}=\{f\in {𝐋}^2(ℝ):(ℱf)(\xi )=0\text{ for almost all }|\xi |>w/2\}={ℱ}^{-1}{\mathscr{H}}^{\prime }$$

is also a Hilbert space.

0.2 Computation of orthonormal basis

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

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

$({ℱ}^{-1}{\varphi }_n)(t)$$={\displaystyle \frac{1}{\sqrt{w}}}{\displaystyle {\int }_{-w/2}^{w/2}}{e}^{-2\pi in\xi /w}{e}^{2\pi i\xi t}𝑑\xi$

$={\displaystyle \frac{1}{\sqrt{w}}}{\displaystyle {\int }_{-w/2}^{w/2}}{e}^{2\pi i\xi (t-n/w)}𝑑\xi$

$={\displaystyle \frac{1}{\sqrt{w}}}\left(w\mathrm{sinc}(w(t-n/w))\right)=\sqrt{w}\mathrm{sinc}(wt-n),$

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

0.3 Expansion by orthonormal basis

Given $f\in \mathscr{H}$, let $g=ℱf\in {\mathscr{H}}^{\prime }$.We can expand $g$ in a Fourier series with respectto the $\{{\varphi }_n\}$:

$$g(\xi )=\sum _{n\in \mathbb{Z}}⟨g,{\varphi }_n⟩{\varphi }_n(\xi ),$$

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

$$f(t)=({ℱ}^{-1}g)(t)=\sum _{n\in \mathbb{Z}}⟨g,{\varphi }_n⟩({ℱ}^{-1}{\varphi }_n)(t)=\sqrt{w}\sum _{n\in \mathbb{Z}}⟨g,{\varphi }_n⟩\mathrm{sinc}(wt-n).$$

Moreover,

$$⟨g,{\varphi }_n⟩=\frac{1}{\sqrt{w}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}g(\xi )e^{2\pi in\xi /w}𝑑\xi =\frac{1}{\sqrt{w}}({ℱ}^{-1}g)\left(\frac{n}{w}\right)=\frac{1}{\sqrt{w}}f\left(\frac{n}{w}\right).$$

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

0.4 Result

Hence, we arrive at the representation:

$$f(t)=\sum _{n\in \mathbb{Z}}f\left(\frac{n}{w}\right)\mathrm{sinc}(wt-n),$$

thereby reconstructing any $f\in \mathscr{H}$ — a square-integrable band-limitedfunction —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 ${𝐋}^2$by construction, but in fact it also converges uniformly and absolutely.To see this, first note that by the Cauchy-Schwarz inequality,

$$\sum _{n\in \mathbb{Z}}|f(\frac{n}{w})||\mathrm{sinc}(wt-n)|\le {\left(\sum _{n\in \mathbb{Z}}{|f(\frac{n}{w})|}^2\right)}^{1/2}{\left(\sum _{n\in \mathbb{Z}}{\mathrm{sinc}}^2(wt-n)\right)}^{1/2}.$$

The series ${\sum }_n{|f(n/w)|}^2$ converges by Parseval’s theorem($w^{-1/2}f(n/w)$ are the Fourier coefficients of $g$).Also, the series ${\sum }_n{\mathrm{sinc}}^2(wt-n)$ is uniformly bounded for all $t\in ℝ$. To prove this, it suffices torestrict to $t$ bounded inside $[0,1/w]$as the function $t\mapsto {\sum }_n{\mathrm{sinc}}^2(wt-n)$ is $1/w$-periodic; and thenit becomes an easy estimate using the fact that .It follows that the series ${\sum }_n|f(n/w)||\mathrm{sinc}(wt-n)|$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 $\mathrm{sinc}$ series with $n=-10,-9\mathrm{\dots },0,\mathrm{\dots },10$. 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 $\mathrm{sinc}(t)$, $\frac{1}{2}\mathrm{sinc}(t\pm 1)$, $\frac{1}{4}\mathrm{sinc}(t\pm 2)$

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