proof of sampling theorem

0.1 Set-up

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

Consider the space of functions:

={g𝐋2():g(ξ)=0 for almost all |ξ|>w/2}

which is clearly seen to be a complex Hilbert spaceMathworldPlanetmath with the usual inner productMathworldPlanetmath for 𝐋2().

Let denote the Fourier transformDlmfMathworldPlanetmath on 𝐋2(), which is a unitaryPlanetmathPlanetmath transform by Plancherel’s theorem. So,

={f𝐋2():(f)(ξ)=0 for almost all |ξ|>w/2}=-1

is also a Hilbert space.

0.2 Computation of orthonormal basis

One orthonormal basisMathworldPlanetmath for consists of the usual Fourier functionsMathworldPlanetmath on the interval [-w/2,w/2], extended to be zero on [-w/2,w/2]:


Mapping these by -1 produces an orthonormal basis for :

(-1ϕn)(t) =1w-w/2w/2e-2πinξ/we2πiξt𝑑ξ

where we have used the fact that the Fourier transform of twsinc(wt) (normalized sinc functionDlmfMathworldPlanetmath) is the rectangular box function of bandwidth w, and vice versa.

0.3 Expansion by orthonormal basis

Given f, let g=f. We can expand g in a Fourier series with respect to the {ϕn}:


with the infinite sum converging in 𝐋2(). Taking -1 of both sides, we obtain:




(Since g is also in 𝐋1, its inverse Fourier transform -1g is a continuous functionMathworldPlanetmath. Provided that we modify f on a set of measure zero, we can assume that f=-1(f)=-1g is continuous. So it is legal to talk about the pointwise values f(n/w).)

0.4 Result

Hence, we arrive at the representation:


thereby reconstructing any f — 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 𝐋2 by construction, but in fact it also converges uniformly and absolutely. To see this, first note that by the Cauchy-Schwarz inequality,


The series n|f(n/w)|2 converges by Parseval’s theorem (w-1/2f(n/w) are the Fourier coefficients of g). Also, the series nsinc2(wt-n) is uniformly bounded for all t. To prove this, it suffices to restrict to t bounded inside [0,1/w] as the function tnsinc2(wt-n) is 1/w-periodic; and then it becomes an easy estimate using the fact that nn-2<. It follows that the series n|f(n/w)||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 functionDlmfMathworldPlanetmathPlanetmath (of the first kind), using a sinc series with n=-10,-9,0,,10. Theoretically exact bandwidth is w=1/π0.32.
Figure 2: Effect of under-sampling and over-sampling beyond the actual bandwidth of the function
Figure 3: The basis functions sinc(t), 12sinc(t±1), 14sinc(t±2)
  • 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