proof of sampling theorem
Let be the (two-sided) bandwidth. The variable below will denote frequency, and the variable will denote time. (Both and are measured in .)
Consider the space of functions:
0.2 Computation of orthonormal basis
Mapping these by produces an orthonormal basis for :
where we have used the fact that the Fourier transform of (normalized sinc function) is the rectangular box function of bandwidth , and vice versa.
0.3 Expansion by orthonormal basis
Given , let . We can expand in a Fourier series with respect to the :
with the infinite sum converging in . Taking of both sides, we obtain:
(Since is also in , its inverse Fourier transform is a continuous function. Provided that we modify on a set of measure zero, we can assume that is continuous. So it is legal to talk about the pointwise values .)
Hence, we arrive at the representation:
thereby reconstructing any — a square-integrable band-limited function — from its samples at every time period of length .
0.5 Uniform and absolute convergence of series
The infinite series for converges in by construction, but in fact it also converges uniformly and absolutely. To see this, first note that by the Cauchy-Schwarz inequality,
The series converges by Parseval’s theorem ( are the Fourier coefficients of ). Also, the series is uniformly bounded for all . To prove this, it suffices to restrict to bounded inside as the function is -periodic; and then it becomes an easy estimate using the fact that . It follows that the series is uniformly bounded for all , and its tail tends to zero uniformly in .
http://aux.planetmath.org/files/objects/8650/sample.pyPython program to produce the three figures
|Title||proof of sampling theorem|
|Date of creation||2013-03-22 16:28:57|
|Last modified on||2013-03-22 16:28:57|
|Last modified by||stevecheng (10074)|