Editing Signal reconstruction (section)

== General principle ==

Let ''F'' be any sampling method, i.e. a linear map from the [[Hilbert space]] of square-integrable functions <math>L^2</math> to [[complex number|complex]] space <math>\mathbb C^n</math>.

In our example, the vector space of sampled signals <math>\mathbb C^n</math> is ''n''-dimensional complex space. Any proposed inverse ''R'' of ''F'' (''reconstruction formula'', in the lingo) would have to map <math>\mathbb C^n</math> to some subset of <math>L^2</math>. We could choose this subset arbitrarily, but if we're going to want a reconstruction formula ''R'' that is also a linear map, then we have to choose an ''n''-dimensional linear subspace of <math>L^2</math>.

This fact that the dimensions have to agree is related to the [[Nyquist–Shannon sampling theorem]].

The elementary linear algebra approach works here. Let <math>d_k:=(0,...,0,1,0,...,0)</math> (all entries zero, except for the ''k''th entry, which is a one) or some other basis of <math>\mathbb C^n</math>. To define an inverse for ''F'', simply choose, for each ''k'', an <math>e_k \in L^2</math> so that <math>F(e_k)=d_k</math>. This uniquely defines the (pseudo-)inverse of ''F''.

Of course, one can choose some reconstruction formula first, then either compute some sampling algorithm from the reconstruction formula, or analyze the behavior of a given sampling algorithm with respect to the given formula.

Ideally, the reconstruction formula is derived by minimizing the expected error variance. This requires that either the signal statistics is known or a prior probability for the signal can be specified.  [[Information field theory]] is then an appropriate mathematical formalism to derive an optimal reconstruction formula.<ref>{{cite web |url=http://www.mpa-garching.mpg.de/ift/ |title=Information field theory |last1= |first1= |last2= |first2= |date= |website= |publisher= Max Planck Society|accessdate=13 November 2014 }}</ref>