In signal processing, reconstruction usually means the determination of an original continuous signal from a sequence of equally spaced samples.
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This article takes a generalized abstract mathematical approach to signal sampling and reconstruction. For a more practical approach based on band-limited signals, see Whittaker–Shannon interpolation formula.
General principle
Let F be any sampling method, i.e. a linear map from the Hilbert space of square-integrable functions
In our example, the vector space of sampled signals
This fact that the dimensions have to agree is related to the Nyquist–Shannon sampling theorem.
The elementary linear algebra approach works here. Let
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.
Popular reconstruction formulae
Perhaps the most widely used reconstruction formula is as follows. Let
although other choices are certainly possible. Note that here the index k can be any integer, even negative.
Then we can define a linear map R by
for each
(This is the usual discrete Fourier basis.)
The choice of range
A similar approach can be obtained by using wavelets instead of Hilbert bases. For many applications, the best approach is still not clear today.