Signal reconstruction

In signal processing, reconstruction usually means the determination of an original continuous signal from a sequence of equally spaced samples.

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.

Let F be any sampling method, i.e. a linear map from the Hilbert space of square-integrable functions ${\displaystyle L^{2}}$ to complex space ${\displaystyle {\mathbb {C}}^{n}}$.

In our example, the vector space of sampled signals ${\displaystyle {\mathbb {C}}^{n}}$ is n-dimensional complex space. Any proposed inverse R of F (reconstruction formula, in the lingo) would have to map ${\displaystyle {\mathbb {C}}^{n}}$ to some subset of ${\displaystyle L^{2}}$. 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 ${\displaystyle L^{2}}$.

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