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 L 2 {displaystyle L^{2}} to complex space C n {displaystyle mathbb {C} ^{n}} . In our example, the vector space of sampled signals C n {displaystyle mathbb {C} ^{n}} is n-dimensional complex space. Any proposed inverse R of F (reconstruction formula, in the lingo) would have to map C n {displaystyle mathbb {C} ^{n}} to some subset of L 2 {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 L 2 {displaystyle L^{2}} . This fact that the dimensions have to agree is related to the Nyquist–Shannon sampling theorem. The elementary linear algebra approach works here. Let d k := ( 0 , . . . , 0 , 1 , 0 , . . . , 0 ) {displaystyle d_{k}:=(0,...,0,1,0,...,0)} (all entries zero, except for the kth entry, which is a one) or some other basis of C n {displaystyle mathbb {C} ^{n}} . To define an inverse for F, simply choose, for each k, an e k ∈ L 2 {displaystyle e_{k}in L^{2}} so that F ( e k ) = d k {displaystyle F(e_{k})=d_{k}} . This uniquely defines the (pseudo-)inverse of F.