Signal reconstruction
Encyclopedia
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
.
of square-integrable functions
to complex
space
.
In our example, the vector space of sampled signals
is n-dimensional complex space. Any proposed inverse R of F (reconstruction formula, in the lingo) would have to map 
to some subset of 
. 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 
.
This fact that the dimensions have to agree is related to the Nyquist–Shannon sampling theorem
.
The elementary linear algebra approach works here. Let
(all entries zero, except for the kth entry, which is a one) or some other basis of 
. To define an inverse for F, simply choose, for each k, an 
so that 
. 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.
is a basis of 
in the Hilbert space sense; for instance, one could use the eikonal
,
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
, where 
is the basis of 
given by
(This is the usual discrete Fourier basis.)
The choice of range
is somewhat arbitrary, although it satisfies the dimensionality requirement and reflects the usual notion that the most important information is contained in the low frequencies. In some cases, this is incorrect, so a different reconstruction formula needs to be chosen.
A similar approach can be obtained by using wavelet
s instead of Hilbert bases. For many applications, the best approach is still not clear today.
Signal processing
Signal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...
, 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
Whittaker–Shannon interpolation formula
The Whittaker–Shannon interpolation formula or sinc interpolation is a method to reconstruct a continuous-time bandlimited signal from a set of equally spaced samples.-Definition:...
.
General principle
Let F be any sampling method, i.e. a linear map from the Hilbert spaceHilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
of square-integrable functions
to complexComplex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
space
.In our example, the vector space of sampled signals
is n-dimensional complex space. Any proposed inverse R of F (reconstruction formula, in the lingo) would have to map to some subset of . 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 .This fact that the dimensions have to agree is related to the Nyquist–Shannon sampling theorem
Nyquist–Shannon sampling theorem
The Nyquist–Shannon sampling theorem, after Harry Nyquist and Claude Shannon, is a fundamental result in the field of information theory, in particular telecommunications and signal processing. Sampling is the process of converting a signal into a numeric sequence...
.
The elementary linear algebra approach works here. Let
(all entries zero, except for the kth entry, which is a one) or some other basis of . To define an inverse for F, simply choose, for each k, an so that . 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.
Popular reconstruction formulae
Perhaps the most widely used reconstruction formula is as follows. Let is a basis of in the Hilbert space sense; for instance, one could use the eikonal,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
, where is the basis of given by(This is the usual discrete Fourier basis.)
The choice of range
is somewhat arbitrary, although it satisfies the dimensionality requirement and reflects the usual notion that the most important information is contained in the low frequencies. In some cases, this is incorrect, so a different reconstruction formula needs to be chosen.A similar approach can be obtained by using wavelet
Wavelet
A wavelet is a wave-like oscillation with an amplitude that starts out at zero, increases, and then decreases back to zero. It can typically be visualized as a "brief oscillation" like one might see recorded by a seismograph or heart monitor. Generally, wavelets are purposefully crafted to have...
s instead of Hilbert bases. For many applications, the best approach is still not clear today.
See also
- Nyquist–Shannon sampling theoremNyquist–Shannon sampling theoremThe Nyquist–Shannon sampling theorem, after Harry Nyquist and Claude Shannon, is a fundamental result in the field of information theory, in particular telecommunications and signal processing. Sampling is the process of converting a signal into a numeric sequence...
- Whittaker–Shannon interpolation formulaWhittaker–Shannon interpolation formulaThe Whittaker–Shannon interpolation formula or sinc interpolation is a method to reconstruct a continuous-time bandlimited signal from a set of equally spaced samples.-Definition:...
- AliasingAliasingIn signal processing and related disciplines, aliasing refers to an effect that causes different signals to become indistinguishable when sampled...