Signal reconstruction

﻿
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.

General principle

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

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

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

The elementary linear algebra approach works here. Let $d_k:=\left(0,...,0,1,0,...,0\right)$ (all entries zero, except for the "k"th entry, which is a one) or some other basis of $Bbb C^n$. To define an inverse for "F", simply choose, for each "k", an $e_k in L^2$ so that $F\left(e_k\right)=d_k$. 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 $\left\{ e_k \right\}$ is a basis of $L^2$ in the Hilbert space sense; for instance, one could use the canonical

:$e_k\left(t\right):=e^\left\{2pi i k t\right\},$,

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

:$R\left(d_k\right)=e_k,$

for each $k=lfloor -n/2 floor,...,lfloor \left(n-1\right)/2 floor$, where $\left(d_k\right)$ is the basis of $Bbb C^n$ given by

:$d_k\left(j\right)=e^\left\{2 pi i j k over n\right\}$

(This is the usual discrete Fourier basis.)

The choice of range $k=lfloor -n/2 floor,...,lfloor \left(n-1\right)/2 floor$ 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 wavelets instead of Hilbert bases. For many applications, the best approach is still not clear today.

* Nyquist–Shannon sampling theorem
* Whittaker–Shannon interpolation formula
* Aliasing

Wikimedia Foundation. 2010.

Look at other dictionaries:

• Reconstruction filter — In a mixed signal system (analog and digital), a reconstruction filter (or anti imaging filter) is used to construct a smooth analogue signal from the output of a digital to analogue converter (DAC) or other sampled data output device.ampled data …   Wikipedia

• 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. Signals of interest can include sound, images, time varying measurement …   Wikipedia

• Reconstruction de Bande Spectrale — ou (SBR) (Spectral Band Replication), est une technique inventée par la société Coding Technologies visant à améliorer la qualité dans le domaine de la compression audio. Cette technique est essentiellement utilisée pour les bas débits et est… …   Wikipédia en Français

• Reconstruction from zero crossings — The problem of reconstruction from zero crossings can be stated as: given the zero crossings of a continuous signal, is it possible to reconstruct the signal (to within a constant factor)? Worded differently, what are the conditions under which a …   Wikipedia

• Signal Corps in the American Civil War — U.S. Army Signal Corps station on Elk Mountain, Maryland, overlooking the Antietam battlefield. The Signal Corps in the American Civil War comprised two organizations: the U.S. Army Signal Corps, which began with the appointment of Major Albert J …   Wikipedia

• Sampling (signal processing) — Signal sampling representation. The continuous signal is represented with a green color whereas the discrete samples are in blue. In signal processing, sampling is the reduction of a continuous signal to a discrete signal. A common example is the …   Wikipedia

• Quantification (signal) — Pour les articles homonymes, voir quantification. En traitement du signal, la quantification est le procédé qui permet d approximer un signal continu (ou à valeurs dans un ensemble discret de grande taille) par des valeurs d un ensemble discret d …   Wikipédia en Français

• Spectral phase interferometry for direct electric-field reconstruction — In ultrafast optics, spectral phase interferometry for direct electric field reconstruction (SPIDER) is an ultrashort pulse measurement technique.The basicsSPIDER is an interferometric ultrashort pulse measurement technique in the frequency… …   Wikipedia

• Surface reconstruction — refers to the process by which atoms at the surface of a crystal assume a different structure than that of the bulk. Surface reconstructions are important in that they help in the understanding of surface chemistry for various materials,… …   Wikipedia

• Peak Signal to Noise Ratio — PSNR (sigle de Peak Signal to Noise Ratio) est une mesure de distorsion utilisée en image numérique, tout particulièrement en compression d image. Il s agit de quantifier la performance des codeurs en mesurant la qualité de reconstruction de l… …   Wikipédia en Français