# Signal extraction

*in a noise background*

A branch of statistical communication theory. The mathematical problems of signal extraction are statistical problems of the theory of stochastic processes (see also Information theory). A number of typical problems in signal extraction are listed below.

A message $s(t)$, which may be a random or a non-random function of a certain structure, is converted into a signal $x(t)=V(s,n)$, where $n(t)$ is a stochastic process (a noise), while $V$ (the communication channel) is an operator which converts the pair $(s,n)$ into the received signal $x$. It is usually assumed that the effect of the noise on the signal is additive: $x(t)=s(t)+n(t)$. In such a situation the problems of signal extraction are as outlined below.

1) Detection of the signal, that is, checking the hypothesis $x(t)=s(t)+n(t)$ (a signal is present) against the alternative $x(t)=n(t)$ (a signal is absent). More involved varieties of the initial hypothesis are also studied: $x(t)=s(t)+n(t)$ starting from some moment, possibly random, of time $\tau$, which is the moment of appearance of the signal. Here arises the problem of estimating $\tau$.

2) Differentiating between signals, that is, checking the hypothesis $x(t)=s(t)+n(t)$, $s\in S_1$, against the hypothesis $x(t)=s(t)+n(t)$, $s\in S_2$, where $S_1$ and $S_2$ are two different sets of signals.

3) Filtering (reconstruction of a signal), that is, finding statistical estimates for the values of the signal $s(t)$ at a point $t$, after $x(t)$, $t\in T$, has been received.

See also Statistical hypothesis; Stochastic processes, filtering of.

#### References

[1] | W.B. Davenport, W.L. Root, "An introduction to the theory of random signals and noise" , McGraw-Hill (1970) |

[2] | A.A. Kharkevich, "Channels with noise" , Moscow (1965) (In Russian) |

#### Comments

#### References

[a1] | J.M. Wozencraft, I.M. Jacobs, "Principles of communication engineering" , Wiley (1965) |

[a2] | C.W. Helstrom, "Statistical theory of signal detection" , Pergamon (1968) |

**How to Cite This Entry:**

Signal extraction.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Signal_extraction&oldid=32901