Signal detection in Gaussian noise of unknown level: An invariance application

Author

Scharf, Louis L. ; Lytle, Dean W.

Volume

17

Issue

4

fYear

1971

fDate

7/1/1971 12:00:00 AM

Firstpage

404

Lastpage

411

Abstract

The concept of invariance in hypothesis testing is brought to bear on the problem of detecting signals of known form and unknown energy in Gaussian noise of unknown level. The noise covariance function is assumed to be $K(t,u) = \\sigma ^2 pho(t,u)$ where $\\rho(t,u)$ is the known form of the covariance function and $\\sigma ^2$ is the unknown level. Classical approaches to signal detection depend on the assumption that $K(t,u)$ is known completely. Then, a correlation-type receiver that is the uniformly most powerful (UMP) test of $H_o$ (signal absent) versus $H_1$ (signal present) can be derived. When $\\sigma ^2$ is unknown, there exists no UMP test. However, it is shown in this paper that there exists a test of $H_o$ versus $H_1$ that is UMP-invariant for a very natural group of transformations on the space of observations. The derived test is found to be independent of knowledge about the noise level $\\sigma ^2$ , since the derived test (receiver) contains an error-free estimate of $\\sigma ^2$ . This utopian conclusion is reconciled by noting that the derived receiver can never be physically realized. It is shown that any physically realizable version of the receiver has a $t$ -distributed test statistic. This permits choice of operating receiver thresholds and evaluation of performance characteristics.

Keywords

Gaussian processes; Signal detection; Filtering theory; Gaussian noise; Gaussian processes; Linearity; Noise level; Reverberation; Scattering; Signal detection; Signal processing; Testing;

fLanguage

English

Journal_Title

Information Theory, IEEE Transactions on

Publisher

ieee

ISSN

0018-9448

Type

jour

DOI

10.1109/TIT.1971.1054668

Filename

1054668