Problems of Information Transmission, Vol. 37, No. 1, 2001, pp. 46–64. Translated from Problemy Peredachi Informatsii, No. 1, 2001, pp. 52–71.
Original Russian Text Copyright
2001 by Trifonov, Korchagin.
METHODS OF SIGNAL PROCESSING
Optimal Detection of a Signal with Unknown
Appearance and Disappearance Times
A. P. Trifonov and Yu. E. Korchagin
Received December 30, 1999; in ﬁnal form, November 23, 2000
Abstract—We obtain maximum likelihood and optimal (Bayesian) algorithms for detection
and measurement of moments of appearance and disappearance of a signal having arbitrary
shape and observed in additive white Gaussian noise. Asymptotic expressions for character-
istics of the maximum likelihood algorithms are obtained. By means of computer modeling,
characteristics of the Bayesian algorithms are found.
The problem of detecting a signal with unknown appearance and disappearance times is of
considerable interest for radar- and hydro-location, radio communication, seismology, and other
ﬁelds of science and technology. The problem of detection of a signal with random appearance and
disappearance times for which the a priori distributions are assumed to be known is considered in .
The algorithms found appear to be extremely complicated from both the viewpoint of hardware
or software implementation and viewpoint of analyzing their eﬃciency. In , this problem is
solved for the case of unknown a priori distributions of signal appearance and disappearance times;
moreover, detection algorithms simpler than those in  are obtained and possibility to analyze
their eﬃciency on the basis of solving the corresponding integral equations is discussed. However,
the results of  are valid for processing of a sequence of independent random variables only. In
addition, in many applications, the problem of estimating the signal appearance and disappearance
times is of great interest.
In this study, we consider detection of a determinate signal with unknown (or random) appear-
ance and disappearance times in Gaussian white noise for continuous-time observation . Such a
restriction on the class of signals allows us to ﬁnd detection and estimation algorithms which are
signiﬁcantly simpler than those in [1, 2], as well as to analyze the performance of the algorithms.
A signal with unknown appearance and disappearance times can be expressed in the form
where f(t) is an a priori known continuously diﬀerentiable function describing the signal shape and
are unknown moments of appearance and disappearance, respectively, which take values
from a priori intervals
] ,i=1,2. (2)
To ensure that a signal cannot disappear before it has appeared, we assume that θ
Assume that f(θ
) = 0, so signal (1) is discontinuous [3; 4, pp. 113–114].
If transmitted information is merely the fact of presence or absence of signal (1) in a ran-
dom process x(t) observed, then it is necessary to solve the problem of detection, which is con-
Supported in part by the Russian Foundation for Basic Research, project no. 01-01-00139.
2001 MAIK “Nauka/Interperiodica”