Problems of Information Transmission, Vol. 38, No. 3, 2002, pp. 203–217. Translated from Problemy Peredachi Informatsii, No. 3, 2002, pp. 45–61.
Original Russian Text Copyright
2002 by Trifonov, Zakharov, Pronyaev.
METHODS OF SIGNAL PROCESSING
Adaptive Detection of a Stochastic Signal
under Parametric a priori Uncertainty
Received August 27, 2001; in ﬁnal form, March 14, 2002
Abstract—We obtain a maximum likelihood algorithm for detecting a Gaussian stochastic
signal with unknown appearance (disappearance) time and average power. Asymptotic expres-
sions for the probabilities of the 1st- and 2nd-kind detection errors are found. Applicability
limits for the derived expressions are found by statistical computer simulation.
The problem of detecting signals against a background of additive noise is one of the classical
problems in the theory of information systems. Detection of a stochastic signal which is a sta-
tionary Gaussian random process with correlation time considerably smaller than the observation
(processing) interval was considered in [1–3, etc.] under the assumption that all parameters of the
stochastic signal are a priori known.
However, information reception and processing is often performed under a priori uncertainty,
where distribution laws for probabilities of random processes observed are known up to a ﬁnite
number of parameters [4, 5]. In particular, parameters of the stochastic signal, such as its appear-
ance time, disappearance time (duration), intensity and bandwidth (average power), etc., may be
unknown. In this case, the results of [1–3] obtained under full a priori deﬁniteness turn out to be
inapplicable. In [5–7], based on the Bayesian approach in continuous and discrete time, algorithms
are synthesized for detection of a stochastic signal with random appearance time and/or duration
under known a priori distributions of these parameters. However, the detection algorithms pro-
posed in [5–7] are extremely complicated and require considerable a priori knowledge; moreover,
for the case of continuous time, attempts of ﬁnding eﬃciency characteristics of the algorithms
fail. In , simpler detection algorithms were synthesized, which do not require any knowledge on
the a priori distributions of unknown appearance and disappearance times of a stochastic signal,
and expressions for characteristics of the synthesized algorithms were obtained. Here, detection is
accomplished in discrete time on the basis of processing independent samples of data observed.
Of certain interest are synthesis and analysis of algorithms for processing stochastic signals
in continuous time. In particular, processing of signals in continuous time allows one to avoid
degradation of detection eﬃciency caused by a partial loss of information on observed data in the
process of its discretization. Obviously, characteristics of detection in continuous time are potential
(limiting) characteristics for the case of discrete time, which are attained as the discretization time
inﬁnitely decreases. In addition, characteristics of signal detection in continuous time often provide
suﬃciently good approximation of the corresponding characteristics for discrete time if samples of
the observed data are essentially correlated.
In the sequel, we consider the problem of maximum likelihood detection of a Gaussian stochas-
tic signal with unknown appearance or disappearance times and unknown intensity and bandwidth
Supported by the CRDF, Grant VZ-010-0, and Ministry of Education of Russian Federation, project
2002 MAIK “Nauka/Interperiodica”