ISSN 0032-9460, Problems of Information Transmission, 2009, Vol. 45, No. 2, pp. 158–167.
Pleiades Publishing, Inc., 2009.
Original Russian Text
A.P. Trifonov, Yu.E. Korchagin, 2009, published in Problemy Peredachi Informatsii, 2009, Vol. 45, No. 2, pp. 91–100.
METHODS OF SIGNAL PROCESSING
Exact Formulas for Computation
of Reception Characteristics of a Signal with
Unknown Appearance and Disappearance Times
A. P. Trifonov and Yu. E. Korchagin
Voronezh State University
Received April 9, 2008; in ﬁnal form, July 10, 2008
Abstract—We consider the performance of the maximum-likelihood algorithm for detection
and measurement of appearance and disappearance times of an arbitrary waveform signal ob-
served against additive white Gaussian noise. We ﬁnd exact expressions for detection error
probabilities and densities of estimated appearance and disappearance times.
The problem of reception of a signal with unknown appearance and disappearance times, being
of considerable interest in radar, sonar, radio, seismology, telemetry, and other areas of science and
technology, was often discussed in the literature. In  there was considered the detection problem
for a signal with random appearance and disappearance times whose prior distributions are assumed
to be known. The obtained detection algorithms were found to be very complicated from the point
of view of both hardware or software implementation and performance analysis. In  this problem
was solved for the case of unknown prior distributions of signal appearance and disappearance times;
detection algorithms simpler than those of  were found, and the possibility for analysis of their
performance based on solution of the corresponding integral equations was discussed. However,
results of  are valid only for processing a sequence of independent random variables.
Reception of a deterministic signal with unknown (or random) appearance and disappearance
times against white Gaussian noise with continuous observation time was considered in . This
conﬁnement of the signal class makes it possible to ﬁnd algorithms for detection and estimation of
appearance and disappearance times that are considerably simpler than those in [1, 2]. However,
performance analysis for algorithms constructed in  was made only approximately.
In the present paper, using results of , we ﬁnd exact performance characteristics of maximum-
likelihood algorithms constructed in .
The setting of the problem of signal detection and estimation of unknown appearance and
disappearance times is borrowed from . Assume that on a time interval [0,T]weobservea
random process x(t)=γ
)+n(t) which is an additive mixture of a useful signal
≤ t ≤ θ
and white Gaussian noise n(t) with one-sided spectral density N
tinuously diﬀerentiable function describing the signal waveform; θ
are unknown appearance
Supported in part by the Russian Foundation for Basic Research, project no. 07-01-00042.