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Reception of a stochastic signal with unknown appearance and disappearance times

“…Following [2,9], we assume that the times θ 01 and θ 02 of the signal appearance and disappearance, respectively, take the values from the specified a priori intervals [θ 1 min ; θ 1 max ] and [θ 2 min ; θ 2 max ], respectively, such that θ 2 min > θ 1 max . We also assume [6,9] that the correlation time of the process ξ(t) is significantly smaller than the duration τ 0 = θ 02 − θ 01 of signal (1), i.e.,…”

“…Then the maximum-likelihood estimate i m of the parameter i 0 is determined as the value i = 0 or i = 1 for which the LRF logarithm attains its absolute (highest) maximum in the regions θ 1 ∈ [θ 1 min ; θ 1 max ] and θ 2 ∈ [θ 2 min ; θ 2 max ] of the possible values of the parameters θ 1 and θ 2 , respectively. According to [6,9], the LRF logarithm can be written in the form…”

“…Allowing for the fact that L(i = 0, θ 1 , θ 2 ) = 0, by analogy with [2,9], we find that the maximum-likely (ML) algorithm for detecting the signal (estimating the parameter i 0 ) can be written in the form…”

“…In [9], simple algorithms for detecting the Gaussian stochastic signal and estimating its unknown disappearance time (duration) are obtained on the basis of the maximum-likelihood method and the characteristics of these algorithms are found analytically. However, the signal appearance (arrival) time can also be unknown in practice in addition to the signal duration.…”

“…However, the signal appearance (arrival) time can also be unknown in practice in addition to the signal duration. In what follows, the results of [9] are generalized to the case of reception of the Gaussian stochastic signal with unknown appearance and disappearance times.…”