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• W2002051255 abstract "The problem of optium reception of M-ary Gaussian signals in Gaussian noise is to specify, in terms of the observable waveform, a scheme for deciding among M alternative mean and covariance functions with minimum error probability. Although much literature on the problem exists, a mathematically rigorous solution has yet to appear. By formulating the problem as optimum discrimination of M Gaussian measures in function space induced by the mean and covariance functions, this paper presents such a solution. Let mk(t) and rk(s,t), k &equal; 1, …, M, be the alternative mean and covariance functions of the Gaussian signal, and let m <inf xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>0</inf> (t) and r <inf xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>0</inf> (s,t) be the mean and covariance functions of the Gaussian noise. If, for each k &equal; 1, …, M, the integral equations, <tex xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>$int r_{0}(s,t)g_{k}(s) ds = m_{k}(t)$</tex> and <tex xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>$int int r_{0}(s,u)h_{k}(u,v)[r_{0}(v,t) + r_{k}(v,t)] du dv = r_{k}(s,t),$</tex> admit a square-integrable solution g <inf xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>k</inf> (t) and a symmetric, square-integrable solution h <inf xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>k</inf> (s,t), then the following decision scheme is optimum: given an observable waveform x(t), choose m <inf xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>k</inf> (t) and r <inf xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>k</inf> (s,t) if I <inf xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>k</inf> (x) is the largest among all I <inf xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>j</inf> (x), j &equal; 1, …, M, where I <inf xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>k</inf> is defined by <tex xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>$I_{k}(x) = {1 over 2} int int x(s)h_{k}(s,t)x(t) ds dt + int x(t)f_{k}(t) dt + c_{k}$</tex> in which <tex xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>$f_{k}(t) = g_{k}(t) - int h_{k}(s,t)[m_{0}(s) + m_{k}(s)] ds$</tex> and c <inf xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>k</inf> is a constant determined by the mean and covariance functions m <inf xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>0</inf> (t), m <inf xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>k</inf> (t), r <inf xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>0</inf> (s,t) and r <inf xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>k</inf> (s,t) as well as the a priori probability associated with m <inf xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>k</inf> (t) and r <inf xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>k</inf> (s,t). The first section introduces and defines the problem and the second presents the solution with pertinent discussions while a precise mathematical treatment is left to the appendix." @default.
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• W2002051255 date "1965-11-01" @default.
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• W2002051255 title "Optimum Reception of M-ary Gaussian Signals in Gaussian Noise" @default.
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• W2002051255 doi "https://doi.org/10.1002/j.1538-7305.1965.tb03164.x" @default.
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