Problems of Information Transmission, Vol. 40, No. 1, 2004, pp. 44–52. Translated from Problemy Peredachi Informatsii, No. 1, 2004, pp. 48–57.
Original Russian Text Copyright
2004 by Burnashev.
METHODS OF SIGNAL PROCESSING
On Optimal Detectors
in Multiuser Detection Problems
M. V. Burnashev
Institute for Information Transmission Problems, RAS, Moscow
Received May 20, 2003
Abstract—The paper is a supplement to . Conditions under which asymptotically opti-
mal detectors are linear are found. It is shown also that if, in contrast to , we consider
not the Bayesian but minimax statement of the problem with unknown coeﬃcients, then opti-
mal detectors are linear (moreover, nonasymptotically). A geometrical meaning of Theorem 1
from  is explained, and it is shown that the theorem follows from some general results [2, 3]
on hypotheses testing. It is also shown that some results of  follow from [1, Theorem 1].
It is assumed that an observed signal X(t) has the form
dX(t)=s(t) dt + σdW
, 0 ≤ t ≤ T, (1)
is a standard Wiener process, σ>0 is a known value, and s(t) is a data signal from
K users, of the form
(t), b =(b
[0,T]isthekth-user waveform. It is assumed that b
independent and identically distributed random variables with P (b
= −1) = 1/2.
Of course, it is assumed that s
> 0, k =1,...,K.
For signals u(t)andv(t)andanysetsA, B ⊆ L
u(t)v(t) dt, u
d(u, v)=u − v,d(A, B)= inf
For linear detection, user 1 chooses an arbitrary function y
(t) ∈ L
[0,T] and makes the following
, where v
(t) dX(t). (3)
Supported in part by the Russian Foundation for Basic Research, project no. 03-01-00098, and INTAS,
2004 MAIK “Nauka/Interperiodica”