ISSN 0032-9460, Problems of Information Transmission, 2008, Vol. 44, No. 4, pp. 315–320.
Pleiades Publishing, Inc., 2008.
Original Russian Text
R.Z. Khasminskii, 2008, published in Problemy Peredachi Informatsii, 2008, Vol. 44, No. 4, pp. 33–38.
METHODS OF SIGNAL PROCESSING
Nonparametric Estimation of Signal Amplitude
in White Gaussian Noise
R. Z. Khasminskii
Institute for Information Transmission Problems, RAS, Moscow
Wayne State University, Detroit, USA
Received August 7, 2008
Abstract—We assume that a transmitted signal is of the form S(t)f(t), where f(t)isaknown
function vanishing at some points of the observation interval and S(t) is a function of a known
smoothness class. The signal is transmitted over a communication channel with additive white
Gaussian noise of small intensity ε. For this model, we construct an estimator for S(t)which
is optimal with respect to the rate of convergence of the risk to zero as ε → 0.
1. PROBLEM STATEMENT
Assume that an observed signal X
(t)=S(t)f(t) dt + εdw(t),X
(0) = 0, 0 ≤ t ≤ 1, (1)
where w(t) is a standard Wiener process and f(t) is a known function with the following properties:
f(t) is continuously diﬀerentiable, it vanishes at a ﬁnite number of points t
<... < t
< 1, and f
) =0,i =1, 2,...,.ByF we denote the class of functions f with these
The estimated function S(t) is assumed to belong to the smoothness class Σ(β,L)(see):
there exist continuous derivatives
(t),t∈ (0, 1),
(t + h) − S
, 0 <α= β − k ≤ 1.
The object of this study is the problem of estimating S(t)fromobservationsX
(t). First of all,
note that the problem has an obvious solution for the domain U
∩[0, 1], where U
since in this case it reduces, by dividing by f(t), to a known (see ) problem of estimating a signal
from its observations in a white Gaussian noise of intensity of order ε. Here we consider the problem
of estimating S(t)fort − t
= o(1) (as ε → 0) and obtain estimators in
which are optimal with
respect to the rate of convergence of risks to zero.
Remark 1. The estimation problem for S(t)inmodel(1)isequivalent(see)totheproblem
of estimating a periodic function S(t)ofperiodT from observations
(t)=S(t)f(t) dt + dw(t),X
(0) = 0, 0 ≤ t ≤ nT, (2)
where f(t) is also a periodic function of period T , n →∞.Inparticular,forf (t)=cos(2πt/T)the
problem reduces to estimating the amplitude of a sinusoidal signal in a neighborhood of the points
=(2i +1)T/4fromobservationsofn periods of the signal in white noise, n →∞.