Min-max detection of weak signals in phi-mixing noise

Detection of weak signals in a special v-mixing noise class is the signals are weak, asymptotic performance is of interest. considered. The detector structure is restricted to sums of memoryless nonlinear transformations of the observations, correlated with the data A common measure of asymptotic performance is the sequence and compared to a fixed threshold. Using the efficacy to measure efficacy. It is well-known that for independent identically performance, the nonlinearity that has min-max performance is derived. distributed (i.i.d.) observations the efficacy is maximized when the nonlinear transformation is given by the locally I. INTR~DLJCTI~N optimum nonlinearity defined by the marginal density. When this density is not known exactly, optimality is often F OR the detection of signals in additive noise, a very defined in a min-max way. Following the ideas of Huber commonly used detector structure consists of a sum on robust estimation and hypothesis testing [l], [2] the of memoryless nonlinear transformations of the observamin-max nonlinearities for detection are derived in [3]-[5] tions, correlated with the signals and compared to a fixed for the i.i.d. case and for densities belonging to an e-conthreshold. When the number of observations is large and tamination class. In [3], [4] the noise densities are assumed to be symmetric. In [5] noise symmetry is assumed inside Manuscript received January 24, 1983; revised June 27, 1983. This work an interval around the origin. was supported in part by the US. Army under Grant DAAG29-82-K0095 and in part by the National Science Foundation under Grant ECS-79All of these approaches assume independent observa18915. tions. For the dependent case, in order to calculate the G. V. Monstakides was with the Department of Electrical Engineering efficacy we must know the bivariate densities of all pairs of and Computer Science, Princeton University, Princeton, NJ 08544; he is now with the Institut de Recherche en Informatique et Systemes Aleatoires, observations. Finding the nonlinearity that maximizes the Rennes, France. efficacy is not easy as in the i.i.d. case. This problem for J. B. Thomas is with the Department of Electrical Engineering and stationary sequences is considered in [6] for the m-depenComputer Science, Princeton University, Princeton, NJ 08544. dent case and in [7] for the q-mixing case. It is shown that, 001%9448/84/0500-0529$01.00 01984 IEEE 530 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. IT-30, NO. 3, MAY 1984 for the m-dependent case, the optimum nonlinearity satisfies a Fredholm integral equation of the second kind. Min-max detection with dependent observations is considered in [8]. Following similar ideas from [9], [lo] the min-max nonlinearity is derived under the assumption that the observations are generated by a moving average process and are weakly dependent. In [ll] the problem of min-max detection of a constant signal in stationary Markov noise is considered. It is shown that, for a special class of Markov noise processes, the min-max nonlinearity is very closely related to the one for the i.i.d. case. Here we consider an extension of [ll]. We consider detection of nonconstant signals in a class of cp-mixing noise processes. The class defined in [ll] is a special case of this q-mixing class. We optimize over structures that consist of sums of memoryless nonlinear transformations. It is important to point out that, even though this structure is optimum for the i.i.d. case, this optimality does not hold under dependency. But, in any case, we would like to see how much the independence-assumption structure changes under dependency and also if the performance changes drastically. II. PRELIMINARIES Let { Ni} be a strictly stationary noise sequence. Denote by &f,” the u-algebra generated by the random variables {N,, Nu+l,. . ., Nh }. Let f(x) be the common marginal density for the random variables Ni. We assume that this density is symmetric, that it has a continuous derivative different from zero almost everywhere with respect to f(x) and that it has finite Fisher’s information for location. For simplicity we will denote random variables with capital letters and sequences of random variables with boldface capital letters. We call the stationary sequence N a cp-mixing sequence if there exists a sequence { T,~ } of real numbers satisfying 1 2 ‘pi 2 (p2 2 . . * 2 0 (1) such that, for each positive integer n, if A an event from J4: and B from Mr+,, then (P(A n B) P(A)P(B)( I pJ(A). (2) This is the q-mixing class defined in [13, p. 1741. We call a q-mixing sequence acceptable if in addition to (1) and (2) it satisfies