Regularization of Invers Problem for M-Ary Channel

The problem of computation of parameters of m-ary channel is considered. It is demonstrated that although the problem is ill-posed, it is possible “turning” of the parameters of the system and transform the problem to well-posed one. Statement of the problem. We analyze the well-known formula for probability of correct identification if orthogonal signal in m-ary channel, which has the form dz z F B g z m B P P q m n s 1 2 ) ( ] 2 / ) ) 1 ( ( [ exp 2 1 ) , , , , ( − ∞ − ∫ − − − = δ π δ ∞ , (1) where [1] ∫ ∞ − − = z dt t z F ) 2 / ( exp 2 1 ) ( 2 π , is «signal to noise» ratio( and are averaged powers of signal and noise), n s P P g / 2 = s P n P B is «base»a of signal (duration of signal multiplied by the specter width), m is dimension of signal, d is cancel interval thickness, I d / = δ is relatively cancel interval thickness. The problem under consideration is formulated as follows: one has to determine a parameter of M-ary channel, if probability * q q = is known (from experiment, analysis of statistics etc.). In other words, one has to solve equation * ) , , , , ( q m B P P q n s = δ with respect to one of the parameters m B P P n s , , , , δ . Observing formula (1), we find that the function ) , , , , ( m B P P q n s δ of the arguments B P P n s , , , δ depends, in fact, on the variable (invariant) B g x ) 1 ( δ − = . (2) and has the form ) ( ) , , , , ( x Q m B P P q m n s = δ . (3) By virtue of (3) and (4), the inverse problem can be written in the terms of the invariant x * ) ( q x Qm = . (4) Results of numerical analysis of formula (1). Plots of function (3) of the argument were drown (using Mathcad software) for various m. The plots are shown at Fig.1. x It is seen from Fig.1 that the problem (4) is unstable with respect to the right-hand side for and for small when m is large (100 and greater). At the same time, we see from Fig.2 that for every m there exists interval where the problem (4) is well-posed. The number for small m, and * q 1 * ≈ q * q ] , [ m m b a 0 = m a 5 3 ≤ ≤ m b . Fig.1. The plots of the function ) (x Q q m = for various m Regularization of the problem by “turning” of parameters of the channel. The original problem (3) is solved with respect to one of the variables B P P n s , , , δ , not with respect to the invariant . If we know interval of possible values of the unknown variable, we can use the remaining variables and give them values such that invariant x ] , [ m m b a x∈ . In this case the problem (4) can be solved with high accuracy with respect to the invariant and then the unknown variable can be computed. x