Regularization of Invers Problem for M-Ary Channel

REGULARIZATION OF INVERS PROBLEM FOR M-ARY CHANNE L N. A. Filimonova Novosibirsk, Russia Abstract. The problem of computation of parameters of m -ary channel is considered. It is demonstrated that although the problem is ill-posed, it is poss ible “turning” of the param eters of the system and transform the problem to well-posed one. Statement of the problem. We analyze the well-known form ula for probability of correct identification if or thogonal 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 inte rval thickness. The problem under consideration is formulat ed as follows: one has to determine a parameter of M -ary channel, if proba bility * q q = is known (from experiment, analysis of statistics etc.). In other wo rds, 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 th e 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 term s of the invariant x * ) ( q x Q m = . (4) Results of numerical anal ysis 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 respec t 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 interv al 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 respe ct to one of the var iables 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 in variant and then the unknown variable can be computed. x Thus, on the set m m b B g a ≤ − ≤ ) 1 ( δ . (5) the problem of determin ing of a param eter of m -ary channel is well-posed. In the technical terms our results m eans the following. In the general, the problem of determining of a para meter of m -ary chann el is ill-posed. But it is well-posed if one tunes devices in the appropriate way. The condition (5 ) is the condition of the appropriate “tuning” of devices forming m -ary channel. The term “turning” in th is paper corresponds to turning of real equipments. R EFERENCES 1. Markhasin A., Kolpakov A., Drozdova V. Optimi zation of the spectral and power efficiency of m -ary channels in wireless and mobile sy stems // Third Intern ational conference in Central Asia on Internet, th e next generation of mob ile, wireless and optical communications. Sept. 26-28, 2007, Tashkent, Uzbekistan.