Regularization of Invers Problem for M-Ary Channel
RREGULARIZATION OF INVERS PROBLEM FOR M-ARY CHANNEL
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 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 dzzFBgzmBPPq mns )( ]2/))1(([exp21),,,,( −∞− ∫ −−−= δπδ ∞ , (1)where [1] - ∫ ∞− −= z dttzF )2/(exp21)( π , - is «signal to noise» ratio( and are averaged powers of signal and noise), ns PPg / = 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, - Id / =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 * qq = is known (from experiment, analysis of statistics etc.). In other words, one has to solve equation *),,,,( qmBPPq ns =δ with respect to one of the parameters mBPP ns ,,,, δ . Observing formula (1), we find that the function ),,,,( mBPPq ns δ of the arguments BPP ns ,,, δ depends, in fact, on the variable (invariant) Bgx )1( δ−= . (2)and has the form )(),,,,( xQmBPPq mns =δ . (3)By virtue of (3) and (4), the inverse problem can be written in the terms of the invariant x *)( qxQ m = . (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 ≈ q * q ],[ mm ba = m a ≤≤ m b . Fig.1.
The plots of the function )( xQq 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
BPP ns ,,, 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 ],[ mm bax ∈ . 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 Thus, on the set mm bBga ≤−≤ )1( δ . (5)the problem of determining of a parameter of m -ary channel is well-posed. In the technical terms our results means the following. In the general, the problem of determining of a parameter of m -ary channel 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 this paper corresponds to turning of real equipments. R EFERENCES Markhasin A., Kolpakov A., Drozdova V. Optimization of the spectral and power efficiency of mm