Linear Filtering of the Sum of Two Known Stochastic Processes

The linear filtering got the great attention of statisticians and applied mathematician; therefore the present study aims at finding the linear filtering of stationary stochastic process and that is when we know the values of the sum of two stochastic processes at all moments of the time and when t ≥ 0, and this requires us to know the spectral density function fXX(λ) for the stochastic processes. In this paper, we opted to take two cases after giving the necessary definitions for all important terms and finding the spectral density function for each stochastic processes (Poisson process and Wide Sense Markov process) ; in the first case we supposed that both of the stochastic processes are stationary Poisson processes and after finding the linear filtering we compute the mean square filtering error ;and in second case we suppose one of the stochastic process is Poisson process and the other is wide sense Markov process also in this case we find the mean square filtering error .


Introduction
Filtering of stochastic process has attracted a lot of attention.One of examples is target tracking, when the target is observed over a discrete time grid, corresponding to the successive passes of radar [5].Any real data or signal measuring process includes some degree of noise from various possible sours.The desired signal may have added noise due to thermal or other physical effects related to the signal generation system, or it may be introduced noise due the measuring system or digital data sampling process.[11] the filtering problem in this paper is applied to two wellknown stochastic process (Poisson process and wide senses stationary Markova process) .wehave to know that a filter is said to be linear if a set of it's inputs is linear and the filter operator is linear therefore we are dealing with linear filter operation.A linear filter is an operation that takes a function X(t), the input and transform it into another function X(t)-φ() the output, such that the transform of a linear combination of functions is equal to same linear combination of transforms of individual function of future values and for identification of interesting features or removal of noise, [3].It is fare to say that the spectral theory of stationary process is most useful in connection with linear filters so we have to find her the spectral representation of the stationary process to be able finding their filters since that the method of linear filtering is based on expression determining the spectral representation of the stochastic process ate the input and output of linear filtering Our work here should be interest to the applied mathematician.The paper Is organized as follows.In Section 2 we collect some important definitions for Markov process, Poisson process and Spectral density function.In Section 3 we find the spectral density function of wide sense Markov Process.In Section 4 we discuss our problem by taking two cases and we show that how can we find linear filtering of the sum of two known stochastic processes with the mean square filtering error for each case.

Basic definitions 1. Wide sense Markov process
The random process {():  ∈ }is said to be a Markov process in the wide sense if

The spectral density function
Every covariance function for stationary process has a Fourier transform (or rather an inverse Fourier transform) called the spectral density function [3]; i.e. if   () is the correlation function(cov.function)for the stochastic process ()then the spectral density function of () can be represented in the following equation See [2] The spectral density function of wide sense Markov Process The auto-covariance function (the correlation function) of wide sense Markov process is exponential type Which can easily be Fourier transformed to find the spectral density function [1] So by using eq.(2.6) we get 3)

The statement of the problem
Let X (t) be a stochastic process and suppose that it is represent a "signal"(telephone message, radio broadcast, etc).But in our daily life and because of the existence of noise (which will also assume as a stochastic process Y (t)) that cams from the new technology or even our voice; X (t) will become spoiled and which is mean that we got a new function which came from the sum of these stochastic processes (signal and noise) : In this paper we'll consider the problem of reconstruction of the value of a known stochastic process (noise) at the time ( + ), where ≥ 0, we'll refer to the problem of linear filtering of stochastic process ,   .

𝜋𝜆
Where The linear filtering problem is equivalent to geometric problem of dropping a perpendicular from the point  1 ( + ) of Hilbert space  onto the linear subspace   () spanned by the set of all vectors ( − ) ,  ≥ 0 [10] so we have to find the vector   () in   () such that: ( 1 ( + ) −   (), ( − )) = 0 for  ≥ 0 (4.6) () satisfied this condition is uniquely defined therefore the random variable   () is the best approximation to  1 ( + ) which depends linearly on ( − ),  ≥ 0 since   () belong to   () that means there is a sequence converges in the mean square to   () and it can be written as the following: Now since we said in our assumption that we know the value of the sum of the noise and signal at all the moments of time so in this case, the spectral characteristic function   () for filtering must satisfy the condition It is clear that eq. ( 4.21) is a Fourier transform and since the function is an absolutely integral function on ℝ whose Fourier transform is identically equal to zero then  We have to find now the mean square filtering error by using eq.(4.27)

2 .
complete suppression of the noise is possible only when the spectra of the signal and the noise do not overlap [10] Linear Filtering of the Sum of wide sense Markov Process and Poisson process We'll consider the case where the signal {();  ≥ 0} is stationary process of Markov type (wide sense Markov process) and the noise();  ≥ 0} is Poisson process then by(5)&(3)   () =   2 +  2 ,   () =   () = () + () (4.1) ́   () =   () −   () to condition (b), the function   () =     () −   ()  () in the upper half-plane.This implies that the zeros of the denominator of   () at the point =0 and  =  must be canceled by zeros of the numerator at the same points, so that   () =  −    () =  − ,   (0) = 0 (4.34)Since the function   () must be analytic in the lower half-plane and the function   () must be analytic in the upper half-plane it follows that   () and   () can have no singularities in common and because of the form (4.33)this implies that   () can have no singularities in any point so that   () =   (4.35)