The Covarience Matrix of Za is obtained by executing the expectations of Za and
ZaZaT from above the equation .So in the presence of the matrix we need to
evaluate the equation .Where we can expresss the noise and the real value in terms
of equations
Ga=Gao+ ▲Ga
And
h=ho+ ▲h
Since Ga0Za0=ho implies that
Ψ=▲h-▲GaZa0
By solving the equations above we can get the covariance matrix
Cov(Za)=(Ga0T Ψ-1Ga0)-1
…………………..5)
So the above equation specifies that the Square error term is been ignored from the
equations 2 and 3 and gives us Covariance of Za Matrix. So the solution of Za
assumes that x,y, r are independent they are not related to each other to relate each
other and incorporate the solution which gives the result with out noise free and
improved estimate value.
By ignoring the bais when the TDOA’s are very small noise then we can get the
Za values are:
Za,1=x0+e1 Za,2=y0+e2 Za,3=r10+e3
Where e1,e2,e3 are estimation errors of Za.
Subtracting za,1 and za,2 by x1 and y1 and squared ,where we will get the
equation
Ψ’=h’-Ga’Za’o
Where
h’=
Ga1=
Where Ψ’ is a vector specified inaccurancies of Za.
Where after substituting the above equations we will get the Covariance of Ψ’
matrix .
Ψ’=4B’cov(Za)B’
---------------------7
Since Ψ is Gaussian ,then Ψ’ is also Gaussian .Thus
Za’=(Ga’T Ψ’-1Ga’)-1Ga’T Ψ’-1h’
----------------------8
Where B1 can be evaluated by using the values of Za, Ψ’ will be executed by
using the values of Za and B1 ,Ga0 is approximately equal to Ga .Where we will
get when the source is near then Za’ values will be verified if it is far then we get
another equation as
Za’=(Ga’T B’-1Ga’Q-1GaB’-1Ga’)-1 (Ga’T B’-1GaQ-1GaB’-1)h1 --------9
Where we can get the final position of x ,y by using the equation of binomial is
Zp=√Za’+[x1;y1] or Zp=-√Za’+[x1;y1]
---------------10
Summary:
1. To find the Values when the Position of Sensors are very near towards the
mobile station then we use the equations which are defined above are:
Za≈ (GaTQ-1Ga)-1GaTQ-1h.
Where the values of Za is used to find out the B equation then the equations
we need to find are
‫=טּ‬E[‫טּטּ‬t]= c2BQB
Za=(GaT‫טּ‬-1Ga)-1GaT‫טּ‬-1h
Za’=(Ga’T Ψ’-1Ga’)-1Ga’T Ψ’-1h’
Zp=√Za’+[x1;y1] or Zp=-√Za’+[x1;y1]
Where we will get the appropriate position of x and y Mobile Station
position with error vector which will be negatioble random Noise.
2. To find the values when the position is Far from the Sensor then we need to
accept the equations defined as ..
Za≈ (GaTQ-1Ga)-1GaTQ-1h.
Where the values of Za is used to find out the B equation then the equations
we need to find are
‫=טּ‬E[‫טּטּ‬t]= c2BQB
Za=(GaT‫טּ‬-1Ga)-1GaT‫טּ‬-1h
Za’=(Ga’T B’-1Ga’Q-1GaB’-1Ga’)-1 (Ga’T B’-1GaQ-1GaB’-1)h1
Zp=√Za’+[x1;y1] or Zp=-√Za’+[x1;y1