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h1. PART 4 : Position Estimation.

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p(. Once the navigation bits from at least 4 satellites have been retrieved from the acquisition/tracking part, it is possible to estimate the desired position of the receiver.

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h2. 1 - Ephemeris.

h3. a - GPS satellite ephemeris data.

p=. !Eph12min.png!

h3. b - GPS satellite position calculation algorithm.

p=. !Alg12min.png!

from GPS Interface Control Document

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h2. 2 - Navigation computation.

h3. a - Reminder about the impairments.

The following figure gives the impairments affecting the range in case of the GPS system as well as the correction process :

p=. !003.PNG!

*Figure 4.1 :* Pseudo-range measurement extracted from *[2]*

h3. b - Demonstration of the Pseudo-ranges with Least Square method.

Starting from the fact that can determine most of the elements within the pseudo-range measurement PR_sat(i) from the information provided by each satellite, we have the equation : 

p=. !Pos1.png!

*Equation 1*

or put in another way,

p=. !Pos2.png!

*Equation 2*

Indeed 4 measurements are needed, providing 4 equations with 4 unknows which are the receiver coordinates and the clock bias of the receiver. As the equation is highly non-linear, it is important to proceed to a linearization such as the Taylor expansion :

p=. !Pos3.png!

*Equation 3*

Hence,

p=. !Pos4.png!

*Equation 4*

or in matrix equation form,

p=. !Pos5.png!

*Equation 5*

which can be expressed as :

p=. !Pos6.png!

*Equation 6*

with the Least Square solution :

p=. !Pos7.png!

*Equation 7*

Thus, it is possible to retrieve the receiver position.

_Note that all unknowns are depicted in red color._

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*References :* 

*[1]* K. Borre, D. M. Akos, N. Bertelsen, P. Rinder, S. H. Jensen, A software-defined GPS and GALILEO receiver

*[2]* Position Estimation Workshop, March 2016