1. Each satetllite sends two signals:
   • 1. The location/3-dimensional coordinates of the satellite in space (based on on-board computer models and updated data from NORAD)

     Signal 1: (x_s, y_s, z_s)

   • 2. A timing signal, based on a "pseudo-random code" (in our class, we consider the signal to contain the precise moment the signal is sent, which can then be compared to when the signal is received)

     Signal 2: time sent

2. The receiver then:
   • Subtracts the timing signals to find the total time it took the signal to reach it

     t_total = t_rcvd - t_sent

   • Multiplies this total by the speed of light to get the distance away that each satellite is

     d = rt(distance = rate x time)

     rate of the signal = c = the speed of light = 186,282.396miles per second

     d_{to satellite} = c * t_total

   • The receiver now has the location of each satellite (think sphere center) and the distance away that each satellite is (think sphere radius). The user holding the GPS receiver must be somewhere on an imaginary sphere around that satellite.
   • Since the satellites are in motion, the GPS receivers must then take into account Doppler data, which represents the relative speeds between the satellites and the receiver. However, for the purposes of this class (and this website), we will consider the system to be stationary.
   • The receiver can now establish equations of spheres that surround each satellite:

     (x - x_s)² + (y - y_s)² + (z - z_s)² = d²_{to satellite}

   • The intersection point of 4 of these spheres, S₁, S₂, S₃, and S₄, will give (x_r, y_r, z_r) coordinates on earth of the reciever (3 satellites is actually enough when you throw out impossible locations)

     (X_r, Y_r, Z_r)

   

   • To solve the system of 4 spheres for their common intersection point, we subtract 3 pairs of equations of spheres from each other, which gives 3 equations of planes(P₁, P₂, P₃) (the pairs of spheres actually intersect in circles, but doing the algebra gives you the equation of the plane in which the circle of intersection lies)

   P₁ = S₁ - S₂
   P₂ = S₁ - S₃
   P₃ = S₁ - S₄
   

   • Solving this system of three equations with three variables can be done by hand or, more efficiently, using matrices to find (x, y, z) coordinates

   (In reality, the receiver must solve a system of equations with 7 unknowns, which include the Doppler information, and does so using a "least-squares" method. For the purposes of this class, we consider how to solve a system of spheres algebraically)

   • (x, y, z) coordinates can then be converted to latitude and longitude using trigonometry and to elevation using the distance formula between the center of the earth and the point (x, y, z) and subtracting the radius of the earth.

     (X,Y,Z) to (Lat, Long, Elev.)

   For example, the latitude could be calculated using the tangent function (or the sine or cosine, for that matter):
   

   As can the longitude:
   
   Elevation is found by finding the distance from the (x, y, z) point on earth to the center of the earth and subtracting the radius of the earth (in reality, the geoid would have to be considered):
   

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Recommended Readings

• Understanding GPS, by Greg French
  • The most readable book on the mathematics and science of GPS, but it's hard to find. I got mine at the Air and Space Museum in D.C.
• GPS, Intertial Navigation, and Integration by Grewal
  • A highly technical, but extremely thorough, treatment of the real mathematics behind GPS and it's integration with intertial navigation systems. This is a book for the technical experts, but it's amazing to look through.
• The Story of Maps by Brown
  • This is a great book (written in 1949!) that gives the whole story of the history of maps. I learned some amazing things from it.

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Contact: abeckwith@colonial.net
This Page Last Updated: 8/1/08
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