1. Least Squares Measurement model Weighted LSQ  Optimal estimates  Linear  Unbiased  Minimum variance

2. White Gaussian noise sequence Random sequence  Sequence of random variable:  Discrete random process  Let White noise sequence  If White Gaussian sequence  If it is white and each is normal

3. Measurement equation Observation eq. Of Linear system   where : the state sequence : the deterministic control sequence : measured data sequence : Constant known matrix : Zero mean white Gaussian measurement noise –With

4. Optimal Estimate Obtain optimal estimate of state vector from the information  What is optimal ?  Minimum estimation error:  Error free case 

5. Cost function We wish to minimize   Where Find such that   And optimum weight Note that if, then ordinary LSQ

6. Least Squares Estimates Let be nonsingular  Then show that (Exercise 1.11)   It has minimum when   Or What happens when is singular ?

7. Optimal Weight W k Define estimate error  Find W k which minimize 

8. W k = R k -1 For positive definite symmetric R k  SVD :  Schwarz inequality  with  Therefore, W k = R k -1  Since,

9. Least Squares Optimal Estimate With W k = R k -1, optimal estimates of x k  Unbiased estimate of x k  Minimum variance estimate of x k  Consistent estimate

10. Application of Least Squares Estimation Least Squares Estimation Definition of GDOP Variation of GDOP: PDOP, HDOP, …. 10

11. WLS at GPS Given Code measurement  or Linearizing at Nominal Point:  Taylor Series:  Code measurement becomes Define and  Then  where is LOS(line of Sight) vector between and Satellite 11

12. WLS at GPS (cont) For m Satellite  Let Covariance of measurement is   Then LSQ gives   where 12

13. WLS at GPS (cont) 13 S4 S2 S1 S3 (X 1,Y 1,Z 1 ) (X 2,Y 2,Z 2 ) (X 3,Y 3,Z 3 ) (X 4,Y 4,Z 4 ) PseudoRange Measurement PR=R TRUE +Bias B BB B R=True Distance between Satellite and User B=Distance Error Due to Clock Bias R1 R2 R3 R4 P0P0 h1h1 h2h2 h3h3 h4h4 P  = A  u

14. WLS at GPS (cont) Covariance of Estimates  Least Square Estimation  Assume performance of each channel is same:  Then Estimation Error is affected by  Receiver performance  C/A code:  Carrier Phase:  User-Satellite Geometry: Quantitative Measure?  14

15. Definition of GDOP Define where Definition of GDOP  Note  Estimation Error can be expressed  15

16. Meaning of Positioning Error = GDOP x UERE  UERE: User Equivalent Ranging Error  Ionospheric delay + Troposheric delay + Multipath + Receiver Noise  GDOP amplifies UERE  Design of Constellation is very important  Selection Satellites which gives small GDOP To reduce Positioning Error Both GDOP and UERE are concerned 16

17. Meaning of GDOP Bad GDOPGood GDOP 17 UERE

18. Coordinate Transformation Transform WGS-84 to ENU   where Covariance Transformation  Thus Errors in ENU 18 x y z N E E

19. GDOP in ENU Definition of GDOP  Other DOPs  19

20. Applications of GDOP Predicts Positioning Error Constellation Design Selection of Satellites 20