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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Meaning of GDOP Bad GDOPGood GDOP 17 UERE

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

GDOP in ENU Definition of GDOP  Other DOPs  19

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