chapter 6 best linear unbiased estimator (blue)

Chapter 6Chapter 6BEST Linear Unbiased Estimator (BLUE)BEST Linear Unbiased Estimator (BLUE)

Shin won jaeShin won jae

Sejong University Dept. Computer Eng.

WDC Lab.Wireless Digital Communication

Contents

1. Introduction

2. Best Linear Unbiased EstimateDefinition

Finding in scalar case

Finding BLUE for Scalar Linear Observation

Applicability of BLUE

Vector Parameter Case

3. TOA Measurement ModelLinearization of TOA Model

TOA vs. TDOA

Conversion to TDOA Model

Apply BLUE to TDOA linearized Model

Apply TDOA Result to Simple Geometry

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Introduction

Motivation for BLUEExcept for Linear Model case, the optimal MVU estimator might:

1. not even exist

2. be difficult or impossible to find Resort to a sub-optimal estimate

BLUE is one such sub-optimal estimate

Idea for BLUE1. Restrict estimate to be linear in data x

2. Restrict estimate to be unbiased

3. Find the best one (i.e. with minimum variance)

Advantage of BLUE: Needs only 1st and 2nd moments of PDF

Disadvantage of BLUE:1. Sub-optimal

2. Sometimes totally inappropriate



Definition of BLUE (scalar case)

Observed data:

PDF: depends on unknown

BLUE constrained to be linear in data:Choose a’s to give: 1. unbiased estimator

2. then minimize variance

4

T= [0] [1] [ 1]x x x N x

( ; )p x

1

BLU

0

[ ] N

nn

a x n

Ta x



Finding the BLUE (Scalar Case)

1. Constrain to be linear:

2. Constrain to be Unbiased:

Q: When can we meet both of these constraints?

A: Only for certain observation models (e.g., linear observations)

5

1

0

[ ]N

nn

a x n

{ }E

1

0

[ ]N

nn

a E x n



Finding the BLUE for Scalar Linear Observation

Consider scalar-parameter Linear observation:

For the unbiased condition we need:

Now … given that these constraints are met…

We need to minimize the variance

Given that C is the covariance matrix of x we have:

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[ ] [ ] [ ] { [ ]} [ ]x n s n w n E x n s n

1

0

{ } [ ]N

nn

E a s n

1Need Ta s

1

0

[ ]N

nn

a x n

var varBLU T Ta x a Ca



Finding the BLUE for Scalar Linear Observation

Goal: minimize subject to

Constrained optimization

Appendix 6A: Use Lagrangian MultipliersMinimize

7

Ta Ca 1Ta s

1)J s T Ta Ca a

Set:10

J

a

Ta s 1

a C s

11

1 1

T T

Ta s s C s

s C s

1

1

T

C sa

s C s

1

TBLU

T 1

T

s C xa x

s C s

1

1var( )

Ts C s



Applicability of BLUE

We just derived the BLUE under the following:

1. Linear observations but with no constraint on the noise PDF

2. No knowledge of the noise PDF other than its mean and cov

8

What does this tell us????

BLUE is applicable to Linear observationsBut... Noise need not be Gaussian!!!

(as was assumed in CH.4 Linear Model)

And all we need are the 1st and 2nd moment of the PDF!!!

But... We’ll see in the Example that wecan often linearize a nonlinear model!!!



Vector Parameter Case: Gauss-Markov Theorem

Gauss-Markov Theorem:If data can be modeled as having linear observations in noise:

Then the BLUE is :

and its covariance is:

9

x Hθ wKnown Matrix

Known Mean & Cov (PDF is otherwise

arbitrary & inknown)

11 1T TBLUE

θ H C H H C x

11T θ

C H C H

Note: If noise is Gaussian then BLUE is MVUE



Ex. 6.3 TDOA-Based Emitter Location

Assume that the i-th RX can measure its TOA:

Then… from the set of TOAs… compute TDOAs

Then… from the set of TDOAs… estimate location

10

it

( , )s sx y

We won’t worry about “how” they do that.Also... there are TDOA systems that never actually estimate TOAs!



TOA Measurement Model

Assume measurements of TOAs at N receivers (only 3 shown above):

TOA measurement model:: Time the signal emitted

: Range from Tx to

: Speed of Propagation (for EM: )

11

There are measurement errors 0 1 1, ,..., Nt t t

0T

iR

c

Rx i

83 10 m/sc

0 / 0,1,..., -1i it T R c i N Measurement Noise : zero-mean, variance , independent (bot PDF unknown)(variance determined from estimator used to estimate

'it s

1/ 22 2Now use: ( ) ( )i s i s iR x x y y

1/ 22 20

1( , ) ( ) ( )i s s s i s it f x y T x x y y

c

NonlinearModel



Linearization of TOA Model

So… we linearize the model so we can apply BLUE:

Assume some rough estimate is available

Now use truncated Taylor series to linearize

12

( , )n nx y

s n sx x x s n sy y y [ ]Tx y θ

know estimate know estimate

( , ) :i n nR x y

Three unknown parameters to estimate:0 , ,s sT x y



TOA Model vs. TDOA Model

Two options now:1. Use TOA to estimate 3 parameters:

2. Use TDOA to estimate 2 parameters:

Generally the fewer parameters the better…everything else being the same.

But… here “everything else” is not the same:

Option 1&2 have different noise models

(Option 1 has independent noise)(Option 2has correlated noise)

In practice… we’d explore both options and see which is best

13

0 , ,s sT x y ,s sx y



Conversion to TDOA Model

14

N-1 TDOA rather than N TOAs



Apply Blue to TDOA Linearized Model

Things we can now do:

1. Explore estimation error cov for different Tx/Rx geometries

Plot error ellopses

2. Analytically explore simple geometries to find trends

See next chart (more details in book)

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chapter 6 best linear unbiased estimator (blue)

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