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Least Squares Estimation

- S-88.4221 Postgraduate Seminar on Signal

Processing

Alexandra Oborina

This presentation is based on 3rd chapter of the

book Dan Simon, Optimal State Estimation

Kalman, Hinf, and Nonlinear Approaches

Content

- Estimation of a constant vector
- Weighted least square estimation
- Recursive least square estimation
- Wiener filtering
- Homework

Estimation of a constant

- Given x-constant, unknown vector, y-noisy

measurement vector. Problem find best estimate

of x.

Example

Weighted LS Estimation

- Given x-constant, unknown vector, y-noisy

measurement vector. The variance of the

measurement noise may be different for each

element of y. Problem find best estimate of x.

Example

Recursive LS Estimation

- What if we get measurements sequentially?
- Suppose is given after k-1 measurements. So

new yk is obtained. How to update the estimate

?

Recursive LS Estimation

Recursive LS Estimation

Recursive LS EstimationAlternative forms

- Using matrix inversion lemma, substitution and

inversion alternative forms for Kk and Pk can be

obtained. - Alternative forms are mathematically identical,

but can be beneficial from computational point of

view.

Recursive LS EstimationAlgorithm

- Initialize the estimator as
- For k1,2,.. perform
- Obtain the measurements with white noise

Recursive LS EstimationAlgorithm

- Update the estimate of x and estimation-error

covariance

Recursive LS Estimation

Example 1

Example 1

- By induction
- If x is known perfectly a priori

Example 1

- If x is completely unknown a priory

Example 1

Example 2 - Linear data fitting

- Suppose we want to fit a straight line to a set

of data points - Problem find linear relation between yk and tk,

that means estimate the constants x1 and x2

Example 2 - Linear data fitting

- Recursive LS initialization
- Using equations (1), (3), (4) perform recursion

Example 3 - Quadratic data fitting

- Suppose, a priory is known that the data is a

quadratic function of time

Wiener Filtering

- Problem design a stable LTI filter to extract a

signal from noise.

Wiener Filtering - parametric filter optimization

- Lets find optimal G(w) as a first order, stable,

causal filter with 1/T BW - Suppose also the following forms for
- Recall
- Substitute everything to E(e2(t)) and

differentiate with respect to t

Wiener Filtering - general filter optimization

- Problem find filter g(t) that minimize E(e2(t))
- Replace g(t) with

Wiener Filtering - noncausal filter optimization

- Noncausal filter means
- So,
- Thus, quantity inside squire brackets must be zero

Wiener Filtering - causal filter optimization

- Causal filter means g(t)0 for tlt0. So,
- Let denote some function a(t), that is 0 for tgt0

and arbitrary for tlt0.

Example

- The signal and noise power spectra are given as

Homework

- For recursive LS estimation decide is estimator

unbiased or not. - For recursive LS estimation show explicitly

calculations of - For Wiener filtering prove that