View by Category

Loading...

PPT – Least Squares Estimation PowerPoint presentation | free to download

The Adobe Flash plugin is needed to view this content

About This Presentation

Write a Comment

User Comments (0)

Transcript and Presenter's Notes

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

About PowerShow.com

PowerShow.com is a leading presentation/slideshow sharing website. Whether your application is business, how-to, education, medicine, school, church, sales, marketing, online training or just for fun, PowerShow.com is a great resource. And, best of all, most of its cool features are free and easy to use.

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

Recommended

«

/ »

Page of

«

/ »

Promoted Presentations

Related Presentations

Page of

Page of

CrystalGraphics Sales Tel: (800) 394-0700 x 1 or Send an email

Home About Us Terms and Conditions Privacy Policy Contact Us Send Us Feedback

Copyright 2014 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

Copyright 2014 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

The PowerPoint PPT presentation: "Least Squares Estimation" is the property of its rightful owner.

Do you have PowerPoint slides to share? If so, share your PPT presentation slides online with PowerShow.com. It's FREE!