Cramer-Rao+Bounds-1+JAN+2012

Cramer-Rao+Bounds-1+JAN+2012 - Random Signals and Noise...

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Unformatted text preview: Random Signals and Noise Jerusalem College of Engineering Autumn Semester 2011 21 August 2011 Professor I. Kalet The Cramer-Rao Bound on an Unbiased Estimate 1 The Cramer-Rao Bound on an Unbiased Estimate The Cramer-Rao bound is a lower bound on the variance of estimators. Specifically in this case we will consider the unbiased estimator. In the work below we find the Cramer-Rao Bound for the unbiased estimator. We also show that if the bound is achieved, it will be achieved by the maximum likelihood estimate ( a ml (r)). An estimator that achieves the Cramer-Rao Bound is called an efficient estimator. An unbiased estimator a(r) means that expected value of a(r) is equal to A. a(r) p(r/A) dr A=0- [a(r)-A] p(r/A) dr= 0- Now we take the derivative of the equation above with respect to A. d/dA { [a(r)-A] p(r/A) dr = 0 }- 2 E{a(r)}= a(r) p(r/A) dr = A- The derivative is given by the equation below. [(a(r)-A) [d/dA{p(r/A)}]-p(r/A)]dr= 0- The integral of p(r/A) over all of the values of r is equal to one, so we can write [(a(r)-A) d/dA{p(r/A)}]dr = 1- Now we make use of the following derivative d/dA[Ln p(r/A)]= [d/dA {p(r/A)}]/p(r/A) and rewrite the previous equation as [(a(r)-A) p(r/A) [d/dA{Ln p(r/A)}]dr = 1...
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Cramer-Rao+Bounds-1+JAN+2012 - Random Signals and Noise...

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