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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