Fisher Matrix

Cramer Rao Lower Bound for Phase Estimation

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Key focus: Derive the Cramer-Rao lower bound for phase estimation applied to DSB transmission. Find out if an efficient estimator actually exists for phase estimation. Problem formulation Consider the DSB carrier frequency estimation problem given in the introductory chapter to estimation theory. A message is sent across a channel modulated by a sinusoidal carrier with … Read more

Efficient Estimators by applying CRLB

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It has been reiterated that not all estimators are efficient. Even not all the Minimum Variance Unbiased Estimators (MVUE) are efficient. Then how do we quantify whether the estimator designed by us is efficient or not? An efficient estimator is defined as the one that is* Unbiased (mean of the estimate = true value of … Read more

Applying Cramer Rao Lower Bound (CRLB) to find a Minimum Variance Unbiased Estimator (MVUE)

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It was mentioned in one of the earlier articles that CRLB may provide a way to find a MVUE (Minimum Variance Unbiased Estimators). Theorem: There exists an unbiased estimator that attains CRLB if and only if, Here \( ln \; L(\mathbf{x};\theta) \) is the log likelihood function of x parameterized by \(\theta\) – the parameter … Read more

Cramer Rao Lower Bound Introduction with Python Examples

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The Cramer Rao lower bound introduction starts with a simple question: how accurately can any unbiased estimator measure an unknown value? The Cramér-Rao lower bound (CRLB) gives a hard answer. It is a floor on the variance of every unbiased estimator, so it tells you what is achievable before you design anything. This post explains … Read more