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 to be estimated, \( I(\theta)\) is the Fisher Information and \( g(x)\) is some function.

Then, the estimator that attains CRLB is given by

## Steps to find MVUE using CRLB:

If we could write the equation (as given above) in terms of Fisher Matrix and some function \( g(x)\) then \(g(x)\) is a Minimum Variable Unbiased Estimator.

1) Given a signal model \( x \), compute \(\frac{\partial\;ln\;L(\mathbf{x};\theta) }{\partial \theta }\)

2) Check if the above computation can be put in the form like the one given in the above theorem

3) Then \(g(\mathbf{x})\) given an MVUE

Let’s look at how CRLB can be used to find an MVUE for a signal that has a DC component embedded in AWGN noise.

## Finding a MVUE to estimate DC component embedded in noise:

Consider the signal model where a DC component – \(A\) is embedded in an AWGN noise with zero mean and variance=\(\sigma \).

Our goal is to find an MVUE that could estimate the DC component from the observed samples \(x[n]\).

$$x[n] = A + w[n], \;\;\; n=0,1,2,\cdots,N-1 $$

We calculate CRLB and see if it can help us find a MVUE.

From the previous derivation

From the above equation we can readily identify \( I(A)\) and \(g(\mathbf{x})\) as follows

Thus,the Fisher Information \(I(A)\) and the MVUE \(g(\mathbf{x})\) are given by

Thus for a signal model which has a DC component in AWGN, the sample mean of observed samples \(x[n]\) gives a Minimum Variance Unbiased Estimator to estimate the DC component.