Question

A parameter θ is estimated from N samples of noisy measured signal: x[k]=θᵏ+n[k],k=0,1,…,N−1 where n[k] is White Gaussian Noise (WGN) with zero mean and variance σ². 1. Determine the Cramer-Rao Lower Bound (CRLB) for the estimate of θ.

Answer 1

The Cramer-Rao Lower Bound (CRLB) for the estimate of θ can be calculated as the inverse of the Fisher Information Matrix (FIM).

Step-by-step explanation:

To determine the Cramer-Rao Lower Bound (CRLB) for the estimate of θ in the given scenario, we can use the Fisher Information Matrix (FIM) which is the negative of the second derivative of the log-likelihood function. The CRLB for the estimate of θ can be calculated as the inverse of the Fisher Information, i.e., CRLB = 1 / FIM. In this case, the log-likelihood function can be derived from the given expression for x[k]. Once the log-likelihood function is obtained, we can find its second derivative and calculate the FIM. Finally, by taking the inverse of the FIM, we can determine the CRLB for the estimate of θ.