Unless stated otherwise, noise sequences are zero-mean, Gaussian, stationary, an

Question

Unless stated otherwise, noise sequences are zero-mean, Gaussian, stationary, and exist for 0 lessthanorequalto n lessthanorequalto N-1. All white noise sequences have variance sigma^2. Let z^i(n) = s(n) + n^i(n) where n^i(n) is white and 1 lessthanorequalto i lessthanorequalto N_ch. var(n^i(n)) = v(i), so the noise variance is different for each channel. The s(n)'s are unknown. (a) Find the LLF for s(m). (b) Find the MLE of s(m). Is the estimate efficient? (c) Find the variance of the estimate.

Explanation / Answer

The parameters of a Gaussian distribution are the mean (µ) and variance ( 2 ). Given observations x1, . . . , xN , the likelihood of those observations for a certain µ and 2 (assuming that the observations came from a Gaussian distribution) is p(x1, . . . , xN |µ, 2 ) = Y N n=1 1 2 exp ( (xn µ) 2 2 2 ) (2) and the log likelihood is L(µ, ) = 1 2 N log(22 ) X N n=1 (xn µ) 2 2 2 (3) We can then find the values of µ and 2 that maximize the log likelihood by taking derivative with respect to the desired variable and solving the equation obtained. By doing so, we find that the MLE of the mean is µˆ = 1 N X N n=1 xn (4) and the MLE of the variance is ˆ 2 = 1 N X N n=1 (xn µˆ) 2 (5)

The variance of the estimator  is 24/n

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