A discrete-time sequence s[n] is transmitted over a noisy channel and retrieved. The received sequence x[n] is modeled as x[n] = a×θ[n]+w[n] where w[n] and a represents the channel noise and attenuation respectively. At a particular time instant n = n0, suppose x[n0], θ[n0] and w[n0] are random variables, which we denote as X, Θ and W respectively. We assume that Θ and W are independent, that W is distributed as a Gaussian N(0,1) and the signal Θ is distributed as a Gaussian N(10,3). Please compute:
1. The LMS estimator of θ given X in function of a.
2. The Linear LMS estimator of θ given X. How are both estimators related?