A classical problem in statistical signal processing involves estimating a weak signal (e. g., the heart beat of a fetus) in the presence of a strong interference (the heart beat of its mother) by making two observations; one with the weak signal present and one without (by placing one microphone on the mother's belly and another close to her heart). The observations can then be combined to estimate the weak signal by canceling out the interference. The following is a simple version of this application.
Let the weak signal X be a random variable with mean µ and variance P. Let the observations be Y1 = X + Z1 and Y2 = Z1 + Z2, where Z1 is the strong interference and Z2 is measurement noise. Assume that Z1 and Z2 are zero mean with variances N1 and N2, respectively. Further assume that X, Z1, and Z2 are uncorrelated. Find the LLSE estimate of X given Y1 and Y2 and the corresponding MSE. Interpret the results.