On Parameter Estimates of the Lossy Wave Equation

The Cramer-Rao bound (CRB) and maximum-likelihood estimates in Gaussian noise for the parameters of a sampled signal that satisfies the lossy wave equation (such as the voltage sampled along a transmission line) are derived. Results for both equally and unequally spaced samples are given. It is shown that for certain values of the wave number (frequency) the estimates are sensitive to the amount of the ldquoreflected waverdquo in the signal, while for other values of the wave number the estimates are insensitive. It is also shown that for certain special cases it is possible to improve on standard frequency estimation (up to 3.6 dB) by reflecting some of the (noise-free) signal energy into a backward-propagating wave and applying the maximum-likelihood estimator (MLE) to the new measured signal.