On the zeros of the linear prediction-error filter for deterministic signals

The zeros of a linear prediction-error filter polynomial for a class of deterministic signals, that are a sum of samples of M exponentially damped/undamped sinusoids, is studied. It is assumed that N samples are available for processing and that they are uncorrupted by noise. It is shown that the exponent parameters of the M signals can be determined from M zeros (called "signal zeros") of an Lth degree prediction-error filter polynomial (L>M) if L lies in between M and N - M (N - M/2 in a special case). The rest of the L - M zeros of the filter polynomial switch are called extraneous zeros, are shown to be approximately uniformly distributed with in the unit circle, regardless of the type of exponentials in the signal, if the prediction filter coefficients are chosen to have minimum Euclidean length. The results obtained provide insight into the estimation problem with noisy data.