Estimation of the parameters of all-pole sequences corrupted by additive observation noise
Johnson, Don H.
Master of Science
Ordinary Least Squares procedures and the equivalent Yule-Walker formulation result in biased estimates of all-pole model parameters when applied to noise corrupted all-pole sequences. This bias is shown to be proportional to the inverse of the signal-to-noise ratio. The algorithm investigated applies an autocorrelation-like operation to the noise corrupted all-pole sequence which increases the signal-to-noise ratio but preserves the pole locations. This operation is applied recursively until acceptable signal-to-noise ratio is obtained. The all-pole parameters are then estimated from the high signal-to-noise ratio sequence using an Ordinary Least Squares estimator. The improvement in signal-to-noise ratio varies for different modes in an allpole sequence with modes corresponding to pole locations close to the unit circle showing the most improvement. A signal-to-noise ratio cutoff exists below which no improvement in signal-to-noise ratio is possible for a given mode. This cutoff is dependent on the radius of the poles of the mode and goes to zero as the pole approaches the unit circle. The signal-to-noise ratio cutoff also corresponds to the point at which the mode’s peak spectral value just equals the level of the noise floor. Estimates of the poles from the high signal-to-noise ratio sequences show reduction in the noise induced bias concomitant with the increased signal-to-noise ratio. Correlations of up to four times are shown to be advantageous. The sensitivity of the successive autocorrelation algorithm to a white observation noise assumption is found to be small. With long correii lation length signals, such as sinusoids, unbiased low variance estimates of the parameters are possible at signal-to-noise ratios of as low as .1.