Pole and Residue Estimation from Impulse Response Data: New Error Bounding Techniques

Estimates of non-random pole and residue parameters from noisy impulse-response data are characterized. Specifically, Barankin-type lower bounds (BB) on the estimation error variance of unbiased estimators are developed for single-input single-output systems with multiple but distinct real poles. Two variants of the Barankin-type bound are compared with the widely-used Cramer-Rao lower bound (CRB) in examples. The BB is found to significantly improve on the CRB when noise levels are high compared to the impulse response signal, indicating limited effectiveness of the CRB for unbiased estimators in this low signal-to-noise regime. In addition, an apparent paradox in the error bounds for low signal-to-noise ratios (SNR), which arises because the bounds are limited to unbiased estimators, is explored. Two simple constructions of biased estimators are then developed, which achieve lower mean square errors than the BB in low SNR regimes in examples.