Performance bounds and a parameter transformation for decay rate estimation

Decay rate estimation has been proposed as an effective method for signal characterization in many application areas, including subsurface sensing using time-domain electromagnetic induction (EMI) sensors. The physical basis for this strategy is that every signal of interest, which corresponds to a target or phenomenon of interest, possesses a unique set of decay rates. In theory, the characteristic decay rates can be estimated from the measured signal, and then utilized for signal detection and subsequent identification. Using this approach, signal discrimination performance is dependent upon decay rate estimation performance. The Cramér-Rao lower bound (CRLB) for decay rate and amplitude coefficient estimates is utilized to investigate the fundamental limits of decay rate estimation accuracy. Previous derivations of the CRLB for decay rate estimates have focused on signals which are linearly sampled beginning at time t = 0. Here, the CRLB is generalized to accommodate any arbitrary sampling method and any initial starting time. A parameter transformation which improves decay rate estimation performance is also presented. Simulation results across a wide range of decay rates and SNRs show that nonlinear least squares estimation of the decay rates via the proposed transformation provides estimates with smaller RMS and bias than can be obtained without the parameter transformation. The parameter transformation also provides decay rate estimates that approach the CRLB. Improvement in estimation performance for this class of signals has important ramifications in signal detection, classification, and identification performance in several geophysical application areas.