Performance Bounds for Estimating Time Delay and Radial Velocity with Multiple Broadband Frequency-Modulated Pulses

Estimation of the location and motion of an object of interest is one of the primary inferential objectives in underwater acoustical remote sensing. In active sensing systems, this often begins with estimation of range through time delay and radial velocity by exploiting the Doppler effect. In systems that project a sequence of pulses, radial velocity can also be estimated from multiple time-delay measurements using waveforms insensitive to Doppler. The focus of this report is on performance bounds for estimation of time delay and radial velocity when using multiple frequency-modulated pulses that are not restricted to being narrowband. An emphasis is placed on the case of estimating radial velocity when time delay is also unknown while using combinations of the basic sonar waveforms: continuous-wave, linear-frequency-modulated, and hyperbolic-frequency-modulated pulses. A review of single-pulse bounds on the variance of unbiased estimators is presented to facilitate development of bounds when combining echoes from multiple pulses. The pulse characteristic time-frequency properties comprising the single-pulse bounds are employed to provide multiple-pulse bounds that are straightforward to evaluate. As might be expected, the case of coherent echoes (i.e., common bulk phase) generally leads to a lower bound on estimation performance than when the echoes are incoherent (i.e., different bulk phases). A number of examples are used to demonstrate multiple-pulse estimation performance. An important theme seen throughout the examples is that diversity across multiple pulses can have an outsize effect on parameter estimation. For similar types of pulses, spectral diversity improves time-delay estimation and temporal diversity aids estimation of radial velocity. Independent of this, diversity in the time-frequency character of the pulses (e.g., up- and down LFM or HFM pulses) can provide a similarly significant improvement over the performance of any of the pulses alone.