On the Impact of Unknown Signals on Delay, Doppler, Amplitude, and Phase Parameter Estimation

The estimation of time delay, Doppler shift, amplitude, and phase is an important fundamental tool in signal processing, which has received extensive study for cases with known transmitted signals, but little study for unknown transmitted signals. We derive the closed-form Cramér–Rao bound (CRB) expressions for joint or separate estimation of time delay, Doppler shift, amplitude, and phase with unknown signals with possibly known structure and possible multiple looks at direct path and reflected path observations. The presented results generalize previous results for known transmitted signals and show how many looks from the direct path and the reflected path we need to derive an accurate estimation of time delay, Doppler shift, amplitude, and phase. The advantages of the known signal format with unknown parameters over totally unknown signals are illustrated. After analysis under a simple white clutter-plus-noise model, extensions to the case with dependent clutter plus noise are discussed. Numerical results show very similar behavior. Numerical calculations of the mean square error from maximum likelihood estimation are provided to support the utility of the unknown signal CRB and the known signal format CRB.