Information Theoretic Bounds for Compressive Time Delay Estimation

Compressive sensing (CS) simplifies software and hardware via sub-Nyquist sampling, widely used in radar signal processing. But research on performance bounds for compressive parameter estimation in CS radar systems is insufficient. This paper proposes the global bounds for compressive time delay estimation in CS radar, utilizing Shannon information theory. Specifically, we define the indicators of the time delay entropy error (TDEE) and time delay information (TDI) to evaluate the compressive time delay estimation performance. The theoretically derived TDEE and TDI provide the global lower and upper bounds for the estimation performance, respectively. Moreover, we deduce the asymptotic lower bound for TDEE and the asymptotic upper bound for TDI. The asymptotic lower bound and the derived Cramér-Rao bound are identical. The theoretical analyses show that the asymptotic lower bound is inversely proportional to the square of the compression ratio (CR), while the asymptotic upper bound is directly proportional to the logarithm of CR, reflecting the trade-off between compression level and estimation performance. Particularly, the TDEE and TDI do not depend on CR in low SNR. Simulation results illustrate the obvious advantages of TDEE over the Cramér-Rao bound on evaluating and predicting the time delay estimation performance for CS radar.