Direct estimation of time difference of arrival from compressive sensing measurements

In many applications such as localization, there is a need to determine the unknown time shift D between a signal, and its time shifted version. Aligning one against the other until the two match will find D. When working with compressive sensing (CS) measurements, only linearly transformed samples of the signal and its time-shifted version are available. These CS samples conceal the explicit time shift relationship between the signals, and D can no longer be found by a simple alignment of the CS measurements. As a result, estimation of the time-difference-of-arrival (TDOA) from CS measurements requires the restoration of the original signals. The nonlinear restoration can be time consuming, and may introduce large errors when noise is present. This paper provides an alternate TDOA estimator that avoids restoration. The key is in making additional measurements to preserve the time shift relationship of the signals. This requires a slight modification of the random modulator pre-integrator, as described in the paper, which also includes a simulation example.