Compressed Maximum Likelihood

One application for time-difference-of-arrival (TDOA) estimation is in emitter localization. A signal from an emitter reaching a group of sensors, each in a separate location, will have different arrival times. Finding the TDOAs between the output of pairs of sensors will provide the necessary measurements for the hyperbolic localization of the emitter. When the sensors acquire the signal by compressed sensing (CS), their outputs are reduced dimension linear transformation of the time samples of the signal. This shuffling of the time samples breaks up their time relation. Thus, a cross correlation of the CS output of two sensors cannot determine the TDOA. To apply cross correlation, it is necessary to reconstruct the time samples. This letter proposes an alternative that uses only the coefficients of the discrete Fourier transform (DFT) of the CS samples. It begins with the derivation of the maximum likelihood (ML) equation and the ML estimator. This estimator requires known values of signal and noise powers. Substituting these values by their estimates lead to the approximate ML estimator. The phase of the product of two DFT coefficients from each sensor is proportional to the unknown TDOA. Hence, these coefficients can provide an estimation of the TDOA. Simulation results show that although ML is the best, as expected, all these estimators have very close performance.