Temporal and spatial sampling influence on the estimates of superimposed narrowband signals: when less can mean more

This paper addresses the influence that the sampling locations have on the estimated frequencies of superimposed sinusoids. This problem has application in harmonic time-series analysis or direction-finding phased-array systems. Generalized mathematical bounds are developed in terms that are independent of the array locations and have an intuitively appealing physical interpretation. They establish the influence of the sampling locations on the variance of the frequency estimate and the limit at which two sources can be resolved using signal subspace estimators. For the resolution criteria, an expression dominated by the fourth central moment of the sensor locations expresses the resolving ability of the sensing array, irrespective of the array aperture or number of sensors. Increasing the fourth central moment increases an array's resolution ability. The commonly accepted notion that resolution necessarily depends on the array aperture is misleading and, indeed, that fewer snapshots from a reduced aperture array can outperform a larger array of more elements. For the estimator variance criteria, it is found that the product of the number of sensors and the second central moment (array variance) characterizes the estimator variance lower bound. The metrics developed demonstrate that the sampling topology is itself an important factor in determining the performance of the sampling system (and not the covariance lags sampled or the aperture spanned). Simulations are used to describe the results.