Direction finding is a hot research area in radar and sonar systems. In the case of q ≥ 2, the 2qth-order cumulant based direction of arrival (DOA) estimation algorithm for the 2q-level nested array can achieve high resolution performance. A virtual 2qth-order difference co-array, which contains O(N2q) virtual sensors in the form of a uniform linear array (ULA), is yielded and the Gaussian noise is eliminated. However, some virtual elements are separated by the holes among the 2qth-order difference co-array and cannot be fully used. Even though the application of the multi-frequency method for minimum frequency separation (MFMFS) can fill the holes with low computation complexity, it requires that the number of frequencies must increase with the number of holes. In addition, the signal spectra have to be proportional for all frequencies, which is hard to satisfy when the number of holes is large. Aiming at this, we further propose a multi-frequency method for a minimum number of frequencies (MFMNF) and discuss the best frequency choice under two specific situations. Simulation results verify that, compared with the MFMFS method, the proposed MFMNF method can use only one frequency to fill all the holes while achieving a longer virtual array and the DOA estimation performance is, therefore, improved.
Neural Network Virtual Sensors for Fuel Injection Quantities with Provable Performance Specifications
