Fourier Domain Gradient Descent Total Least Square/Fourth Algorithm for Efficient Adaptive Direction of Arrival Estimation

Direction-of-arrival (DOA) estimation is formulated within an adaptive-filtering framework that partitions the sensor array into a reference element and an auxiliary array. The auxiliary-array signal is filtered and subtracted from the reference to produce an error, minimized by the complex least-mean-square (LMS) algorithm. Although LMS converges rapidly with a large step size, it exhibits degraded steady-state performance; conversely, the complex least-mean-fourth (LMF) algorithm yields better steady-state accuracy but slower convergence. To combine their strengths, we propose two algorithms: complex LMS/F, which adaptively switches between LMS and LMF algorithms according to a threshold parameter; and complex GD-TLS/F, which employs a gradient-descent total-least-squares criterion to enhance robustness against noisy inputs. We derive the cost functions and weight update rules for both algorithms and introduce a novel computationally efficient Fourier domain approach for DOA estimation from the adaptive filter weights. A comprehensive theoretical analysis that includes a global optimal solution, mean stability, steady-state mean-square performance, and mean-square convergence is presented. Extensive simulation results demonstrate that the proposed algorithms achieve lower estimation error compared to existing adaptive algorithms.