STOCHASTIC AVERAGING ANALYSIS OF A STEEPEST-DESCENT-TYPE ADAPTIVE TIME-DELAY ESTIMATION ALGORITHM

In this paper stochastic averaging analysis tools are used to study an adaptive time-delay estimation algorithm. Analyzing such an algorithm is very difficult because of its nonlinear, infinite-dimensional, and time-variant nature. By stochastic averaging analysis, we show that for the time-invariant delay case, the adaptive algorithm output converges weakly to the solution of an ordinary differential equation. Local convergence is demonstrated by showing that the solution of this differential equation converges exponentially to the true delay under reasonable initial conditions. Implementation of the algorithm is also discussed. Guided by the averaging results, a modified algorithm is proposed to eliminate the bias of the delay estimation. Second-order analysis is carried out and the results provide a theoretical justification of the observations made by other researchers with simulation and heuristic argument. Computer simulations are also included to support the analysis.