Convergence analysis of a data echo canceller with a stochastic gradient adaptive FIR filter using the sign algorithm

Under the assumption that the "errors at the taps" are Gaussian distributed, a new set of recurrence formulae is derived for calculating theoretical convergence process of a data echo canceller equipped with an FIR filter that is adaptively controlled by using the "stochastic gradient sign algorithm" with a binary and white process as the filter input. Convergence curves for the mean squared residual echo based on the recurrence formulae show an excellent agreement with those obtained by simulation. Approximate recurrence formulae that yield a useful, though less accurate, estimation of the residual echo convergence are also proposed. Furthermore, a closed-form solution to the approximate recurrence formulae is derived. And finally, probability density function of the residual echo is depicted as it changes its shape as the echo canceller converges.