Quiver diagrams and signed adaptive filters

A graphical technique for exploring the behavior of deterministic discrete time adaptive algorithms excited by periodic inputs is examined. The constraint to periodic inputs is convenient; the underlying concepts used also apply to stationary inputs drawn from finite alphabets. The resulting graph is based on plotting single-period parameter trajectories in the parameter error space and is essentially a discrete version of flow diagrams associated with continuous-time systems. For the least-mean-square (LMS) adaptive FIR filter with the algorithm step size sufficiently small, the change of adapted parameters from a particular initial value is given by the total update over the input period with the parameterization frozen at its initial value. This results in the familiar elliptical hyperparaboloid-surface steepest-descent interpretation. This method can be extended to the LMS adaptive infinite-input-response (IIR) filter with suitable approximations based on a small step size