Error surfaces of adaptive recursive filters

For an adaptive filter with N adjustable coefficients or weights, the error surface is a plot, in N + 1 dimensions, of the mean squared error vs the N coefficient values. If the adaptive filter is nonrecursive, the error surface is a quadratic function of the coefficients. With recursive adaptive filters, the error surface is not quadratic and may even have local minima. The nature of the recursive error surface and the conditions under which local minima may exist are examined using theory and examples. The effects of the nonquadratic error surface on gradient search algorithms for recursive adaptive filters are discussed.