Gradient-descent adaptation of cascade-form adaptive filters

The behavior of gradient descent algorithms for cascade-form adaptive filters is examined via the geometry of the filter error surface. Relationships are formulated between direct-form filter error surfaces and cascade-form error surfaces by considering properties of the nonlinear transformation between these two parametrizations. Different choices of model set for the cascade-form are discussed together with their implications for gradient-descent behavior. It is demonstrated that in the most practical case, unbounded filter coefficients can result in the adaptive cascade-form filter, even in situations in which the direct-form filter is well-behaved