Resolving delay-free loops in recursive filters using the modified Härmä method

Resolving delay-free loops in recursive filter structures has been a longstanding problem approached in several different ways including signal flow graph manipulation [1], [2] and more recently with Zavalishin's instantaneous response technique [3],[4]. Härmä demonstrates a method for resolving delay-less loops in recursive filter structures [5] but the technique is limited to a specific generic loop topology in which the feedforward branch does not implement signal processing; all processing is implemented in one or more delay-less feedback loops. We modify Härmä's method to accommodate filter processing in the feedforward branch and provide a step-by-step method to resolve delay-less loops in recursive filter structures. We conclude with examples including a new method of synthesizing fourth order filters.