Concise representation and cellular structure for universal maximally flat FIR filters

The universal maximally flat lowpass FIR filters of Baher possess closed-form expressions for their transfer function. The expressions involve binomial coefficients and nested sums. We show that there exists a concise formula for the universal maximally flat low-pass filter H_N,K,d(z) in the form of the Nth power of a linear algebraic operator acting on a finite-length sequence. The linear operator involves the forward shift operator, widely used in numerical analysis and the theory of finite differences. We also employ the formula to develop a hierarchical cellular structure for realization of arbitrary universal maximally flat filters. The structure is comprised of identical cells that are interconnected regularly. The structure is especially suitable for a variable realization where the values of the parameters can be varied by adding or deleting extra cells or changing the value of a single multiplier coefficient.