Implementation of linear digital filters based on morphological representation theory

Recent advances in mathematical morphology have led to a representation theory that covers increasing linear translation invariant (ILTI) filters as a subclass of morphological filters. The representation is based on supremum, minimum, and addition operations, and does not require any multiplications. However, this representation has not been practical because its essence is to evaluate the supremum of a set with an infinite number of elements. Based on the previously developed morphological representation, we present a finite max-min representation for ILTI filters. The new representation does not require the use of multipliers and is practical, in the sense that it consists of a finite number of maximum, minimum, and addition operations. The proposed max-min representation is modified to cover FIR linear shift invariant filters in general. The modified representation requires only one multiplication per output sample and has lower computational complexity than the max-min representation. Other related topics such as duality, roundoff noise, and implementation are also discussed