Volume 9, Issue 3

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Elliptic Rational Functions

# Introduction

Designers of many practical systems searched for a rational function in the variable that has

• the equiripple property and over the interval
• the largest value of for
• the minimal order

The rational function with those properties was found [1, 2] by using the Jacobi elliptic functions [3] and it is referred to as the elliptic rational function [4].

A function has the equiripple property if it oscillates between maximums and minimums of equal amplitude [1]. A quotient of two polynomials is called a rational function in the variable , and the highest power in the polynomials is called the order of the rational function. The minimal value of for is called the discrimination factor and is designated by . In signal processing theory is known as the selectivity factor and can be any real number greater than 1, . Note that is not a rational function in . A typical plot of is shown for and .

We have defined the function EllipticRationalFunction that implements . The algorithm is detailed in the subsequent sections.

We used our symbolic algorithm for the elliptic rational function to optimize the symbolic performance of analog and digital systems. This optimization is not possible using traditional numeric algorithms. We derived closed-form formulas for designing high-speed low-consumption systems known as quadrature mirror filter banks [5]. is extensively used in analog signal processing as the best approximation function [6].

We found a new function, known as Minimum-Q Elliptic [4, 7], by symbolically optimizing the elliptic rational function. Minimum-Q Elliptic became a standard function in manufacturing integrated filters [7]. In addition, again using symbolic optimization, we implemented a very efficient digital signal processing (DSP) system using programmable logic devices and very large-scale integrated circuits [5, 8]. By an efficient DSP system, we mean processing by multiplierless systems that consist of a small number of adders and binary shifters.