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Rational polynomial functions have many applications, such as filter design, system theory, and digital image processing. In particular, rational polynomial functions provide the most common way of representing the z-transform. A rational polynomial function takes the form of the division of two polynomials, as shown by the following equation.

where F(x) is the rational polynomial, B(x) is the numerator polynomial, A(x) is the denominator polynomial, and A(x) cannot equal zero.

The roots of B(x) are the zeros of F(x). The roots of A(x) are the poles of F(x).

The following equations define two rational polynomials used in the following sections.

Rational Polynomial Function Addition

The following equation shows the addition of two rational polynomials.

Rational Polynomial Function Subtraction

The following equation shows the subtraction of two rational polynomials.

Rational Polynomial Function Multiplication

The following equation shows the multiplication of two rational polynomials.

Rational Polynomial Function Division

The following equation shows the division of two rational polynomials.

Negative Feedback with a Rational Polynomial Function

The following figure shows a diagram of a generic system with negative feedback.

For the system shown in the previous figure, the following equation yields the transfer function of the system.

Positive Feedback with a Rational Polynomial Function

The following figure shows a diagram of a generic system with positive feedback.

For the system shown in the previous figure, the following equation yields the transfer function of the system.

Derivative of a Rational Polynomial Function

The derivative of a rational polynomial function also is a rational polynomial function. Using the quotient rule, you obtain the derivative of a rational polynomial function from the derivatives of the numerator and denominator polynomials. According to the quotient rule, the following equation yields the first derivative of the rational polynomial function F₁(x) defined in Equation B.

You can derive the second derivative of a rational polynomial function from the first derivative, as shown by the following equation.

You continue to derive rational polynomial function derivatives such that you derive the j^th derivative of a rational polynomial function from the (j – 1)^th derivative.

Partial Fraction Expansion

Partial fraction expansion involves splitting a rational polynomial into a summation of low order rational polynomials. Partial fraction expansion is a useful tool for z-transform and digital filter structure conversion.

Heaviside Cover-Up Method

The Heaviside cover-up method is the easiest of the partial fraction expansion methods.

The following actions and conditions illustrate the Heaviside cover-up method:

• Define a rational polynomial function F(x) with the following equation.

  (L)

  where m < n, meaning, without loss of generality, the order of B(x) is lower than the order of A(x).

• Assume that A(x) has one repeated root r₀ of multiplicity k and use the following equation to express A(x) in terms of its roots.

  A(x) = an(x – r0)k(x – r1)(x – r2) … (x – rn – k)

  (M)

• Rewrite F(x) as a sum of partial fractions.

  (N)

  (O)

  

  where

  (P)

  (Q)