# Confusion regarding terminology?

While studying the topic of filter design, I came across following terms:

1. Factored form
2. SOS (second order sections)form
3. Rational form
4. Transfer function form

What is the difference between all these terms?

According to my understanding, the first two terms are mutually synonyms, and are implemented in the same way in MATLAB, and the last two terms are mutually synonyms, and are implemented in same way in MATLAB.

Is my understanding valid?

Factored Form
In factored form the transfer function is expressed as a ratio of two factored polynomials. The numerator factors each correspond to one of the $$M$$ zeros and the denominator factors each correspond to one of the $$N$$ poles. $$H(z) = b_0\frac{(1-q_1z^{-1})(1-q_2z^{-1})\cdots(1-q_Mz^{-1})}{(1-p_1z^{-1})(1-p_2z^{-1})\cdots(1-p_Nz^{-1})}$$

Second Order Sections

A transfer function in second order section form has the polynomials factored and grouped such that each factor is second order (with one order 1 term if an odd order for the original polynomial). The numerator and denominator factors are then paired in K "sections." $$H(z) = \left(\frac{1+b_{11}z^{-1}+b_{12}z^{-2}}{1+a_{11}z^{-1}+a_{12}z^{-2}}\right) \left(\frac{1+b_{21}z^{-1}+b_{22}z^{-2}}{1+a_{21}z^{-1}+a_{22}z^{-2}}\right) \cdots \left(\frac{1+b_{K1}z^{-1}+b_{K2}z^{-2}}{1+a_{K1}z^{-1}+a_{K2}z^{-2}}\right)$$

Rational Form

Rational form transfer functions are represented as a ration of numerator and denominator polynomials. $$H(z) = \frac{\sum_{m=0}^M b_mz^{-m}}{\sum_{n=0}^N a_nz^{-n}}$$

Transfer Function Form

A transfer function is simply the ratio of the numerator expression to the denominator expression. In this sense all of the above cases are transfer functions, just expressed differently. $$H(z) = \frac{N(z)}{D(z)}$$