Partial Fraction Decomposition

Any rational transfer function can be written as

$$P(s)=P(\infty )+\sum _{i=1}^n\sum _{j=1}^{n_i}\frac{c_{i,j}}{(s-p_i{)}^j}$$

And in Discrete Space

$$G[z]=G[\infty ]+\sum _{i=1}^n\sum _{j=1}^{n_i}\frac{c_{i,j}}{(z-p_i{)}^j}$$

Where $P(\infty )$ can mean different things in different contexts. But can be a useful characteristic. $n_i$ is the multiplicity of the pole $p_i$ (ie. $(s-3{)}^3$ has a multiplicity of 3).

Found by:

$$P(\infty )=\underset{s\to \infty }{\lim }P(s)$$

Or by just doing the partial fraction decomposition and maybe a constant comes out of it.

What we can tell from $P(\infty )$

The transfer function is:

• strictly proper if $P(\infty )$ is a constant
• proper if $P(\infty )$ is 0
• improper if $P(\infty )$ is $\infty$