# zp2tf

Convert zero-pole-gain filter parameters to transfer function form

## Syntax

`[b,a] = zp2tf(z,p,k)`

## Description

`zp2tf` forms transfer function polynomials from the zeros, poles, and gains of a system in factored form.

`[b,a] = zp2tf(z,p,k)` finds a rational transfer function

$\frac{B\left(s\right)}{A\left(s\right)}=\frac{{b}_{1}{s}^{\left(n-1\right)}+\cdots +{b}_{\left(n-1\right)}s+{b}_{n}}{{a}_{1}{s}^{\left(m-1\right)}+\cdots +{a}_{\left(m-1\right)}s+{a}_{m}}$

given a system in factored transfer function form

$H\left(s\right)=\frac{Z\left(s\right)}{P\left(s\right)}=k\frac{\left(s-{z}_{1}\right)\left(s-{z}_{2}\right)\cdots \left(s-{z}_{m}\right)}{\left(s-{p}_{1}\right)\left(s-{p}_{2}\right)\cdots \left(s-{p}_{n}\right)}$

Column vector `p` specifies the pole locations, and matrix `z` specifies the zero locations, with as many columns as there are outputs. The gains for each numerator transfer function are in vector `k`. The zeros and poles must be real or come in complex conjugate pairs. The polynomial denominator coefficients are returned in row vector `a` and the polynomial numerator coefficients are returned in matrix `b`, which has as many rows as there are columns of `z`.

`Inf` values can be used as place holders in `z` if some columns have fewer zeros than others.

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### Algorithms

The system is converted to transfer function form using `poly` with `p` and the columns of `z`.