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# 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.