# 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.

## Examples

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### Transfer Function of Mass-Spring System

Compute the transfer function of a damped mass-spring system that obeys the differential equation

 

The measurable quantity is the acceleration, , and is the driving force. In Laplace space, the system is represented by

 

The system has unit gain, a double zero at , and two complex-conjugate poles.

z = [0 0]'; p = roots([1 0.01 1]) k = 1; 
p = -0.0050 + 1.0000i -0.0050 - 1.0000i 

Use zp2tf to find the transfer function.

[b,a] = zp2tf(z,p,k) 
b = 1 0 0 a = 1.0000 0.0100 1.0000 

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### Algorithms

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