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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.

## Examples

collapse all

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])```
```p = -0.0050 + 1.0000i -0.0050 - 1.0000i ```
`k = 1;`

Use `zp2tf` to find the transfer function.

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

## Algorithms

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