# tf2zpk

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

## Syntax

[z,p,k] = tf2zpk(b,a)

## Description

tf2zpk finds the zeros, poles, and gains of a discrete-time transfer function.

 Note   You should use tf2zpk when working with transfer functions expressed in inverse powers (1 + z-1 + z-2), which is how transfer functions are usually expressed in DSP. A similar function, tf2zp, is more useful for working with positive powers (s2 + s + 1), such as in continuous-time transfer functions.

[z,p,k] = tf2zpk(b,a) finds the matrix of zeros z, the vector of poles p, and the associated vector of gains k from the transfer function parameters b and a:

• The numerator polynomials are represented as columns of the matrix b.

• The denominator polynomial is represented in the vector a.

Given a single-input, multiple output (SIMO) discrete-time system in polynomial transfer function form

$H\left(z\right)=\frac{B\left(z\right)}{A\left(z\right)}=\frac{{b}_{1}+{b}_{2}{z}^{-1}\cdots +{b}_{n-1}{z}^{-n}+{b}_{n}{z}^{-n-1}}{{a}_{1}+{a}_{2}{z}^{-1}\cdots +{a}_{m-1}{z}^{-m}+{a}_{m}{z}^{-m-1}}$

you can use the output of tf2zpk to produce the single-input, multioutput (SIMO) factored transfer function form

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

The following describes the input and output arguments for tf2zpk:

• The vector a specifies the coefficients of the denominator polynomial A(z) in descending powers of z.

• The ith row of the matrix b represents the coefficients of the ith numerator polynomial (the ith row of B(s) or B(z)). Specify as many rows of b as there are outputs.

• The zero locations are returned in the columns of the matrix z, with as many columns as there are rows in b.

• The pole locations are returned in the column vector p and the gains for each numerator transfer function in the vector k.

## Examples

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### Poles, Zeros, and Gain of an IIR Filter

Design a 3rd-order Butterworth filter with normalized cutoff frequency rad/sample. Find the poles, zeros, and gain of the filter. Plot them to verify that they are where expected.

[b,a] = butter(3,.4); fvtool(b,a,'polezero') [z,p,k] = tf2zpk(b,a) text(real(z)-0.1,imag(z)-0.1,'\bfZeros','color',[0 0.4 0]) text(real(p)-0.1,imag(p)-0.1,'\bfPoles','color',[0.6 0 0]) 
z = -1.0000 + 0.0000i -1.0000 + 0.0000i -1.0000 - 0.0000i p = 0.2094 + 0.5582i 0.2094 - 0.5582i 0.1584 + 0.0000i k = 0.0985