# Documentation

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# sos2tf

Convert digital filter second-order section data to transfer function form

## Syntax

[b,a] = sos2tf(sos)
[b,a] = sos2tf(sos,g)

## Description

sos2tf converts a second-order section representation of a given digital filter to an equivalent transfer function representation.

[b,a] = sos2tf(sos) returns the numerator coefficients b and denominator coefficients a of the transfer function that describes a discrete-time system given by sos in second-order section form. The second-order section format of H(z) is given by

$H\left(z\right)=\prod _{k=1}^{L}{H}_{k}\left(z\right)=\prod _{k=1}^{L}\frac{{b}_{0k}+{b}_{1k}{z}^{-1}+{b}_{2k}{z}^{-2}}{1+{a}_{1k}{z}^{-1}+{a}_{2k}{z}^{-2}}.$

sos is an L-by-6 matrix that contains the coefficients of each second-order section stored in its rows.

$\text{sos}=\left[\begin{array}{cccccc}{b}_{01}& {b}_{11}& {b}_{21}& 1& {a}_{11}& {a}_{21}\\ {b}_{02}& {b}_{12}& {b}_{22}& 1& {a}_{12}& {a}_{22}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ {b}_{0L}& {b}_{1L}& {b}_{2L}& 1& {a}_{1L}& {a}_{2L}\end{array}\right].$

Row vectors b and a contain the numerator and denominator coefficients of H(z) stored in descending powers of z.

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

[b,a] = sos2tf(sos,g) returns the transfer function that describes a discrete-time system given by sos in second-order section form with gain g.

$H\left(z\right)=g\prod _{k=1}^{L}{H}_{k}\left(z\right).$

## Examples

collapse all

Compute the transfer function representation of a simple second-order section system.

sos = [1  1  1  1  0 -1; -2  3  1  1 10  1];
[b,a] = sos2tf(sos)
b =

-2     1     2     4     1

a =

1    10     0   -10    -1

## Algorithms

sos2tf uses the conv function to multiply all of the numerator and denominator second-order polynomials together. For higher order filters (possibly starting as low as order 8), numerical problems due to roundoff errors may occur when forming the transfer function.