# sos2ss

Convert digital filter second-order section parameters to state-space form

## Syntax

``[A,B,C,D] = sos2ss(sos)``
``[A,B,C,D] = sos2ss(sos,g)``

## Description

````[A,B,C,D] = sos2ss(sos)` converts a second-order section representation of a digital filter `sos` to its equivalent state-space form.```

example

````[A,B,C,D] = sos2ss(sos,g)` converts a second-order section representation of a digital filter `sos` to its equivalent state-space form with gain `g`.```

## Examples

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Compute the state-space representation of a simple second-order section system with a gain of 2.

```sos = [1 1 1 1 0 -1 ; -2 3 1 1 10 1]; [A,B,C,D] = sos2ss(sos,2)```
```A = 4×4 -10 0 10 1 1 0 0 0 0 1 0 0 0 0 1 0 ```
```B = 4×1 1 0 0 0 ```
```C = 1×4 42 4 -32 -2 ```
```D = -4 ```

## Input Arguments

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Second-order section representation, specified as a matrix. `sos` is an L-by- 6 matrix of the form

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

The entries of `sos` must be real for proper conversion to state space.

Overall system gain, specified as a real-valued scalar. The function applies the gain to the system as

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

## Output Arguments

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State matrix, returned as a 2L-by-2L matrix.

Input-to-state vector, returned as a 2L-by-1 vector.

Output-to-state vector, returned as a 1-by-2L vector.

Feedthrough matrix, returned as a scalar.

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### Transfer Function

The discrete transfer function in second-order section form is given by

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

## Algorithms

The `sos2ss` function first converts from second-order sections to transfer function using the `sos2tf` function, and then from transfer function to state-space form using the `tf2ss` function.

The single-input, single-output state-space representation is given by

`$\begin{array}{ll}x\left(n+1\right)\hfill & =Ax\left(n\right)+Bu\left(n\right),\hfill \\ y\left(n\right)\hfill & =Cx\left(n\right)+Du\left(n\right).\hfill \end{array}$`