Accelerating the pace of engineering and science

# ss2sos

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

## Syntax

[sos,g] = ss2sos(A,B,C,D)
[sos,g] = ss2sos(A,B,C,D,iu)
[sos,g] = ss2sos(A,B,C,D,'order')
[sos,g] = ss2sos(A,B,C,D,iu,'order')
[sos,g] = ss2sos(A,B,C,D,iu,'order','scale'')
sos = ss2sos(...)

## Description

ss2sos converts a state-space representation of a given digital filter to an equivalent second-order section representation.

[sos,g] = ss2sos(A,B,C,D) finds a matrix sos in second-order section form with gain g that is equivalent to the state-space system represented by input arguments A, B, C, and D. The input system must be single output and real. sos is an L-by-6 matrix

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

whose rows contain the numerator and denominator coefficients bik and aik of the second-order sections of H(z).

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

[sos,g] = ss2sos(A,B,C,D,iu) specifies a scalar iu that determines which input of the state-space system A, B, C, D is used in the conversion. The default for iu is 1.

[sos,g] = ss2sos(A,B,C,D,'order') and

[sos,g] = ss2sos(A,B,C,D,iu,'order') specify the order of the rows in sos, where 'order' is

• 'down', to order the sections so the first row of sos contains the poles closest to the unit circle

• 'up', to order the sections so the first row of sos contains the poles farthest from the unit circle (default)

The zeros are always paired with the poles closest to them.

[sos,g] = ss2sos(A,B,C,D,iu,'order','scale'') specifies the desired scaling of the gain and the numerator coefficients of all second-order sections, where 'scale' is

• 'none', to apply no scaling (default)

• 'inf', to apply infinity-norm scaling

• 'two', to apply 2-norm scaling

Using infinity-norm scaling in conjunction with up-ordering minimizes the probability of overflow in the realization. Using 2-norm scaling in conjunction with down-ordering minimizes the peak round-off noise.

 Note   Infinity-norm and 2-norm scaling are appropriate only for direct-form II implementations.

sos = ss2sos(...) embeds the overall system gain, g, in the first section, H1(z), so that

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

 Note   Embedding the gain in the first section when scaling a direct-form II structure is not recommended and may result in erratic scaling. To avoid embedding the gain, use ss2sos with two outputs.

## Examples

expand all

### Second-Order Section Form of a Filter

Design a Butterworth lowpass filter of order 5 using the function butter. Specify the cutoff frequency to be one-fifth of the Nyquist frequency. Express the output in state-space form. Convert the state-space result to second-order sections. Visualize the magnitude response.

```[A,B,C,D] = butter(5,0.2);
sos = ss2sos(A,B,C,D)
fvtool(sos)
```
```sos =

0.0013    0.0013         0    1.0000   -0.5095         0
1.0000    2.0012    1.0012    1.0000   -1.0966    0.3554
1.0000    1.9968    0.9968    1.0000   -1.3693    0.6926

```

## Diagnostics

If there is more than one input to the system, ss2sos gives the following error message:

`State-space system must have only one input.`

expand all

### Algorithms

ss2sos uses a four-step algorithm to determine the second-order section representation for an input state-space system:

1. It finds the poles and zeros of the system given by A, B, C, and D.

2. It uses the function zp2sos, which first groups the zeros and poles into complex conjugate pairs using the cplxpair function. zp2sos then forms the second-order sections by matching the pole and zero pairs according to the following rules:

1. Match the poles closest to the unit circle with the zeros closest to those poles.

2. Match the poles next closest to the unit circle with the zeros closest to those poles.

3. Continue until all of the poles and zeros are matched.

ss2sos groups real poles into sections with the real poles closest to them in absolute value. The same rule holds for real zeros.

3. It orders the sections according to the proximity of the pole pairs to the unit circle. ss2sos normally orders the sections with poles closest to the unit circle last in the cascade. You can tell ss2sos to order the sections in the reverse order by specifying the 'down' flag.

4. ss2sos scales the sections by the norm specified in the 'scale' argument. For arbitrary H(ω), the scaling is defined by

$\parallel H{\parallel }_{p}={\left[\frac{1}{2\pi }\underset{0}{\overset{2\pi }{\int }}|H\left(\omega \right){|}^{p}d\omega \right]}^{\frac{1}{p}}$

where p can be either ∞ or 2. See the references for details. This scaling is an attempt to minimize overflow or peak round-off noise in fixed point filter implementations.

## References

[1] Jackson, L. B. Digital Filters and Signal Processing. 3rd Ed. Boston: Kluwer Academic Publishers, 1996, chap. 11.

[2] Mitra, S. K. Digital Signal Processing: A Computer-Based Approach. New York: McGraw-Hill, 1998, chap. 9.

[3] Vaidyanathan, P. P. "Robust Digital Filter Structures." Handbook for Digital Signal Processing (S. K. Mitra and J. F. Kaiser, eds.). New York: John Wiley & Sons, 1993, chap. 7.