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

`[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(...)

`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`

.

**Note**

The input state-space system must be single-output and real.

`sos`

is an *L*-by-6 matrix

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

whose rows contain the numerator and denominator coefficients *b _{ik}* and

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

`[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,`

and * 'order'*)

`[sos,g] = ss2sos(A,B,C,D,iu,`

specify
the order of the rows in * 'order'*)

`sos`

, where `'order'`

`'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,`

specifies
the desired scaling of the gain and the numerator coefficients of
all second-order sections, where * 'order'*,

`'scale'`

`'scale'`

`'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, *H*_{1}(*z*),
so that

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

**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.

`ss2sos`

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

It finds the poles and zeros of the system given by

`A`

,`B`

,`C`

, and`D`

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

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

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.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.`ss2sos`

scales the sections by the norm specified in the`'`

`scale`

`'`

argument. For arbitrary*H*(ω), the scaling is defined by$${\Vert H\Vert}_{p}={\left[\frac{1}{2\pi}{\displaystyle \underset{0}{\overset{2\pi}{\int}}{\left|H(\omega )\right|}^{p}d\omega}\right]}^{1/p}$$

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.

[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.