Main Content

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

The `ss2sos`

function uses this four-step algorithm to determine the
second-order section representation for an input state-space system.

Find the poles and zeros of the system given by

`A`

,`B`

,`C`

, and`D`

.Use 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 these rules:Match the poles that are closest to the unit circle with the zeros that are closest to those poles.

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

Continue this process until all of the poles and zeros are matched.

The

`ss2sos`

function groups real poles into sections with the real poles that are closest to them in absolute value. The same rule holds for real zeros.Order the sections according to the proximity of the pole pairs to the unit circle. The

`ss2sos`

function normally orders the sections with poles that are closest to the unit circle last in the cascade. You can specify for`ss2sos`

to order the sections in the reverse order by setting the order input to`'down'`

.Scale the sections by the norm specified by the

`scale`

input. 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. For details, see the references. This scaling is an attempt to minimize overflow or peak round-off noise in fixed-point filter implementations.

[1] Jackson, Leland B. *Digital Filters and Signal
Processing*. Boston: Kluwer Academic Publishers, 1996.

[2] Mitra, Sanjit Kumar. *Digital Signal Processing: A
Computer-Based Approach*. New York: McGraw-Hill, 1998.

[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.