zp2sos (Signal Processing Toolbox)

Conversion of zero-pole-gain to second-order sections.

Syntax

[sos,g] = zp2sos(z,p,k)
[sos,g] = zp2sos(z,p,k,'order')
[sos,g] = zp2sos(z,p,k,'order','scale')
sos = zp2sos(...)

Description

zp2sos converts a zero-pole-gain representation of a given system to an equivalent second-order section representation.

[sos,g] = zp2sos(z,p,k)

finds a matrix sos in second-order section form with gain g equivalent to the zero-pole-gain system represented by input arguments z, p, and k. Vectors z and p contain the zeros and poles of the system H(z), not necessarily in any order:

where n and m are the lengths of z and p, respectively, and k is a scalar gain. The zeros and poles must be real or complex conjugate pairs. sos is an L-by-6 matrix:

whose rows contain the numerator and denominator coefficients b_ik and a_ik of the second-order sections of H(z):

The number of rows L of matrix sos is the maximum of the ceiling of n/2 and the ceiling of m/2.

[sos,g] = zp2sos(z,p,k,'order')

specifies 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)

[sos,g] = zp2sos(z,p,k,'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.

sos = zp2sos(...)

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

Example

Find a second-order section form of a Butterworth lowpass filter:

[z,p,k] = butter(5,0.2);
sos = zp2sos(z,p,k);

Algorithm

zp2sos uses a four-step algorithm to determine the second-order section representation for an input zero-pole-gain system:

1.: It groups the zeros and poles into complex conjugate pairs using the cplxpair function.

2.: It forms the second-order section 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.

: zp2sos 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. zp2sos normally orders the sections with poles closest to the unit circle last in the cascade. You can tell zp2sos to order the sections in the reverse order by specifying the down flag.

4.: zp2sos scales the sections by the norm specified in the 'scale' argument. For arbitrary H(), the scaling is defined by:

: where p can be either or 2. See the references for details on the scaling.

: This scaling is an attempt to minimize overflow or peak round-off noise in fixed point filter implementations.

`cplxpair`	Group complex numbers into complex conjugate pairs.
`sos2zp`	Conversion of second-order sections to zero-pole-gain.
`ss2sos`	Conversion of state-space to second-order sections.
`tf2sos`	Conversion of transfer function to second-order sections.
`zp2ss`	Conversion of zero-pole-gain to state-space.
`zp2tf`	Conversion of zero-pole-gain to transfer function.