zp2sos (Signal Processing Toolbox)

Convert zero-pole-gain filter parameters to second-order sections form

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 discrete-time zero-pole-gain representation of a given digital filter to an equivalent second-order section representation.

Use [sos,g] = zp2sos(z,p,k) to obtain a matrix sos in second-order section form with gain g equivalent to the discrete-time zero-pole-gain filter represented by input arguments z, p, and k. Vectors z and p contain the zeros and poles of the filter's transfer function H(z), not necessarily in any particular 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 L of rows of the matrix sos is the closest integer greater than or equal to the maximum of n/2 and 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.

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

sos = zp2sos(...) embeds the overall system gain, g, in the first section, H₁(z), so that

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

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:

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

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.
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.

See Also

cplxpair, filternorm, sos2zp, ss2sos, tf2sos, zp2ss, zp2tf

References

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

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

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

zerophase zp2ss