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Convert zero-pole-gain filter parameters to second-order sections form
[sos,g] = zp2sos(z,p,k)
[sos,g] = zp2sos(z,p,k,'order')
[sos,g] = zp2sos(z,p,k,'order','scale')
[sos,g] = zp2sos(z,p,k,'order','scale',zeroflag)
sos = zp2sos(...)
zp2sos converts a discrete-time zero-pole-gain representation of a given digital filter to an equivalent second-order section representation.
[sos,g] = zp2sos(z,p,k) creates 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.
[sos,g] = zp2sos(z,p,k,'order','scale',zeroflag) specifies whether to keep together real zeros that are the negatives of each other instead of ordering them according to proximity to poles. Setting zeroflag to true keeps the zeros together and results in a numerator with a middle coefficient equal to zero. The default for zeroflag is false.
sos = zp2sos(...) embeds the overall system gain, g, in the first section, H_{1}(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. |
[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.
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