You are here : matlab → Signal Processing → tf2sos
# tf2sos() - Signal Processing

### Syntax

### Example

### Output / Return Value

### Limitations

### Alternatives / See Also

### Reference

tf2sos converts a transfer function representation of a given digital filter to an equivalent second-order section representation.[sos,g] = tf2sos(b,a) finds a matrix sos in second-order section form with gain g that is equivalent to the digital filter represented by transfer function coefficient vectors a and b.H(z)=B(z)A(z)=b1+b2z−1+⋯+bn+1z−na1+a2z−1+⋯+am+1z−msos is an L-by-6 matrixsos=[b01b11b211a11a21b02b12b221a12a22⋮⋮⋮⋮⋮⋮b0Lb1Lb2L1a1La2L]whose rows contain the numerator and denominator coefficients bik and aik of the second-order sections of H(z).H(z)=g∏k=1LHk(z)=g∏k=1Lb0k+b1kz−1+b2kz−21+a1kz−1+a2kz−2[sos,g] = tf2sos(b,a,'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] = tf2sos(b,a,'order','scale') specifies the desired scaling of the gain and 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 scalingUsing 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 = tf2sos(...) embeds the overall system gain, g, in the first section, H1(z), so thatH(z)=∏k=1LHk(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.

[sos,g] = tf2sos(b,a)[sos,g] = tf2sos(b,a,'order')[sos,g] = tf2sos(b,a,'order','scale')sos = tf2sos(...)

Second-Order Section Implementation of a Butterworth FilterOpen This Example Design a Butterworth 4th-order lowpass filter using the function butter. Specify the cutoff frequency as half the Nyquist frequency. Implement the filter as second-order sections. Verify that the two representations are identical by comparing their numerators and denominators. [nm,dn] = butter(4,0.5); [ss,gn] = tf2sos(nm,dn); numers = [conv(ss(1,1:3),ss(2,1:3))*gn;nm] denoms = [conv(ss(1,4:6),ss(2,4:6)) ;dn] numers = 0.0940 0.3759 0.5639 0.3759 0.0940 0.0940 0.3759 0.5639 0.3759 0.0940 denoms = 1.0000 -0.0000 0.4860 -0.0000 0.0177 1.0000 -0.0000 0.4860 -0.0000 0.0177