# Documentation

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# tf2sos

Convert digital filter transfer function data to second-order sections form

## Syntax

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

## Description

`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`.

`$H\left(z\right)=\frac{B\left(z\right)}{A\left(z\right)}=\frac{{b}_{1}+{b}_{2}{z}^{-1}+\cdots +{b}_{n+1}{z}^{-n}}{{a}_{1}+{a}_{2}{z}^{-1}+\cdots +{a}_{m+1}{z}^{-m}}$`

`sos` is an L-by-6 matrix

`$\text{sos}=\left[\begin{array}{cccccc}{b}_{01}& {b}_{11}& {b}_{21}& 1& {a}_{11}& {a}_{21}\\ {b}_{02}& {b}_{12}& {b}_{22}& 1& {a}_{12}& {a}_{22}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ {b}_{0L}& {b}_{1L}& {b}_{2L}& 1& {a}_{1L}& {a}_{2L}\end{array}\right]$`

whose rows contain the numerator and denominator coefficients bik and aik of the second-order sections of H(z).

`$H\left(z\right)=g\prod _{k=1}^{L}{H}_{k}\left(z\right)=g\prod _{k=1}^{L}\frac{{b}_{0k}+{b}_{1k}{z}^{-1}+{b}_{2k}{z}^{-2}}{1+{a}_{1k}{z}^{-1}+{a}_{2k}{z}^{-2}}$`

`[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 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 = tf2sos(...)` embeds the overall system gain, `g`, in the first section, H1(z), so that

`$H\left(z\right)=\prod _{k=1}^{L}{H}_{k}\left(z\right)$`

### 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

collapse all

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]```
```numers = 0.0940 0.3759 0.5639 0.3759 0.0940 0.0940 0.3759 0.5639 0.3759 0.0940 ```
`denoms = [conv(ss(1,4:6),ss(2,4:6)) ;dn]`
```denoms = 1.0000 0.0000 0.4860 0.0000 0.0177 1.0000 0.0000 0.4860 0.0000 0.0177 ```

## Algorithms

`tf2sos` uses a four-step algorithm to determine the second-order section representation for an input transfer function system:

1. It finds the poles and zeros of the system given by `b` and `a`.

2. It uses 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 the following rules:

1. Match the poles closest to the unit circle with the zeros closest to those poles.

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

3. Continue until all of the poles and zeros are matched.

`tf2sos` 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. `tf2sos` normally orders the sections with poles closest to the unit circle last in the cascade. You can tell `tf2sos` to order the sections in the reverse order by specifying the `'down'` flag.

4. `tf2sos` scales the sections by the norm specified in the `'scale'` argument. For arbitrary H(ω), the scaling is defined by

`${‖H‖}_{p}={\left[\frac{1}{2\pi }\underset{0}{\overset{2\pi }{\int }}{|H\left(\omega \right)|}^{p}d\omega \right]}^{1/p}$`

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.

## References

[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.