sos2ctf

Convert digital filter second-order section parameters to cascaded transfer function form

Since R2024a

Syntax

[b,a] = sos2ctf(sos)

[b,a] = sos2ctf(sos,g)

Description

[b,a] = sos2ctf(sos) computes the cascaded transfer function of the filter system described by the second-order section matrix sos.

example

[b,a] = sos2ctf(sos,g) also specifies the scale values g to perform gain scaling across the sections of the cascaded transfer function of the system.

Examples

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Second-Order Sections to Cascaded Transfer Function

Open Live Script

Convert a second-order sections matrix to the cascaded transfer function form.

sos = [2 4 2 1 0 0;3 2 0 1 1 0];

[ctfB,ctfA] = sos2ctf(sos)

ctfB = 2×3

     2     4     2
     3     2     0

ctfA = 2×3

     1     0     0
     1     1     0

Cascaded Transfer Function of Elliptic Filter

Open Live Script

Obtain the cascaded transfer function of a 10th-order lowpass elliptic filter with a normalized cutoff frequency of 0.25 $π$ rad/sample.

[z,p,k] = ellip(10,1,60,0.25);
[sos,g] = zp2sos(z,p,k);

[b,a] = sos2ctf(sos,g)

b = 5×3

    0.3216    0.2716    0.3216
    0.3216   -0.2850    0.3216
    0.3216   -0.4033    0.3216
    0.3216   -0.4345    0.3216
    0.3216   -0.4432    0.3216

a = 5×3

    1.0000   -1.5959    0.6751
    1.0000   -1.5123    0.8125
    1.0000   -1.4458    0.9225
    1.0000   -1.4169    0.9730
    1.0000   -1.4099    0.9936

Plot the filter response.

filterAnalyzer(b,a)

Input Arguments

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`sos` — Second-order section representation
L-by-6 matrix

Second-order section representation, specified as an L-by-6 matrix, where L is the number of second-order sections. The matrix

$sos = [\begin{matrix} b_{01} & b_{11} & b_{21} & 1 & a_{11} & a_{21} \\ b_{02} & b_{12} & b_{22} & 1 & a_{12} & a_{22} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ b_{0 L} & b_{1 L} & b_{2 L} & 1 & a_{1 L} & a_{2 L} \end{matrix}]$

represents the second-order sections of H(z):

$H (z) = g \prod_{k = 1}^{L} H_{k} (z) = g \prod_{k = 1}^{L} \frac{b_{0 k} + b_{1 k} z^{- 1} + b_{2 k} z^{- 2}}{1 + a_{1 k} z^{- 1} + a_{2 k} z^{- 2}} .$

Example: [z,p,k] = butter(3,1/32); sos = zp2sos(z,p,k) specifies a third-order Butterworth filter with a normalized 3 dB frequency of π/32 rad/sample.

Data Types: single | double
Complex Number Support: Yes

`g` — Scale values
`1` (default) | scalar | vector

Scale values, specified as a real-valued scalar or as a real-valued vector with L+1 elements, where L is the number of second-order sections.

The sos2ctf function applies a gain to the filter sections using the scaleFilterSections function. Depending on the value you specify in g:

Scalar — The function uniformly distributes the gain across all filter sections.
Vector — The function applies the first L gain values to the corresponding filter sections and distributes the last gain value uniformly across all filter sections.

Output Arguments

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`b,a` — Cascaded transfer function coefficients
L-by-3 matrix

Cascaded transfer function coefficients, returned as L-by-3 matrices, where L is the number of second-order sections.

The matrices b and a list the numerator and denominator coefficients of the cascaded transfer function, respectively. See Cascaded Transfer Functions for more information.

More About

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Cascaded Transfer Functions

Partitioning a filter system into sections connected in cascade form offers several advantages, such as easy design, less susceptibility to coefficient quantization errors, and improved stability [1]. A filter system H(z) can be expressed in L sectional transfer functions H₁(z), H₂(z), …, H_L(z), where

$H (z) = \prod_{k = 1}^{L} H_{k} (z) .$

The sos2ctf computes the numerator and denominator coefficients of the cascaded-transfer-function sections from the second-order-section coefficients of the filter system.

The output arguments b and a associate with the matrices

$b = [\begin{matrix} {b^{'}}_{01} & {b^{'}}_{11} & {b^{'}}_{21} \\ {b^{'}}_{02} & {b^{'}}_{12} & {b^{'}}_{22} \\ ⋮ & ⋮ & ⋮ \\ {b^{'}}_{0 L} & {b^{'}}_{1 L} & {b^{'}}_{2 L} \end{matrix}]$

and $a = [\begin{matrix} 1 & a_{11} & a_{21} \\ 1 & a_{12} & a_{22} \\ ⋮ & ⋮ & ⋮ \\ 1 & a_{1 L} & a_{2 L} \end{matrix}]$ .

These matrices contain the second-order cascaded transfer function coefficients of H(z)

$H (z) = \prod_{k = 1}^{L} \frac{{b^{'}}_{0 k} + {b^{'}}_{1 k} z^{- 1} + {b^{'}}_{2 k} z^{- 2}}{1 + a_{1 k} z^{- 1} + a_{2 k} z^{- 2}} .$

References

[1] Parks, Thomas W., and C. Sidney Burrus. Digital Filter Design. Hoboken, NJ: John Wiley & Sons, 1987, pp. 233–239.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Version History

Introduced in R2024a

sos2ctf

Syntax

Description

Examples

Second-Order Sections to Cascaded Transfer Function

Cascaded Transfer Function of Elliptic Filter

Input Arguments

sos — Second-order section representation L-by-6 matrix

g — Scale values 1 (default) | scalar | vector

Output Arguments

b,a — Cascaded transfer function coefficients L-by-3 matrix

More About

Cascaded Transfer Functions

References

Extended Capabilities

C/C++ Code Generation Generate C and C++ code using MATLAB® Coder™.

Version History

See Also

`sos` — Second-order section representation
L-by-6 matrix

`g` — Scale values
`1` (default) | scalar | vector

`b,a` — Cascaded transfer function coefficients
L-by-3 matrix

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.