ctf

Convert digital filter to coefficients in cascaded transfer function format

Since R2026a

Syntax

[B,A] = ctf(filtObj)

[B,A,sv] = ctf(filtObj)

Description

[B,A] = ctf(filtObj) returns the numerator and denominator coefficients in the Cascaded Transfer Functions (CTF) format for the digitalFilter object or filter System object™.

Using a filter System object requires a DSP System Toolbox™ license.

example

[B,A,sv] = ctf(filtObj) returns the scale values in addition to the numerator and denominator coefficients.

Examples

collapse all

Get Coefficients in CTF Format for `digitalFilter` Object

Open Live Script

Design a lowpass IIR filter with the passband frequency of 0.65π rad/sample and the stopband frequency of 0.80π rad/sample. The passband ripple is 1 dB and the stopband attenuation is 50 dB. Use the designfilt function to create a digitalFilter object with these specifications.

digiFilt = designfilt("lowpassiir", ...
    PassbandFrequency=0.65,StopbandFrequency=0.80, ...
    PassbandRipple=1,StopbandAttenuation=50)

digiFilt = 
 digitalFilter with properties:

   Coefficients:
              Numerator: [6×3 double]
            Denominator: [6×3 double]

   Specifications:
      FrequencyResponse: 'lowpass'
        ImpulseResponse: 'iir'
    NormalizedFrequency: 1
    StopbandAttenuation: 50
         PassbandRipple: 1
      StopbandFrequency: 0.8000
      PassbandFrequency: 0.6500
           DesignMethod: 'butter'

 Use filterAnalyzer to visualize filter
 Use designfilt to edit filter
 Use filter to filter data

Get coefficients in the CTF format for this filter.

[num,den] = ctf(digiFilt)

num = 6×3

    0.6865    1.3729    0.6865
    0.5694    1.1388    0.5694
    0.4953    0.9907    0.4953
    0.4498    0.8996    0.4498
    0.4250    0.8500    0.4250
    0.6459    0.6459         0

den = 6×3

    1.0000    0.9601    0.7857
    1.0000    0.7964    0.4812
    1.0000    0.6928    0.2885
    1.0000    0.6291    0.1701
    1.0000    0.5944    0.1056
    1.0000    0.2917         0

Get Coefficients in CTF Format for `dsp.SOSFilter` Object

This example uses:

DSP System Toolbox DSP System Toolbox

Open Live Script

Design a lowpass IIR filter of order 8 using the designLowpassIIR function. Set the SystemObject flag to true to generate a dsp.SOSFilter object. Set HasScaleValues to true.

sosFilt = designLowpassIIR(FilterOrder=8,HasScaleValues=true,...
    SystemObject=true)

sosFilt = 
  dsp.SOSFilter with properties:

            Structure: 'Direct form II transposed'
    CoefficientSource: 'Property'
            Numerator: [4×3 double]
          Denominator: [4×3 double]
       HasScaleValues: true
          ScaleValues: [0.1287 0.1051 0.0922 0.0865 1]

  Show all properties

Use the ctf function to obtain the filter coefficients in the CTF format. The function also outputs the scale values.

[sosnum,sosden,sv] = ctf(sosFilt)

sosnum = 4×3

     1     2     1
     1     2     1
     1     2     1
     1     2     1

sosden = 4×3

    1.0000   -1.2428    0.7575
    1.0000   -1.0153    0.4359
    1.0000   -0.8906    0.2595
    1.0000   -0.8351    0.1810

Get Coefficients in CTF Format for `dsp.AllpassFilter` Object

This example uses:

DSP System Toolbox DSP System Toolbox

Open Live Script

Create a dsp.AllpassFilter object and specify the allpass polynomial coefficients.

apFilt = dsp.AllpassFilter(AllpassCoefficients=[0   0.5539   0   0.2503;
                                                0  -0.4364   0   0.2218;
                                                0   0.3781   0  -0.3933])

apFilt = 
  dsp.AllpassFilter with properties:

                    Structure: 'Minimum multiplier'
          AllpassCoefficients: [3×4 double]
    TrailingFirstOrderSection: false

Use the ctf function to obtain coefficients in the CTF format and the scale values.

[apnum,apden,sv] = ctf(apFilt)

apnum = 3×5

    0.2503         0    0.5539         0    1.0000
    0.2218         0   -0.4364         0    1.0000
   -0.3933         0    0.3781         0    1.0000

apden = 3×5

    1.0000         0    0.5539         0    0.2503
    1.0000         0   -0.4364         0    0.2218
    1.0000         0    0.3781         0   -0.3933

sv = 
1

Get Coefficients in CTF format for Single-Rate Filter Cascade

This example uses:

DSP System Toolbox DSP System Toolbox

Open Live Script

Create a dsp.FilterCascade object with these filter stages:

dsp.FIRFilter object
dsp.SOSFilter object
dsp.FourthOrderSectionFilter object
dsp.IIRFilter object
A scalar value of 3

Create a dsp.FIRFilter object using the designLowpassFIR function.

firFilt = designLowpassFIR(FilterOrder=8,SystemObject=true);

Create a dsp.SOSFilter object using the designLowpassIIR function.

sosFilt = designLowpassIIR(FilterOrder=8,HasScaleValues=true,...
    SystemObject=true);

Create a dsp.FourthOrderSectionFilter object. Specify the filter coefficients.

fosNum =[0.0135    0.0541    0.0812    0.0541    0.0135
         0.0080    0.0319    0.0478    0.0319    0.0080];
fosDen =[1.0000   -2.2581    2.4552   -1.3108    0.3302
         1.0000   -1.7257    1.1842   -0.3779    0.0470];
fosFilt = dsp.FourthOrderSectionFilter(fosNum,fosDen);

Create a dsp.IIRFilter object. Specify the filter coefficients.

sosNum = [0.1287    0.2574    0.1287
          0.1051    0.2103    0.1051
          0.0922    0.1844    0.0922
          0.0865    0.1729    0.0865];
sosDen = [1.0000   -1.2428    0.7575
          1.0000   -1.0153    0.4359
          1.0000   -0.8906    0.2595
          1.0000   -0.8351    0.1810];
iirFilt = dsp.IIRFilter(sos2tf([sosNum,sosDen]));

Cascade these filters into a single dsp.FilterCascade object using the cascade function.

filtCascadeObj = cascade(firFilt,cascade(sosFilt,fosFilt),...
    3,iirFilt)

filtCascadeObj = 
  dsp.FilterCascade with properties:

         Stage1: [1×1 dsp.FIRFilter]
         Stage2: [1×1 dsp.FilterCascade]
         Stage3: 3
         Stage4: [1×1 dsp.IIRFilter]
    CloneStages: true

Use the ctf function to obtain the filter coefficients and scale values of the overall filter cascade in the CTF format.

[cascNum,cascDen,cascSV] = ctf(filtCascadeObj)

cascNum = 9×9

    0.0000    0.0191    0.1019    0.2309    0.2963    0.2309    0.1019    0.0191    0.0000
    1.0000    2.0000    1.0000         0         0         0         0         0         0
    1.0000    2.0000    1.0000         0         0         0         0         0         0
    1.0000    2.0000    1.0000         0         0         0         0         0         0
    1.0000    2.0000    1.0000         0         0         0         0         0         0
    0.0135    0.0541    0.0812    0.0541    0.0135         0         0         0         0
    0.0080    0.0319    0.0478    0.0319    0.0080         0         0         0         0
    3.0000         0         0         0         0         0         0         0         0
    0.0001    0.0009    0.0030    0.0060    0.0076    0.0060    0.0030    0.0009    0.0001

cascDen = 9×5

    1.0000         0         0         0         0
    1.0000   -1.2428    0.7575         0         0
    1.0000   -1.0153    0.4359         0         0
    1.0000   -0.8906    0.2595         0         0
    1.0000   -0.8351    0.1810         0         0
    1.0000   -2.2581    2.4552   -1.3108    0.3302
    1.0000   -1.7257    1.1842   -0.3779    0.0470
    1.0000         0         0         0         0
    1.0000    0.1000         0         0         0

cascSV = 10×1

    1.0000
    0.1287
    0.1051
    0.0922
    0.0865
    1.0000
    1.0000
    1.0000
    1.0000
    1.0000

Input Arguments

collapse all

`filtObj` — Digital filter
`digitalFilter` object | filter System object

Digital filter, specified as one of these filter objects:

digitalFilter — Use designfilt to generate a digital filter based on frequency-response specifications.
dsp.FIRFilter (DSP System Toolbox)
dsp.IIRFilter (DSP System Toolbox)
dsp.SOSFilter (DSP System Toolbox)
dsp.FourthOrderSectionFilter (DSP System Toolbox)
dsp.AllpassFilter (DSP System Toolbox)
dsp.FilterCascade (DSP System Toolbox)

Output Arguments

collapse all

`B,A` — Cascaded transfer function (CTF) coefficients
row vector | matrix

Cascaded transfer function (CTF) coefficients, returned as a row vector or matrix. B and A list the numerator and denominator coefficients of the cascaded transfer function, respectively.

The sizes for B and A are L-by-(m+1) and L-by-(n+1), respectively. The function returns the first column of A as 1, thus A(1) = 1 when A is a row vector.

L represents the number of filter sections.
m represents the order of the filter numerators.
n represents the order of the filter denominators.

Each row of the B and A matrices contains the respective polynomial coefficients of each CTF section.

For more information about the cascaded transfer function format and coefficient matrices, see Return Digital Filters in CTF Format.

Data Types: single | double

`sv` — Scale values
scalar | column vector

Scale values, returned as a scalar or a column vector of length L+1, where L is the number of filter sections. The scale values represent the distribution of the filter gain across the sections of the cascaded filter representation.

Data Types: single | double

More About

collapse all

Cascaded Transfer Functions

Partitioning an IIR digital filter into cascaded sections improves its numerical stability and reduces its susceptibility to coefficient quantization errors. The cascaded form of a transfer function H(z) in terms of the L transfer functions H₁(z), H₂(z), …, H_L(z) is

$H (z) = \prod_{l = 1}^{L} H_{l} (z) = H_{1} (z) \times H_{2} (z) \times \dots \times H_{L} (z) .$

Return Digital Filters in CTF Format

Get the filter coefficients by specifying to return B and A. That way, you can have digital filters in CTF format for analysis, visualization and signal filtering.

Filter Coefficients

When you specify to return the numerator and denominator coefficients in the CTF format, the L-row matrices B and A are returned as

$B = [\begin{matrix} b_{11} & b_{12} & \dots & b_{1, m + 1} \\ b_{21} & b_{22} & \dots & b_{2, m + 1} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ b_{L 1} & b_{L 2} & \dots & b_{L, m + 1} \end{matrix}], A = [\begin{matrix} 1 & a_{12} & \dots & a_{1, n + 1} \\ 1 & a_{22} & \dots & a_{2, n + 1} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 1 & a_{L 2} & \dots & a_{L, n + 1} \end{matrix}],$

such that the full transfer function of the filter is

$H (z) = \frac{b_{11} + b_{12} z^{- 1} + \dots + b_{1, m + 1} z^{- m}}{1 + a_{12} z^{- 1} + \dots + a_{1, n + 1} z^{- n}} \times \frac{b_{21} + b_{22} z^{- 1} + \dots + b_{2, m + 1} z^{- m}}{1 + a_{22} z^{- 1} + \dots + a_{2, n + 1} z^{- n}} \times \dots \times \frac{b_{L 1} + b_{L 2} z^{- 1} + \dots + b_{L, m + 1} z^{- m}}{1 + a_{L 2} z^{- 1} + \dots + a_{L, n + 1} z^{- n}},$

where m ≥ 0 is the numerator order of the filter and n ≥ 0 is the denominator order.

Note

To filter signals using cascaded transfer functions, use the ctffilt function.
To analyze filters represented as cascaded transfer functions, use the Filter Analyzer app. You can also use these Signal Processing Toolbox™ functions to visualize and analyze filters in CTF format:
- Time-Domain Responses — impzlength, impz, and stepz
- Frequency-Domain Responses — freqz, grpdelay, phasedelay, phasez, zerophase, and zplane
- Filter Exploration — filtord, islinphase, ismaxphase, isminphase, and isstable

References

[1] Lyons, Richard G. Understanding Digital Signal Processing. Upper Saddle River, NJ: Prentice Hall, 2004.

Version History

Introduced in R2026a

ctf

Syntax

Description

Examples

Get Coefficients in CTF Format for `digitalFilter` Object

Get Coefficients in CTF Format for `dsp.SOSFilter` Object

Get Coefficients in CTF Format for `dsp.AllpassFilter` Object

Get Coefficients in CTF format for Single-Rate Filter Cascade

Input Arguments

`filtObj` — Digital filter
`digitalFilter` object | filter System object

Output Arguments

`B,A` — Cascaded transfer function (CTF) coefficients
row vector | matrix

`sv` — Scale values
scalar | column vector

More About

Cascaded Transfer Functions

Return Digital Filters in CTF Format

References

Version History

See Also

Functions

Objects

ctf

Syntax

Description

Examples

Get Coefficients in CTF Format for digitalFilter Object

Get Coefficients in CTF Format for dsp.SOSFilter Object

Get Coefficients in CTF Format for dsp.AllpassFilter Object

Get Coefficients in CTF format for Single-Rate Filter Cascade

Input Arguments

filtObj — Digital filter digitalFilter object | filter System object

Output Arguments

B,A — Cascaded transfer function (CTF) coefficients row vector | matrix

sv — Scale values scalar | column vector

More About

Cascaded Transfer Functions

Return Digital Filters in CTF Format

References

Version History

See Also

Functions

Objects

Get Coefficients in CTF Format for `digitalFilter` Object

Get Coefficients in CTF Format for `dsp.SOSFilter` Object

Get Coefficients in CTF Format for `dsp.AllpassFilter` Object

`filtObj` — Digital filter
`digitalFilter` object | filter System object

`B,A` — Cascaded transfer function (CTF) coefficients
row vector | matrix

`sv` — Scale values
scalar | column vector