ctf2sysobj

Create System object from cascaded transfer functions

Since R2025a

Syntax

ctf = ctf2sysobj(Num,Den)

ctf = ctf2sysobj(Num,Den,g)

ctf = ctf2sysobj(___,Name=Value)

Description

ctf = ctf2sysobj(Num,Den) creates a System object™ from a cascaded transfer function (CTF) with numerator coefficients Num and denominator coefficients Den.

ctf = ctf2sysobj(Num,Den,g) also uses the scale values g of a digital filter.

ctf = ctf2sysobj(___,Name=Value) uses additional options specified in the name-value arguments. Use this syntax with any of the input arguments in previous syntaxes.

Examples

Convert IIR Filter in CTF Format to System Object

Design a sixth-order bandpass elliptic filter. Obtain the numerator and denominator coefficients in CTF format. The size of the filter coefficients matrices indicate three fourth-order sections.

[Num,Den] = ellip(6,3,50,[0.3 0.6],"bandpass","ctf")

Num = 3×5

    0.2275   -0.0435   -0.1999   -0.0435    0.2275
    0.2275   -0.1070    0.1733   -0.1070    0.2275
    0.2275   -0.1188    0.2423   -0.1188    0.2275

Den = 3×5

    1.0000   -0.6236    1.6521   -0.5161    0.6936
    1.0000   -0.5779    1.3890   -0.5327    0.8714
    1.0000   -0.5568    1.2684   -0.5468    0.9715

Convert the filter in the CTF format to a System object. The ctf2sysobj function returns a dsp.FourthOrderSectionFilter object.

ctfObj = ctf2sysobj(Num,Den)

ctfObj = 
  FourthOrderSectionFilter with properties:

      Numerator: [3×5 double]
    Denominator: [3×5 double]

Visualize the magnitude response of the System object and of the filter representation in numerator and denominator coefficients. Both filter representations give the same magnitude response.

fa = filterAnalyzer(ctfObj);
addFilters(fa,Num,Den)

Convert Cascade of FIR Filters to System Object

Define a CTF numerator array for a cascade of two FIR filters of order 100. Convert the cascade of filters to a System object.

Num = [designLowpassFIR(FilterOrder=100); ...
       designHighpassFIR(FilterOrder=100)];

Hsys = ctf2sysobj(Num)

Hsys = 
  dsp.FilterCascade with properties:

         Stage1: [1×1 dsp.FIRFilter]
         Stage2: [1×1 dsp.FIRFilter]
    CloneStages: true

Cascade of Second-Order Section Filters with Scale Values

Customize the conversion from CTF to System object and add filter cascade stages.

Design an eight-order Butterworth peak IIR filter. The number of columns in the filter coefficients, B and A, indicate that the peak filter is divided in second-order sections. The vector of scale values, g, contains the gain for the four stages and the overall system gain.

[B,A,g] = designNotchPeakIIR(FilterOrder=8, ...
     QualityFactor=5,HasScaleValues=true)

B = 4×3

    1.0000    2.0000    1.0000
    1.0000   -2.0000    1.0000
    1.0000    2.0000    1.0000
    1.0000   -2.0000    1.0000

A = 4×3

    1.0000    0.2738    0.8880
    1.0000   -0.2738    0.8880
    1.0000    0.1067    0.7455
    1.0000   -0.1067    0.7455

g = 5×1

    0.1479
    0.1479
    0.1380
    0.1380
    1.0000

Convert the filter coefficients to a System object.

sosSys = ctf2sysobj(B,A,g)

sosSys = 
  dsp.SOSFilter with properties:

            Structure: 'Direct form II transposed'
    CoefficientSource: 'Property'
            Numerator: [4×3 double]
          Denominator: [4×3 double]
       HasScaleValues: true
          ScaleValues: [5×1 double]

  Show all properties

By default, the ctf2sysobj function generates a dsp.SOSFilter System object for IIR filters of the order 2 or less. To generate a dsp.FilterCascade System object, specify the ForceCascade name-value argument as true. You can add filter stages in the next step.

Convert the filter coefficients to a dsp.FilterCascade System object.

ctfSys = ctf2sysobj(B,A,g,ForceCascade=true)

ctfSys = 
  dsp.FilterCascade with properties:

         Stage1: [1×1 dsp.SOSFilter]
    CloneStages: true

Add two stages to the dsp.FilterCascade System object. The stages comprise an allpass filter and a polyphase FIR sample-rate conversion filter. Display the System object. The resulting dsp.FilterCascade System object has three stages.

addStage(ctfSys, dsp.AllpassFilter)
addStage(ctfSys, dsp.FIRRateConverter(3,2,triang(9)))
ctfSys

ctfSys = 
  dsp.FilterCascade with properties:

         Stage1: [1×1 dsp.SOSFilter]
         Stage2: [1×1 dsp.AllpassFilter]
         Stage3: [1×1 dsp.FIRRateConverter]
    CloneStages: true

Plot the impulse response discrete-time Fourier transform (DTFT) of the three-stage dsp.FilterCascade System object.

freqzmr(ctfSys)

Input Arguments

Num — CTF numerator coefficients

matrix | row vector | scalar

CTF numerator coefficients, specified as a matrix, vector, or scalar.

Num must be of size L-by-(m + 1), where:

• L represents the number of filter sections.

• m represents the order of the filter numerators.

For more information about the CTF format and coefficient matrices, see Specify Digital Filters in CTF Format.

Data Types: single | double
Complex Number Support: Yes

Den — CTF denominator coefficients

1 (default) | matrix | vector | scalar

CTF denominator coefficients, specified as a matrix, vector, or scalar.

Den must be of size L-by-(n + 1), where:

• L represents the number of filter sections.

• n represents the order of the filter denominators.

For more information about the CTF format and coefficient matrices, see Specify Digital Filters in CTF Format.

Note

If any element of Den(:,1) is not equal to 1, then ctf2sysobj normalizes the filter coefficients by Den(:,1). In this case, Den(:,1) must be nonzero.

Data Types: single | double
Complex Number Support: Yes

g — Scale values

1 (default) | vector | scalar

Scale values, specified as a real-valued scalar or as a real-valued vector with L + 1 elements, where L is the number of filter sections. The scale values represent the distribution of the filter gain across sections of the cascaded filter representation.

The ctf2sysobj function applies a gain to the filter sections using the scaleFilterSections function depending on how you specify g:

• Scalar — The function distributes the gain uniformly 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.

Data Types: single | double

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Example: ctf = ctf2sysobj([2 4 2;1 0 0],ForceCascade=true) specifies an FIR digital filter with option to wrap each filter stage into System object.

ForceCascade — Option to force single-stage cascades into dsp.FilterCascade System object

false or 0 (default) | true or 1

Option to force single-stage cascades into a dsp.FilterCascade System object, specified as one of these:

• false (0) — ctf2sysobj returns each filter stage as one of these formats:

  • dsp.SOSFilter — Second-order sections (SOS) format.

  • dsp.FourthOrderSectionFilter — Fourth-order sections (FOS) format.

  • dsp.IIRFilter — Infinite impulse response (IIR) filter format.

  • dsp.FIRFilter — Finite impulse response (FIR) filter format.

• true (1) — ctf2sysobj wraps and returns each stage into a dsp.FilterCascade object.

  • Single-stage dsp.FilterCascade object, for filters in SOS and FOS format.

  • Multiple-stage dsp.FilterCascade object, for FIR and IIR filters in CTF format.

Data Types: logical

ParseColumnVectorsAsRows — Option to treat column vectors as rows (single-stage CTF)

true or 1 (default) | false or 0

Option to treat column vectors (Num and Den) as row vectors (single-stage CTF), specified as on these values:

• true (1) — When you specify Num and Den as column vectors, ctf2sysobj assumes that Num and Den list the coefficients of a single-stage filter.

• false (0) — When you specify Num and Den as column vectors, ctf2sysobj assumes that each row of Num and Den relates to a separate filter stage.

  To specify ParseColumnVectorsAsRows as false (0), you must specify Num and Den as vectors with the same number of rows and Den must end with nonzero elements.

Data Types: logical

Output Arguments

ctf — CTF filter System object

dsp.FilterCascade object | dsp.IIRFilter object | dsp.SOSFilter object | dsp.FourthOrderSectionFilter object | dsp.FIRFilter object

CTF filter System object, returned as one of these System objects:

• dsp.SOSFilter — Second-order sections (SOS)

• dsp.FourthOrderSectionFilter — Fourth-order sections (FOS)

• dsp.IIRFilter — Infinite impulse response (IIR) filter

• dsp.FIRFilter — Finite impulse response (FIR) filter

• dsp.FilterCascade — Cascade of dsp.SOSFilter, dsp.FourthOrderSectionFilter. dsp.IIRFilter and dsp.FIRFilter System objects

The ctf2sysobj identifies filter stages (SOS, FOS, IIR, or FIR) from the coefficients specified in Num and Den.

• While an SOS or FOS filter stage can have multiple sections, each IIR or FIR filter stage has one section.

• The order of each filter stage (stage order) is the maximum between the numerator order and denominator order.

Depending on the number of stages, maximum order of each stage, and the value specified in ForceCascade, ctf2sysobj returns one of the System objects in this table.

Stage Impulse Response Type	Stage Order	ForceCascade=false (Single-stage CTF)	ForceCascade=false (Multiple-stage CTF)	ForceCascade=true (Single-stage and multiple-stage CTF)
IIR	1 or 2	dsp.SOSFilter	Does not apply	Single-stage dsp.FilterCascade
IIR	3 or 4	dsp.FourthOrderSectionFilter	Does not apply	Single-stage dsp.FilterCascade
IIR	Greater than 4	dsp.IIRFilter	dsp.FilterCascade	dsp.FilterCascade
FIR	Any	dsp.FIRFilter	dsp.FilterCascade	dsp.FilterCascade
Combination of the above		Does not apply	dsp.FilterCascade	dsp.FilterCascade

More About

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

$$H(z)={\displaystyle \prod _{l=1}^L{H}_l(z)}=H_1(z)\times H_2(z)\times \cdots \times H_L(z).$$

Specify Digital Filters in CTF Format

You can specify digital filters in the CTF format for analysis, visualization, and signal filtering. Specify a filter by listing its coefficients B and A in Num and Den, respectively. You can also include the filter scaling gain across sections by specifying a scalar or vector g.

Filter Coefficients

When you specify the coefficients as L-row matrices,

$$B=\left[\begin{array}{cccc}{b}_{11}& b_{12}& \cdots & b_{1,m+1}\\ b_{21}& b_{22}& \cdots & b_{2,m+1}\\ ⋮& ⋮& \ddots & ⋮\\ b_{L1}& b_{L2}& \cdots & b_{L,m+1}\end{array}\right],A=\left[\begin{array}{cccc}{a}_{11}& a_{12}& \cdots & a_{1,n+1}\\ a_{21}& a_{22}& \cdots & a_{2,n+1}\\ ⋮& ⋮& \ddots & ⋮\\ a_{L1}& a_{L2}& \cdots & a_{L,n+1}\end{array}\right],$$

it is assumed that you have specified the filter as a sequence of L cascaded transfer functions, such that the full transfer function of the filter is

$$H\left(z\right)=\frac{b_{11}+b_{12}{z}^{-1}+\cdots +b_{1,m+1}{z}^{-m}}{a_{11}+a_{12}{z}^{-1}+\cdots +a_{1,n+1}{z}^{-n}}\times \frac{b_{21}+b_{22}{z}^{-1}+\cdots +b_{2,m+1}{z}^{-m}}{a_{21}+a_{22}{z}^{-1}+\cdots +a_{2,n+1}{z}^{-n}}\times \cdots \times \frac{b_{L1}+b_{L2}{z}^{-1}+\cdots +b_{L,m+1}{z}^{-m}}{a_{L1}+a_{L2}{z}^{-1}+\cdots +a_{L,n+1}{z}^{-n}},$$

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

• If you specify both B and A as vectors, it is assumed that the underlying system is a one-section IIR filter (L = 1), with B representing the numerator of the transfer function and A representing its denominator.

• If B is scalar, it is assumed that the filter is a cascade of all-pole IIR filters with each section having an overall system gain equal to B.

• If A is scalar, it is assumed that the filter is a cascade of FIR filters with each section having an overall system gain equal to 1/A.

Coefficients and Gain

If you have an overall scaling gain or multiple scaling gains factored out from the coefficient values, you can specify the scale values in g. Scaling filter sections is especially important when you work with fixed-point arithmetic to ensure that the output of each filter section has similar amplitude levels, which helps avoid inaccuracies in the filter response due to limited numeric precision.

The gain can be a scalar overall gain or a vector of section gains.

• If the gain is scalar, the value applies uniformly to all the cascade filter sections.

• If the gain is a vector, it must have one more element than the number of filter sections L in the cascade. Each of the first L scale values applies to the corresponding filter section, and the last value applies uniformly to all the cascade filter sections.

If you specify the coefficient matrices and gain vector as

$$B=\left[\begin{array}{cccc}{b}_{11}& b_{12}& \cdots & b_{1,m+1}\\ b_{21}& b_{22}& \cdots & b_{2,m+1}\\ ⋮& ⋮& \ddots & ⋮\\ b_{L1}& b_{L2}& \cdots & b_{L,m+1}\end{array}\right],A=\left[\begin{array}{cccc}{a}_{11}& a_{12}& \cdots & a_{1,n+1}\\ a_{21}& a_{22}& \cdots & a_{2,n+1}\\ ⋮& ⋮& \ddots & ⋮\\ a_{L1}& a_{L2}& \cdots & a_{L,n+1}\end{array}\right],g=\left[\begin{array}{ccccc}{g}_1& g_2& \cdots & g_L& g_{\text{S}}\end{array}\right],$$

it is assumed that the transfer function of the filter system is

$$H\left(z\right)=g_{\text{S}}\left(g_1\frac{b_{11}+b_{12}{z}^{-1}+\cdots +b_{1,m+1}{z}^{-m}}{a_{11}+a_{12}{z}^{-1}+\cdots +a_{1,n+1}{z}^{-n}}\times g_2\frac{b_{21}+b_{22}{z}^{-1}+\cdots +b_{2,m+1}{z}^{-m}}{a_{21}+a_{22}{z}^{-1}+\cdots +a_{2,n+1}{z}^{-n}}\times \cdots \times g_L\frac{b_{L1}+b_{L2}{z}^{-1}+\cdots +b_{L,m+1}{z}^{-m}}{a_{L1}+a_{L2}{z}^{-1}+\cdots +a_{L,n+1}{z}^{-n}}\right).$$

References

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

Version History

Introduced in R2025a