rational

Perform rational fitting on complex frequency-dependent data

Description

Use the rational object to perform rational fitting on complex frequency-dependent data. This object uses the AAA algorithm [1] to construct a rational fit with complex frequencies s. The AAA algorithm uses a barycentric form to address the scaling challenges encountered by other rational fitting methods, notably the widely-used vector fitting algorithm. This enables AAA algorithm to converge rapidly to a near-optimum rational function fit. The fit of the each element of the rational object is given by this equation:

$$\begin{array}{l}F(s)={\displaystyle \sum _{k=1}^n\frac{C_k}{s-A_k}}+D,\\ \text{where:}\\ s=j\times 2\pi f\\ C-R\text{esidues}\\ A-P\text{oles}\\ D-D\text{irect term}\end{array}$$

Creation

Syntax

fit = rational

fit = rational(freq,data)

fit = rational(s)

fit = rational(filename)

fit = rational(___,tol)

fit = rational(___,Name=Value)

[fit,errdb] = rational(___)

Description

fit = rational creates a non-reflective one-port rational object with default properties.

fit = rational(freq,data) performs rational fitting for the given frequency vector freq and network parameter data data.

fit = rational(s) performs rational fitting for the N-port S-parameters object s.

fit = rational(filename) performs rational fitting of network data in the file specified in filename.

fit = rational(___,tol) strives to satisfy the relative error tolerance tol when performing rational fit.

fit = rational(___,Name=Value) specifies options to control aspects of the fit or sets properties of the rational object. You can also set the NumPoles, Poles, Residues, and DirectTerm properties (since R2023a) using name-value arguments. For example, fit = rational(s,MaxPoles=1002) sets the maximum number of poles for the fit. Specify name-value arguments after any of the input arguments from the previous syntaxes.

[fit,errdb] = rational(___) also returns the error of the fit errdb.

Input Arguments

freq — Nonnegative frequencies

vector

Nonnegative frequencies, specified as a vector of nonnegative frequencies in Hz.

Data Types: double

data — Network parameter data

vector | 2-D array | 3-D array

Network parameter data, specified as a vector, a 2-D array or a 3-D array. The length of the data values must equal the length of the frequency values freq.

s — N-port S-parameters

sparameters object

N-port S-parameters, specified as a sparameters object.

filename — Network data file name

character vector

Network data file name, specified as a character vector. The data file must have .snp file extension (a Touchstone file).

tol — Relative error tolerance

-40 (default) | scalar

Relative error tolerance, specified as a scalar less than or equal to 0 dB.

Note

Do not specify the tol input argument and Tolerance name-value argument when specifying number of poles of the fit using the NumPoles property.

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.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: TendsToZero = true | 'TendsToZero',true

Tolerance — Relative error tolerance

-40 (default) | scalar

Relative error tolerance, specified as a scalar less than or equal to 0 dB.

Note

Do not specify the tol input argument and Tolerance name-value argument when specifying number of poles of the fit using the NumPoles property.

Example: Tolerance=-30

Data Types: double

TendsToZero — Behavior of fit for large S-parameters

true or 1 (default) | false or 0

Behavior of fit for large S-parameters, specified as 1 (true) or 0 (false). When true, the direct term in the fit is set to zero so that the rational fit F(S) tends to zero as S approaches infinity. When false, a nonzero direct term is allowed.

Example: TendsToZero=true

Data Types: logical

MaxPoles — Maximum number of poles

1000 (default) | scalar nonnegative integer

Maximum number of poles, specified as a scalar nonnegative integer.

Example: MaxPoles=1002

Data Types: double

ErrorMetric — Error metrics in rational object

'default' (default) | 'Relative'

Error metrics in the rational object, specified as one of the following:

• If you specify 'ErrorMetric' as 'default', the rational object distributes the error evenly.

• If you specify 'ErrorMetric' as 'Relative', the rational object fits both peaks and valleys or gets smaller error for smaller values.

Example: ErrorMetric='Relative'

Data Types: char

NoiseFloor — Ignores low-level noise in data

-inf (default) | scalar

Ignores low-level noise in the data, specified as a scalar.

Example: NoiseFloor=-60

Data Types: double

Causal — Pole stability

true or 1 (default) | false or 0

Pole stability, specified as a logical true (1) or false (0). When you specify true, all the poles of the fit are stable. When you specify false,the poles can be anywhere in the complex plane.

Example: Causal=false

Qlimit — Maximum value of quality factor

1000 (default) | positive scalar

Maximum value of the quality factor of the poles of the fit, specified as a positive scalar.

Example: Qlimit=1100

Data Types: double

ColumnReduce — Data reduction

true or 1 (default) | false or 0 | logical

Data reduction, specified as a logical 1 (true) or 0 (false). When you specify true, the function reduces the data for the fitter to save memory and computation time. Reduce data for the fitter in order to save memory and computation time, specified as a logical true or false.

Example: ColumnReduce=false

PoleSharing — Type of pole sharing

All (default) | Column | Individual

Since R2024a

Type of pole sharing to share the poles of the S-parameters, specified as one of the following:

• All — Share all the poles of the S-parameters.

• Column — Share the poles by S-parameter column terms.

• Individual — Share the poles individually by S-parameter terms.

Data Types: char | string

Display — Display options for fitting algorithm

'off' (default) | 'on' | 'plot' | 'both'

Display options for the fitting algorithm of the rational object, specified as one of the following:

• 'off'— No display

• 'on'— Printed information

• 'plot'— Plots of the interpolation progress

• 'both'— Both printed information and plots.

Example: Display='on'

Data Types: char

Output Arguments

fit — Computed rational fit

rational object

Computed rational fit of a complex frequency-dependent data, returned as a rational object.

errdb — Error between fit and original data

scalar

Error between the fit and the original data, returned as a scalar in dB.

Properties

NumPorts — Number of ports

Read-only: vector

This property is read-only.

Number of ports in the original S-parameter data, returned as a 1-by-2 vector. The value of the NumPorts is determined from the S-parameter data.

NumPoles — Number of poles

nonegative integer

Number of poles in the fit, specified as a nonegative integer (since R2023a). If you do not specify the NumPoles property, then the object sets the property automatically.

Note

Do not specify the tol input argument and Tolerance name-value argument when specifying number of poles of the fit using the NumPoles property.

Example: 10

Poles — Poles of fit

row vector

Poles of the fit (A), specified as a row vector of length NumPoles-by-1. When you specify this property, the object sets the Residues, DirectTerm, and ErrDB properties or you can set them separately (since R2023a).

If you do not specify the Poles property, then the object sets the property automatically.

Note

When setting the poles of the fit using the Poles property, the NoiseFloor, Causal, ColumnReduce, MaxPoles, NumPoles, and Qlimit arguments are ignored.

Example: [-5.4e+10 + 4.7e+10i;-5.4e+10 - 4.7e+10i] specifies poles for a two-pole fit

Data Types: double

Residues — Residues in fit

3-D matrix

Residues in the fit or complex values of C in the equation provided in the description, specified as a 3-D matrix. The first two dimension of Residues must be of the same length as the first two dimension of data and the last dimension of Residues must be of the same length as Poles. If you specify this property, the object sets the DirectTerm and ErrDB properties or you can set them separately (since R2023a).

If you do not specify the Residues property, then the object sets the property automatically as a 3-D matrix of length NumPorts-by-NumPorts-by-NumPoles.

Data Types: double

DirectTerm — Direct term in fit

2-D matrix

Direct term in the fit or the value of D in the equation provided in the description, specified as a 2-D matrix. The length of this matrix must be the same as the first two dimensions of the data. If you specify this property, the object sets the ErrDB property (since R2023a).

Note

You must specify this argument with Poles and Residues.

If you do not specify this property, then the object sets the property automatically as a 2-D matrix of length NumPorts-by-NumPorts.

Example: [0 0;0 0]

Data Types: double

ErrDB — Error between fit and original data

Read-only: scalar

This property is read-only.

Error between the fit and the original data, returned as a scalar in dB.

Object Functions

timeresp	Time response for rational objects
stepresp	Step-signal response for rational object and rationalfit function object
freqresp	Frequency response of rational object and rationalfit function object
pwlresp	Calculate time response of piecewise linear input signal
impulse	Impulse response for rational function object
ispassive	Return true if rational fit output is passive at all frequencies
makepassive	Enforce passivity of rational fit
passivity	Plot passivity of N-by-N rational fit output
generateSPICE	Generate SPICE file from rationalfit of S-parameters
abcd	Construct state-space matrices from rational object
zpk	Compute zeros, poles, and gain of rational object

Examples

Fit passive.s2p File

Create S-Parameters from the file named passive.s2p.

S = sparameters('passive.s2p');

Perform rational fitting of the S-parameters.

fit = rational(S);

Zeros, Poles, Gain, and DC Gain of Fit

Create an S-Parameters object from the file named default.s2p. Perform rational fitting of the S-Parameters.

S = sparameters('default.s2p');
fit = rational(S,Display='plot')

fit = 
  rational with properties:

      NumPorts: 2
      NumPoles: 52
         Poles: [52×1 double]
      Residues: [2×2×52 double]
    DirectTerm: [2×2 double]
         ErrDB: -22.6872

Calculate the zeros, poles, gain, and DC gain of the rational object.

[z,p,k,dcgain] = zpk(fit)

z=2×2 cell array
    {51×1 double}    {51×1 double}
    {51×1 double}    {51×1 double}

p=2×2 cell array
    {52×1 double}    {52×1 double}
    {52×1 double}    {52×1 double}

k = 2×2
1010 ×

    1.0544   -0.0194
    0.9158    0.0377

dcgain = 2×2

    0.1289   -0.0838
   -0.1209    0.7649

Fit Peaks and Valleys in rational Object

Create an S-parameters object from the specified S2P file.

S = sparameters('sawfilterpassive.s2p');
f = S.Frequencies;

Create a rational object with the tolerance of -40 dB.

fit = rational(S,Tolerance=-40);

Compare the fit to the data. You can see the deviations at the smaller values on a semi-log plot.

dresp = freqresp(fit,f);
plot(f,20*log10(abs(squeeze(S.Parameters(2,1,:)))),...
    f,20*log10(abs(squeeze(dresp(2,1,:)))),f,20*log10(abs(squeeze(S.Parameters(2,1,:)-dresp(2,1,:)))))
title('Default Fitting for sawfilterpassive.s2p');
ylabel('dB');
xlabel('Frequency (Hz)');
legend('Data','Fit','Error','Location','northwest');

Create a rational object with 'ErrorMetric' set to 'Relative' to fit both peaks and valleys.

rfit = rational(S,Tolerance=-40,ErrorMetric='Relative');

Compare the fit to the data. The peaks and valleys are fitted.

rresp = freqresp(rfit,f);
figure(2)
plot(f,20*log10(abs(squeeze(S.Parameters(2,1,:)))),...
    f,20*log10(abs(squeeze(rresp(2,1,:)))), f, 20*log10(abs(squeeze(S.Parameters(2,1,:)-rresp(2,1,:)))));
title('Relative Error Fitting for sawfilterpassive.s2p');
ylabel('dB');
xlabel('Frequency (Hz)');
legend('Data','Fit','Error','Location','northwest');

Ignore Low-Level Noise in Fitter Data

Configure the S-parameter file reader for Touchstone 2.0 files to parses comments in the Touchstone 2.0 file and outputs errors if it detects if invalid characters in the comments of the Touchstone file.

setTouchstoneReader('ParseComments',true) 

Create an S-parameters object from the specified S2P file.

S = sparameters('passive.s2p');
f = S.Frequencies;
data = S.Parameters;

Set one of the data entries to zero.

data(2,2,:) = 0;

Create an rational object with the tolerance of -40 dB.

fit = rational(f,data,Tolerance=-40);

Compare the fit to the data. The fit and the data match closely.

xresp = freqresp(fit,f);
figure(3)
plot(f,abs(squeeze(data(2,1,:))),f,abs(squeeze(xresp(2,1,:))))

Add noise to the data and create a rational object.

rng(1);
noisyData = data + 1e-4 * rand(size(S.Parameters));
nfit = rational(f,noisyData,Tolerance=-40);

Compare the fit to the data with noise. Noise cannot be fitted because this is a data with a higher order fit with a worse error metric.

nresp = freqresp(nfit,f);
figure(4)
plot(f,abs(squeeze(noisyData(2,2,:))),f,abs(squeeze(nresp(2,2,:))))

Create a rational object with the noise floor of -60 dB and plot the fit. The fitter ignores low-level noise.

ffit = rational(f,noisyData,Tolerance=-40,NoiseFloor=-60);
fresp = freqresp(ffit,f);
figure(5)
plot(f,abs(squeeze(noisyData(2,2,:))),f,abs(squeeze(fresp(2,2,:))))

Control Poles in Rational Object

Since R2023a

Create an S-parameter object.

S = sparameters('passive.s2p');

Perform a default fit.

fit = rational(S)

fit = 
  rational with properties:

      NumPorts: 2
      NumPoles: 8
         Poles: [8×1 double]
      Residues: [2×2×8 double]
    DirectTerm: [2×2 double]
         ErrDB: -40.4597

Limit the number of poles to four, and note that and that the error in fit1 is higher than the error in fit.

fit1 = rational(S,NumPoles=4)

fit1 = 
  rational with properties:

      NumPorts: 2
      NumPoles: 4
         Poles: [4×1 double]
      Residues: [2×2×4 double]
    DirectTerm: [2×2 double]
         ErrDB: -29.1219

Force the rational object to use twice the number of poles in fit. Note that the error in fit2 is lower than fit.

fit2 = rational(S,NumPoles=2*fit.NumPoles)

fit2 = 
  rational with properties:

      NumPorts: 2
      NumPoles: 16
         Poles: [16×1 double]
      Residues: [2×2×16 double]
    DirectTerm: [2×2 double]
         ErrDB: -42.4669

References

[1] Nakatsukasa, Yuji, Olivier Sète, and Lloyd N. Trefethen. “The AAA Algorithm for Rational Approximation.” SIAM Journal on Scientific Computing 40, no. 3 (January 2018): A1494–1522. https://doi.org/10.1137/16M1106122.

Version History

Introduced in R2020a

R2024a: Share poles of S-parameters using rational object

You can now use the new PoleSharing name-value argument of the rational object to share all poles of the S-parameters or share the poles individually by S-parameter terms, or by S-parameter column terms. The rational object by default shares all poles of the S-parameters.

R2023b: Recommended over rationalfit

rational is recommend over rationalfit because it enables faster simulation and improved fitting of complex frequency-dependent data.

R2023a: Specify poles and residues in rational object

The rational object now allow you to specify:

• Number of poles

• Poles

• Residues

• Direct term