rational

Perform rational fitting to complex frequency-dependent data

Since R2020a

Description

Use the rational object and an interpolative algorithm to create a rational fit to frequency-dependent data.

The fit is given by the equation:

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

Creation

Syntax

fit = rational(freq,data)

fit = rational(s)

[fit,error] = rational(___)

fit = rational(___,Name=Value)

Description

fit = rational(freq,data) returns a rational object with complex frequencies using the given frequency vector and network parameter data.

fit = rational(s) returns a rational object for N-port S-parameters.

[fit,error] = rational(___) also returns the error of the fit. Use any of input argument combinations from the previous syntaxes.

fit = rational(___,Name=Value) specifies options to control aspects of fit or sets properties. You can 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.

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.

s — N-port S-parameters

N-by-N matrix of elements of S

N-port S-parameters, specified as an N-by-N matrix of elements of S sharing identical poles.

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 zero in dB.

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

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

Properties

NumPorts — Number of ports

scalar integer

This property is read-only.

Number of ports in the original S-parameter data, returned as a scalar integer. The value of the NumPorts is determined from the S-parameter data.

NumPoles — Number of poles

nonegative integer

Number or length 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.

Example: 10

Poles — Poles of fit

row vector

Poles of the fit or complex values of A in the equation provided in the description, 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.

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 t2-D matrix of length NumPorts-by-NumPorts.

Example: [0 0;0 0]

Data Types: double

ErrDB — Error between fit and original data

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 rationalfit output is passive at all frequencies
makepassive	Enforce passivity of rationalfit output or a rational object
passivity	Plot passivity of N-by-N rationalfit function 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: [52x1 double]
      Residues: [2x2x52 double]
    DirectTerm: [2x2 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
    {51x1 double}    {51x1 double}
    {51x1 double}    {51x1 double}

p=2×2 cell array
    {52x1 double}    {52x1 double}
    {52x1 double}    {52x1 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

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 a 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('default.s2p');

Perform a default fit.

fit = rational(S)

fit = 
  rational with properties:

      NumPorts: 2
      NumPoles: 52
         Poles: [52x1 double]
      Residues: [2x2x52 double]
    DirectTerm: [2x2 double]
         ErrDB: -22.6872

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: [4x1 double]
      Residues: [2x2x4 double]
    DirectTerm: [2x2 double]
         ErrDB: -14.7945

Force the rational object to use twice the number of poles in fit. Note that the error in fit2 is lower than fit1.

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

fit2 = 
  rational with properties:

      NumPorts: 2
      NumPoles: 52
         Poles: [52x1 double]
      Residues: [2x2x52 double]
    DirectTerm: [2x2 double]
         ErrDB: -18.7258

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

R2023a: Specify poles and residues in rational object

The rational object now allow you to specify:

• Number of poles

• Poles

• Residues

• Direct term