# IIRrational: Rational Function fitting for System ID¶

Release v2.5.1. (Installation)

IIRrational is an function-fitting library for signal processing and system identification. Fitting both poles and zeros for a rational function is a nonlinear optimization problem and typical formulations require an initial guess for convergence.

This library uses a two stage approach, where the second stage is a typical gradient descent optimization, but the first is a linear technique with fast and reliable convergence, but requires gratuitous over-fitting.

## Features and Conveniences¶

No initial guess required (most of the time)

Automatic order reduction (using heuristics)

guaranteed stable poles or zeros upon request.

uses SNR weight vector

generates fisher matrix / covariance matrix of parameterizations

propagates errorbars to the fit

S-domain for implementable filters

matlab support through the msurrogate package

see What to Expect and quickstart

```
import IIRrational.v2
from IIRrational.testing import IIRrational_data
dataset = IIRrational_data('simple2')
results = IIRrational.v2.data2filter(
data = dataset.data,
F_Hz = dataset.F_Hz,
SNR = dataset.SNR,
F_nyquist_Hz = 16384,
)
axB = results.investigate_fit_plot()
axB = results.investigate_order_plot()
```

and to choose the order which should be used, the order plot will show the residuals

## Contents¶

## Example Output¶

Running the example above from the command line

```
IIRrationalV2fit :simple2: simple2_fit.mat --plot_fit simple2_fit_example.png --plot_order simple2_fit_example_order.png --choose baseline
```

This uses a special file :testdata: to get its data. It can import data from matlab data files, hdf5, yaml, json and CSV using the command line interface. Learn more through help:

```
IIRrationalV2fit -h
```

Which outputs on the commandline the logging and foton filter:

```
Arguments:
F_Hz: [(1000,) float64 array]
F_nyquist_Hz: None
SNR: 10
data: [(1000,) complex128 array]
emphasis: None
mode: full
xfer: [(1000,) complex128 array]
------------:rational fitting:
4P 0.01 chebychev rational fit
3P 1.13 Initial Order: (Z=20, P=20, Z-P=0)
3P 1.72 Fastdrop Order: (Z=12, P=12, Z-P=0)
4P 2.32 mag fitting and phase patching
------------:sample variance (from magnitude):
3A 3.78 Weight Scaling determined: 0.9740626973315305
3A 3.78 Weight Scaling Used 0.9869461471283681
------------:Q-ranked order reduction:
4P 5.84 order reduced annealing
------------:selective order reduction:
5P 14.96 order reduced to 11, residuals=1.82e+01
5P 20.46 order reduced to 9, residuals=1.83e+01
5P 28.16 order reduced to 8, residuals=9.71e+00
5P 40.08 order reduced to 7, residuals=2.17e+00
5P 42.43 order reduced to 7, residuals=9.66e-01
5P 43.46 order reduced to 6, residuals=9.64e-01
2A 44.41 Baseline fit residuals: 9.64e-01, at order 6
------------:successive order reduction:
5P 45.45 order reduced to 4, residuals=4.05e+00
5P 46.23 order reduced to 4, residuals=1.02e+01
5P 46.81 order reduced to 3, residuals=6.40e+00
5P 47.02 order reduced to 3, residuals=3.16e+01
5P 47.15 order reduced to 2, residuals=4.22e+01
------------:investigations:
2I 47.16 max(z, p) ChiSq.
order avg. res. med. res. max. res.
----------- ----------- ----------- -----------
2 42.1719 39.7239 704.977
3 6.40444 2.9071 745.896
4 4.05173 1.85803 693.821
6 0.963768 1.28386 13.4802
plotting for -R, --plot_order=simple2_fit_example_order.png
-L, --LIGO_foton=Sf output:
%%%
ZPK([
-0.104504805261 + 1.09613226443*i; -0.104504805261 - 1.09613226443*i;
-0.98417325191 + 0.169021580444*i; -0.98417325191 - 0.169021580444*i;
],[
-0.00956418703895 + 0.998886697835*i; -0.00956418703895 - 0.998886697835*i;
-0.0981823932483 + 2.99528020808*i; -0.0981823932483 - 2.99528020808*i;
-0.114527880324; -0.0817740982447;
], 39.3478656882, "f")
%%%
plotting for -r, --plot_fit=simple2_fit_example.png
```

You may notice that the initial residuals from the automated fit are poor. Try calling it with the addiational argument –relative-degree -2 and the intial guess will have converged better.

Note

- when the library is installed, the command line tool may also be called via::
python -m IIRrational.v2 [args…]

## Alternatives¶

Vectfit [matlab] https://www.sintef.no/projectweb/vectfit/

python vectfit [python] https://github.com/PhilReinhold/vectfit_python

fdident [matlab] https://www.mathworks.com/products/connections/product_detail/product_35570.html

TFestimate [matlab] https://github.com/Nikhil-Mukund/TFestimate/blob/master/README.md