Without knowing details of your 'complex communication systems', only some general method can be described here.

Transfer function models describe the relationship between the inputs and outputs of a system using a ratio of polynomials. The model order is equal to the order of the denominator polynomial. The roots of the denominator polynomial are referred to as the model poles. The roots of the numerator polynomial are referred to as the model zeros.

The parameters of a transfer function model are its poles, zeros and transport delays.

Continuous-Time Representation

In continuous-time, a transfer function model has the form:

Y(s)=num(s)/den(s)U(s)+E(s)

Where, Y(s), U(s) and E(s) represent the Laplace transforms of the output, input and noise, respectively. num(s) and den(s) represent the numerator and denominator polynomials that define the relationship between the input and the output.

Discrete-Time Representation

In discrete-time, a transfer function model has the form:

y(t)=num(q^-1)/den(q^-1)*u(t)+e(t)

num(q^-1)=b0+b1*q^-1+b2*q^-2+…

den(q^−1)=1+a1*q^−1+a2*q^−2+…

The roots of num(q^-1) and den(q^-1) are expressed in terms of the lag variable q^-1.

If you take the Z-transform, the transfer function has the form:

Y(z^−1)=num(z^−1)/den(z^−1)*U(z^−1)+E(z^−1)

num(z^−1)=b0+b1*z^−1+b2*z^−2+…

den(z^−1)=1+a1*z*−1+a2*z^−2+…

Where, Y(z^-1), U(z^-1) and E(z^-1) represent the Z-transforms of the output, input and noise, respectively. z^-1 is the Z-transform of the lag operator.

Mathworks provides an option for convenient platform for identification of the transfer function. MatLab also provides a convenient platform for doing so.

See, e.g.

https://www.mathworks.com/help/ident/ug/how-to-estimate-transfer-function-models-in-the-system-identification-app.html

Using the User Interface you pick the transfer function model (continuous or discrete), define the number of poles and zeros. Using your input and output data, you get an estimation of your transfer function parameters.

Following

Dear Alexander Kolker,

Thank you for your sharing your thoughts about this question. Based on your answer, it seems that a possible way to address this problem is through System Identification theory and, specifically, data-driven modeling. I'm new to this topic, but I've read that there are some statistical/probabilistic techniques to estimate the transfer function parameters (poles and zeros) of a system. Interestingly, some of those techniques are also used in Machine Learning, such as the Maximum Likelihood and Bayesian estimators. However, the methods that I've read about so far are focused on LTI (Linear Time-Invariant) systems, which delimit their applicability to many complex systems.

Cesar A. Gomez

The concept of system transfer function as the ratio of the Laplace transform of the output to that of input signal (Z-transform for discrete signals) is mostly applicable to linear systems.

If the system is non-linear the identification is much more mathematically involved.

The most frequently used approach is linearization of the dynamic equations of the non-linear system around some basic steady-state solution by expansion in the Taylor series and leaving only linear terms of expansion. Of course, this approach has its natural limitations.

Without knowing details of your specific system it is hardly to say whether linearization will or will not work for you.

Dear Cesar A. Gomes and Profs.

These are System Identification Theories that suit linear and nonlinear Systems. There are also the recent Neural Networks and Backstepping Method. Many softwares support such theories. Problem is when a system is considered there is a very broad set of totally different applications that appear on screen. See book by Ljung for Identification Potential application. Classes of systems are linear / nonlinear and continuous / discrete and there comes again a huge collection of methods and research. I hope this helps. Good luck it is all very interesting work.

Based on Alcy Rodolfo dos Santos Carrara recommendation and after searching for more documentation on the application of Machine Learning in System Identification, I want to share these books that I've found insightful:

O. Nelles, Nonlinear System Identification: From Classical Approaches to Neural Networks and Fuzzy Models. Springer Science & Business Media, 2013.

F. Giri and E.-W. Bai, Block-oriented Nonlinear System Identification. Springer, 2010.

I hope researchers new to this topic and who are following this thread find the books helpful, as well.

Good afternoon Mr. Cesar A. Gomez. Super. I will try to get what you have shared. This way I will be able to follow you as much as my time allows.

Best Regards,

Prof. A. Rodolfo S. Carrara, Ph.D. - UFRGS / DEMEC / GPFAI

Porto Alegre - RS - Brazil