# Applications of polynomial common factor computation in signal processing and systems theory

We consider the problem of computing the greatest common divisor of a set of univariate polynomials and present applications of this problem in system theory and signal processing. One application is blind system identification: given the responses of a system to unknown inputs, find the system. Assuming that the unknown system is finite impulse response and at least two experiments are done with inputs that have finite support and their Z-transforms have no common factors, the impulse response of the system can be computed up to a scaling factor as the greatest common divisor of the Z-transforms of the outputs. Other applications of greatest common divisor problem in system theory and signal processing are finding the distance of a system to the set of uncontrollable systems and common dynamics estimation in a multi-channel sum-of-exponentials model.

## References

- A. Fazzi, N. Guglielmi, and I. Markovsky. Matrix polynomials approximate common factors computation. Technical report, Dept. ELEC, Vrije Universiteit Brussel, 2018.
- A. Fazzi, N. Guglielmi, and I. Markovsky. An ODE based method for computing the approximate greatest common divisor of polynomials. Numerical algorithms, 2018.
- I. Markovsky, A. Fazzi, and N. Guglielmi. Applications of polynomial common factor computation in signal processing. In Latent Variable Analysis and Signal Separation, Lecture Notes in Computer Science, pages 99–106. Springer, 2018.