Free convolution
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Free deconvolution is a signal processing concept. It enables to compute the eigenvalues of involved models of sum or product of random matrices using combinatorial techniques. It has some some strong connections with other works on G-estimation of Girko.
As a straightforward example, suppose that A and B are independent large square Hermitian (or symmetric) random matrices, then under some very general conditions, free deconvolution enables to :
- Deduce the eigenvalue distribution of A from those of A+B and B.
- Deduce the eigenvalue distribution of A from those of AB and B.
The concept is even broader as it provides a method to retrieve the eigenvalue distribution of A from any functional f(A,B) and B (f(A,B) is a function of the tzo matrices A and B).
The applications in wireless communications have provided a useful framework when the number of observations is of the same order as the dimensions (time, frequency or space) of the system. Recent applications in finance and biology have shown the potential of the tool.
References
- "Free Deconvolution for Signal Processing Applications", O. Ryan and M. Debbah, ISIT 2007, pp. 1846-1850