# autocorrelation of multiple signals

Problem: I am looking at an adaptive filtering application where the eigenvaluespread of the autocorrelation matrix $$R$$ is important for the convergence of the algorithm. For a single channel system the autocorrelation matrix $$R$$ for iterationstep $$n$$ can be calculated by $$R=E\{ x(n) x^H(n)\}$$ where $$x(n)$$ is the input signal of the adaptive filter at iterationstep $$n$$ consisting of a number of samples $$N$$ recorded over a timespan. The calculation of the eigenvalues is straight forward.

Question: What is the "multichannel equivalent" for $$R$$ in the case of e.g. an adaptive filtering multichannel application? Do I need to calculate some sort of autocorrelation tensor?

You apply the same formula, but instead of using a scalar $$x(n)$$, you will have $$x(n) \in \mathbb{C}^{M \times 1}$$, where $$M$$ is the number of channels, and $$x^{H}(n)$$ is the $$1 \times M$$ matrix whose entries are the complex conjugate of the entries in $$x(n)$$. As a result $$R \in \mathbb{C}^{M \times M}$$ is an Hermitian matrix, and its eigenvalues are all real.
• But wouldnt $R \in {\mathbb{C}}^{M \times M \times 2N-1}$ ? Mar 9, 2021 at 8:50
• It seemed too easy to be correct hehe. I interpreted E as the expected value,x(n) the covariance of one sample. You meant the matrix R, where each element $R_{i,j} = E\{x(n-i)x^H(n-j)\}$ for random vectors? By equalizing you mean to whiten the signal, to remove the correlation between different samples?