Efficient tracking of the cross-correlation coefficient

In many (audio) processing algorithms, involving manipulation of discrete-time signals, the performance can vary strongly over the repertoire that is used. This may be the case when the signals from the various channels are allowed to be strongly positively or negatively correlated. We propose and analyze a general formula for tracking the (time-dependent) correlation between two signals. Some special cases of this formula lead to classical results known from the literature, others are new. This formula is recursive in nature, and uses only the instantaneous values of the two signals, in a low-cost and low-complexity manner; in particular, there is no need to take square roots or to carry out divisions. Furthermore, this formula can be modified with respect to the occurrence of the two signals so as to further decrease the complexity, and increase ease of implementation. The latter modification comes at the expense that not the actual correlation is tracked, but, rather, a somewhat deformed version of it. To overcome this problem, we propose, for a number of instances of the tracking formula, a simple warping operation on the deformed correlation. Now we obtain, at least for sinusoidal signals, the correct value of the correlation coefficient. Special attention is paid to the convergence behavior of the algorithm for stationary signals and the dynamic behavior if there is a transition to another stationary state; the latter is considered to be important to study the tracking abilities to nonstationary signals. We illustrate tracking algorithm by using it for stereo music fragments, obtained from a number of digital audio recordings.