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The definition of the normalized cross-correlation of a stochastic process is (,) = (,) () = [() ¯] () If the function is well-defined, its value must lie in the range [,], with 1 indicating perfect correlation and −1 indicating perfect anti-correlation. For jointly wide-sense stationary stochastic processes, the definition is ...
Cross-covariance may also refer to a "deterministic" cross-covariance between two signals. This consists of summing over all time indices. For example, for discrete-time signals f [ k ] {\displaystyle f[k]} and g [ k ] {\displaystyle g[k]} the cross-covariance is defined as
Let (,) represent a pair of stochastic processes that are jointly wide sense stationary with autocovariance functions and and cross-covariance function . Then the cross-spectrum Γ x y {\displaystyle \Gamma _{xy}} is defined as the Fourier transform of γ x y {\displaystyle \gamma _{xy}} [ 1 ]
A weaker form of stationarity commonly employed in signal processing is known as weak-sense stationarity, wide-sense stationarity (WSS), or covariance stationarity. WSS random processes only require that 1st moment (i.e. the mean) and autocovariance do not vary with respect to time and that the 2nd moment is finite for all times.
In the case of a time series which is stationary in the wide sense, both the means and variances are constant over time (E(X n+m) = E(X n) = μ X and var(X n+m) = var(X n) and likewise for the variable Y). In this case the cross-covariance and cross-correlation are functions of the time difference: cross-covariance
The parameter belongs to the set of positive-definite matrices, which is a Riemannian manifold, not a vector space, hence the usual vector-space notions of expectation, i.e. "[^]", and estimator bias must be generalized to manifolds to make sense of the problem of covariance matrix estimation.
Pearson's correlation coefficient is the covariance of the two variables divided by the product of their standard deviations. The form of the definition involves a "product moment", that is, the mean (the first moment about the origin) of the product of the mean-adjusted random variables; hence the modifier product-moment in the name.
If the code is correlated with the signal at any time offset other than zero, the correlation should be as close to zero as possible. This is referred to as auto-correlation and is used to reject multi-path interference. [18] [19] An analogy to the problem of multiple access is a room (channel) in which people wish to talk to each other ...