is a Hermitian matrix for complex random vectors and a symmetric matrix for real random vectors. The autocorrelation matrix is a positive semidefinite matrix, i.e. for a real random vector, and respectively in case of a complex random vector. All eigenvalues of the autocorrelation matrix are real and non-negative. The auto-covariance matrix is related to the autocorrelation matrix as follows:Respectively for complex random vectors: Auto-correlation of deterministic signals In signal processing, the above definition is often used without the normalization, that is, without subtracting the mean and dividing by the variance. When the autocorrelation function is normalized by mean and variance, it is sometimes referred to as the autocorrelation coefficient or autocovariance function. Auto-correlation of continuous-time signal Given a signal , the continuous autocorrelation is most often defined as the continuous cross-correlation integral of with itself, at lag . where represents the complex conjugate of . Note that the parameter in the integral is a dummy variable and is only necessary to calculate the integral. It has no specific meaning. Auto-correlation of discrete-time signal The discrete autocorrelation at lag for a discrete-time signal is The above definitions work for signals that are square integrable, or square summable, that is, of finite energy. Signals that "last forever" are treated instead as random processes, in which case different definitions are needed, based on expected values. For wide-sense-stationary random processes, the autocorrelations are defined as For processes that are not stationary, these will also be functions of , or . For processes that are also ergodic, the expectation can be replaced by the limit of a time average. The autocorrelation of an ergodic process is sometimes defined as or equated to These definitions have the advantage that they give sensible well-defined single-parameter results for periodic functions, even when those functions are not the output of stationary ergodic processes. Alternatively, signals that last forever can be treated by a short-time autocorrelation function analysis, using finite time integrals. (See short-time Fourier transform for a related process.) Definition for periodic signals If is a continuous periodic function of period , the integration from to is replaced by integration over any interval of length : which is equivalent to Properties In the following, we will describe properties of one-dimensional autocorrelations only, since most properties are easily transferred from the one-dimensional case to the multi-dimensional cases. These properties hold for wide-sense stationary processes. A fundamental property of the