a function of the time-lag, and that this would be an even function of the lag . This gives the more familiar forms for the auto-correlation function and the auto-covariance function: In particular, note that Normalization It is common practice in some disciplines (e.g. statistics and time series analysis) to normalize the autocovariance function to get a time-dependent Pearson correlation coefficient. However, in other disciplines (e.g. engineering) the normalization is usually dropped and the terms "autocorrelation" and "autocovariance" are used interchangeably. The definition of the auto-correlation coefficient 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 a wide-sense stationary (WSS) process, the definition is . The normalization is important both because the interpretation of the autocorrelation as a correlation provides a scale-free measure of the strength of statistical dependence, and because the normalization has an effect on the statistical properties of the estimated autocorrelations. Properties Symmetry property The fact that the auto-correlation function is an even function can be stated as respectively for a WSS process: Maximum at zero For a WSS process: Notice that is always real. Cauchy–Schwarz inequality The Cauchy–Schwarz inequality, inequality for stochastic processes: Autocorrelation of white noise The autocorrelation of a continuous-time white noise signal will have a strong peak (represented by a Dirac delta function) at and will be exactly for all other . Wiener–Khinchin theorem The Wiener–Khinchin theorem relates the autocorrelation function to the power spectral density via the Fourier transform: For real-valued functions, the symmetric autocorrelation function has a real symmetric transform, so the Wiener–Khinchin theorem can be re-expressed in terms of real cosines only: Auto-correlation of random vectors The (potentially time-dependent) auto-correlation matrix (also called second moment) of a (potentially time-dependent) random vector is an matrix containing as elements the autocorrelations of all pairs of elements of the random vector . The autocorrelation matrix is used in various digital signal processing algorithms. For a random vector containing random elements whose expected value and variance exist, the auto-correlation matrix is defined by where denotes transposition and has dimensions . Written component-wise: If is a complex random vector, the autocorrelation matrix is instead defined by Here denotes Hermitian transposition. For example, if is a random vector, then is a matrix whose -th entry is . Properties of the autocorrelation matrix The autocorrelation matrix