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| {{Correlation and covariance}} | |||
| {{context|date=July 2018}} | |||
| {{technical|date=July 2018}} | |||
| {{R with history}} | |||
| The '''auto-correlation matrix''' (als called second moment) of a ] <math>\mathbf{X} = (X_1,\ldots,X_n)^{\rm T}</math> is a <math>n \times n</math> matrix containing as elements the correlations of all pairs of elements of the random vector <math>\mathbf{X}</math>. The autocorrelation matrixis used in various digital signal processing algorithms. | |||
| ==Definition== | |||
| For a ] <math>\mathbf{X} = (X_1,\ldots,X_n)^{\rm T}</math> containing ]s whose ] and ] exist, the '''auto-correlation matrix''' is defined by<ref name=Papoulis>Papoulis, Athanasius, ''Probability, Random variables and Stochastic processes'', McGraw-Hill, 1991</ref>{{rp|p.190}} | |||
| {{Equation box 1 | |||
| |indent = | |||
| |title= | |||
| |equation = {{NumBlk||<math>\operatorname{R}_{\mathbf{X}\mathbf{X}} \triangleq\ \operatorname{E}</math>|{{EquationRef|Eq.1}}}} | |||
| |cellpadding= 6 | |||
| |border | |||
| |border colour = #0073CF | |||
| |background colour=#F5FFFA}} | |||
| where <math>{}^{\rm T}</math> denotes transposition and has dimensions <math>n \times n</math>. | |||
| Written component-wise: | |||
| :<math>\operatorname{R}_{\mathbf{X}\mathbf{X}} = | |||
| \begin{bmatrix} | |||
| \operatorname{E} & \operatorname{E} & \cdots & \operatorname{E} \\ \\ | |||
| \operatorname{E} & \operatorname{E} & \cdots & \operatorname{E} \\ \\ | |||
| \vdots & \vdots & \ddots & \vdots \\ \\ | |||
| \operatorname{E} & \operatorname{E} & \cdots & \operatorname{E} \\ \\ | |||
| \end{bmatrix} | |||
| </math> | |||
| If <math>\mathbf{Z}</math> is a ], the autocorrelation matrix is instead defined by | |||
| :<math>\operatorname{R}_{\mathbf{Z}\mathbf{Z}} \triangleq\ \operatorname{E}</math>. | |||
| Here <math>{}^{\rm H}</math> denotes ]. | |||
| ==Example== | |||
| For example, if <math>\mathbf{X} = \left( X_1,X_2,X_3 \right)^{\rm T}</math> is a random vectors, then <math>\operatorname{R}_{\mathbf{X}\mathbf{X}}</math> is a <math>3 \times 3</math> matrix whose <math>(i,j)</math>-th entry is <math>\operatorname{E}</math>. | |||
| ==Properties== | |||
| * The autocorrelation matrix is a ] for complex random vectors and a ] for real random vectors.<ref name=Papoulis />{{rp|p.190}} | |||
| * The autocorrelation matrix is a ]. | |||
| * The autocorrelation matrix is a positive semidefinite matrix<ref name=Papoulis />{{rp|p.190}}, i.e. <math>\mathbf{a}^{\mathrm T} \operatorname{R}_{\mathbf{X}\mathbf{X}} \mathbf{a} \ge 0 \quad \text{for all } \mathbf{a} \in \mathbb{R}^n</math> for a real random vector respectively <math>\mathbf{a}^{\mathrm H} \operatorname{R}_{\mathbf{Z}\mathbf{Z}} \mathbf{a} \ge 0 \quad \text{for all } \mathbf{a} \in \mathbb{C}^n</math> in case of a complex random vector. | |||
| * All eigenvalues of the autocorrelation matrix are real and positive. | |||
| * The ''autocovariance matrix'' is related to the autocorrelation matrix as follows: | |||
| :<math>\operatorname{K}_{\mathbf{X}\mathbf{X}} = \operatorname{E})(\mathbf{X} - \operatorname{E})^{\rm T}] = \operatorname{R}_{\mathbf{X}\mathbf{X}} - \operatorname{E} \operatorname{E}^{\rm T}</math> | |||
| : Respectively for complex random vectors: | |||
| :<math>\operatorname{K}_{\mathbf{Z}\mathbf{Z}} = \operatorname{E})(\mathbf{Z} - \operatorname{E})^{\rm H}] = \operatorname{R}_{\mathbf{Z}\mathbf{Z}} - \operatorname{E} \operatorname{E}^{\rm H}</math> | |||
| == References == | |||
| * Hayes, Monson H., ''Statistical Digital Signal Processing and Modeling'', John Wiley & Sons, Inc., 1996. {{ISBN|0-471-59431-8}}. | |||
| * Solomon W. Golomb, and Guang Gong. . Cambridge University Press, 2005. | |||
| * M. Soltanalian. . Uppsala Dissertations from the Faculty of Science and Technology (printed by Elanders Sverige AB), 2014. | |||
| ] | ] | ||
| ] | |||
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