Cross-correlation matrix: Difference between revisions

Revision as of 12:24, 30 December 2018

Revision as of 14:05, 30 December 2018

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|equation = <math>\operatorname{R}_{\mathbf{X}\mathbf{Y}} \stackrel{\mathrm{def}}{=}triangleq\ \operatorname{E}[\mathbf{X} \mathbf{Y}^{\rm T}]</math>

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If <math>\mathbf{Z} = (Z_1,\ldots,Z_m)^{\rm T}</math> and <math>\mathbf{W} = (W_1,\ldots,W_n)^{\rm T}</math> are [[complex random vector|complex random vectors]], each containing random variables whose expected value and variance exist, the cross-correlation matrix of <math>\mathbf{Z}</math> and <math>\mathbf{W}</math> is defined by

:<math>\operatorname{R}_{\mathbf{Z}\mathbf{W}} \stackrel{\mathrm{def}}{=}triangleq\ \operatorname{E}[\mathbf{Z} \mathbf{W}^{\rm H}]</math>

where <math>{}^{\rm H}</math> denotes [[Hermitian transpose|Hermitian transposition]].

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: Respectively for complex random vectors:

:<math>\operatorname{K}_{\mathbf{Z}\mathbf{W}} = \operatorname{E}[(\mathbf{Z} - \operatorname{E}[\mathbf{Z}])(\mathbf{W} - \operatorname{E}[\mathbf{W}])^{\rm H}] = \operatorname{R}_{\mathbf{Z}\mathbf{W}} - \operatorname{E}[\mathbf{Z}] \operatorname{E}[\mathbf{W}]^{\rm H}</math>

==References==

{{reflist}}

* 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. [http://www.cambridge.org/us/academic/subjects/computer-science/cryptography-cryptology-and-coding/signal-design-good-correlation-wireless-communication-cryptography-and-radar Signal design for good correlation: for wireless communication, cryptography, and radar]. Cambridge University Press, 2005.

* M. Soltanalian. [http://theses.eurasip.org/theses/573/signal-design-for-active-sensing-and/download/ Signal Design for Active Sensing and Communications]. Uppsala Dissertations from the Faculty of Science and Technology (printed by Elanders Sverige AB), 2014.

==See also==

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*[[Rate distortion theory#Rate–distortion_functions|Rate distortion theory]]

*[[Radial distribution function]]

==References==

{{reflist}}

==Further reading==

* 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. [http://www.cambridge.org/us/academic/subjects/computer-science/cryptography-cryptology-and-coding/signal-design-good-correlation-wireless-communication-cryptography-and-radar Signal design for good correlation: for wireless communication, cryptography, and radar]. Cambridge University Press, 2005.

* M. Soltanalian. [http://theses.eurasip.org/theses/573/signal-design-for-active-sensing-and/download/ Signal Design for Active Sensing and Communications]. Uppsala Dissertations from the Faculty of Science and Technology (printed by Elanders Sverige AB), 2014.

{{DEFAULTSORT:Correlation Function}}