Complex random variable

In probability theory and statistics, complex random variables are a generalization of real-valued random variables to complex numbers, i.e. the possible values a complex random variable may take are complex numbers.^[1] Complex random variables can always be considered as pairs of real random variables: their real and imaginary parts. Therefore, the distribution of one complex random variable may be interpreted as the joint distribution of two real random variables.

Some concepts of real random variables have a straightforward generalization to complex random variables—e.g., the definition of the mean of a complex random variable. Other concepts are unique to complex random variables.

Applications of complex random variables are found in digital signal processing,^[2] quadrature amplitude modulation and information theory.

^ Eriksson, Jan; Ollila, Esa; Koivunen, Visa (2009). Statistics for complex random variables revisited. 2009 IEEE International Conference on Acoustics, Speech and Signal Processing. Taipei, Taiwan: Institute of Electrical and Electronics Engineers. pp. 3565–3568. doi:10.1109/ICASSP.2009.4960396.
^ Lapidoth, A. (2009). A Foundation in Digital Communication. Cambridge University Press. ISBN 9780521193955.

[1] Eriksson, Jan; Ollila, Esa; Koivunen, Visa (2009). Statistics for complex random variables revisited. 2009 IEEE International Conference on Acoustics, Speech and Signal Processing. Taipei, Taiwan: Institute of Electrical and Electronics Engineers. pp. 3565–3568. doi:10.1109/ICASSP.2009.4960396.

[2] Lapidoth, A. (2009). A Foundation in Digital Communication. Cambridge University Press. ISBN 9780521193955.

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