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In [[Modulation|modulation theory]] and [[stochastic process|stochastic processes theory]], '''Random modulation''' is the [[Quadrature modulation]] of two stochastic [[Baseband_signal#Baseband_signal|baseband signals]] (whose [[frequency spectrum]] is non-zero only for <math>f \in [-B/2,B/2]</math>) <math>x_c(t)</math> and <math>x_s(t)</math> on a carrier frequency <math>f_0</math> (with <math>f_0 > B/2</math>) to form the signal <math>x(t)</math>:
:<math>x(t)=x_c(t)\cos(2 \pi f_0 t)-x_s(t)\sin(2 \pi f_0 t)= \Re \left \{ \underline{x}(t)e^{j 2 \pi f_0 t}\right \}</math>
where <math>\underline{x}(t)</math> is the [[Baseband_signal#Equivalent_baseband_signal|equivalent baseband representation]] of the modulated signal <math>x(t)</math>
:<math>\underline{x}(t)=x_c(t)+j x_s(t)</math>
In the following we can assume that <math>x_c(t)</math> and <math>x_s(t)</math> are two real jointly [[Wide_sense_stationary#Weak_or_wide-sense_stationarity|WSS]] processes.
It can be shown that <math>x(t)</math> is WSS [[iff]] <math>\underline{x}(t)</math> is circular complex, i.e. iff <math>x_c(t)</math> and <math>x_s(t)</math> are such that
:<math>R_{x_c x_c}(\tau)=R_{x_s x_s}(\tau) \qquad R_{x_c x_s}(\tau)=-R_{x_s x_c}(\tau)</math>
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