Random modulation: Difference between revisions

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In the theories of [[modulation]] and of [[stochastic process]]es, '''random modulation''' is the creation of a new signal from two other signals by the process of [[quadrature modulation]]. For applications, the two original signals need have a limited freqency range, and these are used to modulate a third sinusoidal (carrier) signal whose frequency is above the range of frequencies contained in the original signals.

{{Cleanup|date = March 2011}}

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>:

==Details==

:<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>

InThe [[Modulation|modulation theory]] and [[stochastic process|stochastic processes theory]], '''Randomrandom modulation''' isprocedure thestarts [[Quadrature modulation]] ofwith two stochastic [[Baseband_signal#Baseband_signal|baseband signals]], <math>x_c(t)</math> and <math>x_s(t)</math>, whose [[frequency spectrum]] is non-zero only for <math>f \in [-B/2,B/2]</math>). <math>x_c(t)</math>It andapplies <math>x_s(t)</math>[[quadrature onmodulation]] to combine these with a carrier frequency <math>f_0</math> (with <math>f_0 > B/2</math>) to form the signal <math>x(t)</math>: given by

:<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.

In the following it is assumed that <math>x_c(t)</math> and <math>x_s(t)</math> are two real jointly [[Wide_sense_stationary#Weak_or_wide-sense_stationarity|wide sense stationary]] processes. It can be shown{{cn|date=August 2011}} that the new signal <math>x(t)</math> is WSSwide sense stationary [[iff|if and only if]] <math>\underline{x}(t)</math> is circular complex, i.e. iffif and only if <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 \text{and }\qquad R_{x_c x_s}(\tau)=-R_{x_s x_c}(\tau).</math>

{{Cleanupmorefootnotes|date = MarchAugust 2011}}

== Bibliography ==

*{{en}}{{cite book |title=Probability, random variables and stochastic processes |last1= Papoulis|first1= Athanasios|authorlink1= Athanasios Papoulis|first2=S. Unnikrishna|last2= Pillai |year= 2002|publisher= McGraw-Hill Higher Education|edition= 4th|chapter=Random walks and other applications|pages=463–473}}