Random modulation: Difference between revisions

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{{Short description|Concept in signal processing}}

{{New unreviewed article|source=ArticleWizard|date=December 2010}}

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 amplitude modulation]]. In particular, the two signals are considered as being [[random process]]es. For applications, the two original signals need have a limited frequency 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.

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==Details==

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.

InThe [[Modulation|modulation theory]] and [[stochastic process|stochastic processes theory]], '''Randomrandom modulation''' isprocedure thestarts [[Quadrature modulation]] ofwith two stochastic [[Baseband_signalBaseband signal#Baseband_signalBaseband 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

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(t)=x_c x_c}(t)\taucos(2 \pi f_0 t)=R_{-x_s x_s}(t)\tausin(2 \pi f_0 t)= \qquadRe R_\left \{x_c x_s\underline{x}(\taut)=-R_e^{x_sj x_c2 \pi f_0 t}(\tau)right \} ,</math>

where <math>\underline{x}(t)</math> is the [[Baseband_signalBaseband signal#Equivalent_baseband_signalEquivalent 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 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{{citation needed|date=August 2011}} that the new signal <math>x(t)</math> is wide sense stationary [[iff|if and only if]] <math>\underline{x}(t)</math> is circular complex, i.e. if 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>

{{more footnotes|date=August 2011}}

== Bibliography ==

*{{en}}{{cite book |title=Probability, random variables and stochastic processes |last1= Papoulis|first1= Athanasios|authorlink1author-link1= 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}}

*{{it}}{{cite book |title=Segnali, Processi Aleatori, Stima |last1= Scarano|first1= Gaetano|year= 2009|publisher= Centro Stampa d'Ateneo|language=it}}

*{{Cite journal | last1 = Papoulis | first1 = A. | doi = 10.1109/TASSP.1983.1164046 | title = Random modulation: A review | journal = IEEE Transactions on Acoustics, Speech, and Signal Processing | volume = 31 | pages = 96–105| year = 1983 }}

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[[Category:SignalStatistical signal processing]]

[[Category:Time series analysis]]

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