Contents

Random modulation

The random modulation procedure starts with two stochastic baseband signals, ${\displaystyle x_{c}($t)} and ${\displaystyle x_{s}($t)} , whose frequency spectrum is non-zero only for ${\displaystyle f\in [-B/2,B/2]}$. It applies quadrature modulation to combine these with a carrier frequency ${\displaystyle f_{0}}$ (with ${\displaystyle f_{0}>B/2}$) to form the signal ${\displaystyle x($t)} given by

${\displaystyle x($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^{j2\pi f_{0}t}\right\},}

where ${\displaystyle {\underline {x}}($t)} is the equivalent baseband representation of the modulated signal ${\displaystyle x($t)}

${\displaystyle {\underline {x}}($t)=x_{c}(t)+jx_{s}(t).}

In the following it is assumed that ${\displaystyle x_{c}($t)} and ${\displaystyle x_{s}($t)} are two real jointly wide sense stationary processes. It can be shown^{[citation needed]} that the new signal ${\displaystyle x($t)} is wide sense stationary if and only if ${\displaystyle {\underline {x}}($t)} is circular complex, i.e. if and only if ${\displaystyle x_{c}($t)} and ${\displaystyle x_{s}($t)} are such that

${\displaystyle R_{x_{c}x_{c}}(\tau )=R_{x_{s}x_{s}}(\tau )\qquad {\text{and }}\qquad R_{x_{c}x_{s}}(\tau )=-R_{x_{s}x_{c}}(\tau ).}$

This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations. (August 2011) (Learn how and when to remove this message)

• Papoulis, Athanasios; Pillai, S. Unnikrishna (2002). "Random walks and other applications". Probability, random variables and stochastic processes (4th ed.). McGraw-Hill Higher Education. pp. 463–473.
• Scarano, Gaetano (2009). Segnali, Processi Aleatori, Stima (in Italian). Centro Stampa d'Ateneo.
• Papoulis, A. (1983). "Random modulation: A review". IEEE Transactions on Acoustics, Speech, and Signal Processing. 31: 96–105. doi:10.1109/TASSP.1983.1164046.

This signal processing-related article is a stub. You can help Wikipedia by expanding it.

This statistics-related article is a stub. You can help Wikipedia by expanding it.