Random modulation

In modulation theory and stochastic processes theory, Random modulation is the Quadrature modulation of two stochastic baseband signals (whose frequency spectrum is non-zero only for $f\in [-B/2,B/2]$) $x_{c}(t)$ and $x_{s}(t)$ on a carrier frequency $f_{0}$ (with $f_{0}>B/2$) to form the signal $x(t)$:

$x(t)=x_{c}(t)\cos(2\pi f_{0}t)-x_{s}(t)\sin(2\pi f_{0}t)=\Re \left\{{\underline {x}}(t)e^{j2\pi f_{0}t}\right\}$

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

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

In the following we can assume that $x_{c}(t)$ and $x_{s}(t)$ are two real jointly WSS processes.

It can be shown that $x(t)$ is WSS iff ${\underline {x}}(t)$ is circular complex, i.e. iff $x_{c}(t)$ and $x_{s}(t)$ are such that

$R_{x_{c}x_{c}}(\tau )=R_{x_{s}x_{s}}(\tau )\qquad R_{x_{c}x_{s}}(\tau )=-R_{x_{s}x_{c}}(\tau )$

Bibliography

• 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.
• Template:ItScarano, Gaetano (2009). Segnali, Processi Aleatori, Stima. Centro Stampa d'Ateneo.