Random modulation

In the theories of modulation and of stochastic processes, 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.

Details

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

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

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

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

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.