# Analytical spectral density of a On/Off modulation defined by a Bernoulli process

Consider a narrow band signal (laser) that I can modulate digitally with a on/off switch controlled by a digital pseudo random number generator. The resulting signal features a linewidth broadened by an amount given by the timebase of the number generator.

How do I figure out the analytical expression for spectral density of the modulated signal for a more general case below?

"on" and "off" are the only two possible states of a modulator and the transitions between states are Bernoulli process with probability $$p$$ or even more general a two states Discrete Markov Process with time base $$\tau$$.

Approximated solutions are sufficient, but exact solutions would be desirable.

The extreme cases of the Bernoulli process have intuitive results:

• $$p=0$$ the modulator is permanently "on" therefore the spectral density is unchanged
• $$p=1$$ the modulator switches regularly in a square wave pattern, a side band is generated at the clock frequency.

I have interest on the general case for Bernoulli process with $$0 or the Discrete Markov Process.

• Begin by searching for "semi-random telegraph signal" and "random telegraph signal" These were discussed in various editions of a well-known textbook "Probability, Random Variables, and Stochastic Processes" by A. Papoulis (don't know about the latest edition, though). – Dilip Sarwate May 14 at 18:17