Lecture slide SLide#10, mentions how to generate Pseudo Random binary sequence (PRBS). I says that we can take the sign of a white, zero-mean Gaussian noise signal to form the PRBS. What will be the probability density function of PRBS?

Are there any examples for system identification using PRBS as the excitation input? I googled a lot but found no examples fro linear dynamical systems.


1 Answer 1


Let a sequence of white-noise, zero mean normal random variables

$$u_i \sim N(0, \sigma^2),\;\; i=1,...,n$$

To "take the sign" of the realizations of the sequence, we consider the sequence of indicator functions $I_{\{u_i>0\}}$. The element of the sequence is a dichotomous random variable (it takes just two values), and more over, since these values are $\{0,1\}$ it is a Bernoulli random variable (one could use any dichotomous r.v. here). The Bernoulli random variable is fully characterized by a single parameter, the probability of observing, say, the value $1$, denote it $p$, which is also its expected value. So

$$p= E\left[I_{\{u_i>0\}}\right] = \Pr(u_i>0) = \frac 12$$

given how $u_i$ is specified.

  • $\begingroup$ Thank you for your reply. If the values taken are {-1,1}, will the pdf still be Bernoulli? Also, I could not follow the terms like indicator function. Could you probably post a reference where I can understand this in detail? $\endgroup$
    – Srishti M
    Dec 5, 2014 at 22:47
  • 1
    $\begingroup$ It would be Rademacher, en.wikipedia.org/wiki/Rademacher_distribution. Introductory texts about Indicator functions are spread all over the web, you can also start at wikipedia, en.wikipedia.org/wiki/Indicator_function $\endgroup$ Dec 5, 2014 at 22:50

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.