# What is the probability density function of this arbitrary signal

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.

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