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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.

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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.

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  • $\begingroup$ 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 '14 at 22:47
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    $\begingroup$ 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 $\endgroup$ – Alecos Papadopoulos Dec 5 '14 at 22:50

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