Pseudorandom binary sequence

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A binary sequence (BS) is a sequence of $N$ bits,

$a_j$ for $j=0,1,...,N-1$ ,

i.e. $m$ ones and $N-m$ zeros. A BS ispseudo-random (PRBS) if its autocorrelation function:

$C(v)=\sum_{j=0}^{N-1} a_ja_{j+v}$

has only two values:

$C(v)=\begin{cases}m, \mbox{ if } v\equiv 0\;\; (\mbox{mod}N)\\ mc, \mbox{ otherwise }\end{cases}$

where

$c=\frac{m-1}{N-1}$

is called the duty cycle of the PRBS.

A PRBS is random in a sense that the value of an $a_j$ element is independent of the values of any of the other elements, similar to real random sequences.

It is 'pseudo' because it is deterministic and after $N$ elements it starts to repeat itself, unlike real random sequences, such as sequences generated by radioactive decay or by white noise. The PRBS is more general than the n-sequence, which is a special pseudo-random binary sequence of n bits generated as the output of a linear shift register. An n-sequence always has a 1/2 duty cycle and its number of elements $N = 2^k-1$ . PRBS's are used intelecommunication, encryption, simulation, correlation technique and time-of-flight spectroscopy.