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| A [[binary sequence]] (BS) is a [[sequence]] <math>a_0,\ldots, a_{N-1}</math> of <math>N</math> bits, i.e.
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| :<math>a_j\in \{0,1\}</math> for <math>j=0,1,...,N-1</math>.
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| A BS consists of <math>m=\sum a_j</math> ones and <math>N-m</math> zeros.
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| A BS is a '''[[pseudorandomness|pseudo-random]] binary sequence''' (PRBS) if its [[autocorrelation function]]:
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| :<math>C(v)=\sum_{j=0}^{N-1} a_ja_{j+v}</math>
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| has only two values:
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| :<math>C(v)=
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| \begin{cases}
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| m, \mbox{ if } v\equiv 0\;\; (\mbox{mod}N)\\
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| \\
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| mc, \mbox{ otherwise }
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| \end{cases}</math>
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| where
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| :<math>c=\frac{m-1}{N-1}</math>
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| is called the ''duty cycle'' of the PRBS, similar to the [[duty cycle]] of a continuous time signal.
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| A PRBS is 'pseudorandom', because, although it is in fact deterministic, it seems to be random in a sense that the value of an <math>a_j</math> element is independent of the values of any of the other elements, similar to real random sequences.
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| A PRBS can be stretched to infinity by repeating it after <math>N</math> elements, this in contrast to most random sequences, such as sequences generated by [[radioactive decay]] or by white noise, that are 'infinite' by nature. The PRBS is more general than the [[maximum length sequence]], which is a special pseudo-random binary sequence of N bits generated as the output of a linear shift register. A maximum length sequence always has a 1/2 duty cycle and its number of elements <math>N = 2^k-1</math>. PRBS's are used in [[telecommunication]], [[encryption]], [[simulation]], [[correlation]] technique and time-of-flight [[spectroscopy]].
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| == Practical implementation ==
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| Pseudorandom binary sequences can be generated using [[linear feedback shift register]]s.<ref>Paul H. Bardell, William H. McAnney, and Jacob Savir, "Built-In Test for VLSI: Pseudorandom Techniques", John Wiley & Sons, New York, 1987.</ref>
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| == See also ==
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| * [[Pseudorandom number generator]]
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| * [[Gold code]]
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| * [[Complementary sequences]]
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| * [[Bit Error Rate Test]]
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| ==References==
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| {{reflist}}
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| == External links ==
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| * http://www.scriptwell.net/correlation.htm
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| {{Refimprove|date=January 2008}}
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| [[Category:Pseudorandomness]]
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| [[Category:Binary sequences]]
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