State the properties of maximal length sequences.
Answers
maximum length sequences have the following properties, as formulated by Solomon Golomb.
Balance propertyEdit
The occurrence of 0 and 1 in the sequence should be approximately the same. More precisely, in a maximum length sequence of length {\displaystyle 2^{n}-1} there are {\displaystyle 2^{n-1}} ones and {\displaystyle 2^{n-1}-1} zeros. The number of ones equals the number of zeros plus one, since the state containing only zeros cannot occur.
Run propertyEdit
A "run" is a sub-sequence of consecutive "1"s or consecutive "0"s within the MLS concerned. The number of runs is the number of such sub-sequences.[vague]
Of all the "runs" (consisting of "1"s or "0"s) in the sequence :
One half of the runs are of length 1.
One quarter of the runs are of length 2.
One eighth of the runs are of length 3.
... etc. ...
Correlation propertyEdit
The circular autocorrelation of an MLS is a Kronecker delta function[5][6] (with DC offset and time delay, depending on implementation). For the ±1 convention:
{\displaystyle R(n)={\frac {1}{N}}\sum _{m=1}^{N}s[m]\,s^{*}[m+n]_{N}={\begin{cases}1&{\text{if }}n=0,\\-{\frac {1}{N}}&{\text{if }}0<n<N.\end{cases}}}
where {\displaystyle s^{*}} represents the complex conjugate and {\displaystyle [m+n]_{N}}represents a circular shift.
The linear autocorrelation of an MLS approximates a Kronecker delta.