Editing Maximum length sequence (section)

==Properties of maximum length sequences==
MLS have the following properties, as formulated by [[Solomon Golomb]].<ref name="golumb">{{cite book |first=Solomon W. |last=Golomb |title=Shift register sequences |url=https://books.google.com/books?id=LqtMAAAAMAAJ |year=1967 |publisher=Holden-Day |isbn=0-89412-048-4}}</ref>

===Balance property===
The occurrence of 0 and 1 in the sequence should be approximately the same. More precisely, in a maximum length sequence of length <math>2^n-1</math> there are <math>2^{n-1}</math> ones and <math>2^{n-1}-1</math> zeros. The number of ones equals the number of zeros plus one, since the state containing only zeros cannot occur.

===Run property===
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|date=February 2018}}

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 property===
The circular [[autocorrelation]] of an MLS is a [[Kronecker delta]] function<ref>{{Cite book|url=https://books.google.com/books?id=Sq6uFqlHg1gC|title=Fundamentals of General Linear Acoustics|last1=Jacobsen|first1=Finn|last2=Juhl|first2=Peter Moller|date=2013-06-04|publisher=John Wiley & Sons|isbn=978-1118636176|language=en|quote=A maximum-length sequence is a binary sequence whose circular autocorrelation (except for a small DC-error) is a delta function.}}</ref><ref>{{Cite journal|last1=Sarwate|first1=D. V.|last2=Pursley|first2=M. B.|date=1980-05-01|title=Crosscorrelation properties of pseudorandom and related sequences|journal=Proceedings of the IEEE|volume=68|issue=5|pages=593–619|doi=10.1109/PROC.1980.11697|s2cid=6179951 |issn=0018-9219}}</ref> (with DC offset and time delay, depending on implementation).  For the ±1 convention, i.e., bit value 1 is assigned <math>s = +1</math> and bit value 0 <math>s = -1</math>, mapping XOR to the negative of the product:

<math>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}</math>

where <math>s^*</math> represents the complex conjugate and <math>[m+n]_N</math> represents a [[circular shift]].

The linear autocorrelation of an MLS approximates a Kronecker delta.