# Correlation property for sequences

Complex-valued $n$- periodic sequences (i.e. sequences $a = ( a _ {i} ) _ {i = 0 } ^ \infty$ with $a _ {i + n } = a _ {i}$, $a _ {i} \in \mathbf C$) with "good" correlation properties have many applications in signal processing (spread spectrum and code division multiple access communication systems, see [a10]). A good survey is [a9]. Periodic sequences are recurring or shift register sequences, see [a2].

If $a$ and $b$ are $n$- periodic sequences, one defines $C _ {a,b } ( t ) = \sum _ {i = 0 } ^ {n - 1 } a _ {i} { {b _ {i + t } } bar }$. These numbers are called the periodic correlation coefficients (in case $a = b$, periodic autocorrelation coefficients). Sometimes also the aperiodic correlation $A _ {a,b } ( t ) = \sum _ {i = 0 } ^ {n - 1 - t } a _ {i} { {b _ {i + t } } bar }$ is of interest. The odd correlation of $( a _ {i} )$ and $( b _ {i} )$ is $O _ {a,b } ( t ) = A _ {x,y } ( t ) - A _ {x,y } ( t - n )$, $t = 1 \dots n$.

The goal is the design of large sets ${\mathcal S} = \{ a ^ {( 1 ) } \dots a ^ {( m ) } \}$ of $n$- periodic sequences such that

$$m ( {\mathcal S} ) = \max \left \{ { \left | {C _ {a ^ {( i ) } ,b ^ {( j ) } } ( t ) } \right | } :\right .$$

$$\left . {t = 0 \dots n - 1, i,j = 1 \dots s, t \neq 0 \textrm{ if } i = j } \right \}$$

(or the respective value for the aperiodic correlation) is small. Sequences with $C _ {a,a } ( t ) = 0$, $t = 1 \dots n - 1$, are called perfect. Perfect sequences whose entries are $t$ th roots of unity ( $t > 2$) are known for many periods $n$, but no example of a $\pm 1$- sequence with period $n > 4$ is known (the perfect sequence conjecture: there are no perfect binary sequences with $n > 4$). In the aperiodic case, the $\pm 1$- sequences with odd period $n$ and aperiodic correlation coefficients $0$ and $\pm 1$( Barker sequences) have been classified [a12]. If $n$ is even, the existence of an $n$- periodic Barker sequence implies the existence of an $n$- periodic perfect $\pm 1$- sequence.

If one assumes that the sequences are normalized, i.e. $C _ {a ^ {( i ) } ,a ^ {( i ) } } = n$, then it is known that for $k \geq 0$,

$$m ( {\mathcal S} ) ^ {2k } \geq { \frac{1}{nm ( nm - 1 ) } } \left \{ { \frac{n ^ {2k } ( nm ) ^ {2} }{\left ( \begin{array}{c} {k + n - 1 } \\ {n - 1 } \end{array} \right ) } } - mn ^ {2k + 1 } \right \}$$

and, if the entries of the sequences are just $\pm 1$:

$$m ( {\mathcal S} ) ^ {2} > ( 2k + 1 ) ( n - k ) + { \frac{k ( k + 1 ) }{2} } - { \frac{2 ^ {k} n ^ {2k + 1 } }{m ( 2k ) ! \left ( \begin{array}{c} n \\ k \end{array} \right ) } }$$

for $0 \leq k < { {2n } / 5 }$. If the entries of the sequences are $t$ th roots of unity ( $t > 2$), then the bound is

$$m ( {\mathcal S} ) > { \frac{k + 1 }{2} } ( 2n - k ) - { \frac{2 ^ {k} n ^ {2k + 1 } }{m ( k! ) ^ {2} \left ( \begin{array}{c} {2n } \\ k \end{array} \right ) } } , k \geq 0.$$

The first bound is due to L.R. Welch, the second two bounds are due to V.M. Sidel'nikov. In all cases, $k$ is an integer. Some improvements of these bounds are known. Similar bounds hold for the maximum odd correlation coefficient [a7].

Important classes of sequences are derived from so-called $m$- sequences $( a _ {i} ) = ( { \mathop{\rm trace} } ( \alpha ^ {i} ) )$, where $\alpha$ is a primitive element of the finite field ${ \mathop{\rm GF} } ( q ^ {e} )$ and ${ \mathop{\rm trace} } ( x ) = \sum _ {i = 0 } ^ {e - 1 } x ^ {q ^ {i} }$. The sequence $( a _ {i} )$ is defined over a finite field. The complex-valued sequences corresponding to such finite field sequences are $( \chi ( a _ {i} ) )$, where $\chi$ is a homomorphism from the additive group of the field into the multiplicative group of $\mathbf C$. The $m$- sequences yield complex sequences with autocorrelation coefficients $- 1$. The $s$- decimation of an $n$- periodic sequence is $a _ {s \cdot i ( { \mathop{\rm mod} } n ) }$. Systematic investigations of the correlation properties between $m$- sequences and their decimations can be found in [a4] and [a11]. The $m$- sequences are recurring sequences of maximum period length. Their good autocorrelation properties show that they have good "random" properties (pseudo-noise sequences).

In many cases, variations of the construction of $m$- sequences yield sets of sequences with optimal correlation properties with respect to the Welch and Sidel'nikov bounds: The cases $m \approx n$ and $m \approx \sqrt n$ have been investigated intensively. Regarding binary sequences, both the Gold sequences ( $m = n + 2 = 2 ^ {e} + 1$) and the Kasami sequences ( $m = \sqrt {n + 1 } = 2 ^ {e/2 }$) are asymptotically optimal. (One says that a family ${\mathcal S} _ {n}$ of sets of $n$- periodic sequences is asymptotically optimal if ${\lim\limits } _ {n \rightarrow \infty } { {m ( {\mathcal S} _ {n} ) } / {\sqrt n } } = b$, is best possible. If $| { {\mathcal S} _ {n} } | \approx n$, the lower bounds show that $b \geq \sqrt 2$ for binary and $b \geq 1$ for arbitrary complex valued sequences. For $| { {\mathcal S} _ {n} } | \approx \sqrt n$, one has $b \geq 1$.) Sequences whose entries are $t$ th roots of unity ( $t \neq 2$) and which are optimal have been constructed ( $m \approx n$, $t$ prime [a5]; $m \approx n$, $t = 4$[a1]). The latter construction is of particular interest in view of its connection with Kerdock and Preparata codes (cf. Kerdock and Preparata codes) and their linearity as codes over $\mathbf Z _ {4}$, see [a3]. More sequences with good correlation properties where $t$ is not a prime number can be constructed. In these constructions, sequences over Galois rings instead of finite fields are converted via homomorphisms $\chi$ into complex-valued sequences. A systematic investigation of shift register sequences over rings can be found in [a6].

Other important classes of sequences with good correlation properties are bent function sequences, [a8].

How to Cite This Entry:
Correlation property for sequences. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Correlation_property_for_sequences&oldid=46527
This article was adapted from an original article by D. JungnickelA. Pott (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article