Circulant Matrix Representation of PN-sequences with Ideal Autocorrelation Property
aa r X i v : . [ c s . I T ] D ec Circulant Matrix Representation of PN-sequenceswith Ideal Autocorrelation Property
Mohammad J. Khojasteh, Morteza H. Shoreh, and Jawad A. Salehi,
Fellow, IEEE
Optical Networks Research LaboratoryDepartment of Electrical EngineeringSharif University of Technology,Tehran, IranEmail: [email protected]
Abstract —In this paper, we investigate PN-sequences with idealautocorrelation property and the consequences of this propertyon the number of + s and − s and run structure of sequences.We begin by discussing and surveying about the length of PN-sequences with ideal autocorrelation property. From our discus-sion and survey we introduce circulant matrix representation ofPN-sequence. Through circulant matrix representation we obtainsystem of non-linear equations that lead to ideal autocorrelationproperty. Rewriting PN-sequence and its autocorrelation prop-erty in { , } leads to a definition based on Hamming weight andHamming distance and hence we can easily prove some resultson the PN-sequences with ideal autocorrelation property. Index Terms —PN-sequence, ideal autocorrelation property,balance property, run structure, circulant matrix representation.
I. I
NTRODUCTION P SEUDO noise sequences (PN-sequences) are codes thatare considered to have correlation and spectrum propertiessimilar to random sequences, although they are determinis-tically generated. There are many versions of PN-sequenceswith different definitions, approaches and applications such as,maximal-length sequences (m-sequences) [1], Gold codes [2],zero correlation zone sequences (ZCZ) [3], etc. In general,m-sequences are among the most important PN-sequencessince they satisfy randomness postulates stated by Golomb[4], namely, ideal autocorrelation property , balance property ,and run property . In further work by Golomb he makesthe following conjecture [4], which is still considered open:“ The only binary sequences satisfying the three randomnesspostulates are m-sequences. ” [4].The correlation between all non-zero cyclic shifts of an m-sequence is almost zero (ideal autocorrelation property) [5], sothey can be used as sequences with excellent autocorrelationfunction. Sequences with ideal autocorrelation property arein one-to-one correspondence with Paley-Hadamard differencesets [6]. A general algorithm for constructing these classes ofsequences for any arbitrary length n is not known so far.Golomb states another conjecture on the existence of Paley-Hadamard difference sets that is if n , the length of Paley-Hadamard difference sets, is equal to 4 k +
3, then it shouldbe either a prime number, or n must be the product of twinprimes or it should be in the form of 2 k −
1, where k is a Part of this paper was supported by Iran National Science Foundation(INSF). positive integer [7]. To the best of our knowledge when n is aprime number only Legendre sequences [8] and sextic residueconstruction [9] are known. The other known sequences withideal autocorrelation property are; Jacobi symbol [10] for n = p ( p + ) , and m-sequences [1], Gordon-Mills-Welch (GMW)sequences [11] and miscellaneous instances [12] for n = k − n = n = n = n = k − k ≤
10, and proposeda few more conjectures on the general construction of thesesequences and their corresponding difference sets.In many applications generalizing the length of PN-sequences is critical such as in spectrum fragmented cognitiveradio networks [14,15], where the sequences should have awide range of lengths because of the number of availablesub-carriers differ in various conditions. Hence in many ad-vanced communications systems, codes with various lengthsare needed.In generalizing the length of PN-sequences we begin byproposing the circulant matrix representation of PN-sequences.The idea of using circulant matrix representation to constructa desired sequence was first used by Alem and Salehi in[16] in order to represent Optical Orthogonal Code (OOC).In [16] the search space is spectrally classified using circulantmatrix representation of OOCs, followed by a group actionthat introduces an efficient partitioning algorithm.The rest of this paper is organized as follows; in Section II,circulant matrix representation of PN-sequences is proposed.In Section III, based on circulant matrices representation, asystem of n non-linear equations is proposed that can beused to justify ideal autocorrelation property of PN-sequences.Then a new perspective arises by transferring circulant matrixof PN-sequences to { , } domain which leads to a betterunderstanding of these sequences discussed in Section IV.The run structure of the desired sequences are investigatedin Section V. Finally, Section VI, summarizes the results andconcludes the paper. II. C
IRCULANT M ATRIX R EPRESENTATION OF
PN-
SEQUENCES
Lets denote a PN-sequence via a codeword, x =( x , x , . . . , x n − ) . In most technical literature a codeword x is said to have ideal autocorrelation property if it has thefollowing autocorrelation function [4,13] R x ( t ) = (cid:26) n for t ≡ n − otherwise (1)where R x ( t ) is defined as R x ( t ) = n − (cid:229) l = x l x l ⊕ t . (2)and ⊕ is n-module addition.Herein, we recognize that in bipolar codewords, ±
1s av-erage out each other in order to construct an impulse shapeautocorrelation function [17]. In general PN-sequences withideal autocorrelation property are similar to OOCs, sinceboth have cyclic structure with cyclic ideal autocorrelationproperty. The idea of using outer product matrix to design anew searching algorithm to obtain OOC codewords was firstproposed in [18] by Charmchi and Salehi, where the authorsattempt, successfully, remove the bottleneck of designing andgenerating OOCs with certain code lengths. In [16], in orderto develop search algorithm in designing OOCs the authors doan in depth search for finding appropriate types of matricesto representing the characteristics of OOCs. In the followingdefinitions, the circulant matrix representation of PN-sequenceis introduced, as in [16], whereby displaying all possible cyclicshifts of a codeword in a circulant matrix.
Definition 1:
The circulant matrix representation of everycodeword x = ( x , x , . . . , x n − ) as a binary PN-sequence ( x l ∈{± } for 0 ≤ l ≤ n −
1) is defined as follows A x = A ( x , x ,..., x n − ) = x x . . . x n − x n − x . . . x n − ... ... . . . ... x x . . . x . (3)Every row of a circulant matrix is a cyclic shift of it’s aboverow [19]. From (1), (2) and (3) it becomes evident that thecondition of ideal autocorrelation for x = ( x , x , . . . , x n − ) andits circulant matrix A x is presented as follows; A x A Tx = nI n + E n = A ( n , − ,..., − ) (4)where I n represents the identity matrix of order n and if J n denotes an n × n all-ones matrix (every element of J n is equalto 1) then E n = − J n + I n . (5) Example 1: If x = ( − , − , + ) (m-sequence of length 3) then A x = − − + + − − − + − (6) and A x A Tx = + − − − + − − − + = I + ( − J + I ) (7)III. P ROPERTIES OF C IRCULANT M ATRICES AND THECORRESPONDING N ON -L INEAR S YSTEM OF E QUATIONS
The properties of circulant matrices are well known andeasily derived in [20]. The matrix in (3) has eigenvectors, andeigenvalues that are as follows; v m = √ n ( , e − j p mn , . . . , e − j p m ( n − ) n ) T (8) l x m = n − (cid:229) l = x l e − j p mln (9)where, m = , , . . . , n − U n is an n × n matrix that has the eigenvectors as columnsplaced in order (Fourier unitary matrix) and Y = diag ( l x m ) then A x = U n Y x U ∗ n . Also matrices that have this eigenvectormatrix are circulant [21].In order to proceed further we need one more prop-erty about circulant matrix. If x = ( x , x , . . . , x n − ) and y =( y , y , . . . , y n − ) then A x A y = A y A x = U n Y U ∗ n (10)where, Y = diag ( l x m l y m ) and A x A y is also circulant matrix. If y = ( x , x n − , . . . , x ) , then A x A Tx = A x A y (11)So by (10) A x A Tx = U n Y U ∗ n (12)and l x m × l y m calculated in (15). From (4) and (12) we have U n Y U ∗ n = nI n + E n (13)Since U n is unitary matrix ( UU ∗ = I ) so Y = U ∗ n ( nI n + E n ) U n = nI n + U ∗ n E n U n Y − nI n = U ∗ n E n U n (14) Y − nI n (the left hand side of (14)) is obtained as in (16), and(17).There is a fact about orthogonality of the complex expo-nentials [20] n − (cid:229) m = e j p mln = (cid:26) n l mod n = otherwise . (18)So if n is a prime number then we can easily rewrite theright hand side of (14) by substituting E n from (5) U ∗ n E n U n = U ∗ n ( − J n + I n ) U n = I n − U ∗ n J n U n (19) l x m l y m = ( x + x e − j p mn + x e − j p m ( ) n + ··· + x n − e − j p m ( n − ) n )( x + x n − e − j p mn + x n − e − j p m ( ) n + ··· + x e − j p m ( n − ) n )= x + x + ··· + x n − + (cid:229) l > r x l x r cos ( p mn ( l − r )) (15) Y − nI n = n − (cid:229) l = x l + (cid:229) l > r x l x r − n ... n − (cid:229) l = x l + (cid:229) l > r x l x r cos ( p n ( l − r )) − n ... ... n − (cid:229) l = x l + (cid:229) l > r x l x r cos ( p ( n − ) n ( l − r )) − n (16) = (cid:229) l > r x l x r ...
00 2 (cid:229) l > r x l x r cos ( p n ( l − r )) ... ... (cid:229) l > r x l x r cos ( p ( n − ) n ( l − r )) (17) thus, Y − nI n = − n . . .
00 1 . . . . . . (20)which leads to the following system of non-linear equations (cid:229) l > r x l x r = − n (cid:229) l > r x l x r cos ( p n ( l − r )) = . (cid:229) l > r x l x r cos ( p ( n − ) n ( l − r )) = . ( n + ) / (cid:229) l > r x l x r = − n (cid:229) l > r x l x r cos ( p n ( l − r )) = . (cid:229) l > r x l x r cos ( p ( n − ) n ( l − r )) = . ( n − (cid:229) l = x l ) = (cid:229) l > r x l x r + n − (cid:229) l = x l = n − (cid:229) l = x l = ± Corollary:
The ideal autocorrelation property leads tobalance property.In order to find sequences with ideal autocorrelation prop-erty, we need to search balanced {± } n and find codewordssatisfying equations in (22). Example 2 : As an example for n = d i where i = , . . . , d = x x + x x + x x + x x + x x + x x d = x x + x x + x x + x x + x x d = x x + x x + x x + x x d = x x + x x + x x d = x x + x x d = x x (24)reduce to; cos ( p ) . . . cos ( × p ) ... ... ... cos ( × p ) . . . cos ( × p ) d ... d = . . (25)Due to the property of cosine function, the first, second, andthird columns and rows of above 6 × cos ( p ) cos ( p ) cos ( p ) cos ( p ) cos ( p ) cos ( p ) cos ( p ) cos ( p ) cos ( p ) d + d d + d d + d = . . . (26)Multiplying the inverse of the 3 × d + d = x x + x x + x x + x x + x x + x x + x x = − d + d = x x + x x + x x + x x + x x + x x + x x = − d + d = x x + x x + x x + x x + x x + x x + x x = − ?? ) and ( ?? ) in balance n -tuples is suffi-cient for finding sequences with ideal autocorrelation property.On the other hand, this equations are the multiplication ofcodeword with first, second and third circular shift, respec-tively. Corollary:
As expected the sequences with ideal autocor-relation property are solutions to the following non-linearequation system in balanced n -tuples of {± } n − (cid:229) l = x l x l ⊕ = − n − (cid:229) l = x l x l ⊕ n − = − RANSFORMATION TO D OMAIN OF { , } In this section, we investigate PN-sequences by transferringthe {± } to { , } , and then discuss the corresponding con-sequences. If we define the following mapping; q : {− , } n → { , } n ( x ′ , . . . , x ′ i , . . . , x ′ n − ) = q ( x , . . . , x i , . . . , x n − )= ( − x , . . . , − x i , . . . , − x n ) (29)Then the autocorrelation function of x can be written asfollows [13,22]; R x ( t ) = n − (cid:229) l = x l x l ⊕ t = n − (cid:229) l = ( − ) x ′ l + x ′ l ⊕ t (30) = n − w ( x ′ ⊕ T t ( x ′ )) where w ( x ′ ) denotes the Hamming weight of x ′ , and T t represents t cyclic shift to the left. Hence w ( x ′ ⊕ T t ( x ′ )) = (cid:26) t ≡ n n + otherwise (31)thus, every two different rows of A x ′ in n + columns havedifferent value and in n − columns have the same value. If x =( x , x , . . . , x n − ) satisfies ideal autocorrelation property, thenthe sequence y = ( y , y , . . . , y n − ) = ( − x , − x , . . . , − x n − ) also satisfies this property. So without loss of generalitysuppose n − (cid:229) i = x i = −
1. Hence in the columns of every twodifferent rows of A x ′ , the ( , ) pairs appears once more than ( , ) pairs. Eventually there are n − pairs of ( , ) in columnsof every two different rows of A x ′ . Example 3: If x = ( − , − , − , , − , , ) , then x ′ =( , , , , , , ) and T x ′ = ( , , , , , , ) have four pairsof ( , ) , two pairs of ( , ) and one pair of ( , ) in theircolumns.From the above discussion the following results can beobtained. Corollary 1:
There is no sequences with ideal autocorrela-tion property of the length 2 k or 4 k + Corollary 2:
The ideal autocorrelation property is given by A x ′ A Tx ′ = n + I n + n + ( J n − I n )= A ( n + , n + ,..., n + ) (32) corollary 3: A PN-sequence of length n with ideal Au-tocorrelation can be seen as a family of codewords, { , } n , TABLE IPN-
SEQUENCES WITH IDEAL AUTOCORRELATION PROPERTY OF LENGTHLESS THAN
31n Sequence Type3 ( 1, 1,-1) m-sequence7 (-1,-1,-1,1,-1, 1, 1) m-sequence(-1,-1,-1, 1, 1,-1, 1) m-sequence11 (-1,-1,-1, 1,-1,-1, 1,-1, 1, 1, 1) Legendre(-1,-1,-1, 1 -1, 1, 1,-1, 1, 1, 1) Legendre15 (-1,-1,-1,-1, 1, 1, 1,-1, 1, 1,-1,-1, 1,-1, 1) m-sequence(-1,-1,-1,-1, 1,-1, 1,-1,-1, 1, 1,-1, 1, 1, 1) m-sequence19 (-1,-1,-1,-1, 1,-1, 1,-1, 1, 1, 1, 1,-1,-1, 1,-1,-1, 1, 1) Legendre(-1,-1,-1,-1, 1,-1, 1,-1, 1, 1, 1, 1,-1,-1, 1, 1,-1, 1, 1) Legendre23 (1,1,1,1,-1,1,-1,1,1,-1,-1,1,1,-1,-1,1,-1,1,-1,-1,-1,-1,-1) Legendre(1,-1,1,-1,-1,1,1,-1,-1,1,1,-1,1,-1,1,1,1,1,-1,-1,-1,-1,-1) Legendre weighting n + that are cyclic shifts of each other with Ham-ming distances equals to n + amongst each other.V. R UN S TRUCTURE
Consider the codeword x = ( x , x , . . . , x n − ) , a run of length f is a block of consecutive 1s or −
1s in codeword that is notcontained in a larger block of 1s or − R f . Furthermore, let N ( R f ) to denote the number of the runsof length f . The codeword x has the run property [4], if ⌊ n f + ⌋ ≤ N ( R f ) ≤ ⌈ n f + ⌉ . (33)The ideal autocorrelation property and the run property areknown to be independent for more than few decades until in2009 Cai [22] by thinking about autocorrelation run by runinstead of symbol by symbol proved that these two propertiesare related. The main result of his work can be presented inthis relation [22] R x ( t ) = n − tg − (cid:229) f + f + ... + f l < t ( − ) l ( t − i ) N ( R f R f . . . R f l ) (34)where i = f + f + . . . + f l , g is the total number of runsand R f , R f , . . . , R f s represent consecutive runs of lengths f , f , . . . , f s in x .Two special cases that can be obtained easily and wouldgive us some understanding of run structure are R x ( ) = n − g and R x ( ) = n − g + N ( R ) . Therefore, sequences with idealautocorrelation property have n + number of runs in which n + number of them are of length one. With this in mind,it may be true that the only sequences with ideal autocorre-lation property that satisfy (33) are m-sequences (Golomb’sconjecture about m-sequences) but all the sequences that haveideal autocorrelation property are not too far from satisfyingthe conditions in (33). Example 4: If n =
11 then (33) implies that 1 ≤ N ( R ) ≤
3, 1 ≤ N ( R ) ≤
2, and 0 ≤ N ( R f ) ≤ f = , . . . ,
10. The codeword x =( − , − , − , , − , − , , − , , , ) , which has the idealautocorrelation property follows (38) in all cases except f = VI. C
ONCLUSION
We investigated PN-sequences with ideal autocorrelationproperty and the consequence of this property on the numberof +
1s and −
1s and run structure of sequences. A new per-spective was introduced using circulant matrix representationof PN-sequences. We derived a system of non-linear equationswhich led to ideal autocorrelation property from this point ofview. Rewriting PN-sequence and its autocorrelation propertyin { , } led in a definition based on Hamming weight andHamming distance and easily proved a number of results onPN-sequences with ideal autocorrelation property.R EFERENCES[1] S. W. Golomb, “Shift-Register Sequences.”
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