Cross-correlation of binary m and GMW arrays - Semantic Scholar

IEEE 7'h Int. Symp. on Spread-Spectrum Tech. & Appl., Prague, Czech Republic, Sept. 2-5, 2002

Cross-Correlation of Binary m and GMW Arrays A.Z.Tirkel, E.l.Krengel and TE.Hall Department of Mathematics and Statistics~ Monash University. PO Box 28M. Victoria 3800, Australia Ph j Fax +61-3-95922206, [email protected] KEDAH ELECTRONICS ENGINEERING, Moscow. Russia Ph +7-095-5300102 Fax +7-095-5348654, [email protected]

Abstract - Sequences and arrays with good auto and cross-correlation are useful in communications and

digital , ..'atermarking respectively. Two-valued autocorrelation is relatively easy to achieve. However, sets of pSfudonoise sequences with good cross-correlation arc small and difficult to find. Here, the cross-correlation for particular shifts of m-sequences and GM\V sequences is analysed by looking at these sequences as two-dimensional arrays. The particular shifts are natural in the context of the array decomposition. "square" arrays and arrays ·where the column length is 3 yield new results for m-sequences. The array interpretation also detel'mines the conditions under which the cross-correlation of GIV1\V sequences is identical to the cross-correlation of the m-sequences used in the construction of the GI\1_1. In ILl) the shift sequence) describing the order of cyclic shifts is derived. All m-sequences of the same length can be derived from the decimations of a ''prototype'' msequence. Some decimations result in the array representations of the two m-sequences having the same m-sequence colunms. but different shift sequences. Cross-correlation betv\.'een such sequences is obtained from matching entries in the shift sequence. tn 1l.2 \\'e describe the constructioll of GMW nrrays by USing the shift sequences from the msequence decomposition, by substituting the msequence colunms in the m-array by other sequences with the same autocorrelation and balance property as m-sequences. Cross-correlation of such arrays is identical to that of their parent m-arrays. In Section nI, we examine two palticular decompositions: the· "square" array and the array with column length 3. For the square array cross-correlations for purely veltical nnd purely horizontal shifts for a range of decimations arc found by using the shift sequence. This includes the location of previously unexplained crosscorrelafion peaks. Decomposition of m-sequences into arrays with columns of length 3 allows aU m-arrays to be described by the shift sequence. For many decimations, it is possible to deduce cross-correlation peaks for purely vertical shifts. The results in the two

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cases are npplied to GMW arrays in Section IV. II INTRODliCTION In th1s paper, we investigate m and GMW sequences of length 211_1 where n is composite. Such sequences c;:m be represented as arrays \vhose colunms are cyclic shifts of a single shorter balanced sequence, with ideal autoconelatioll or null columns.' Cross-correlation between arrays with the same types of columns becomes a summation of auto correlations of sequence columns, aUlocorrelations of null columns and crossc01Teiations between the two. The first is 2 valued, whilst the latter are both single valued. Cross-. correlation between the arrays becomes a simple addition of column correlations. 11.1 M-Sequence Decomposition Consider an m-sequence s of length 2H_1 where n is composite (n=kmj. where sj=d , ;=0,/ ..... 2"-2 and where (.t is any primitive element of GF(2'1). A binary sequence can be obtained by a trace mapping to the base field GF(2) given by

lSi

= Tr,"(a') . The

sequence can be mapped to subfields GF(2"') or GF(2') by

m-sequence

"'s, =Tr"~(a') or's, =Tr;(a c~a1§91be

-m- array of T= 2 -1

i

).

The

\\Titten as a two dimensional columns of length 2"'_1 or

2" -1 2 -1

T=-,-- colunms of length 2k_1. In either case, the columns are short m~sequences (of length 2"'-1 or i' -1)

or null columns.

·\im-~

S:!"_3 Sl"_2

Si""_2

Fig 1 The long m~sequence of length 2'1_1 appears along a diagonal of either arrny. A sequence in its characteristic phase is embedded down the right diagonal of a matrix as shown in Fig I. See, for example [1]. Actually, this matrix has 4 diagonals, representing 4 different m-sequences. whose

603

IEEE 7'" Int. Symp. on Spread-Spectrum Tech. & Appl., Prague, Czech Republic, Sept. 2-5, 2002

relationship is described in [2]. The leading entry in

each

column

is shown in Fig 2. R = (2'" _1)[(2'" _1)011'1-' Illlld(T)] and I/J is the

Euler Totient Function (3).

locations of the null COllUllllS form a difjer~nce set modulo T and that for any non-zero horizontal shift

Q= T -

2,,-2 -

2n - 4 null

columns match. The cross-

correlation is then:

(2m -1)(M +Q)-(T -.M -Q)=2"'(M +Q)-T.

Fig2. When the array colunms are of length 2tn_J an interpretation of the array is obtained by applying the trace mapping

m Si

= Tr~: (a

i

).

Then the columns are

either 0 or cyclic shifts of an Ill-sequence over GF(2) [4]. KnO\vledge of a single entry per column is sufficient to specify the colUlml because

11.2 GMW Array Construction \Vhen 1J=km, \,vith k and In prime, only two decompositions are possible and the theory above can be applied to In or k. ·When k or 111 is composite, more decompositions are possible. Given a shift sequence for an array decomposition of an m-sequence, the short l11-sequence COiUllIDS can be substituted by any other pseudonoise column of the same length, \'vith off-peak autocorrelation of -1 and balance of -1, without affecting the autocolTelation of the long sequence. Pseudo noise sequences derived by this technique are called Gordon. Mills, Welch (GMW) sequences [5-7]. Balance is required to ensure that cross-correlation of pseudo-noise columns with null columns yields -1, the same as tile out of phase autocorrelation of the pseudonoise colunms. In this context. the sequences are applied as exponents of a primitive root of unity: i.e. () is translated to (-l)Oalld 1 to

(-I)' and the

Tr;(a')E GF(2 m ) and each non-zero element of GF(2tn) occurs exactly once in each non-zero column. Therefore, each entry in a column is a power of a primitive element r of GF(2n~ and in fact, each nonzero column is a list of all nOll-zero elements of GF(2 r11 ) in ascending po\',/ers of '1, starting with some pmver f. The ordered list or exponents e lur all the msequence columns is called the shift sequence. The

shift sequence is e j =illdr(Tr~l(aRi»where ind(x) refers to the index (logarithm) function to base yand i refers to the column number. \Vherever the trace maps to 0, null columns occur, denoted by blank...;; in the shift sequence. The shift sequence obtained from the decomposition of an tn-sequence, together \vith the column sequence is sufficient to specify the long sequence and explains cross-correlation between 111sequences as follO\vs: The leading entry in column i of

the array over GF(2

m )

is Tr,,~m (a

Ri

)

A decimation of the long diagonal sequence by d, which preserves the array columns except rOf shifts, results in an array whose leading row is a decimation of the original array by dR=d(mod T). A corresponding . TI~II"" (a Rd,,) . entry .In tl'1e d· eClinated array IS

!" )" - )

autocorrelation is defined as

acf(r) =

SjS;+(

i=O

Cascaded GMW sequences are obtained fro111 msequences by several steps involving arrays of different sizes and substitl!tion of colunms at each step [8]. The issues of equivalence of GMVl sequences are addressed in [6,8].

Theorem 1 Cross-correlation between two m or GMW sequences with the same colwnn llature. when in alTa), [ann, but different shift sequences, can be obtained by counting lnatching entries in the shift sequence. Matching entries result in 2m_J \vhilst all mismatches result in 1. All m-sequences of length 21'-1 can be obtained from proper decimations d of one prototype sequence. When d=j mod(2f11-1) the column nature is the same. Here, the shift sequences are decimations of each other by d{1nod T) or its multiplicative inverse mod T.

III RESULTS T\\'o special cases arise when k=2.

II!.t "Square" Array

This array has 2in +l columns of length 2In_I. it has a single Ilull column at i=O and its shift sequence is palindromic [9]. Consider the original array shifted by k columns and { We calculate the cross-correlation values for purely rows. The tirst entry in colunm i of the decimated vertical and purely horizontal array shifts. (i) For purely vertical shifts the null coluITU1 matches array becomes: a ITTr",I,m ( a R{i-tkl) '. whilst matches between Ill-sequence colunms nllmber () or 2 The non-zerO columns match \lv'hen the respective first entries of the columns are equal i.e.: (it) For purely horizontal shifts by k colullllls the number of mnlching cohmms is the number of clTTr~'m (aR(hAI).= Tr km (al?d R i) Let M be the If! nt solutions to two Diophantine Equations: number of matching columns. It can be shown that the

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604

IEEE 7 rnt. Symp. on Spread~Spectrum Tech. & Appl., Prague, Czech Republic, Sept. 2-5, 2002 th

d mod ri ::::: -i - k where modulo T arithmetic applies. The tirst equation is obvious: column i is mapped onto column i by the decimation process and since there is no vertical shiftJ the columns must mntch. The second case arises because colunms i and -i have the same phase. The first of these equations can have many solutions when T has large factors. This leads to cross-correlations with high values, so that pairs related by these values of d should be avoided. It has bcen demonstrated that con·elations between m tGMW) sequences related by decimations or the roml J := r (2 I) + I where r is n

-

p

an integer less than p have high peaks for relative shifts of r(2" -1) where r~O,l, ....{J-I [10,11]. The p

lower p, the higher the peaks and the lower the frequency of occurrence. An extreme ex-ample, p=3 has been solved by Helleseth [12]. Here, we examine cases which have not been studied before and which have hil!h cross-correlations. Here we restrict ourse1ves~ to purely horizontal shifts and decirnations which are coluum preserving i.e. dc=i. This means that the shift sequences for these arrays are permutations. Therefore. when all purely horizontal shifts are considered, each m-sequence column must meet its counterpart once and its image once.

Examplel:

+ I Valid val Lies or rare 5 2.3,4. Consider r=2. Therefore d=1639 widl d,=i dr =I4. Collmms match \vhen: 14i~i+k (I) or 14i~ -i+k. (2) Equation (I) has solutions: 0.5,10..... 60 ror k~O, 1,6, 1I,16, ... ,61 for k=13etc where k is a multiple afl3. Equation (2) has solutions for k~O: i~O.13,26,39,52. The match at column 0 is conIDlon, so there are 4 extra columns in agreement due to the second equation i.e. 17 in total. Hence, the number of columns matching is 17,13.13,13,13. Therefore, for purely horizontal shifts .. the cross~correlation behveen the arrays exhibits a peak at characteristic phasc alignment of +]023 and 4 equally spaced peaks of +767. Equation (2) has 5 solutions Jor 12 horizontal shifts:5, 10.... ,60. The crosscon-elation for these is +255. For the remaining 48 horizontal shifts the cross-correlation is ~ 1, Example2: (.) n=20, p~5, d~209716 with d,~I, d,~"I". 6l6i~i+k (1) & OJ6i~ -i+k(2) Equation (1) has solutions: 0,5,10, .. ,1020 for k~O; 2.7.12./5.... ,/022 for k~205; 4,9,14,19, .... 1024 for k~410; 1,6.11,16, .... 1021 fork~615; 3,8,13,18, .. ,,1023 for k=820. Equation (2) has one solution for k=O: i"'"'"(). Therefore 205 COllUll11S match for k=(), resulting in a con·elation peak of 209715-820·=208895. \VhCll k is a multiple of 205. Equation (2) has one solution. n~I2

p~5

d

r(2" -I)

0-7803 -7627-7/02/$17.00(C)20021EEE

For these 4 peaks, 206 columns match. Thus cer In fact, Equation (2) has a single solution for all nOll-zero horizontal shifts becatlse (flJ7,1025)=1. Therdore one column matches for all ho'rizontal shifts between the peaks of ccf=· ~ 1. (b) 1t~20, (J~25, d~4l944 with d,.~ I, d,. ~944. peak~210738-819~2(!9919.

944i~i+k

(I)

944i~-i+k

(2)

Equalion (1) has solutions 0.25.50... ,,1000 for k~O ;12.37.61, ",,1012 for k~41 etc for multiples thereof There are 25 peaks with at least 41 columns matching in each. Equation (2) has 4 new solutions k=t): 205,410, 615, 820. The peak at k=O has 45 matching colunms and ccf=+45V55. For the 20 peaks where k is not divisible by 5 Equation (2) has no solutions, so the total number of matching columns remains at 41 and ccf-=--+40959. For the remaining 4 peaks Equation (2) has 5 solutions. none of \>;'hich are solutions to Equation (I). For these peaks a total of 4f1 columns match, so ccf~46(}7Y. In facl, Equation (2) has 5 solutions for all k \\'hich are multiples of 5: so there are 200 equally spaced minor peaks due to 5 column matches, with ccf::c+4095. The remaining horizontal shifts result in no columns mntching and ccf= -1025. These results apply equally to the long sequences embedded along Ihe diagonals of the arrays, This technique allows a CDMA designer to avoid decimations or phase shifts which lead to high crosscorrelations. In this case, the highest of these is abollt 0,25 of the autocolTelation peak! The analysis applies to higher values ofp, which is not restricted to primes, as shown in Example 2(b). As p increases, there is more likelihood of the predicted peaks being exceeded. Theorem 2 n Let x be an m·sequence of length 2 -1. where n =mk, m>2. k>f. Let z be a GMW sequence Clfsame length obtained from the two-dimensional an-ay of the sequence x with colunm length 2111-1 by substitution of columns by another balanced sequence with ideal autocorrelation and the same shifts. Let y and H' be m and GMW sequences respectively, connected with x and z by decimation dEl mod 2m_J. Then crosscorrelation between x and y is equal to the cr05Scorrelation bet"\veen z: and w. Consequence: Let n=mlkr=m2k:. Let gcd(d, 2!l_l}=/, mod 2ml - I and dEE] mod 21112 -1. Let x be an m-sequence of length 2"-1 and a. b be GMW sequences of the same length constructed from the shift sequence of m-sequence x expressed as arrays of

dE]

column lengths 2 U11 -1 and 2111: -1 respectively. Then C(j(.\'.x,I}=ccFu. ud~('(f(b, h,Il. Example Lei 11=18 and d=r(2'I-f)/3+1, r=1, 2. Define {GMW p/'} and {GM\V)/} as sets of non-equivalent classes of GMW sequences with parameters m=6 and 9. The power of the each class is equal 7776. From [6.13] ,ve have that sets {GMW1~t-} and {GMW\~?} contain 11 (lnd 239 classes respectively. Moreover. these sets have only onc common class. Let

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IEEE 71h Int. Symp. on Spread-Spectrum Tech. & AppL, Prague, Czech Republic, Sept. 2-5, 2002

OE {GMWI'~('} and bE IGMWI~'i}. Then the cross correlation spectrum cc/(a,adJ is the same as cross cOlTclation spectrum cC;/(b,b d). i.e. 6-level [12]. IlI.2 Columns of length 3. When an In sequence is embedded along a diagonal of an array of size 3xT~ the columns of length 3 are cyclic shifts or the only m-sequence or nulls. T Jnd 3 must be relatively prime, so n t;: 3 . Consider the three phases {A.B,C} of the m-sequence colulTIns to occur 1I..J • nB, nc times respectively. The rows of the array then have weights of n.t+ fiB, IIA+ llc, I1B _ lie respectively. In [14] it is shov.ll that the weight of (n,k) codewords is at most two valued. If nA+ nl/= n,4+ ne, then, nll = !lc. The general array autocorrelation behaviour is summarised in the Table 1 Lemma I The weight distribution satisfies two independent

n,.t+2n e =2 11 - 2

equations:

J/~+211{~=

( I)

2]sequences, We show that in many cases the crosscorrelation of certain classes of pseudo noise correliltion of GM'N sequences is independent of this sequences", Radio Engineering, No.6, 8·13, 20(0) substitution. [12] T. Helleseth, "Some results nbout cross· REFERENCES con-elation function between two maximal linear [I] F.J, MacWilliams, N ..f.A, Sloane, "PseudoSequences" Discrete /vfathematics, VoL 16, 209Random Sequences and Arrays'" Proceedings of 232, 1976. rhe IEEE. Oec.1976, 64 (12), 1715-1729. [13] J-I-I. Kim and H- Y. Song. "Existence Of Cyclic [2] AZ.Tirkcl, T.E.Hall, C.F.Osbomc, "A New Cia" Hndamard Dift"erence Sets And Its Relation To of Spreading Sequences" fSSSTA '98 Sun City Binary Sequences With Ideal Correlation". South Africa, Vol I, 46-50.0.H. Green. Journal 0/ Communic(ftions and Networks, vol. 1, "Structural Propcliics of Pseudorandom Arrays and No.1, 14-18. March, 1999. Volumes and Their Related Sequence~;". lEE IU.McEliece, "Weights of [14] LO.Balllllert, Proceedillgs - E, 132, (3), 133-145, May 1985. [rreducible cyclic codes" il1/ormation & Control [3] D.K Green. "Stnlcrural Properties of 1972.20,158-175. Pseudorandom "-frays and V01LUllCS and Their NOll-zt'm Hmizontal Shift & ;JflY vertical

Zero Horizomal Shift

T - 2 2(2k-ll =T-2

Matching null columns Null Columns overlying M·st'quenct'~

overlying

l\l·~eCJuen\.·es

:Vk,