0 IEE 2002
29 November 2001
Electronics Letters Online No: 20020256 D o l : 10.I049/el:20020256
4 5
6 7 8
= L(t)
Theorem: Given a Legendre sequence, a new binary sequence family S can be constructed, which concludes two subsets, M and N as follows:
References
3
+
Form (2): Ldec(t)= 1 L(t) for all t from 1 to P - 1 and L,,(t) for t = 0: when dec E QNRp.
Lemma 1 is self-evident and Lemma 2 follows directly from Lemma 1. The main theorem of this Letter is as follows.
E-mail:
[email protected]
2
= L(t), for all t from 0 to P - 1, when dec E QRP
Form (1): L,,(t)
Hong-Teuk Kim, Dae-Hyun Kim, Youngwoo Kwon and Kwang-Seok Seo (School of Electrical Engineering, Seoul National Universiv, Shillini-Dong, Kwanalc-Ku, Seoul I51-742, Korea)
1
Lemma 2: Let Ldcc(f) denote the decimation by dec of sequence L(t) with period P, then LcIec(t)only takes one of the following two forms:
ADDRICOS, C., BAHL, I.J., and GRIFFIN, E.L.:
‘C-band 6-bit GaAs monolithic phase shifter’, IEEE Trans. Microw. Theor?, Tech., 1985, 33, (12), pp. 1591-1596 LUCYSZYN, s., and ROBERTSON, I . D : ‘Two-octave bandwidth monolithic analogue phase shifter’, IEEE Microw. Guid. Wave Lett., 1992, 2, (8), pp. 343-345 BOIRE, D.C., and MARION, R.: ‘A high performance 6-18 GHz five bit MMIC phase shifter’. GaAs IC Symp. Dig., San Diego, Califomia, USA, 1995, pp. 267-270 LUCYSZYN, S., ROBERTSON, I.D., and AGHVAMI, H.: ‘24 GHz serrodyne frequency translator using a 360” analogue CPW MMIC phase shifter’, IEEE Microw. Guid. Wave Lett., 1994, 4, ( 3 ) , pp. 71-73 LUCYSZYN, S., ROBERTSON, I.D., and AGHVAMI, H.: ‘Monolithic analogue phase shifter and cascode FET amplifier using uniplanar techniques’. 24th European Microwave Conf. Proc., Cannes, France, 1994, pp. 554559 KIM, H.T., KO, W., KIM, D.H., KWON, Y., and SEO, K.S : ‘CPW MMIC coupler based on offset broadside air-gap coupling fabricated by standard airbridge process’, Electron. Lett., 2001,37, (6), pp. 358-359 LUCYSZYN, S., and ROBERTSON, 1.D.: ‘Synthesis techniques for high performance octave bandwidth 180” analogue phase shifters’, IEEE Trans. Microw. Theory Tech., 1992, 40, (4), pp. 731-740 CHUNG, Y., RYU, G.H., KIM, D.H., LEE, J.H, HONG, W.H., and SEO, K.S.: ‘Characterization of source-to-drain capacitance (CdJ effect of GaAs PHEMT for microwave and millimeter-wave switch’. IEEE MTT-S Dig., 2000, pp. 173-176
S=MUN
N=N’UL
. . . , P - l ) , k = l , 2 ( . . . ,( P - 1 ) / 2
M=(TkL,,(t)+L(r).t=O,l,
N’= (TkLL,,(t) + L ( t ) , t = 0 , 1, . . . ,P - I ] , k
= (P+ 1)/2, . . . , P - 1
where L is a Legendre sequence with prime period P, P = 3 (mod 4), Ldec denotes the decimation by dec of the sequence L , and denotes the operator which shifts vectors cyclically to the left by k place. The sequence family S has the following properties: (1) Each sequence within set S has the period P (2) Subset A4 and N have ( P - 1)/2 and ( P 1)/2 elements (sequences), respectively. ( 3 ) Each sequence within A4 and N has balance property when Ldec takes the form (1) or near balance property when Ldec takes the form (2).
+
Property (1) and (2) is obvious.
Proof of property (3): Obviously, sequence L within subset N is balanced; the following discussion excludes this sequence. Let x(t) be any sequence within subset M or N, then x(t) can be written as x(t) = TkL,,(t)
+L(t)
t = 0, 1, . . . , P
-
1
for some k from 1 to P - 1, and the difference between symbol 0 and 1 within x(t) can be expressed as
~ ( k=)
Pseudonoise codes constructed by Legendre sequence
P- I
(-
1)Tkb&)+L(r)
i=O
If L,/,, takes the first form, Le. Ldec(t)= L(t), then
Zhang Guohua and Zhou Quan
B(k) = RL(k) = - 1
A new family of binary sequences is constructed by Legendre sequence with period P, P = 3 (mod 4). The new signal set includes
two subsets consisting of (P- 1)/2 and ( P t 1)/2 codes with good balance property. A conjecture about the maximum over all (nontrivial) autocorrelation and crossconelation values of sequence contained within the two subsets is proposed.
Introdtiction: In code-division multiple access (CDMA) and spreadspectrum communication systems, pseudonoise binary codes are used as signature sequences and spreading codes. Desirable features of a family of binary codes for such applications include long period, low out-of-phase autocorrelation values, low crosscorrelation values, balance of symbols, large family size, and ease of implementation [l]. In this Letter we present a new family of sequences with the abovementioned properties, which is constructed by Legendre sequence. First, we give some definitions and lemmas. We use Z p to represent the set of remainders of all integers on division p where p is a prime integer, i.e. Zp= (0, 1, 2, . . , , p - I}. Z: represents the set of nonzero elements in Z,, i.e. @= { I , 2, . . . ,p - 1}. Definition 1: Let QRp be the set of quadratic residues, modp, i.e. those elements of Zp* which have square roots in @. Let QNRp be the set of quadratic nonresidues, modp, i.e. those elements of Zp* which do not have square roots in Z,*.
where RL denotes the autocorrelation of sequence L. If takes the second form, then P- I
B(k) =
( - ~ ) ~ l + L l ‘ + k ) l + L l ~ )+ (- l)LlO)+L(P-k)
t#P-k i=o
2m 150C
g- 100g
50
Dejinition 2: The Legendre sequence, L(t), of period P, P prime, is defined as follows:
L(0) = 0 . L(t) = 0 , t
E
QRp, L(t) = 1, t
E
-501
QNR,
0
37 6
100
150
200
250
correlation time
Lemma 1: Let xo, x1E QRp,y o , y , E QNRp. Then xOyo E QNRp, x o x l , Y O Y IE QRP.
50
Fig. 1 Autocorrelation of sequence (P = 263, dec = I , k= 26) within set M
ELECTRONICS LETTERS
11th April 2001
Vol. 38 No. 8
[4], which achieve the Welch bound, the new constructed sequence set has two important advantages. First, the sequence period is not constrained by 2", and thus the period selection is more flexible. For example, in all integers smaller than 1000, only eight integers can be used as sequence period in Bent or No cases, while more than 80 integers can be used in our new sequence case. Secondly, the newly presented sequences have better balance property than Gold [5] and No sequence, Table 1 shows the comparison of several different sequences.
250!
2ooi '
Table 1: Comparison of Gold, No, Bent and newly designed
501
sequences
-5OL
50
100
150
ZOO
Maximum
Range of
correlation
imbalance
250
correlation time Fig. 2 Crosscorrelation of two sequences (P = 263, dec = I , k = 26, within set M
soor I
design
pnme
Conclusion: We have constructed by Legendre sequence a new family of pseudonoise codes. Compared with the famous Gold, Bent and No sequences, our new codes have a more flexible period and similar or better balance property. In addition, the new sequences have a conjectured maximal correlation value, which is near-optimal with respect to the Welch bound.
0 IEE 2002 Electronics Letters Online No: 20020220
23 October 2001
References 1
400 -
It seems that the correlation property of set M or N is very good. Figs. 1-4 show the correlation property of sequences within subset M o r N. Furthermore, our experimental results enable US to ProDOse the following conjecture about the correlation property of subset M and N. Conjecture: For any arbitrary prime f: P = 3 (mod 4) and proper forms (Form (1) or Form (2)) which Ldrc takes, the maximum correlation magnitude R,,, (defined in [l]) of set A4 and N is given by .
R,,,
= 4[2&/4J
H. Bissessur, G. Charlet, w. Idler, C. Simonneau, L. Pierre, R. Dischler, c. De Barros and Tran
50 GHz-spaced 80 x 40 Gbit/s WDM transmission over 300 km of TeraLightTMfibre is demonstrated without polarisation demultiplexing, using the PSBT format. A record capacity of 3.2 Tbit/s is thus achieved over the C-band only, with a spectral efficiency of 0.8 bit/s/Hz.
+1
1 lth April 2002
s. Borne,
I
where 1x1 denotes the maximal integer not greater than x. The validity of this conjecture is verified for all prime P, p = 3 (mod 4) from 7 to 1031. Until now we have not found any counter example for this conjecture; therefore it is likely true. is near-optimal compared with the Welch The conjectured Lax bound P I . Although the correlation Property of the new sequence family is not as good as that of the Bent sequence [31 or No s w m c e
ELECTRONICS LE7TERS
and SONG, H.-Y.: 'New COnStrUCtion for families of binary sequences with optimal correlation properties', IEEE Truns. In$ Theov, 1997, IT-43, pp. 1596-1602
NO, J.-S., YANG, K., CHUNG, H.,
Introduction: Wavelength division multiplexing (WDM) transmission systems at 40 Gbit/s bit-rate per channel are becoming very attractive, due to potential cost reduction by decreasing the number of terminals and spares, and the possibility to implement systems with a high spectral efficiency. Therefore, several experiments have been carried out to achieve this goal: without polarisation demultiplexing (but with polarisation interleaving at the transmitter), 0.64 bit/s/Hz have been
Vol. 38 No. 8
377
