Ternary Zero Correlation Zone Sequence Sets for Asynchronous DS-CDMA

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1 Communications and Network, 4, 6, 9-7 Published Online November 4 in SciRes. ernary Zero Correlation Zone Sequence Sets for Asynchronous DS-CDMA enattou Fassi, Ali Debbari, Abdelmalik aleb-ahmed elecommunications and Digital Signal Processing Laboratory, Dillali Liabes University of Sidi el Abbes, Sidi el Abbes, Algeria LAMIH UMR CNRS 853, University of Valenciennes and Hainaut-Cambresis (UVHC), le Mont Houy, France fassibenattou@yahoo.fr, adebari@yahoo.fr, abdelmalik.taleb-ahmed@univ-valenciennes.fr Received August 4; revised September 4; accepted 8 October 4 Copyright 4 by authors and Scientific Research Publishing Inc. his work is licensed under the Creative Commons Attribution International License (CC Y). Abstract In this paper we propose a new class of ternary Zero Correlation Zone (Z) sequence sets based on binary Z sequence sets construction. It is shown that the proposed ternary Z sequence sets can reach the upper bound on the Z sequences. he performance of the proposed sequences set in asynchronous Direct Sequence-Code Division Multiple Access (DS-CDMA) system is evaluated. In the simulation we used two types of channels: Additive White Gaussian Noise (AWGN) and frequency non-selective fading with AWGN noise. he proposed ternary Z sequence sets show better results, in term of it Error Rate (ER), than Hayashi s ternary Z sequence sets. eywords Hadamard Matrix, Zero Correlation Zone Sequences, Correlation, Asynchronous DS-CDMA, ER. Introduction In Code Division Multiple Access (CDMA) systems, the number of spreading sequences determines the number of users and their correlation properties have a significant effect on anti-interference performance of the system []. Different types of codes used in communications systems have been studied in order to reduce Multiple Access Interference (MAI) [] [3]. For an interference-free communication, spreading codes should have zero auto-correlation and zero cross-correlation functions at out-of-phase state. So, spreading sequences with good correlation properties can be used to improve the performance of CDMA systems [3]. One class of spreading sequences called Zero Correlation Zone (Z) sequences possesses good correlation properties but only in specific zones called Zero Correlation Zone (Z ). here are several intensive studies of CDMA systems using Z How to cite this paper: Fassi,., Debbari, A. and aleb-ahmed, A. (4) ernary Zero Correlation Zone Sequence Sets for Asynchronous DS-CDMA. Communications and Network, 6,

2 sequences sets [] [3]-[6]. Various classes of ternary Z sequences sets have been constructed [3] [4] and [7]-[3]. ernary Z sequences have the advantage over binary Z sequences that is, for a given sequence length, the set has longer Z lengths and more sequences, and we may employ such hardware in binary Z sequence sets system [3]. Any ternary Z sequences set Z ( L, M, Z ) could be characterized by the sequence length L, the number of sequences M and the zero correlation zone length Z. An optimal Z set is the one that provides the maximum number of codes for a given Z and sequences lengths. he proposed ternary Z sequences set with Z ( L, M, Z ) is derived from a binary Z sequence set with Z ( LM,, Z ). When compared with previous works on ternary Z sequence sets [3] [4] and [7]-[3], our proposed Z sequence set approaches optimality. he remainder of the paper is organized as follows. After a review of preliminary considerations in Section, the proposed design for sequence construction is explained in Section 3. Example of new Z sequence sets are presented in Section 4. he properties of the proposed sequence sets are explained in Section 5. In Section 6, we consider the performance of the proposed ternary Z sequence sets compared with those in [3] and [] for the asynchronous DS-CDMA system in both AWGN and nonselective fading with AWGN noise channels. At the end, we draw the concluding remarks.. Preliminaries.. Definition For a pair of sequences X and X v of length L, the aperiodic correlation function (ACF) ϕ as follows [4] [5]: L τ xi, x,( ), if τ < L v i+ τ i= ϕ ( ) ( τ ) = τ X, X < < v, if τ L ( ) ( ) X, Xv τ is defined L+ τ x,( ) xv,, i τ τ if L () i= he periodic correlation function (PCF) between X and X v at a lag τ is determined by [5] [6]: ( ) ( ) ( ) ( L ) ( ) (( L) L) X, Xv X, Xv X, Xv τ, θ τ = ϕ τ mod + ϕ τ mod () L τ, θ τ = x x and τ ( ) ( ) X,,( ) mod, X i v i+ v ( L) i= θ τ = θ τ (3) ( ) ( ) ( ) ( ) X, Xv Xv, X.. Definition A set of M sequences { X M } { X, X,, X } = = M is called zero correlation zone sequence set if the periodic correlation functions satisfy [5] [6]: and 3. Proposed Sequence Construction < τ, θ τ = (4) Z cz X X ( ) ( ), ( ) τ θ ( ) ( τ ) v, Z cz, = (5) X X he construction procedure of the new ternary sequence sets is presented. he construction is accomplished across the following three steps: Step : he th row of the Hadamard matrix H of order n is indicated by h = h,, h,,, hn,. A set of n sequences d, each of length n, is constructed as follows [7]: For < n, v

3 d + = h, h (6) d + = h, h (7) Step : For the first stage p =, and for a fixed integer value n, we can generate, based on the schema for n and { } n of n sequences as follows: sequence construction in [5], a series of sets { } = = oth sequences sets { } n and { } n are constructed from the sequences set { } n quences + and = = p + + of length L ( n) sequence pair ( d, d ) d =. A pair of se- = are constructed by applying the interleaving operation of a + + in Equations (6) and (7) [7] and a pair of sequences + and + are cond +, d + and from padding Z, which are zeros of structed by the interleaving operation of a sequence pair length, as follows: For < n, + = d +,, d +,, d +,, d +,,, d +,n, d +,n (8) + = d +,, d +,, d +,, d +,,, d +,n, d +,n (9) + = d +,, Zd, +,, Zd, +,, Zd, +,,, d +,n, Zd, +,n, Z () + = d +,, Z, d +,, Z, d +,, Z, d +,,, d +,n, Z, d +,n, Z () p+ p+ + he length of a pair sequences + and + in Equations () and () is L ( n n) member size of a pair sequence sets { } and { } is M = M = n. Step 3: For p >, we may recursively build a new series of sets { } n and { } n = = actual sets { } n and { } n respectively. Sets { } n and { } n are generated as follows: For < n, = = = = = + and the by interleaving + = +,, +,, +,, +,,, +,4n, +,4n () + = +,, +,, +,, +,,, +,4n, +,4n (3) + = +,, +,, +,,,, +,,4 ( ( )), + n + +,4 ( n( + ) ) (4) + = +,, +,, +,, +,,,,4 ( ( )), + n + +,4 ( n( + ) ) (5) he length of both sequences + and length of both sequences + and 4. Example of Construction ) For n = and p =, the { } 3 + in Equations () and (3) is equal to ( p + n) p+ p++ + is equal to ( n+ n). = is generated as follows [7]: [,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,] + = + = [,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, ] + = [,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, ] + = [,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,] Figure shows the periodic auto-correlation function (PACF) (, v) Figure shows the periodic cross-correlation function (PCCF) (, v) +. he PACF and PCCF confirm that { } ) For n = and p =, the proposed { } 3 = is a Z (3, 4, 4) sequence set. is generated as follows: [7] and the = given in Equation (3) of +, and given in Equation (3) of + with

4 + Figure. PACF of +. PCCF of + with + Figure. PCCF of + with +.,,,,,,,,,,,,,,,,,,,,,,,,,,,, + =,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, + =,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, + =,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, + =,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, Figure 3 shows the PACF of +, and Figure 4 shows the PCCF of + with +. is a Z (64, 4, ) sequences set. he PACF and PCCF confirm that { }

5 PACF of + Figure 3. PACF of +. PCCF of + with + Figure 4. PCCF of + with he Features of the Proposed Sequence p+ he inary Z sequence set with Z (,, ) ( p LM Z n, n, ) p+ p++ Z sequences [7]. he length of in Equations (4) and (5), equals ( L n n) of in Equations () and (). Let ( p + F = ) and ( p + Z p ) with Z (,, ) ( p + p + +,, p p + L M Z n n n ) ( FM ZpM, M, Z Zp) = is optimal or approach optimal binary = +, is twice that =, the proposed ternary Z sequence set = + + = + + is derived from the L, M, Z = FM, M, Z. If the obtained ternary Z sequence set is op- M = L Z + [9] []. If p = (the first iteration), the obtained sequence set is an optimal ternary Z sequence set. M ( Z ) Proof: Let + R =, we calculate the following ratio: L binary Z sequence set Z ( ) ( ) timal, it satisfies the ratio ( ) If p =,, R =. ( Z ) p p M R = = L + p+ p+ 3

6 If of the padded zeros tends to the infinite, the obtained Z sequences set (see able for n = ) is asymptotically optimal. M ( Z + ) + Proof: R = = and lim ( R) = L Noted that for this case, after spreading, the power of the symbol will decrease sharply, it is mandatory to compensate it, but this requirement increases Peak-to-Average Power Ratio (PAPR) and dynamic range of the transmitted signal [3]. From able we can see that the proposed sequence set in this paper can provide certain benefits. he length of sequences and Z will increase together while the number of sequences remains unchanged. In the asynchronous DS-CDMA system, the time delay is typically in some chips, and for this, we can increase the Z to reduce MAI, but the member size will be relatively petty. For a given member size M, we can find various sets of sequences with different lengths L and Z. As an example, assuming that M = 8 and = 3 (see able 3 for n = 4), we can draw upper bounds of our code performance and compare it with the Hayashi s approach. he Hayashi s ternary sequence sets Z- (( ) p+,, p+ n+ n ) in [] based on Hadamard matrix, the n member size of the sequence set is of theoretical upper bound. n + It is clear from able 3, the proposed construction is one of the better type in the constructions mentioned in able. Compared with the Hayashi s work, our proposed, in all cases, is optimal or approximate optimal Z p and the number ( ) able. he parameters of the proposed ternary Z sequence set and the ratio R. p L' Z R able. Comparison of three types of Z sequence sets. Constructed Z sequence sets Length of sequences Sequence element Number size Z Performance parameter R p+ p Z-( n, n, ) in [7] p n p + ( + ) inary n p p+ p+ p+ p+ Z- (( n+ ), n, ) in [] ( n ) + p+ ernary n n n + p+ p+ + p p+ p+ p+ + p Z-( n+ n, n, + ) in this paper n+ n ernary n p+ + p p p p+ + able 3. he parameters of two constructions of ternary Z sequence sets. Sequence sets Hayashi s sequence set Proposed sequence set L Z R

7 sequence set. Consequently it has a higher Z and better performance parameter R than that given by Hayashi s construction. 6. he Performance of the Asynchronous DS-CDMA System Using the Proposed ernary Z Sequence Set In this section, we consider the performance of the proposed ternary Z sequence set used as spreading sequences for the DS-CDMA system shared by M asynchronous users simultaneously. In order to show this performance, the ER of an asynchronous DS-CDMA system over a frequency non-selective fading channel with AWGN noise estimated in [8] [9] is used: ER P = Q N P γ ( γ ) σ A ( i) where P is the common received power, is the symbol duration, the Q function in Equation (6) is given by ( ) Q z y = e d y [], the term π z denote the faded component power from the user i and ( ) (6) N σ n = 4 is the variance for the AWGN noise, the term ( P γ 4) σ i A is the global (non-faded) interference MAI power for the required i-th user. Let γ =, the ER of an asynchronous DS-CDMA system over AWGN channels is: P ER = Q N + σ A ( i) 4 he MAI variance for the required i-th user can be calculated as [8] [9] σ A ( i) (7) P M = 3 m r = mi, L m (8) i where r mi, in Equation (8) is the interference term caused by all other users m except the user i. he term r mi, from Equation () can be written as [8] [9]: ϕτ = ϕ τ Let ( ) ( ) ( ) Xm, Xi L L ϕτ ( ) ϕτϕτ ( ) ( ) (9) = + + rmi, τ= L τ= L In Figure 5 and Figure 6 we compared the ER performance of the asynchronous DS-CDMA system employing Hayashi s Z (4, 8, 3), Hayashi s Z (, 8, ) and the proposed ternary Z sequence sets with parameter Z (3, 8, 3). At ER = 4 in Figure 5, the system using constructed Z can attain d and 6 d gains over the same system employing Hayashi s Z (4, 8, 3) and Hayashi s Z (, 8, ) in AWGN respectively. he ER performance, in Figure 6 was simulated assuming a frequency nonselective fading channel with AWGN noise with the common faded power ratio. sequence sets show better performance than Hayashi s Z sequence sets. At ER =.5 the system can attain d and d gains over the system employing Hayashi s Z (4, 8, 3) and Hayashi s Z (, 8, ) respectively. he amelioration over comparable Hayashi s ternary Z sequences is due to the correlation properties of the proposed ternary sequence set. 7. Conclusion γ =. As we can see in Figure 6, the proposed Z A new construction method to create ternary Z sequences set based on binary Z sequence sets was proposed in this paper. his ternary Z sequences set is either optimal or asymptotically optimal and their con- 5

8 Performance of Asynchronous DS-CDMA -5 ER Proposed Z (3,8,3) over AWGN Hayashi Z (4,8,3) over AWGN Hayashi Z (,8,) over AWGN SNR [d] Figure 5. ER performance of Asynchronous DS-CDMA for different Z over AWGN. - Performance of Asynchronous DS-CDMA Proposed Z (3,8,3) over a Fading AWGN Hayashi Z (4,8,3) over a Fading AWGN Hayashi Z (,8,) over a Fading AWGN - ER -3-4 Figure 6. ER performance of Asynchronous DS-CDMA for different Z over a Fading AWGN. struction is more flexible than other ternary Z constructions. he asynchronous DS-CDMA using the proposed ternary Z sequences shows better ER performance in both AWGN and frequency non-selective fading channel with AWGN noise. References SNR [d] [] Zhang, Z.Y., Ge, L.J., Yang, X.G., Zeng, F.X. and Xuan, G.X., AISS (3) Construction of Multiple Mutually Orthogonal Z Subsets for CDMA Communication Systems. 5, [] ai, Z.Q., Zhao, F., Wang, C.H. and Wang, C.-X., IJCS (3) Multiple Access Interference and Multipath Interference Analysis of Orthogonal Complementary Code-ased Ultra-Wideband Systems over Multipath Channels. 6

9 [3] Suk-Hoon, N. (5) On Z Sequences and Its Application to MC-DS-CDMA. Master of Science, Yonsei University, Seoul. [4] Donelan, H. and O Farrell,. () Large Families of ernary Sequences with Aperiodic Zero Correlation Zones for a MC-DS-CDMA System. Proc. of 3th. IEEE Intl. SPIMRC, 5, [5] arthikeyan, V. and Jeganathan Viayalakshmi, V., IJEEE (3) Analysis of Carrier Frequency Selective Offset Estimation Using Zero-IF and Z in MC-DS-CDMA., [6] Huang, J.C., Matsufui, S., Matsumoto,. and uroyanagi, N., IJCS () A Z-CDMA System with FS Modulation. 5, [7] Wu, D., Spasoevié, P. and Seskar, I. ernary Zero Correlation Zone Sequences for Multiple Code UW. WINLA, Rutgers University, [8] Wu, D., Spasoevié, P. and Seskar, I. (3) ernary Complementary Sets for Orthogonal Pulse ased UW. WIN- LA, Rutgers University, [9] Hayashi,. and Matsufu, S. (6) On Optimal Construction of wo Classes of Z Codes. IEICE RANS, 89, [] Hayashi,. (3) A Class of ernary Sequence Sets Having a Zero-Correlation Zone for Even and Odd Correlation Functions. Proc. IEEE ISI, 434. [] Xu, S. and Li, D. (3) ernary Complementary Orthogonal Sequences with Zero Correlation Window. Proc. IEEE PIMRC, 49, [] akatsukasa,., Matsufui, S., Watanabe, Y., uroyanagi, N. and Suehiro, N. () ernary Z Sequence Sets for Cellular CDMA System. IEICE RANS, 85, [3] Cha, J.S., Electronics Letters () Class of ernary Spreading Sequences with Zero Correlation Duration. 37, [4] Renghui, S., Xiaoqun, Z. and Li, L.Z. JCI () Research on Construction Method of Z Sequence Pairs Set. 6, 5-3. [5] Maeda,., anemoto, S. and Hayashi,. () A Novel Class of inary Zero-Correlation Zone Sequence Sets. Proc. IEEE ENCON, [6] Hayashi,. (9) A Class of Zero-Correlation Zone Sequence Set Using a Perfect Sequence. Signal Processing Letters, IEEE, 6, [7] Fassi,., Debbari, A., aleb-ahmed, A. and Dayoub, I., IOSR-JECE (3) A New Class of inary Zero Correlation Zone Sequence Sets. 5, 5-9. [8] Vladeanu, C. (5) Optimum Chaotic Quantized Sequences for Asynchronous DS-CDMA System. University of ucharest, Romania. [9] oulanger, C., Loubet, G., Lequepeys, J.R. and Ouvry, L. (999) Direct Sequence Spread Spectrum Sequences. raitement du Signal, 6, [] Proakis, J.G. () Digital Communications. 4th Edition, McGraw-Hill, New York. 7

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