NEW BCH-DERIVED SEQUENCES FOR CDMA SYSTEMS Cesaltina N. E. Ricardo Escola Náutica Infante D. Henrique * and FCT *** *Av. Bonneville Franco, Paço D’Arcos, 2780 Oeiras, Portugal. E-mail:
[email protected] José A. L. Inácio Escola Náutica Infante D. Henriqu* e and INESC E-mail:
[email protected] José António B. Gerald Instituto Superior Técnico** and INESC **R. Alves Redol, 9, 2º 1000 – 029 Lisboa, Portugal. E-mail:
[email protected] Manuel D. Ortigueira Faculdade de Ciências e Tecnologia (FCT) *** ***Campus da FCT da Universidade Nova de Lisboa, Quinta da Torre 2825 – 114 Monte da Caparica, Portugal. E-mail:
[email protected] Abstract - New BCH-derived PN-EB (pseudo-noise even balanced) sequences suitable to be used in CDMA systems are presented in this paper. It is assumed a new definition for Processing Gain, which better accounts for the system performance regarding the narrow band noise rejection, and it is shown how to obtain high processing gain values, namely, by using zero mean spreading signals in channels with selective noise. The new sequences have low autocorrelation levels, exist in large numbers and can provide higher processing gain with selective noise.
I. INTRODUCTION Here, we will focus our attention into Direct Sequence Spread Spectrum (DS-SS) communication system. We shall not be concerned with the PSK modulation features, but only with the spreading effect and its benefits. So, we will consider henceforth a base-band DS-SS system, as shown in figure 1. In this system, the signal b(t) is a pseudo-noise (PN) binary signal composed by rectangular pulses (the socalled chips), which are usually generated by a linear shift register generator with a chip rate Rc, that is much greater than the bit rate Rb of the information signal x(t). In this way, the signal transmitted has a bandwidth that is much greater than the bandwidth of the signal, x(t). This constitutes the called spectral spreading effect. At the receiver, the received signal is dispreading by b(t) (see figure 1) to recover the original signal x(t).
Note that, the signals with spread spectrum modulation have two important characteristics: they provide low probability of intercept and reduced interference in a CDMA environment, due to the randomness properties of the PN sequences [1][2]. Also, these properties have a strong influence on the performance (measured by the processing gain) of the spread spectrum systems, as will see later. This paper is fundamentally dedicated to the study of a new family of even sequences, the BCH-derived pseudonoise even balanced (PN-EB) sequences, which have a very superior performance than that obtained with the classic odd sequences of Gold [1][2], BCH [1], Gold-Like [1][2], large set of Kasami and m-sequences [1][2]. The PN-EB sequences constitute a new class of sequences which is specifically attractive in frequency-selective channels. Next, we show how to improve the processing gain, we refer the methods to derive the new BCH-derived sequences, compare the new sequences with other classic sequences (inclusive the odd sequences above-mentioned) and present the conclusions.
II. IMPROVING THE PROCESSING GAIN In fig. 1 (shown at the top of next page) it is represented an ideal base-band DS-SS communication system, where x(t) is the information signal, with bit rate Rb, and b(t) is the periodic PN sequence, with chip rate Rc>>Rb. The low-pass filter has the same bandwidth Bx of x(t). Later, we’ll also
n(t)
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x(t)+ n (t)
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x(t) + n (t) f
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b(t) Figure 1 Ideal Base-band DS-SS System.
assume that b(t), the spread spectrum signal, is formed by N pulses of duration τ at each period T. Here, the performance of the system depends only on the spread noise power because, ideally, we do recover the original signal x(t). In this case, the signal power with (Ss) or without (So) spreading, is the same at the output system, that is Ss = So. As the noise output power depends on the spreading, the output signal to noise ratio with, and without spreading are different. So, it makes sense to analyse the performance of the system by means of its Processing Gain (PG), defining it as the quotient between the output signal to noise ratio with spreading, Ss/Ns, and without spreading, So/No, according to [3]: PG = (Ss/Ns)/(So/No).
(1)
Using Ss = So in (1), we will obtain the final result: PG =
No . Ns
frequencies of 2π/T. For a base-band band-limited noise, if T is low enough the spectral replicas have no aliasing. This happens when T≤1/(Bx+Bn). Here, there will be only one replica of the noise signal, Gn(ω), located at the zero frequency. In this situation, the ratio between the noise power, before and after spreading, inside the band Bx will be given by: PG =
1 1 = 2 τ r A 0 ( ) [0 ] A(0) 2 T
where r0= τ/T, is the Fourier coefficient of zero order of the periodic signal b(t) with period T (being bn = rn A(n), its Fourier coefficients). A(n) is the DFT (Discrete Fourier Transform) of the amplitude of sequence ai, i = 0,1,2,…,N-1. If T is greater than 1/(Bn+ Bx) we have aliasing inside the reference band, and we can show that the PG will result in [3]:
(2)
Remark that this definition of PG is different from the usual, in which the PG is defined as the quotient between the spread signal and original signal bandwidths [4][5]. Next, to compute the PG defined in (2), we will do the analysis of the spreading effect over the band-limited noise. The case of the non band-limited noise has been analysed in [3]. In this situation, it is easy to prove that PG ≈ 1 and, therefore, the spreading has no effect on the white noise. So, it is useless. In the case of interest of the band-limited noise, let us consider any noise signal n(t), with power spectrum Gn(ω) and bandwidth Bn ≤ Bx. After spreading, the power spectrum of the noise signal ns(t), Gs(ω), is formed by a sequence of repetitions of Gn(ω) located at multiple
(3)
PG =
Bs Bx
(4)
which is the classic PG definition [4][5]. However, the result stated in (3) is very interesting since it allows us to increase the PG. In fact, in an ideal situation, using a spreading signal with a zero mean value, we have PG = ∞. This is rather strange, but not impossible to conceive. In the case of using an m-sequence [1] or a similar oddbalanced sequence, there are two ways of obtaining such situation: a) to increase the period of the odd sequence adding one symbol in order to make the number of ones and zeros equal; or,
Here, we use the two new methods presented in [6] and [7], based on the above referred option a), in order to generate new sequences with zero mean value, and consequently, may allow higher processing gain. The methods are called “Ranging Criterion” (RC) [6] and “Generators Ranging Criterion” (GRC) [7]. Next, we analyse the results obtained when RC and GRC are applied to the odd balanced BCH sequences [1]. Also, we compare the PG obtained with new even BCH sequences generated by application of the RC and GRC methods, and the PG obtained with others classic sequences.
III. SIMULATION RESULTS Table I presents the results obtained with RC and GRC criteria applied to a small set of odd balanced Dual-BCH sequences [1] of degree n=6. We can verify that both methods allow a high number of sequences with low peak correlation levels when compared with the Even BCH sequences with the extra zero insert at the end of sequence period. We can also note that, the GRC gives slightly better results in this case. Figures 2 and 3 show the results obtained for the processing gain as a function of the period, for the new and classic spreading sequences, namely, the odd sequences of Gold [1], large set of Kasami sequences [1], BCH [1], Goldlike [1] and m-sequences [1]; and the even sequences of Walsh/Hadamard [5] and Manchester (obtained from the msequences with Manchester pulses), both having zero mean value as the new even sequences BCH with GRC. All the simulated sequences are balanced excepting the odd BCH and Gold-Like sequences. Table I The Results Obtained with RC and GRC Applied to Odd BCH Sequences of Degree N = 6
In the simulations, we used the model of base-band DSSS system of the fig. 1, and a PN signal with a chip rate Rc = 400 Kchips/s. We also assumed a base-band noise signal with bandwidth Bn equal to the bandwidth signal, that is Bn = Bx = 1 KHz. The processing gain was calculated using the result (2). In these conditions, the aliasing occurs when the sequence period has a number of chip pulses that is superior to 200. Figure 2 shows that the Gold, Kasami, m-sequences and the non-balanced set of odd sequences of BCH and GoldLike have a low processing gain, which is worst to the nonbalanced group of BCH and Gold-Like sequences. However, we expect similar results to the processing gain when the BCH and Gold-like sequences are balanced. In figure 3, we can observe that all the sequences with zero mean value, inclusively the even BCH sequences generated by the GRC, present a very high processing gain compared to the odd sequences, in case of no aliasing. This confirms our previous results discussed above. Relatively to the RC, the results obtained are identical to those obtained with GRC, and therefore, we don’t include them here.
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b) to use a Manchester pulse instead of a rectangular one, paying all the double speed requirements needed. For this reason, this method was not chosen here.
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Number of Sequences by Correlation Level Maximum Correlation levels
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Figure 2 Performance of Classic Odd Sequence.
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theoretical results and represent a high motivation for the study of new even spreading sequences suitable for SS systems.
350 Manchester Hadamard BCH with GRC
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V. ACKNOWLEDGMENT
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The authors acknowledge at the Foundation for Science and Technology (FCT)- “Programa de Apoio `a Comunidade Científica”- for their financial contribution that makes possible this paper to participate in the MWSCAS03 conference.
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VI. REFERENCES
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Figure 3 Performance of Classic Even Sequences and Even Optimized BCH Sequences (with GRC).
Another aspect which concerns the generating process of the even BCH sequences is related with their facility of implementation in hardware, once, they can be generated either by a linear shift register generator (as the odd sequences) and adding the extra zero symbol, or by a cyclically read memory table.
IV. CONCLUSIONS In this paper, we have analysed several sequences, which due to the fact that they present good correlation properties (excepting the Hadamard sequences) and exist in a high number make them useful to use in CDMA systems. The results obtained for processing gain gave an indication of the degree of interference reduction that these sequences can provide in the receiver. Here, we can see that the classic odd Gold, Kasami and m-sequences have identical processing gain, but these values are very inferior to those presented by the even balanced sequences, inclusively to the new even BCH sequences generated by the GRC, in conditions of the narrow band noise. These results are important in frequency-selective channels, where the classical odd sequences present a minor efficiency. To note that, the results already obtained confirm the
[1] Sarwate, D., V., and Pursley, B., Michael, “Crosscorrelation Properties of Pseudorandom Sequences and Related Sequences”, Proceeding of the IEEE, Vol. 68, No. 5, May 1980. [2] Dinan, E., H., and Jabbari, B., “Spreading Codes for Direct Sequence and Wideband CDMA Cellular Networks”, IEEE Comm. Magazine, September 1998. [3] Ortigueira, M., D., Gerald, J., A., and Inácio, J., A., L.,“Higher Processing Gains with DS Spread Spectrum”, Anais do “XV Simpósio Brasileiro de Telecomunicações, September 1997, pp 207-210. [4] Cooper, G., R., and McGillen, C., D., “Modern Communication and Spread Spectrum”, McGraw-Hill International, 1986. [5] Proakis, J, G., “Digital Communications”, McGraw-Hill International, 1995. [6] Inácio, J., A., L., Gerald, J., A, Ortigueira, M., D., “New PN Even Balanced (PN-EB) Sequences for High Processing Gain DS-SS Systems”, Proceeding of “42nd Midwest Symposium on Circuits and Systems - MWSCAS’99”, Las Cruces, New Mexico, USA, August 1999. [7] Inácio, J., A., L., Gerald, J., A., and Ortigueira, M., D., “Design of New PN Even Balanced (PN-EB) Sequences for High Processing Gain DSSS Systems”, Proceeding of “The International Conference on Signal Processing Applications and Techonology-ISCPAT’99”, Orlando, Florida, U.S.A, November 1999.
