# Approximate bit error probability analysis for DS-CDMA systems.

Introduction

Quality of service is measured in terms of BER. System model is described in section 2. Sections 3 and 4, deal with standard Gaussian approximation and simplified Gaussian approximation. In section 5 computationally simple method of probability analysis is discussed.

System Model If there are K active users in the system; kth user received signal at Base Station BS is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where [b.sub.k](t) and [c.sub.k](t) are data and spreading sequence respectively, [P.sub.k] is the received power, and [omega].sub.c] the carrier frequency. [[tau].sub.k] is the delay of user k relative to some reference user 0. [[phi].sub.k] is the carrier phase offset of user k relative to reference user 0. Since [[tau].sub.k]and [[phi].sub.k] are relative terms, we can define [[tau].sub.0] = 0 and [[phi].sub.k] = 0. The data signal [b.sub.k](t) and spreading sequence [c.sub.k](t) are sequences of unit amplitude (positive and negative) rectangular pulses. The data signal [b.sub.k](t) and [c.sub.k](t) are independent random processes with outcomes [+1, -1] with equal probability.

For detecting the desired user data (k=0) the decision statistics at receiver is given By

[Z.sub.0] = [I.sub.0] + [??] + [eta] (2)

where [I.sub.0] is contribution from the desired user (k =0) to decision statistic, [??] is multiple access interference from all co-channel users, and [eta] is the noise contribution. MAI contribution [??] is given by

[??] = [K-1.summation over(k=1)][I.sub.k] (3)

where [I.sub.k] are independent and identically distributed random variables given by

[I.sub.k]=[2([S.sup.2.sub.k]-[S.sub.k])(2B+1)+N][cos.sup.2][phi] (4)

where [S.sub.k] is the chip delay of [k.sup.th] interfering user signal from the nearest chip of the desired user signal chip (k =0). The quantity B in the equation (4) is the number of times the desired user spreading code changes it's state either from -1 to +1 or from +1 to -1, within a single data bit duration. If the user spreading code is random with odd number of chips N within one data bit duration, then B is binomially distributed with minimum value zero, average value of (N-1)/2 and maximum value of (N-1) [2,4,8].

Standard Gaussian Approximation

Standard Gaussian approximation can be used to determine the BER probability for DS-CDMA based systems. BER probability is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where Q(.) function is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

Simplified Improved Gaussian Approximation

The above discussion is valid only if the number of active users is large otherwise it leads poor accuracy. Simplified improved Gaussian approximation requires the evaluation of [mu] and variance [[sigma].sup.2] of decision statistics [??]. The probability of BER becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The mean [mu] and variance [[sigma].sup.2] are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

Since [mu] is proportional to K and [sigma] is proportional to [square root of K] the SIGA converges to SGA as K increases [5, 6].

Modified Siga

It can be noticed that [square root of 3[sigma]] can be greater than [mu] for smaller values of K. This results in imaginary argument to Q function and hence c can not be determined. Under such conditions c can be neglected without the loss of accuracy . If the quantity B, is set to a value (N-1)/2 without significant loss in accuracy, [[sigma].sup.2] can be evaluated as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

Many times, it would be required to find BER when all users' carriers and/or chips are synchronized. BS station of CDMA cellular communication systems modulates several messages on to a single carrier. In this case carriers of all messages will be aligned. Further BS can be configured to synchronize the chips of all messages. Hence four different situations can exist as given below

* chips aligned and carrier phase aligned

* chips aligned and random carrier phases

* random chip delays and carrier phases aligned

* random chip delays and random carrier phases.

Figure 1 shows the probability of BER for these four different situations, where as figure 2 shows the performance of the system for different values of B .

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

Conclusion

Gaussian approximation can be further simplified in section 5. When K increases the modified approximation approaches the Gaussian approximation. This modified approximation method is computationally simple and needs less CPU time; hence it can be used in real time systems. As shown in figure 1, synchronization between the carriers of different messages and/or chip delays affects the BER performance of CDMA systems.

References

 Ilkka Saarinen, Aarne Mammela, Pertii Jarvensivu, and Keijo Ruotsalainen, 2001, "Power Control in Feedback Communications Over a Fading Channel" IEEE Transactions on Vehicular Technology, vol. 50, no.5, pp. 1231 1240,.

 Jay E. Padgett, Christoph G. Gunther, and Takeshi Hattori, 1995, "Overview of Wireless Personal Communications." IEEE Communication Magazine, vol. 34, no. 01, pp. 28-41.

 Jerry D Gibson, 1997, The Mobile Communications Hand Book,Second Edition, IEEE Press, Chap. 17.

 J. G. Proakis, 1995, Digital Communications, 3rd Edition McGraw-Hill.

 Julian Chang, and Norman C. Beaulieu, 2002, "Accurate DS-CDMA Bit-Error probability calculation in Rayleigh fading," IEEE Transactions on Wireless Communications, vol. 01, no. 01, pp. 248-1252.

 Mingxi Fan, Ceilidh Hoffmann, and Kai-Yeung Siu, 2003, "Error Rate Analysis for Multirate DS-CDMA Transmission Schemes" IEEE Transactions on Communications vol. 51, no.11, pp. 1897-1910.

 Norman C. Beaulieu, and Julian Cheng, 2004, "Precise Error-Rate Analysis of Bandwidth-Efficient BPSK in Nakagami Fading and Cochannel Interference", IEEE Transactions on Communications vol. 52, no.1,, pp. 149-159.

 W. C. Y. Lee, 1991, "Overview of cellular CDMA." IEEE Transactions on Vehicular Technology, vol. 40, no. 02, pp. 291-302.

D.N. Kyatanavar (a), *,Rekha S. Patil (b), M.S. Patil (c) and R.G. Zope (d)

(a,c,d) S R E S' College of Engineering, Kopargaon-423 603 (M.S.) India Email: kyatanavar@yahoo.co.in, (b) J. J. Mugdum College of Engineering, Jaysingpur, Kolhapur (M.S.) India
Title Annotation: Printer friendly Cite/link Email Feedback direct sequence-code division multiple access Kyatanavar, D.N.; Patil, Rekha S.; Patil, M.S.; Zope, R.G. International Journal of Applied Engineering Research Report 1USA Jun 1, 2008 1388 A survey of voltage and current harmonics in various industries connected to a state electrical grid. Solution of linear equations and performance analysis on desktop computing systems. CDMA CDMA technology Combinatorial probabilities Geometric probabilities Probabilities Probability theory Telecommunications transmission errors