Theoretical and Approximate Derivation of Bit Error Rate in DS-CDMA Systems under Rician Fading Environment

Transcription

1 660 IEICE TRANS. FUNDAMENTALS, VOL.E8 A, NO. DECEMBER 999 PAPER Special Section on Spread Spectrum Techniques and Applications Theoretical and Approximate Derivation of Bit Error Rate in DS-CDMA Systems under Rician Fading Environment Fumihito SASAMORI, Student Member and Fumio TAKAHATA, Member SUMMARY The transmission quality in mobile wireless communications is affected by not only the thermal noise but also the multipass fading which changes drastically an amplitude and a phase of received signal. The paper proposes the theoretical and approximate methods for deriving an average bit error rate in DS-CDMA systems under the Rician fading environment on the assumption of the frequency non-selective fading, as parameters of the number of simultaneous access stations, the maximum Doppler frequency and so on. It is confirmed from the coincidence of theoretical and approximate results with simulation ones that the proposed approach is applicable to a variety of system parameters. key words: DS-CDMA, bit error rate, frequency non-selective fading, Rician fading. Introduction It is one of key themes in direct sequence code division multiple access DS-CDMA systems to evaluate an average bit error rate BER performance. In time division multiple access TDMA systems and frequency division multiple access FDMA systems, stations signals are completely isolated in time and frequency domains respectively, while those collide in both time and frequency domains in CDMA systems. The Gaussian approximate method [], [] has been generally used for calculating easily the average BER under the thermal noise environment. The transmission quality in mobile wireless communications is affected by not only the thermal noise but also the multipath fading which changes drastically an amplitude and a phase of the received signal. The multipath fading is classified into the frequency nonselective fading and the frequency selective fading. An attention has to be paid for system design to the kind of fadings, depending on system parameters. In general, the frequency non-selective fading and frequency selective fading could be considered and overcome in the narrow and wide band mobile communications, respectively. Several reports present methods for deriving BER under the frequency non-selective fading environment [3], [4]. But those methods assume the quasi-static fading environment that the receiving station doesn t move and the Doppler frequency shift can be ignored. Manuscript received April 9, 999. Manuscript revised June 6, 999. The authors are with the Graduate School of Science and Engineering, Waseda University, Tokyo, Japan. Taking account of the above situation, it would be important to derive the methods for giving BER under several fading environments with the Doppler frequency shift. We already derived the theoretical and approximate methods for giving an average BER under the Rayleigh fading environment in terms of the Doppler frequency shift on the assumption of frequency non-selective fading [5]. The paper proposes the theoretical and approximate derivation methods of an average BER under the Rician fading environment, which is featured by a basic model for line-of-sight transmission path such as indoor wireless and mobile satellite communications, on the assumption of frequency nonselective fading as parameters of the number of simultaneous access stations to the same frequency band, the maximum Doppler frequency and so on. Concretely speaking, we will derive the theoretical equation which is applicable to the system where any spread spectrum code and any band limitation filter are used, and the approximate equation which is applicable to the system where Gold codes are assigned to stations at random as spread spectrum SS code, the root cosine roll-off filter is used as the band limitation. It is confirmed from the coincidence of theoretical and approximate results with simulation ones that the proposed approach is applicable to a variety of system parameters. The model of DS-CDMA systems and the derivation method of the average BER are presented in Sect.. The average BER is evaluated with the comparison between results obtained by the derived equations and the computer simulation in Sect. 3.. Derivation of Average Bit Error Rate The system model will be shown to express the signal transmission in the Rician fading, and then the theoretical and approximate equations to give the average BER will be derived by introducing statistical characteristics of signals.. System Model Figure shows the model of DS-CDMA systems expressed in equivalent low-pass system, where K transmitting stations access simultaneously to the same frequency band on CDMA basis and each receiving station recovers a desired information signal by demultiplexing

2 SASAMORI and TAKAHATA: THEORETICAL AND APPROXIMATE DERIVATION OF BIT ERROR RATE 66 Fig. System model of DS-CDMA systems. Fig. Output signals from matched filter in station #i. the signals sent from all transmitting stations. Speaking in detail, the information signal b k t of each transmitting station #k k K with the form of binary data sequence is carried out the differential encoding, the spreading by SS code a k t and the band limitation in a modulation block. In a transmission block, the signal is added the delay t k and the phase offset θ k because respective stations access asynchronously to the system. In the Rician fading, there exists a dominant stationary non fading signal component such as a line-of-sight propagation path, and random multipath signal components arriving at different angles are superimposed on the stationary dominant signal. The SS signal of each transmitting station is fed to two paths to simulate the Rician fading. One is the dominant stationary signal s dk t, which we call here a direct wave. The other is the random multipath signal s rk t, which we call here a Rayleigh wave. And we call the signal s k t =s dk t+s rk t a Rician wave. The direct and Rayleigh waves from all transmitting stations are combined and added the thermal noise nt, and respective stations receive the combined signal zt = K k= s kt +nt. In a demodulation block, the received signal zt is carried out the band limitation, the code matching by means of the matched filter, the differential detection, and then the desired information signal b k t is demodulated. Other assumptions to the system configuration are as follows,. a period of SS code is equal to the bit period of information bit,. each signal s k t from each transmitting station is received with the same power at the receiver on an average, and has an equal Rician factor the ratio between the deterministic signal power and the variance of the multipath, 3. the receiver ideally recovers the clock timing of the direct wave s dk t, and 4. the modulation is based on the differential encoding binary phase shift keying DBPSK and the differential detection is used so as to overcome the phase shift caused by fading.. Theoretical Derivation of BER Respective stations receive the signal zt which consists of the direct and Rayleigh waves from all transmitting stations and the thermal noise. The theoretical equation to give the average BER is derived by introducing statistical characteristics of the received signal. The received signal zt is the same at all receiving stations and is given by zt = {s dk t+s rk t} + nt k= We consider the recovery of the desired information signal b i t in the receiving station #i. In a demodulation block, the signal zt is carried out the band limitation and the code matching by means of the matched filter. Let z i t be the signal which is obtained after filtering the received signal zt in the receiving station #i. Figure shows the kinds of signals included in the output signal z i t from the matched filter. As shown in the figure, the signal z i t is composed of the desired signal, the undesired signals and the thermal noise. The desired signal consists of the direct wave d di t and the Rayleigh wave d ri t related the signal s i t sent from the transmitting station #i, and the undesired signals consist of the direct waves u dj t and the Rayleigh waves u rj t related to the signals s j t sent from the transmitting stations #j j K, j i. In the next subsections, we will discuss the mean values and variances of the desired and undesired signals and the thermal noise shown in the figure. Since the output signal from the differential decoder is obtained by detecting a phase difference between the signal z i t t=mtb = z m at t = mt b and the previous signal z i t t=m Tb = z m at t =m T b, the output signal v m from the decoder at the decision timing t = mt b T b : bit period is given by [6] v m =Rez m zm = { zm + z m z m z m }. 4 A bit error occurs in BPSK when v m is minus in spite

3 66 IEICE TRANS. FUNDAMENTALS, VOL.E8 A, NO. DECEMBER 999 of successive information bits in in-phase or0 0 or when plus in spite of those bits in anti-phase 0or0. Here we define new parameters x and x as follows, x = z m + z m x = z m z m, 3 then Eq. is expressed as Fig. 3 Signal vectors of direct waves in space diagram. v m = 4 x x. 4 Comparing an amplitude of x with that of x, the BER P e is given by P e = Prob v m < 0 = Prob x < x. 5 Equation 5 can be concretely calculated by obtaining the mean values and variances of x and x based on statistical characteristics of the received signal. Since the signals z m and z m relate to the direct and Rayleigh waves from all transmitting stations and the thermal noise, the probability density functions PDFs of amplitudes of z m and z m follow the Rician distribution. In addition, since the PDFs of amplitudes of x and x are obtained from the addition and subtraction between z m and z m respectively, they follow the Rician distribution, too... Mean Values of x and x Since the mean values of the Rayleigh waves d ri t, u rj t and the thermal noise nt are equal to zero, the mean values of the direct waves d di t, u dj t are meaningful. Let A be an amplitude of the direct wave d di t in the desired signal, v ij and w ij be interference signal components from the undesired signal to the desired signal at t =m T b and t = mt b, respectively. They are normalized by A, and amplitudes of the direct waves u dj t in the undesired signals. Furthermore, let θ ij be a phase difference between the direct waves d di t, u dj t in the desired and undesired signals. Figure 3 shows the signal vectors of the direct waves d di t, u dj t in a space diagram at t = mt b. Considering that the mean values of the Rayleigh waves d ri t, u rj t in the desired and undesired signals equal to zero, the mean values of real and imaginary parts of z m are derived from the summation of cosine and sine components of the direct waves d di t, u dj t, respectively. Therefore the mean values of x and x are given by E[Re x ] = E[Re z m + z m ] =A+ w ij A cos θ ij +A+ = A{+ w ij + v ij cos θ ij } AI x v ij A cos θ ij E[Im x ] = E[Im z m + z m ] = w ij A sin θ ij + v ij A sin θ ij = A w ij + v ij sin θ ij AQ x E[Re x ] = E[Re z m z m ] = A+ w ij A cos θ ij A = A w ij v ij cos θ ij AI x E[Im x ] = E[Im z m z m ] = w ij A sin θ ij v ij A sin θ ij = A v ij A cos θ ij w ij v ij sin θ ij AQ x, 6 where E[ ] is an expect value of [ ], K is the number of simultaneous access stations, I x and Q x are the mean values of the real and imaginary parts of the variable x normalized by A, respectively... Variances of x and x Since the variances of the direct waves d di t, u dj t are equal to zero, the variances of the Rayleigh waves d ri t, u rj t and the thermal noise nt are meaningful. The real part of the variance is equal to the imaginary part of the variance because of the uniform distribution of phase shift by the Rayleigh fading and the thermal noise. Therefore the variances of x and x are given by E[{Re x AI x } ]=E[{Im x AQ x } ] = E[{Re z m AI zm } ] +E[{Re z m AI zm } ] +E[{Re z m AI zm }{Re z m AI zm }] = σd m + σu m + σd m + σu m +σn +σ dm σ dm J 0 πf D T b +σ um σ um J 0 πf D T b

4 SASAMORI and TAKAHATA: THEORETICAL AND APPROXIMATE DERIVATION OF BIT ERROR RATE 663 E[{Re x AI x } ]=E[{Im x AQ x } ] = E[{Re z m AI zm } ] +E[{Re z m AI zm } ] E[{Re z m AI zm }{Re z m AI zm }] = σd m + σu m + σd m + σu m +σn σ dm σ dm J 0 πf D T b σ um σ um J 0 πf D T b, 7 Fig. 4 Waveforms of direct wave d di t in desired signal. where σ d m and σ u m are variances of the Rayleigh wave d ri t in the desired signal and the Rayleigh waves u rj t in the undesired signals at t = m T b respectively m = m or m = m in Eq. 7, and σ n is the variance of the thermal noise. Above equations are derived from the facts that a co-variance becomes zero between the independent variables with the mean values of zero and that a time-correlation of the Rayleigh wave is given by J 0 πf D T b, where f D is the maximum Doppler frequency and J 0 x is a Bessel function of the first kind of zeroth order...3 Mean Values and Variances of Interference Signal Components Figures 4 and 5 illustrate the waveforms of the direct wave d di t in the desired signal and the direct wave u dj t j i in the undesired signal at the receiving station #i, respectively. The former is called an autocorrelation and the latter is called a cross-correlation. The waveforms are band-limited and are normalized by the maximum absolute value at the bit timing of the direct wave d di t in the desired signal. Since it is assumed that the receiver ideally recovers the clock timing of the direct wave d di t and as a result it precisely detects the peak of the auto-correlation, the amplitude A of the direct wave d di t and the variances σd m, σd m of the Rayleigh wave d ri t in the desired signal are time-invariant. Then the variances are expressed as σ d m = σ d m = σ d 8 On the other hand, since each transmitting station accesses asynchronously to the system and as a result the cross-correlation asynchronously collides the autocorrelation, the amplitudes v ij, w ij of the direct wave u dj t and the variances σu m, σu m of the Rayleigh wave u rj t j i in the undesired signal are timevariant. As shown in Fig. 5, the cross-correlation shows two different waveforms; an even-correlation in the case of successive two information bits of the undesired signal in in-phase and an odd-correlation in the case of those bits in anti-phase. Therefore it is necessary to consider four kinds of combinations of interference v ij at t =m T b and interference w ij at t = mt b from the direct wave u dj t in the undesired signal. In other words, the possible combination of {v ij, w ij } are ex- Fig. 5 Waveforms of direct wave u dj t in undesired signal. pressed by {even-correlation, even-correlation}, {evencorrelation, odd-correlation}, {odd-correlation, evencorrelation} and {odd-correlation, odd-correlation}. Furthermore, as shown in Fig. 4, there are two cases in which successive two information bits of the desired signal are in in-phase and in anti-phase. As a result, we have to consider eight kinds of combinations as the interference to the direct wave d di t in the desired signal from the direct wave u dj t in the undesired signal. Now we define v ij, w ij and σu m, σu m as v ijl n, w ijl n and σu n, ml σ u m l n respectively, where l is the number of the combination of interference l 8 and n is the sampling point in one bit duration T b, then v ijl n, w ijl n, σu n and ml σ u n m l are given by {v ij n,w ij n} = {α ij n,α ij n} {v ij n,w ij n} = {α ij n,β ij n} {v ij3 n,w ij3 n} = {β ij n,α ij n} {v ij4 n,w ij4 n} = {β ij n,β ij n} {v ij5 n,w ij5 n} = { α ij n,α ij n} {v ij6 n,w ij6 n} = { α ij n,β ij n} {v ij7 n,w ij7 n} = { β ij n,α ij n} {v ij8 n,w ij8 n} = { β ij n,β ij n} 9 σ u mln = σ u m l n = K w ijlnσ d vijl nσ d, l 8, 0 where α ij n and β ij n are the values of evencorrelation and odd-correlation at the sampling point n, respectively. The values in the cross-correlations are given by d di t u dj t t=nts i j, T s : sampling period, : convolution.

5 664 IEICE TRANS. FUNDAMENTALS, VOL.E8 A, NO. DECEMBER Average BER The PDFs of amplitudes of x and x follow the Rician distribution as mentioned above. Defining amplitudes of x and x as r and r respectively, the mean values B, C and variances σr, σ r ofr and r are obtained from Eq. 6 to 0. B = A Ix + Q x = C = A Ix + Q x = σ r =S 0+ + S i+ σ d +σ n σd ξi x + Q x σd ξi x + Q x σr =S 0 + S i σd +σn S 0+ =+J 0 πf D T b S 0 = J 0 πf D T b S i+ = {vijl n+w ijl n +v ijl nw ijl nj 0 πf D T b } S i = {vijl n+w ijl n v ijl nw ijl nj 0 πf D T b }, where ξ is the Rician factor which is given by ξ = A /σd. The PDFs of r and r are given by pr = r σr exp r + B Br σr I 0 σr pr = r σr exp r + C σ r I 0 Cr σ r. The BER P e in Eq. 5 shows the probability that the value of r is larger than that of r. The BER P e i, K in the receiving station #i when the number of simultaneous access stations is K, is given by P e i, K = Prob r <r = pr pr dr dr 0 r C B = Q, σ r + σr σ r + σr σr σ + r = Q exp B +C σr +σr I 0 BC σ r +σ r I x + Q x ξσ d /σ n S 0+ + S 0 + S i+ + S i σ d /σ n +4, I x + Q x ξσ d /σ n S 0+ + S 0 + S i+ + S i σ d /σ n +4 S 0 +S i σ d /σ n + S 0++S i+σ + d /σ n + exp I x +Q x +Ix +Q x ξσd /σ n S 0+ +S 0 +S i+ +S i σd /σ n+4 I I x + Q x Ix + Q x ξσd /σ n 0 S 0+ + S 0 + S i+ + S i σd /σ n +4, 3 where Qx, y is a Marcum Q-function and I 0 x isa modified Bessel function of the first kind of zeroth order. In the equation, the following integral formulas are utilized. exp p x I 0 axxdx y = a a p exp p Q p,py exp p x Qb, axi 0 cxxdx 0 = c bp {exp p p Q p + a, ac p p + a a c p + a exp p b } abc p + a I 0 p + a. 4 The term σd /σ n in Eq. 3 implies the average C/N of the Rayleigh wave, and then the average C/N of the Rician wave is related by C/N Rice =C/N Direct +C/N Rayleigh =ξ + C/N Rayleigh, 5 where C/N is equal to E b /N 0 in the BPSK, and the relation between C/N of the Rayleigh wave and E b /N 0 of the Rician wave is given by C/N Rayleigh = E b/n 0 Rice. 6 ξ + The values of v ijl n and w ijl n for calculating I x, Q x, I x, Q x and S i± in Eq. 3 vary depending on the kind of interference l and the sampling point n. In addition to that, the BER also vary depending on the receiving station #i. Therefore Eq. 3 has to be averaged by the variables of l, n and i. The average BER P e K as parameters of the number of simultaneous access stations K is given by P e K = 8N s K 8 N s i= l= n=0 P e i, K, 7 where N s is the number of sampling point in one bit duration and is given by the multiplication of the length of the SS code N and the number of sampling point in one chip duration S. The above theoretical equation to give the average BER is applicable to DS-CDMA systems where any spread spectrum code and any band limitation filter are

6 SASAMORI and TAKAHATA: THEORETICAL AND APPROXIMATE DERIVATION OF BIT ERROR RATE 665 used, though it takes a long computer time to calculate the average BER in the case of the large value of K and N..3 Approximate Derivation of BER The approximate equation to give the average BER easily is derived in DS-CDMA systems where Gold codes are assigned to stations at random as SS code and the root cosine roll-off filter is used as the band limitation. The derivation method is based on the assumption that the interference from the undesired signals is estimated as the Gaussian noise for reasons that the values of the cross-correlation between a pair of any Gold code take one of three values and the generation probabilities of the three values are known. The approximate equation is derived by utilizing the SNIR in which the interference from the undesired signals estimated as the Gaussian noise is added to the signal-to-noise ratio SNR of the desired signal []. The theoretical equation in the case where only one station K = accesses to the system is utilized to derive the approximate equation. The average BER in K = is derived by setting v ijl n and w ijl n to zero in I x, Q x, I x, Q x and S i± of Eq. 3, P e = exp ξσ d /σ n {+J 0πf DT b }σ d /σ n + { J 0πf DT b }σ d /σ n + + σd /σ n +, 8 where the characteristics Q0,y = exp y / and I 0 0 = are utilized. The SNIR is given by [] { K SNIR = α + N } 0, 9 N 4 E b where α is the roll-off factor of the root cosine roll-off filter. The relation among C/N of the Rayleigh wave, E b /N 0 of the Rician wave and SNIR of the Rician wave in the BPSK is given by C/N Rayleigh = E b/n 0 Rice ξ + = SNIR Rice. 0 ξ + The approximate equation to give the average BER is derived by inserting Eqs. 9 and 0 to 8. P e K = exp {+J 0πf DT b } SNIR ξ+ + + { J 0πf DT b } SNIR ξ+ + ξ SNIR ξ+ SNIR ξ+ +. There exists a disadvantage that the equation cannot approximate the average BER precisely in the case where the interference from the undesired signals cannot be estimated as the Gaussian noise in the region Table Modulation Detection SS code Length of SS code N Band limitation Maximum Doppler frequency normalized by bit rate f DT b Rician factor ξ Number of sample point in one chip duration S Number of sample point in one bit duration N s Phase difference θ ij Carrier frequency recovery Symbol timing recovery Table Length of SS code 3 63 System parameters. DBPSK differential detection Gold code 3 and 63 [chip] root cosine roll-off filter of α=.0, 0.5 Butterworth filter of sixth order 0 5,5 0 0, 3, 6, 0 [db] 4 N S random ideal ideal Primitive polynomials of M-sequence. Primitive polynomial f ax =x 5 + x + f b x =x 5 + x 4 + x 3 + x + f ax =x 6 + x + f b x =x 6 + x 5 + x + x + that the effects of the Rayleigh fading and the thermal noise are ignored in the high E b /N 0 and ξ and then the central limit theorem isn t satisfied. There is, however, an advantage that the approximate equation can calculate the average BER easily by inserting the values of E b /N 0, N, K, etc. to Eq.. 3. Evaluation of Average Bit Error Rate The theoretical and approximate equations are evaluated by comparing the results obtained by the equations with simulation ones. Tables and show the system parameters and primitive polynomials of M- sequence for generating Gold codes, respectively. 3. Fixed Assignment of Gold Codes to Stations It is assumed in the evaluation that Gold codes are assigned fixedly to stations. The Gold code {Z k,i } to be assigned to the station #k is obtained from the following exclusive-or between M-sequences {X i } and {Y i } which are generated from primitive polynomials f a x and f b x, respectively. Z k,i =X i Y i+k mod N i=,,,n. Figure 6 shows performances of E b /N 0 versus the average BER obtained by the theoretical equation and by the computer simulation as parameters of the length of Gold code, the number of simultaneous access stations, the maximum Doppler frequency normalized by the bit rate and so on. It is confirmed from the coincidence of theoretical results with simulation ones that

7 666 IEICE TRANS. FUNDAMENTALS, VOL.E8 A, NO. DECEMBER 999 the proposed theoretical equation is applicable to the system where Gold codes are assigned fixedly to stations. It is observed from Fig. 6 b that the average BER is hardly degraded by the Doppler frequency f D at the large Rician factor ξ because the Doppler frequency shift is almost ignored in the case where the power of the direct wave is much larger than that of the Rayleigh wave. On the other hand, the average BER is degraded by the increase of the Doppler frequency f D at the small Rician factor ξ, in particular at K =, because the existence of Rayleigh waves has a dominant factor to degrade the BER performance. Furthermore the average BER at K = becomes worse as compared with that at K = because the cross correlation between Gold codes degrades the BER performance considerably. Of course the increase of the Doppler frequency f D degrades the BER performance. Relative BER degradation due to the increase of the Doppler frequency f D at K = is seen to be smaller than that at K = because the average BER at K = is inherently worse than that at K =. It is observed from Fig. 6 c that the average BER in the system where the Butterworth filter is used as the band limitation is degraded as compared with that in the system where the root cosine roll-off filter is used because of the intersymbol interference in the Butterworth filter. Using the roll-off filter, the average BER with the roll-off factor α =.0 is slightly better than that with α =0.5 because the ripple of the filter impulse response with α =.0 is smaller than that with α =0.5 and then the interference due to the cross correlation is suppressed. However, as shown in Fig. 6 d which indicates the average BER at the small Rician factor ξ and the large Doppler frequency f D, there is no much difference among the average BERs with the root cosine roll-off filters α =.0 and α =0.5 and the Butterworth filter because the effect of the fading determines the BER performance and the interference due to the impulse response of the band limitation filter has a minor contribution to degrade the BER performance. a α =.0, f D T b =0 5, ξ = 0[dB] b N = 3, α =.0 c N = 3, f D T b =0 5, ξ = 0[dB] 3. Random Assignment of Gold Codes to Stations It is assumed in the evaluation that Gold codes are assigned to stations at random. The Gold code {Z k,i } to be assigned to the station #k is selected among the N + kinds of Gold codes at random, and the K kinds of Gold codes are used in the system. Figure 7 shows performances of E b /N 0 versus the average BER obtained by the theoretical equation, by the approximate equation and by the computer simulation. It is confirmed from the coincidence of theoretical results with simulation ones that the proposed theoretical equation is applicable to the system. It is found from Fig. 7 a that the approximate results coincide with simulation ones in the case of the d N = 3, f D T b =5 0, ξ =3[dB] Fig. 6 Average bit error rate fixed assignment. small Rician factor ξ the lines of the theoretical and approximate results overlap each other, while the difference between those results occurs in the case of the

8 SASAMORI and TAKAHATA: THEORETICAL AND APPROXIMATE DERIVATION OF BIT ERROR RATE 667 a N = 63, α =.0, f D T b =0 5 Fig. 8 Average bit error rate N = 047, α =.0, ξ = 3 [db]. figure that the average BER is little degraded by the Doppler frequency shift in the large K because the effect of the interference from the undesired signals has a larger contribution to the degradation of the average BER than the effect of the Doppler frequency shift. 4. Conclusion Fig. 7 b N = 3, K = Average bit error rate random assignment. large Rician factor ξ. The small Rician factor ξ means the large contribution of the Rayleigh fading to the BER performance, where it is known that the Rayleigh distribution is equal to the complex Gaussian distribution [3] and then the interference from the undesired signals can be estimated as the Gaussian noise irrespective of the number of stations K. On the other hand, the large Rician factor ξ means the small contribution of the Rayleigh fading and the interference among direct waves determines the BER performance. In that case, it is known that the central limit theorem isn t satisfied when the number of access stations K is small [7]. However the larger the number of stations K is, the less the difference between the approximate and simulation results is, because the central limit theorem is satisfied in the large K. The approximate equation, however, is an effective means to calculate the average BER easily and roughly in the system where Gold codes are assigned to stations at random. It is observed from Fig. 7 b that the effect of the Doppler frequency f D is small in the large Rician factor ξ comparing dashed lines with one-dotted lines according to the same comment for Fig. 6 b. It is observed from Fig. 7 that the average BER is improved in proportion to the increase of the Rician factor. Figure 8 shows the performance of E b /N 0 versus the average BER obtained by the approximate equation in the case of N = 047. It is observed from the We propose the theoretical and approximate derivation methods of the average BER in DS-CDMA systems under the Rician fading environment. The performance evaluation is carried out for the purpose of justifying the proposed derivation methods. It is confirmed from the coincidence of theoretical and approximate results with simulation ones that the proposed approach is applicable to a variety of system parameters. In addition to that, the average BER performances obtained by those methods are discussed briefly. References [] M.B. Pursley, Performance evaluation for phase-coded spread-spectrum multiple-access communication Part I: System analysis, IEEE Trans. Commun., vol.com-5, no.8, pp , Aug [] T. Sibata, M. Katayama, and A. Ogawa, Performance of asynchronous band-limited DS/SSMA systems, IEICE Trans. Commun., vol.e76-b, no.8, pp.9 98, Aug [3] T.S. Rappaport, Wireless communications principles and practices, pp.7 74, pp.84 88, IEEE PRESS, 996. [4] K. Tsukakoshi, A study on multiplexing transmission of multi-level modulation channels in DS-CDMA, IEICE Gen. Conf. 97, Japan, B-5-55, March 997. [5] F. Sasamori, Y. Yamane, and F. Takahata, Theoretical derivation of bit error rate performance in DS-CDMA systems under Rayleigh fading environments, Technical Report of IEICE, SST98-, SAT98-7, June 998. [6] M. Schwartz, W.R. Bennett, and S. Stein, Communication systems and techniques, pp , pp , McGraw- Hill Inc., 966. [7] R.K. Morrow, Jr. and J.S. Lehnert, Bit-to-bit error dependence in slotted DS/SSMA packet systems with random signature sequences, IEEE Trans. Commun., vol.com-37, no.0, pp.05 06, Oct. 989.

9 668 IEICE TRANS. FUNDAMENTALS, VOL.E8 A, NO. DECEMBER 999 Fumihito Sasamori received the B. Eng. and M. Eng. degrees from Waseda University, Tokyo in 994 and 996, respectively. He is currently pursuing the D. Eng. degree at the Major in Electronics, Information and Communication Engineering, Graduate School of Science and Engineering, Waseda University. His current interests are in spread spectrum system and digital satellite communications. Fumio Takahata received the B. Eng., M. Eng. and D. Eng. degrees from Waseda University, Tokyo in 97, 974 and 980, respectively. He joined the KDD Research and Development Laboratories, Tokyo in 974, where he was engaged in the research and development of future satellite communications systems. Since April 988, he has been a Professor in the School of Science and Engineering, Waseda University. His current interests are in digital transmission technologies for wireless communications.