Diversity Performance of Multi-Code Spread Spectrum CDMA System Sami A. Khorbotly and Okechukwu C. Ugweje Department of Electrical and Computer Engineering The University of Akron, Akron, OH 44325-3904 Tel: (330) 972-7168, Fax: (330) 972-6487, E-mail:
[email protected] Abstract— In this paper, multi-code spread spectrum CDMA system is analyzed. The transmitter and receiver needed for this system are described and its performance is evaluated for different parameters and different diversity techniques. The channel is assumed to be a multipath slowly fading Nakagami channel. The performance of the system is evaluated in terms of its average bit error rate and is compared to that of the conventional single-code spread spectrum CDMA system. The performance of the system is obtained for three diversity schemes namely, selection diversity, equal gain combining and maximal ratio combining. The effect of changing the number of substreams, users, diversity branches, power level and Nakagami parameter are also investigated.
I. I NTRODUCTION In the wireless technology era, emerging wireless communication systems are expected to provide a wide range of services that require high data rate transmission of high quality voice, as well as data, images, video and multimedia information. Second-generation (2G) systems designed exclusively for voice transmission are not of much use for the increasing demands for data and multimedia systems. The third-generation (3G) standards, on the other hand, are supposed to support higher data rate systems. 3G wireless communication systems based on code division multiple access (CDMA) implemented by direct-sequence spread-spectrum modulation are currently being fielded through out the world. Achieving multimedia communication will require systems that transmit at higher data rates with low average bit error rate (BER). One of the suggested solutions is the use of adaptive equalization at the receiver [1]. Such an equalizer reduces channel fading but the required number of delay taps at high data rate may be very high, which makes the system complex and non-realizable. A spread spectrum approach with a Rake receiver may also be used to overcome channel fading effects. But such a system requires large bandwidth for high transmission rates. Another possible solution is the antenna diversity and space time coding techniques suggested by the multiple input multiple output (MIMO) communication systems [2], [3]. However, the complexity of these systems is limiting factors. The multi-carrier CDMA [4], [5], or multi-tone CDMA (MT-CDMA) system [6], [7], may also be a viable solution since it converts the high rate bit stream into several lower rate substreams modulating several subcarriers. The main
drawback is that the design (or cost) of modulating and demodulating several carriers can be quite complex (or high) at high data rates. In this paper, we employ a multirate data transmission technique known as multi-code spread spectrum CDMA (MCSS/CDMA) system. Although there are many ways of implementing multirate transmission, multi-code systems are the most prevalent. A multi code system is similar to the MTCDMA system since it converts the high rate bit stream into several lower rate substreams with each stream differently encoded. For example, for a user with rate R bits per second (bps), one code would be assigned, for a user with rate 2R bps, two codes would be assigned, etc. The sum of the data modulated codes would then be spread, modulated and transmitted. The main difference is that in MC-SS/CDMA, orthogonal signals are used to code the different substreams that modulate only one carrier (assuming single carrier multi-code system), which significantly reduces the system complexity. MC-CDMA system affords several advantages including increased power efficiency, increased capacity, effective suppression of multiuser and multipath interferences. It also provides means for adjustable transmission rate and allows packet data transmission. The MC-SS/CDMA system has an excellent resource sharing capability (being a CDMA modulation technique); it has a great anti-jamming capability (because of the SS nature of the signal); it is capable of transmitting at high bit rates (because of the serial-to-parallel (S/P) conversion of the bit stream); it has simplified antenna structure (compared to MIMO); and is less complex than the MT-CDMA (because only one carrier is being modulated and demodulated). II. S YSTEM M ODEL Consider the system with K users and J substreams shown in Fig. 1. The data signal corresponding to the kth user is given by µ ¶ ∞ X T bk (t) = bik P TJ t − i , J i=−∞ where Px is a rectangular pulse of duration x. This data signal having a bit rate of J/T is S/P converted into J substreams each of a lower bit rate 1/T . The j th data
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hk (t) = Serial-to-Parallel converter
bk(t)
l=1
#
a1(t)
Σ
#
aj(t)
e
jωct
ck(t)
aJ(t)
Fig. 1. Multi code MC-SS/CDMA system model.
After the S/P conversion, these J data substreams are coded by a set of orthogonal signals to reduce the inter substream interference (ISSI). The orthogonal signal coding the j th substream is given by aj (t) =
N X i=1
aij PTc (t − iTc ).
ck (t) =
N X i=1
cik PTc (t − iTc ).
In order to maintain orthogonality of the coding signals, the maximum number of substreams J is limited to N = T/Tc . Note that aij , bijk and cik ∈{-1,1} with probabilities p(1) = p(−1) = 1/2. The transmitted j th substream signal corresponding to the th k user can be written as √ ¤ £ (1) sjk (t) = 2P aj (t)ck (t) Re bjk (t)ej2πfc t+θk ,
where P is the carrier power (assuming perfect power control); fc is the carrier frequency; and θk is a uniformly distributed random variable in [0, 2π]. The total transmitted signal, representing the sum of the signals corresponding to all the J substreams of all the K users can be written as K X J X √ ¤ £ 2P aj (t)ck (t) Re bjk (t)ej2πfc t+θk . (2)
2 ³ m ´m 2m−1 − β e Γ(m) Ω
The signal corresponding to the j substream of the k user propagates through in the multipath channel with transfer function hkj (t) is given by hkj (t) =
L X l=1
jψjkl
β jkl e
th
δ(t − τ jkl ).
(3)
³
mβ 2 Ω
´
,
(4)
where Ω is the second moment of β, m is the Nakagamim fading parameter and Γ(m) is the gamma function. The Nakagami model is used here because it is the most convenient model since it can easily describe the Rayleigh model by setting m = 1, and it can also approximate the Ricean model with a high degree of accuracy for values of (m > 1) [11]. In the receiver structure shown in Fig. 2, J groups of correlators are used to detect the J substreams. Every group consists of Q correlators to detect Q path signals such that Q ≤ L. The Q path signals at the receiver are combined by the diversity combiner to obtain J received substreams, which are subsequently parallel-to-serial (P/S) converted back into one main stream.
∫
T
∫
T
0
a1(t1) ck(t1)
e− jωct1 +φk1 0
a1(tQ) ck(tQ) e− jωctQ +φkQ T
0
aJ(t1) ck(t1)
(.)dt
# (.)dt
# ∫
k=1 j=1
th
β kl ejψkl δ(t − τ kl ),
where ψkl is the phase of the lth path of the kth user and is uniformly distributed over [0, 2π], τ kl is the time delay of the lth path of the kth user and is uniformly distributed over [0, T ]. The parameter β kl corresponds to the path gain. Different distribution functions can be used to model this path gain depending on the nature of the channel and the propagation environment. The Ricean distribution is sometimes used to model a fading channel when the signal is received via a direct wave and many scattering components [8]. Rayleigh distribution may be also used to model the channel path gain when no line-of-sight (LOS) wave exist [9]. Another distribution that is more popular in modeling fading channel path gain is the Nakagami-m distribution with probability density function (pdf) [10] fβ (β) =
The coded substreams are also multiplied by the pseudo-noise (PN) signature sequence corresponding to the kth user, which can be written as
sT (t) =
L X
e− jωct1 +φk 1
∫
T
0
(.)dt
# (.)dt
Parallel-to-serial converter
i=−∞
bijk PT (t − iT ).
But since all the J substreams propagate through the channel in the same time, the channel impulse response can be simplified to
Diversity combiner for the 1st substream
bjk (t) =
∞ X
Diversity combiner for the jth substream
substream corresponding to the kth users signal is given by
ym
aJ(tQ) ck(tQ) e− jωctQ +φkQ
Fig. 2. Receiver structure of the MC-SS/CDMA system.
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th and bRe substream of the bth ab,0 denotes the real part of the a Im user’s current data bit and bab,−1 denotes the imaginary part of the ath substream of the bth user’s previous data bit.
The received signal may be written as r(t) = sT (t) ~ hk (t) K X J X L X √ = 2P β kl aj (t-τ kl )ck (t-τ kl )
III. S IGNAL -T O -N OISE R ATIO
k=1 j=1 l=1
i h × Re bjk (t-τ kl )ej(2πfc t+φkl ) + n(t)
(5)
where ~ denotes convolution, n(t) is the additive white Gaussian noise (AWGN) and φkl = θk + ψkl − 2πfc τ kl . At the receiver, the signal is despread and correlated to obtain the decision variable. The receiver assumed in this paper is a synchronous receiver designed to detect the j th substream of the 1st user signal propagating via the 1st path. If perfect synchronization is assumed, the output of the correlator can be written as Z T r(t)aj (t − τ 11 )c1 (t − τ 11 )ej2πfc t+φ11 dt yj1 = 0
= yDS + yKP I + yISSI + yKU I + η,
(6)
It is easy to show that the signal power, S, is given by "r #2 P T 2 β 211 P 2 Re S = (yDS ) = β 11 bj1 T = . (7) 2 2 To find the signal-to-noise ratio (SNR), we need to compute the noise variance. The noise variance is the sum of the variances of all the interference terms and the AWGN noise component. For simplicity, we assume that all the terms are zero mean and statistically independent random variables. It is shown in [14] that h i 2 buv,wz (τ kl ) = 2T , var [Ruv,wz (τ kl )] + var R 3N J
and since var [cos φkl ] = var [sin φkl ] = 12 , we can show that the noise variances are given by where r " L # 2 P X Re P T β b T, yDS = var β 1l , σ 2MP I = 2 11 j1 3NJ r l=2 L " L # P X X yKP I = β 1l P T 2 (J − 1) 2 2 var β 1l , and σ ISSI = 3NJ n l=2 h i l=1 Re b " L # × cos φ1l bRe j1,−1 Rjj,11 (τ 1l ) + bj1,0 Rjj,11 (τ 1l ) 2 X P T h io var β 1l . σ2MUI = J(K − 1) Im b , + sin φ1l bIm 3NJ j1,−1 Rjj,11 (τ 1l ) + bj1,0 Rjj,11 (τ 1l ) l=1
yISSI
=
yKUI
and η=
= var [η] = No T /4, the total noise and Also, since J L P X X interference variance is given by β 1l 2 " L " L # # i=1,i6=j l=1 X X PT2 P T 2 (J-1) h n i 2 var var β 1l + β 1l σT = Re b × cos φ1l bRe 3NJ 3N J i1,−1 Rji,11 (τ 1l ) + bi1,0 Rji,11 (τ 1l ) l=2 l=2 h io " L # Im b X , + sin φ1l bIm PT2 No T i1,−1 Rji,11 (τ 1l ) + bi1,0 Rji,11 (τ 1l ) var . (8) +J(K − 1) β 1l + 3N J 4 l=1 r K J L P XXX Also, we considered the exponentially decaying multipath = β kl cos φkl 2 intensity profile (MIP), with a decay factor δ [15] such that k=2 i=1 l=1 h i · Re Re b × bik,−1 Rji,1k (τ kl ) + bik,0 Rji,1k (τ kl ) P T 2 Ω e−δ − e−δL (J − 1)(1 − e−δL ) 2 + σT = £ 3NJ 1 − e−δ 1 − e−δ + sin φkl bIm ik,−1 Rji,1k (τ kl ) ¸ −δL i J(K − 1)(1 − e )) 3NNo J Im b + + . (9) +bik,0 Rji,1k (τ kl ) , 1 − e−δ 4P T Ω r
Z
0
σ2η
Hence, the SNR can be written as
T
n(t)aj (t − τ 11 )c1 (t − τ 11 ) cos(2πfc t)dt,
buv,wz (τ kl ) denoting the partial correwith Ruv,wz (τ kl ) and R lation functions defined in [12] as Z τ kl au (t)av (t-τ kl )cw (t)cz (t-τ kl )dt, Ruv,wz (τ kl ) = 0 Z T buv,wz (τ kl ) = au (t)av (t-τ kl )cw (t)cz (t-τ kl )dt, R τ kl
SN R = γ 11 = β 211 R, where R =
·
1 − e−δL e−δ − e−δL + (J − 1) −δ 1−e 1 − e−δ ¸−1 J(K − 1)(1 − e−δL ) 3NNo J + + , 1 − e−δ 4Eb 3NJ 2Ω
and Eb = P T Ω is the bit energy.
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(10)
IV. B IT E RROR R ATE P ERFORMANCE
and
In the presence of AWGN, the instantaneous probability of error of a coherent detection system is given as √ 1 (11) Pe (γ) = erf c( γ). 2 To compute the average BER, P¯e , the instantaneous BER is averaged over the pdf of the output SNR such that Z ∞ P¯e = fγ (γ)Pe (γ)dγ (12) 0
where fγ (γ) is the pdf of the output SNR. The pdf of the output SNR, γ, depends on the pdf of the input SNR,γ i and on the method of diversity combining employed, as will be shown below.
PT γ egc =
For selection diversity, the overall output SNR γ is given by ¤ £ (13) γ = max γ 1 , γ 2 , ..., γ Q ,
Q P
q=1
βq
!2
2Q × var [N ]
where var[N ] = σ2T and X =
=
Q P
q=1
P T 2X 2 , 2Q × var [N ]
(16)
βq .
It is shown in [10] that the sum of Q Nakagami distributed random variables with parameters (m, Ω), is also a Nakagami distributed random variable with parameters (mQ, Ω(1 − 1 2 5m )Q ). This implies that the pdf of X can be written as 2mQ−1
fX (x) =
2x Γ(mQ)
P¯e
µ
m 1 Ω(1- 5m )Q
¶mQ
Ã
−
e
mx2 Ω 1− 1 Q 5m
(
)
!
.
λm m−1 −λγ γ e , Γ(m)
µ
¶mQ m 1 Ω(1- 5m )Q 0 q µ ¶ P 2 mx 2T exp erf c p 2 x dx. (17) 1 Ω(1- 5m )Q σT Q Z∞
=
where γ i , i = 1, 2, ..., Q are the SNR per diversity branch. Each input SNR, γ i , is gamma distributed with pdf
x2mQ−1 Γ(mQ)
C. Maximal Ratio Combining
m where λ = ΩR . If the pdf and the cdf are assumed independent and identically distributed, i.e.,
fγ Fγ
Ã
Hence, from (12) the average BER becomes
A. Selection Diversity
fγ i (γ) =
2
= fγ 1 = fγ 2 = ... = fγ Q , = Fγ 1 = Fγ 2 = ... = Fγ Q ,
then the output cdf is given by ¶Q µ ¡ ¢Q 1 G (m, λγ) Fγ (γ) = Fγ 1 (γ) = , Γ(m)
where G(m, λγ) is the incomplete gamma function of the second kind. Correspondingly, the pdf of the output becomes ¶Q−1 m µ m−1 1 λ G (m, λγ) γ e−λγ , (14) fγ (γ) = Q Γ(m) Γ(m)
from which the average BER is given by ¶Q Z ∞ µ m−1 Q 1 ¯ Pe = (G (m, λγ))Q−1 λm γ 2 Γ(m) 0 √ ×e−λγ erf c( γ)dγ. (15) B. Equal Gain Combining
To find the SNR at the output of the equal gain combining (EGC), we follow the same procedure as in [16]. Let Yi and γ i , denote the signal and SNR at the ith input diversity branch respectively. And let Y and γ denote the resultant signal and SNR at the output of the combiner. Hence, r Q Q Q X X X P bT Yq = βq + Nq , Y = 2 q=1 q=1 q=1
For maximal ratio combining (MRC), the signal at the output of the combiner is given by r Q Q Q X X X P 2 Y = b β q Yq = βq + β q Nq , qT 2 q=1 q=1 q=1
and the output SNR is
1
γ mrc =
var [N ]
Q P
q=1
This implies that
√ γ mrc = T
when Z =
s
Q P
q=1
β 2q
s
ÃQ !2 P 2 X 2 × T βq . 2 q=1
Z = TZ 2σ 2T
s
(18)
P , 2σ2T
β 2q .
Substituting in (11), the instantaneous BER becomes à s ! 1 P √ Pe ( γ mrc ) = erf c T Z . 2 2σ2T If β i are Nakagami distributed random variables with parameters (m, Ω), Z is also a Nakagami distributed random variable with parameters (mQ, ΩQ) [10]. The average error will be à s ! Z∞ 2mQ-1 ³ ´mQ ³ 2 ´ mZ Z P m e Ω erf c T Z dZ. P¯e = Γ(mQ) Ω 2σ 2T 0
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V. R ESULTS AND D ISCUSSION
Average Bit Error Rate
TABLE 1. S AMPLE VALUES FOR N UMERICAL P LOTS Bit rate of original stream 1 kilo bit per sec. Processing Gain N1 = TTc = 64 No. of substreams per use J = 1, 2, 4, 8, 16, 32 No. of users K = 10 No. of paths L=3 Exponential MIP δ=0 Nakagami-m fading parameter m ≥ 0.5 No. of resolvable paths Q=2 Local mean received power Ω =10 dB 10
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Fig. 4. Performance of MC-SS/CDMA systems as a function of number of substreams and diversity.
k=1 k=2 k=4 k=8 k=16 k=32
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Average Bit Error Rate
The performance of the MC-SS/CDMA system is shown in terms of plots of the average BER as a function of several system parameters. In the plots, one or more of the parameters are varied while assuming constant values for others. Unless otherwise stated, we assume the numerical values shown in Table 1. Using these values, Fig. 3 shows the average BER against the Eb /No for different values of J for the selection diversity receiver. For all values of Eb /No , the multi code SS/CDMA systems have lower average BER compared to single code SS/CDMA system.
MRC has the best performance; EGC has a slightly lower performance while SD is clearly the worst. The graph also shows that a system with a low number of users (K = 5) performs better than the same system with a higher number of users (K = 20).
50
o
Fig. 3. SD performance of SC-SS/CDMA and MC-SS/CDMA system for several number of substreams.
Figure 4 shows the average BER for the three diversity receivers and for J = 4 and 32. Notice that for both large and small number of substreams, the MRC performs better than the EGC and SD with the SD having the worst performance. This classification of performance is true for both large number of substreams J = 32, and low number of substreams J = 4. In Fig. 5, the number of substreams is fixed at J = 16 and the plots are generated for two different values of the number of users (K = 5 and K = 20). This graph also shows that
In Fig. 6, the number of substreams is J = 16 and the number of users is K = 10. The performance of the system is tested for different values of the Nakagami parameter m. The plot shows that the less the fading in the channel (higher values of m), the better is the performance of the system. The plot is generated for m = 1 which represents Rayleigh faded channel model, m = 2 and m = 4 for less faded channels. Notice that the less the channel is faded, the more the EGC and MRC perform better than the SD technique. Figure 7 compares the performance of the system for a different numbers of diversity branches at the receiver. The plot is generated for values of Q = 1, 4 and 8. Obviously, for Q = 1, no diversity combining takes place since only one branch is available. Hence, the performances of the three diversity techniques are the same. However, for a higher number of diversity branches Q = 4, the figure indicates some difference in the performance of the different techniques. As expected, when the number of diversity branches is increased (Q = 8 in the plot), the performances of MRC and EGC become much better than the performance of the SD system. VI. C ONCLUSION We have analyzed the performance of the multi-code SS/CDMA system, in terms of average bit error rate and for different diversity techniques. We also considered the number of substreams, number of users, different power levels, and different fading conditions. The results indicate overall better performance of multi-code SS/CDMA system compared to the single-code SS/CDMA system. This illustrates the
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efficiency of this technique, which is expected to get more and more significance in the 3G and the upcoming 4G wireless communication systems. 10
10
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Fig. 5. BER comparison of MC-SS/CDMA systems for different number of users and diversity.
[5] S. Kondo and L. B. Milstein, “Performance of multicarrier DS CDMA systems,” IEEE Trans. Commun., vol. 44, pp. 238-246, Feb. 1996. [6] L. Vanderdorpe, “Multitone Spread Spectrum multiple access communications system in a multipath rician fading channel, ” IEEE Trans. Veh. Technol., vol. 44, pp. 327-337, May 1995. [7] S. Matin and O. Ugweje, “Performance of phase-coded multitone CDMA system in Nakagami multipath fading,” Third generation wireless commun. and beyond, pp. 792-797, May-June 2001. [8] J. G. Proakis, Digital Communications, 3rd edition. McGraw-Hill, New York, 1995. [9] A. Sathyendran, K. W. Sowerby and M. Shafi, “Interference modelling in DS/CDMA cellular systems operating in a general fading environment,” Wireless Personal Commun., vol. 5, issue 3, pp. 279–301, 1997. [10] N. Nakagami, “The m-distribution, a general formula for intensity distribution of rapid fading,” In Statistical Methods in Radio Wave Propagation, W. G. Hoffman, Ed., UK: Pergamon, pp.3-35, 1960. [11] O. C. Ugweje, “Selection diversity for wireless communications in Nakagami fading with arbitrary parameters,” IEEE Trans. Veh. Tech., vol. 50, no. 6, pp. 1437-1448, Nov. 2001. [12] M. B. Pursley, “Performance evaluation for phase-coded spread spectrum multiple-access communication-Part I: system analysis,” IEEE Trans. Commun., vol. 25, pp. 795-799, Aug. 1977. [13] M. Kavehrad, “Performance of nondiversity receiver for spread spectrum in indoor wireless communications,” AT&T Tech. Jour., vol. 64, pp. 1181-1210, Jul.-Aug. 1985. [14] S. A. Khorbotly, Performance Analysis of Multi-Code Spread Spectrum Modulation. Masters Thesis, The University of Akron, May 2003. [15] J. H. Han and S. W. Kim, “Capacity of DS/CDMA communication systems with optimum spectral overlap,” IEEE Commun. Letters, vol. 2, no. 11, pp. 298-300, Nov. 1998. [16] S. A. Matin, Performance of Multitone CDMA Communication System With Diversity, Narrowband Signaling and Coding. Masters Thesis, The University of Akron, May 2001.
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Fig. 6. BER comparison of MC-SS/CDMA systems for different diversity schemes under different fading condition.
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Fig. 7. System performance with different order and type of diversity techniques.
R EFERENCES [1] J. G. Proakis, “Adaptive equalization for TDMA digital mobile radio,” IEEE Trans. Veh. Tech., vol. 40, no. 2, pp. 333-341, May 1991. [2] A. Paulraj and D. Gore, “Optimal antenna selection in MIMO with space-time block codes,” IEICE Trans. Commun. vol. E84-B, no. 7, pp.1713-1719, July 2001. [3] K. Wong, R. Murch and K. Letaief, “Optimizing time and space MIMO antenna system for frequency selective fading channels,” IEEE J. Select. Areas Commun., vol. 19, no. 7, pp. 1395-1407, July 2001. [4] E. Sourour and M. Nakagawa, “Performance of multicarrier CDMA in a multipath fading channel,” IEEE Trans. Commun., vol. 44, no. 3, pp. 356-367, Mar. 1996.
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