On Carrier Spacing in Multicarrier CDMA Systems - Individual ...

IEICE TRANS. COMMUN., VOL.E87–B, NO.3 MARCH 2004

768

LETTER

On Carrier Spacing in Multicarrier CDMA Systems Ha H. NGUYEN† , Member and Ed SHWEDYK†† , Nonmember

SUMMARY This letter considers multicarrier CDMA systems using rectangular chip waveform, where each user’s data stream is serial to parallel converted to a number of lower rate streams. Each lower rate stream is then spread by a random spreading code and a suitable chip rate before modulating orthogonal carriers. It is shown that, for a fixed system bandwidth and a given number of carriers, there exists an optimal carrier spacing that minimizes the multiple access interference. Numerical examples also show that the multicarrier DS-CDMA system previously proposed in [1] performs very close to the muticarrier CDMA system using the optimal carrier spacing. key words: CDMA, multicarrier-CDMA, carrier spacing

1.

Introduction

Recently, a number of multicarrier code-division multiple access (CDMA) systems has been proposed (see [2] and the references therein). Among these systems, both multitone CDMA (MT-CDMA) [3] and multicarrier direct-sequence CDMA (MC-DS-CDMA) [1] combine time domain spreading and multicarrier modulation, as opposed to the combination of frequency domain and multicarrier modulation of other systems. In essence, in these systems, each user’s data stream is first serial to parallel converted to a number of lower rate streams. Each lower rate stream is then spread by a user-specific spreading sequence before modulating the orthogonal carriers. The distinction between the two above mentioned schemes is due to the carrier spacing. The distance between the two adjacent carriers equals the bit rate (after serial-to-parallel conversion) in MT-CDMA system, whereas it equals the chip rate in MC-DS-CDMA system. In [4] the authors obtain optimal spacing for the frequency tones in frequency shift keying (FSK) multiple access systems. Motivated by the work in [4], this letter generalizes the MT-CDMA and MC-DS-CDMA systems by considering an arbitrary multiple of bit rate for the carrier spacing. Although an arbitrary chip waveform can be used in multicarrier CDMA systems, only rectangular chip pulse is considered in this letter. This allows one to concentrate on the effect of carrier spacing and to readily compare the performance to both MT-CDMA and MC-DS-CDMA systems∗ . Analysis of multiple access interference (MAI) under a fixed null-to-null system bandwidth and assuming random Manuscript received July 25, 2003. Manuscript revised September 15, 2003. † The author is with the Department of Electrical Engineering, University of Saskatchewan, Saskatoon, SK, S7N 5A9 Canada. †† The author is with the Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, MB, R3T 5V6 Canada.

signature sequences shows that it is possible to optimize the carrier spacing so that the MAI is minimized. Numerical examples also show that there is a very little degradation in performance of the MC-DS-CDMA system compared to that of the generalized multicarrier CDMA system using optimal carrier spacing. The letter is organized as follows. Section 2 describes the multicarrier CDMA system model. Multiple access interference analysis is carried out in Sect. 3. Numerical examples are given in Sect. 4. Finally, conclusions are drawn in Sect. 5. 2.

Multicarrier CDMA Systems

Consider a multicarrier CDMA system with K users. At the transmitter, each user’s bit stream with bit duration T b is serial-to-parallel converted into M lower rate streams. The new bit duration on each lower rate stream is T = MT b . Each lower rate bit stream is spread by a random signature sequence before it is modulated onto M orthogonal carriers that are equally spaced. The kth user transmits the following signal over the same channel, yk (t − δk ) =

∞ √ M



2P

m=1 i=−∞

× bk,m (i)sk (t − iT − δk ) cos(2π fm t + ϕk,m )

(1)

In (1), P is the signal power, δk is the delay (uniform over [0, T ]) and ϕk,m is the random phase of the mth carrier (uniform over [0, 2π] and independent for all k and m). The sequence {bk,m (i)} is the binary data sequence of the mth bit stream of the kthuser. The signature waveform of user k is N−1 sk (i)ψ(t − iT c ), where the rectangular given by sk (t) = i=0 chip waveform ψ(t) is defined as ψ(t) = 1 for 0 ≤ t ≤ T c and 0 otherwise. Both {bk,m (i)} and {sk (i)} are modeled as sequences of independent and identically distributed random variables taking values in {−1, +1} with equal probability. Furthermore the signature sequence {sk (i)} is periodic with period N = T/T c . The orthogonal carriers are related by fm = f1 +(m−1)∆, m = 1, 2, . . . , M, where ∆ is the space between  two adjacent carriers. The received signal then equals K yk (t−δk )+n(t), where n(t) is additive white Gausr(t) = k=1 sian noise with two-sided power spectral density of N0 /2. As mentioned earlier, the carrier spacing in multicarrier ∗ Note that rectangular chip waveform is also used in both MTCDMA and MC-DS-CDMA systems.

LETTER

769

CDMA systems considered in this letter is a multiple of the bit rate, i.e., ∆ = q/T with q is some integer number. For a fixed null-to-null system bandwidth W, it can be shown that the processing gain N and parameter q are related by [4] N = MN1 −

(M − 1) q 2

(2)

where N1 is the processing gain of the single carrier CDMA (SC-CDMA) system (M = 1) of the same bandwidth, given by N1 = T b /(2/W) = WT b /2. As special cases, when q = 1 the MT-CDMA system of [3] is realized and when q = N the multicarrier CDMA system becomes the MC-DS-CDMA system of [1]. The processing gains of the MT-CDMA and MC-DS-CDMA systems are determined as follows. Let N2 be the processing gain of the MT-CDMA system, then N2 can be obtained from (2) by setting q = 1, i.e., N2 = MN1 − (M − 1)/2. Let N3 be the processing gain of the MC-DS-CDMA system. Since the carrier spacing in this system is N3 /T , then it can be shown that N3 = 2MN1 /(M + 1). The receiver of user k employs a bank of M matched filters, each of which detects the bits transmitted on a particular carrier. 3.

Interference Analysis

Consider the detection of the bit stream associated with the lth carrier of the first user. Since only relative delays and phases are important one can set δ1 = 0 and ϕ1,l = 0. The decision statistic at the output of the lth carrier’s matched filter of user one is given by  T Z1,l = r(t)s1 (t) cos(2π fl t)dt 0

= D + η + I1 + I2 + I3 (3) √ where D = P/2b1,l (0)T is the desired signal component, η is a Gaussian random variable with zero mean and variance N0 T /4. The last three terms in (3) account for the following types of interference. The interference from other carriers of user one is given by, M  T √
2Pb1,m (0) I1 = m=1 m
l

0

× s21 (t) cos(2π fm t + ϕ1,m ) cos(2π fl t)dt M √
= 2Pb1,m (0) m=1 m
l



δk

R1,k (m, l, δk , ϕk,m ) = 1,k (m, l, δk , ϕk,m ) = R



0 T δk

 ×

T

cos(2π fm t + ϕ1,m ) cos(2π fl t)dt

(4)

0

Due to the orthogonality of the carriers, the integral in (4) is zero, hence I1 = 0. The interference from the same carrier l of other K − 1 users can be expressed as follows.    δk K
P I2 = s1 (t)sk (t + T − δk )dt bk,l (−1) 2 0 k=2   T + bk,l (0) s1 (t)sk (t − δk )dt cos(ϕk,l ) (5) δk

Note that I2 can be treated as the MAI in a SC-CDMA system (with carrier frequency fl ). Thus it is well-known [5] that I2 has zero mean and the following variance.  2 PT K − 1 (6) var(I2 ) = 2 3N The last term in (3) is the interference from all other carriers (m
l) of This interference can be exKK − 1M users. (k,m) , where I3(k,m) is the interferpressed as I3 = k=2 m=1 I3 m
l ence from the mth carrier of the kth user. This interference is given by   P (k,m) I3 = bk,m (−1)R1,k (m, l, δk , ϕk,m ) 2  1,k (m, l, δk , ϕk,m ) + bk,m (0)R (7) 1,k (m, l, δk , ϕk,m ), respectively, where R1,k (m, l, δk , ϕk,m ) and R are given in (8) and (9) on bottom of this page. The random variable I3(k,m) has zero mean, while its variance can be computed as follows.

  P var I3(k,m) = var R1,k (m, l, δk , ϕk,m ) 2   1,k (m, l, δk , ϕk,m ) + var R (10) 1,k (m, l, δk , ϕk,m ) The variances of R1,k (m, l, δk , ϕk,m ) and R can be found in the same way as in Appendix II of [4] and are given by  

 1,k (m, l, δk , ϕk,m ) var R1,k (m, l, δk , ϕk,m ) = var R    2π(m − l)q NT 2 = 2 1 − sinc (11) N 4π (m − l)2 q2 where sinc(x) = sin x/x. Note that I3(k,m) are uncorrelated for all k (2 ≤ k ≤ K)

  (m − l)q t + ϕk,m dt, 0 ≤ δk ≤ T s1 (t)sk (t + T − δk ) cos 2π T   (m − l)q t + ϕk,m dt, 0 ≤ δk ≤ T s1 (t)sk (t − δk ) cos 2π T

(8) (9)

IEICE TRANS. COMMUN., VOL.E87–B, NO.3 MARCH 2004

770

and all m (m
l) since they incorporate independent phase angles. Thus the variance of I3 is, var(I3 ) =

M K

k=2

·

 var I3(k,m)

m=1 m
l

(K − 1)NPT 4π2 q2



M 2


2π(m − l)q N (m − l)2



1 − sinc

m=1 m
l

(12)

It is not hard to see that I2 , I3 and η are all uncorrelated, thus the total variance of MAI-plus-noise is [var(I2 ) + var(I3 ) + N0 T/4]. Typically, the multiple access channel is interference-limited, not noise-limited. It is therefore useful to concentrate on the variance of MAI. Furthermore, from (12) it can be seen that the amount of MAI depends on the particular carrier fl under consideration. Intuitively, the MAI is largest for the center carrier(s) (l = M2 , M2 + 1 when M is even, and l = M+1 2 when M is odd) and decreases as l moves towards the edge carriers. Thus to evaluate the effect of carrier spacing on MAI variance, define the following normalized average MAI variance.  −1
M PT 2 [var(I2 ) + var(I3 )] γ(M, q) = (K − 1)M 2 l=1 =

N 1 + 3N 2Mπ2 q2



2π(m − l)q M 1 − sinc M

N × 2 (m − l) l=1 m=1

 (13)

m
l

For a given a number of carriers M, it is desired to optimize the spacing parameter q to minimize the MAI. Unfortunately, the optimal value of q is not analytically tractable but can only be obtained by numerical computations. This will be illustrated in Sect. 4 through examples. For comparison purpose, let γ1 , γ2 and γ3 be the normalized average MAI variances produced by the SCCDMA, MT-CDMA and MC-DS-CDMA systems respectively. These parameters can be obtained by setting q and N in (13) appropriately and are given by 1 (14) γ1 = 3N1   2π(m − l) M
M 1 − sinc
N2 1 N2 γ2 = + 3N2 2Mπ2 l=1 m=1 (m − l)2 m
l

(15) γ3 =

4.

1 1 + 3N3 2Mπ2 N3

M M

l=1

m=1 m
l

1 (m − l)2

(16)

Numerical Examples

In this section, the effects of both the number of carriers

Fig. 1 Interference reduction compared to SC-CDMA system, N1 = 128 (Note: γ1 , γ2 , γ3 and γ(M, q) are the normalized MAI variance in SCCDMA, MT-CDMA, MC-DS-CDMA and generalized MC-CDMA systems respectively).

and the carrier spacing on multiple access interference are investigated. With rectangular chip waveform, the multicarrier systems under consideration have a total null-to-null bandwidth of W = 256/T b , i.e., the processing gain of conventional single carrier system is N1 = 128. Figure 1 compares multiple access interference of different multicarrier CDMA systems with that of the SCCDMA system when the number of carriers varies. More precisely, plotted in that figure are the ratios γ2 /γ1 , γ3 /γ1 and γ(M, q)/γ1 . For the generalized multicarrier CDMA systems, several values of q are selected to calculate γ(M, q). As can be seen from Fig. 1, the MAI produced in MTCDMA system is almost identical (it is actually slightly larger) to that produced in the SC-CDMA system, regardless of the number of carriers. This observation agrees with the results in [6] where the authors show that using more carriers does not increase the multiple access capacity of MTCDMA system† . By contrast, the MAI in a MC-DS-CDMA system is smaller than that in a SC-CDMA system and it decreases as M increases. For this particular example, using M = 7 carriers is a good choice for the MC-DS-CDMA system since there is a negligible reduction of MAI for larger M. It can be clearly observed from Fig. 1 that increasing the number of carriers has an effect on the MAI of the generalized multicarrier systems and that this effect depends on the specific value of q that is chosen. Depending on the value of q, the MAI may increase, decrease or first decrease and then increase as M increases. From Fig. 1 it also seems that the MC-DS-CDMA system always achieves the minimum MAI. However this is not clear due to the following two reasons. † We would like to point out that Eq. (15) given in this letter is a more explicit expression for MAI variance of MT-CDMA system than Eq. (18) of [6].

LETTER

771

5.

Fig. 2 Influence of carrier spacing q on the normalized average MAI variance, N1 = 128.

Conclusions

A generalized multicarrier CDMA system employing rectangular chip waveform and where the carrier spacing is a multiple of the bit rate is considered in this letter. A closedform expression for multiple access interference (MAI) has been given. This expression allows one to find the optimal carrier spacing that minimizes MAI. Comparisons to previously proposed multicarrier CDMA systems under the same null-to-null bandwidth and the same bit rate are also made. In particular, it has been shown that the MC-DS-CDMA system in [1] performs almost identically to the muticarrier CDMA system using the optimal carrier spacing. Finally, as the result obtained in this letter is only valid for rectangular chip waveform, extending the analysis in this letter to an arbitrary chip waveform is another interesting research topic. Acknowledgement

First, each point in the curve of γ3 /γ1 is achieved by a different value of q. In particular, for M = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19}, the corresponding values of q are q = {128, 192, 213, 224, 230, 234, 237, 240, 241, 243}. Secondly, the curves γ(M, q)/γ1 are only plotted for some selected values of q. To illustrate the existence of the optimal carrier spacing in generalized multicarrier CDMA systems, Fig. 2 plots the normalized MAI variance γ(M, q) versus q for some values of M. Also shown in Fig. 2 are the values of γ1 , γ2 and γ3 for comparison. It can be determined from Fig. 2 that, for M = {3, 5, 7, 15} the corresponding optimal carrier spacings of the generalized multicarrier systems are q = {195, 215, 225, 240}, compared to q = {192, 213, 224, 240} of MC-DS-CDMA systems. Thus it is clear that the MCDS-CDMA system can be considered to be optimal in terms of minimizing MAI. Finally, all the above observations and discussions also hold for systems with other different values of the null-to-null bandwidth.

This work was supported by NSERC Discovery Grants. References [1] V.M. DaSilva and E.S. Sousa, “Multicarrier orthogonal CDMA signals for quasi-synchronous communications systems,” IEEE J. Sel. Areas Commun., vol.12, no.5, pp.842–852, June 1994. [2] S. Hara and R. Prasad, “Overview of multicarrier CDMA,” IEEE Commun. Mag., vol.20, no.1, pp.126–133, Dec. 1997. [3] L. Vandendorpe, “Multitone spread spectrum multiple access communications system in a multipath rician fading channel,” IEEE Trans. Veh. Technol., vol.44, no.2, pp.327–337, May 1995. [4] L.L. Yang and L. Hanzo, “Overlapping M-ary frequency shift keying spread-spectrum multiple-access systems using random signature sequences,” IEEE Trans. Veh. Technol., vol.48, no.6, pp.1984–1995, Nov. 1999. [5] E. Geraniotis and B. Ghaffari, “Performance of binary and quaternary direct-sequence spread-spectrum multiple-access systems with random signature sequences,” IEEE Trans. Commun., vol.39, no.5, pp.713–724, May 1991. [6] K.W. Yip, X. Zhang, T.S. Ng, and J. Wang, “On the multiple access capacity of multitone-CDMA communications,” IEEE Commun. Lett., vol.4, no.2, pp.40–42, Feb. 2000.