Spectral Efficiency of Equal-Rate DS-CDMA Systems ... - IEEE Xplore
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Spectral Efficiency of Equal-Rate DS-CDMA Systems with Multiple Transmit Antennas Husheng Li, Member, IEEE, and H. Vincent Poor, Fellow, IEEE
Abstract— The spectral efficiency of equal-rate DS-CDMA systems with multiple transmit antennas is analyzed. Explicit expressions for the spectral efficiency are obtained in the low and high energy regions and numerical results are presented for the moderate energy region. The performance gains resulting from the use of multiple transmit antennas are discussed. An equation characterizing the diversity-multiplexing tradeoff in this situation is derived. A conclusion that adding more transmit antennas does not necessarily improve the performance with the constraint of equal rate is drawn. Index Terms— Direct sequence code division multiple access (DS-CDMA), multiple-input multiple-output (MIMO), spectral efficiency.
I. I NTRODUCTION
T
HE use of multiple transmit antennas plays a major role in high data rate wireless communication systems due to the resulting two-fold improvement in the system performance arising from multiplexing and diversity. For multiplexing, it was shown in [10] that the channel capacity in the high energy region is scaled by the number of transmit antennas, provided that the number of receive antennas is sufficiently large. In this context, the well known BLAST receiver [2] has been shown to achieve high data transmission rate for practical applications. Space-time codes were proposed in [9] as a means to achieve full transmit diversity, which enhances the transmission reliability. The tradeoff between multiplexing gain and diversity gain is studied for single-user systems in [15] and for multiple access channels in [12]. However, the analysis in the above papers is based on narrow band systems. For direct sequence code division multiple access (DS-CDMA) systems with multiple transmit antennas, the sum capacity is studied in [5] and the interesting conclusion is drawn that the performance gains in equal power uplink CDMA systems are more sensitive to the number of receive antennas than to the number of transmit antennas. This observation motivates the study in this paper of the performance gain obtained by adding multiple antennas to the transmitting terminals in uplink CDMA systems. Our discussion is focused on equal-rate systems without channel state information at transmitters (CSIT), which is well-suited for practical systems and can be extended to multi-rate systems.
Manuscript received January 26, 2005; revised January 14, 2006; accepted March 29, 2006. The associate editor coordinating the review of this paper and approving it for publication was M. Sawahashi. This research was supported in part by the National Science Foundation under Grant No. ANI-03-38807, and in part by the New Jersey Center for Wireless Telecommunications. H. Li is with Qualcomm Inc., CA (email: [email protected]). H. V. Poor is with the Department of Electrical Engineering, Princeton University, Princeton, NJ (email: [email protected]). Digital Object Identifier 10.1109/TWC.2006.05057.
We consider both full-multiplexing and optimal systems. We will examine the performance of such systems as measured by the corresponding spectral efficiency, particularly in the low and high energy regions. In each case, we develop analytical expressions for the spectral efficiency parameters, namely the minimum energy per bit and spectral slopes in the low and high energy regions [14], and discuss the performance gain provided by multiple transmit antennas. Similarly to the diversity-multiplexing tradeoff discussed in [15] and [12], we also develop equations that describe the diversity-multiplexing tradeoff for full-multiplexing systems and optimal systems, respectively. It should be noted that our discussion is based on asymptotic analysis, which means that the system size tends to infinity while the system load is fixed. In summary, we tackle the following two aspects of multiple transmit antenna CDMA systems: • The spectral efficiency of equal-rate systems without CSIT and the performance gain due to the use of multiple transmit antennas • The diversity-multiplexing tradeoff in equal-rate systems The remainder of this paper is organized as follows. The signal model and spectral efficiency parameters are introduced in Section II. Full-multiplexing systems and optimal systems are discussed in Section III and Section IV, respectively. Simulation results and conclusions are given in Section V and Section VI, respectively. II. S IGNAL M ODEL AND S PECTRAL E FFICIENCY A. Signal Model In this paper, we consider synchronous single-cell uplink DS-CDMA systems with nt transmit antennas per user, nr receive antennas, K active users, spreading gain N , and system load β = K N . The system is illustrated in Fig. 1. We denote the binary spreading code used for the n-th transmit antenna of the k-th user by skn , which satisfies skn 2 = 1. We assume that signals are transmitted through frequency flat fading channels and the complex channel amplitude gain between the n-th transmit antenna of user k and the m-th receive antenna is denoted by gkmn . These gains are assumed to be identically distributed and mutually independent with unit variance. The probability density function (PDF) and cumulative distribution function (CDF) of channel power gain |gkmn |2 are denoted by f and F , respectively. To simplify the analysis, we can convert the multiple receive antenna system into an equivalent system with a single-antenna receiver, by merely replacing f and F with fnr and Fnr , which denote the PDF and CDF of the sum of nr independent random variables distributed with PDF f . This equivalence can be
c 2006 IEEE 1536-1276/06$20.00
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Single cell. Usually, there are multiple cells in cellular systems. However, the inter-cell interference can be incorporated into noise on assuming that soft handoff is not used. Using the above notation and simplifications, the outputs of a chip matched filter at the equivalent single-antenna receiver during a symbol period can be sampled at the chip rate to give an N -vector r of observations: •
User 1
...
1
nt ...
User K
...
1
...
1
nr
r =
nt
=
nt K
gkn skn xkn + n
k=1 n=1 K
Hk xk + n,
(1)
k=1
Fig. 1.
Illustration of uplink MIMO CDMA systems.
justified by the resource pooling effect in [3]. It should be noted that we do not confine our discussion to any specific fading distribution. We assume that CSIT is not available; thus all users transmit with the same power, regardless of the channel conditions. In this paper, we consider only the case of assigning independent random spreading codes to the different transmit antennas of a given user. Thus, adding more transmit antennas has a two-fold impact, namely increasing the number of degrees of freedom in the fading channels and decreasing the number of degrees of freedom in the spreading code space. Thus, as we will see, adding more transmit antennas does not necessarily imply an improvement of system performance, as there is a tradeoff between these two effects. It should be noted that we adopt some assumptions which may not be exact in practical systems. However, these assumptions simplify the analysis and provide insight for practical systems. Some comments on the validity of these assumptions are given as follows. • Fading channels. A typical CDMA system usually experience frequency selective fading channels, instead of frequency flat fading channels. However, it is difficult to obtain explicit expressions for frequency selective fading channels. Therefore, we adopt the assumption of frequency flat channels simplifying the analysis, similar to [5] and [7]. • Synchronicity. For many CDMA systems, such as CDMA2000 and W-CDMA, the transmissions of different users in the uplink are asynchronous. Again, we assume the synchrony in order to simplify the analysis, similar to [3][5][7][9]. It should be noted that the assumption of synchrony is valid for time division duplexing (TDD) systems, such as TDS-CDMA systems. • Random spreading code. This assumption is also used in [5][6]. An alternative scheme is to use the same spreading code for different transmit antennas of the same user. Then, fewer spreading codes are used at the cost of causing more interference (the same as narrow band systems) across the signals transmitted from different antennas of the same user. The discussion of the same code scheme is beyond the scope of this paper.
where gkn is the channel amplitude gain of the equivalent model, n is additive complex white Gaussian noise with variance σn2 , {xkn } are the transmitted channel symbols and1 Hk xk
= =
(gk1 sk1 , ..., gknt sknt ) , T
(xk1 , ..., xknt ) ,
k = 1, ..., K,
k = 1, ..., K,
1 where the constraint on xk is given by E{xk xH k } = nt Int ×nt . Thus, the transmitting signal-to-noise ratio (SNR) is given by γ = σ12 . Throughout this paper, we assume that the receiver n has perfect channel state information, thus enabling coherent detection and decoding. Note that the supports of f and F are in [0, ∞). To analyze the performance with small outage probability in the high energy region, it is of interest to study the behavior of fm (x) and Fm (x) in the vicinity of x = 0. Supposing that f (x) is infinitely differentiable at x = 0, we define Θf min n : f [n] (0) = 0 ,
where f [n] denotes the n-th derivative of f . It is easy to verify that ΘFm = Θfm + 1. Then we have the following lemma Lemma 2.1: For any m > 0, Θfm = m (Θf + 1) − 1 . (2) ΘFm = mΘF Proof: We can actually show a more generous conclusion. Let X and Y be two independent nonnegative random variables distributed according to the PDFs f and h, respectively. Denoting the PDF of random variable Z = X +Y by g, we can show Θg = Θf + Θh + 1. The proof is straightforward by applying the fact that s g(s) = f (t)h(s − t)dt. 0
Taking the n-th derivative on both sides, we have s n−1 [n] [n−1−k] [k] g (s) = f (s)h (0) + f (t)h[n] (s − t)dt. 0
k=0
Thus, we have g [n] (0) =
n−1
f [n−1−k] (0)h[k] (0),
k=0 1 The superscript T denotes transposition and the superscript H denotes conjugate transposition.
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which implies Θg = Θf + Θh + 1. So, by induction on m, we have Θfm = m (Θf + 1) − 1. The conclusion for ΘFm follows immediately since ΘFm = Θfm + 1. Thus when x is sufficiently small, we have the following approximations due to Taylor’s expansion:
fm (x) = C1 xm(Θf +1)−1 +o xm(Θf +1)−1 , Fm (x) = C2 xm(Θf +1) + o xm(Θf +1)
(3)
where C1 and C2 are positive constants. It is easy to show that for both Rayleigh and Ricean fading channels, Θf = 0, and for Nakagami-d fading channels, Θf = d−1. In the following sections, we denote |g|2 by g˜ for notational simplicity, since we are only concerned with the the channel power gain, and the corresponding PDF and CDF are given by fnr and Fnr .
B. Spectral Efficiency Spectral efficiency is defined as C=
K R = βR, N
(4)
where R is the information transmission rate (in nats per symbol). We denote the spectral efficiency (measured in nats/s/Hz) as a function of the transmit SNR γ as C(γ). The relationship among the energy per information bit, the spectral efficiency and SNR is given by βγ log 2 Eb , = N0 C
(5)
where N0 is the one-sided noise spectral density which equals σn2 . An explicit expression for C as a function of β and Eb N0 is difficult to obtain. However, the following important parameters can be computed with the tools developed in [14]2 :
Eb Eb • = β˙log 2 , which represents the minimal N N0 0 C(0) min required for reliable communications. 2 C˙ (0) , which quantifies the increasing slope of • S0 = −2 ¨ C(0) the spectral efficiency in the low energy region. ˙ ), which quantifies the increasing • S∞ = limP →∞ P C(P slope of the spectral efficiency in the high energy region. Since our discussion on spectral efficiency is based on asymptotic analysis, we adopt the following notations of asymptotic equality and inequality. • •
•
log f (x) log h(x)
→ 1 as x → ∞;
(x)) ˙ ˙ f (x)>h(x) and f (x)1 log(h(x))
log(f (x)) and lim log(h(x)) < 1, respectively;
(x)) ˙ ˙ ≥1 f (x)≥h(x) and f (x)≤h(x) means lim log(f log(h(x))
log(f (x)) and lim log(h(x)) ≤ 1, respectively. f (x)=h(x) ˙ means
2 Throughout
this paper, logarithms are base e.
III. F ULL - MULTIPLEXING S YSTEMS In this section, we assume that the information symbols are multiplexed, channel coded and transmitted independently by different transmit antennas of a given user, which is similar to BLAST systems; thus the multiple transmit antennas are used exclusively for multiplexing. We call this a fullmultiplexing system. Since the signals from some transmit antennas experience deep fading, this system demands fairly high transmitter power to achieve reliable communications for all users. Thus, it is reasonable that the receiver decodes only the users with good channel conditions, and announces the rest of the users as experiencing an outage [8]. We denote this outage probability by q. In the single-antenna systems, a user experiences outage when its channel power gain is lower than a cutoff channel gain 3 . However, as we will see, determining an outage is more complicated for multiple-antenna systems. Since each transmit antenna operates independently, the multi-antenna system is equivalent to a single-antenna system with system load nt β, transmitting SNR nγt , outage probability gcut ), q and cutoff channel gain g˜cut , which satisfies q = Fnr (˜ except for the outage probability. A user experiences outage if any of its antennas falls in the outage set of the equivalent single-antenna system, which implies that q = 1 − (1 − q )nt . Thus we consider single-antenna systems first by applying the conclusions in [4], and then extend the results to the multiantenna case using these equivalences. Applying Theorem 2 in [8], the maximum transmission rate of each user (instead of each transmit antenna) with the constraint of equal rate is given by Ψ ((1 − q )(1 − x)) − Ψ(1 − q ) , 0<x≤1 (1 − q )βx
Rmax (q) = inf
(6)
where Ψ ((1 − Pout )(1 − x)) denotes the sum capacity (nor )(1 − x), which is malized by N ) of users 0 to 1 − (1 − Pout given by
= −
Ψ ((1 − q )(1 − x)) ψ(x,q ) g˜γη(γ, x) log 1 + g) nt β dFnr (˜ nt 0 log η(γ, x) + η(γ, x) − 1,
(7)
(1 − (1 − q )(1 − x)) and the multiuser where ψ(x, q ) Fn−1 r efficiency η(γ, x) is given by solving the following Tse-Hanly equation [11] ψ(x,q ) g˜η(γ, x)γ dFnr (˜ η(γ, x) + nt β g ) = 1. (8) n ˜η(γ, x)γ t+g 0 A. Low Energy Region In the low energy region, η(γ, x) → 1 as γ → 0. It is shown in [4] that the infimum of (6) is achieved at x = 0, and is given by
˙ Ψ((1 − q )(1 − x)) x=0 Rmax (q) = (1 − q )β η(γ, 0)˜ gcut γ = nt log 1 + , (9) nt 3 It should be noted that the outage here is defined for individual users, not for the whole system.
LI and POOR: SPECTRAL EFFICIENCY OF EQUAL-RATE DS-CDMA SYSTEMS WITH MULTIPLE TRANSMIT ANTENNAS
where η(γ, 0) is also a function of γ, which is determined by (8). ˙ Then we have C(0) = (1 − q)β˜ gcut , which results in Eb log 2 = , N0 min (1 − q)˜ gcut log 2
. (10) = 1 (1 − q)Fn−1 1 − (1 − q) nt r Due to
(9),
dη 2 dγ
we
can obtain
¨ C(0)
=
(1 −
− g˜cut . Taking the derivative on
g˜ dη both sides of (8), we have dγ = −β 0 cut g˜dFnr (˜ g ),
q)β˜ gcut
γ=0,x=0
γ=0,x=0
thus resulting in S0 =
2β(1 − q)˜ gcut . g˜cut 2β 0 g˜dFnr (˜ g ) + g˜cut
(11)
Eb An interesting conclusion drawn from (10) is that N 0 min increases monotonically with nt since g˜cut decreases with nt , which means that adding more transmit antennas decreases the system performance in the low energy region. Thus the negative impact of multiple transmit antennas on the spreading code space dominates, and the single-antenna system is optimal for the full-multiplexing transmitter in the low energy region. Also, the result implies the necessity of joint channel coding across different transmit antennas of a given user. An intuitive explanation for this phenomenon is based on the fact that, in full-multiplexing systems, the transmission reliability of a given user is determined by the transmit antenna with the weakest channel power gain, when the channel encoding is independent across different antennas. Thus, as nt increases, the expectation of the weakest channel power gain of a given user decreases, demanding more transmitting power to sustain reliable transmission and thereby decreasing the spectral efficiency. B. High Energy Region For the high energy region, we discuss the following two cases for the outage probability q, namely fixed positive q and arbitrarily small q which tends to 0 as γ → ∞. In the former case, the reliability of communication is not enhanced with the increase of transmit SNR γ, thus achieving no diversity gain; in the latter case, part of the gain of multiple transmit antennas is spent on the diversity gain. 1) Fixed Positive Outage Probability: We discuss the equivalent single-antenna systems first, and we assume that nt βq < 1. As γ → ∞, we can obtain that •
•
•
if nt β(1 − (1 − q )(1 − x)) < 1, then η → 1 − nt β(1 − (1 − q )(1 − x)) and Ψ((1 − q )(1 − x))=n ˙ t β(1 − (1 − q )(1 − x)) log γ;
if nt β(1 − (1 − q )(1 − x)) = 1, then η = O √1γ and Ψ((1 − q )(1 − x))= ˙ log γ;
if nt β(1 − (1 − q )(1 − x)) > 1, then η = O γ1 and also Ψ((1 − q )(1 − x))= ˙ log γ.
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Due to the assumption that nt βq < 1, we have Ψ(1 − q )=n ˙ t q β log γ. Substituting these expressions into (6), we can obtain that If β < n1t , then the infimum of (6)is attained at any 0 < x ≤ 1 and we have ˙ t log γ Rmax =n ; (12) S∞ = (1 − q)βnt
•
•
If β > n1t , then the infimum of (6) is achieved at x = 1. Thus, we have ⎧ t βq ⎨ Rmax = ˙ 1−n (1−q )β log γ . (13) ⎩ S∞ = (1−q)(1−n t βq ) (1−q ) When q is small enough, we have q ≈ nqt . Thus S∞ ≈ (1−q)(1−βq) for small outage probabilities. 1− q nt
An interesting observation on these results is that the spectral slope S∞ does not increase monotonically with respect to the number of transmit antennas. When nt > β1 , adding more transmit antennas undermines the system performance. This is due to the fact that S∞ for sum capacity increases linearly with respect to β and saturates at S∞ = 1 for overloaded systems (β > 1), as observed in [14]. Essentially, this phenomenon is due to our use of independent spreading codes for different transmit antennas. For overloaded systems, the negative impact on the spreading code space dominates, thus undermining the system performance. A conclusion of interest is that a single-antenna transmitter attains optimal spectral efficiency for overloaded equal-rate systems in the high energy region. 2) Arbitrarily Small Outage Probability: For the case of arbitrarily small outage probability, we define the diversity gain D (only receive diversity and no transmit diversity due to full multiplexing) and the multiplexing gain r by q = ˙ γ1D and ˙ log γ. The relationship between D and r is revealed Rmax =r by the equation in the following proposition. Proposition 3.1: For full-multiplexing systems with nt β < 1, the diversity gain D and the multiplexing gain r satisfy r D + = 1. nt (Θf + 1) nr Proof: From the definition of g˜cut , we have log g˜cut
= = ˙ = ˙ = ˙
(14)
log Fn−1 (q ) r
(q) log Fn−1 r 1 log Fn−1 r γD D log γ, − nr (Θf + 1)
where the last equation is due to equation (3). Since nt β < 1, we have η ≥ 1 − nt β > 0. When x > 0, with γ large enough, the first term in (7) dominates the quantity of Ψ((1 − q )(1 − x)), thus resulting in Ψ ((1 − q )(1 − x)) =n ˙ t β(1 − (1 − q )(1 − x)) log γ. Thus the increasing slope of Ψ (1 − (1 − q )(1 − x)) with respect to x is equal to nt , which means r = nt . Therefore, the multiplexing gain r is determined at x = 0. ˙ 1 , Ψ((1 − When D ≥ nr (Θf + 1), or equivalently g˜cut ≤ γ q )(1 − x)) = o(log γ) in the vicinity of x = 0, which
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implies no multiplexing gain. Thus, we consider only D < nr (Θf + 1), which means g˜cut γ → ∞ as γ → ∞. When γ is sufficiently large, (7) can be rewritten as Ψ ((1 − q )(1 − x)) = ˙ nt β(1 − (1 − q )(1 − x)) log γ ψ(x,q) 1 + nt β + g˜ Fnr (˜ log g )d˜ g. γ 0
−
D
In the moderate energy region, it is intractable to find an analytical expression for Ω(x, q). Thus, we discuss only the low and high energy regions. A. Low Energy Region
1 ) dψ (x, Pout = . ) f (˜ dx (1 − Pout nr gcut )
Eqn. (14) follows immediately. Using the above results, the spectral slope in the high energy region is given by D S∞ = n t β 1 − . (15) (Θf + 1) nr When nt β > 1, we can combine the above results and obtain 1 D , nt 1 − r = min . (16) β (Θf + 1) From (14), we can see that the maximum diversity gain in the full-multiplexing system is (Θf + 1) nr , which means that the multiple transmit antennas achieve no transmit gain for diversity (all diversity gain is from the multiple receive antennas), since the transmit antennas of a given user operate independently.
In the low energy region, η(γ, x) → 1 as γ → 0, which imnt log 1 + g˜n γη(γ,x) by plies that we can approximate n=1 nt γ nt ˜n . Thus the multi-antenna transmitter is equivalent n=1 g nt to a single-antenna one, whose channel power gain is equal nt to n1t n=1 g˜n . Hence, on denoting the CDF of equivalent channel power gain by Fˆ , where Fˆ (x) = Fnt nr (nt x), we have
1 nt −1 Ω(x, q) = {˜ gn }
gn < Fˆ (1 − (1 − x)(1 − q)) , nt n=1
nt and the users with n1t n=1 gn < Fˆ −1 (q) are announced as experiencing outage. We call this condition the sum criterion in this paper since the outage is determined by the sum of channel power gains of a given user. We can obtain results similar to (10) and (11) as follows. log 2 Eb , (19) = N0 min (1 − q)Fˆ −1 (q) and S0 =
IV. O PTIMAL S YSTEMS In the preceding section, we assume that each antenna of a given user operates independently, and thus all transmit antennas are used for multiplexing. However, for optimal systems without CSIT, cooperation, namely joint channel encoding, across different transmit antennas is indispensable; thus the multi-antenna transmitter cannot be converted into an equivalent single-antenna one. The sum capacity of users 1 to K is given by γ Ψsum = log det I + HHH , nt where H = (H1 , ..., HK ). Thus, the sum capacity is equivalent to that of a single-antenna system with nt K users. Similar to (6), the maximum transmission rate is given by Ψ (Ω(x, q)) − Ψ (Ω(0, q)) , (1 − q)βx
(18)
and the Tse-Hanly equation nt nt g˜n η(γ, x)γ (dFnr (˜ gn )) = 1. η(γ, x) + β ˜n η(γ, x)γ n=1 Ω(x,q) n=1 nt + g
where the second asymptotic equation is due to
0<x≤1
log η(γ, x) + η(γ, x) − 1,
Ω(x, q) = arg min Ψ(D) P (D) = 1 − (1 − x)(1 − q) ∈ Rnt ,
Consequently, the maximum rate is determined by
d
dx (Ψ((1 − q )(1 − x))) x=0 Rmax = β(1 − q ) = ˙ nt log γ + nt log g˜cut Dnt = ˙ nt log γ − log γ, (Θf + 1) nr
Rmax (q) = inf
where for any Borel set D ∈ Rnt , Ψ(D) is the sum capacity (normalized by N1 ) of users in D, which is given by nt nt g˜n γη(γ, x) log 1 + (dFnr (˜ gn )) Ψ(D) = β nt D n=1 n=1
(17)
2β(1 − q)Fˆ −1 (q) . Fˆ −1 (q) 2β 0 gdFˆ (g) + Fˆ −1 (q)
(20)
It is obvious that the spectral efficiency increases in the low energy region as nt increases. This means that optimal systems fully make use of the positive effect of multiple transmit antennas by alleviating the fluctuations in channel power gain, whereas the full-multiplexing systems cannot do this. B. High Energy Region 1) Fixed positive outage probability: For any fixed positive q, using the same approach as in Section III, we can obtain the following results: 1 • If β < n , we obtain the same result as in (12). t 1 • If β > n , we have t t βq ˙ 1−n Rmax = (1−q)β log γ . (21) S∞ = 1 − nt βq
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2) Arbitrarily small outage probability: For an arbitrarily small q, we assume that nt β < 1. Since η(x) > 1 − nt β, the last two terms in (18) can be neglected in the high energy region; thus we can obtain the optimal outage user set Ω(x, q) in the high energy region as follows:
gn }) < FS−1 (1 − (1 − x)(1 − q)) , (22) Ω(x, q) = {˜ gn } S({˜
t log γ1 + g˜n and FS is the CDF of where S({˜ gn }) = nn=1 the random variable S({˜ gn }). When γ is sufficiently large, the optimal outage set comprises the users with small product of channel power gains and we call this the product criterion. In this case, Ψ(Ω(x, q)) is given by Ψ(Ω(x, q))
= ˙ +
nt β(1 − (1 − q)(1 − x)) log γ FS−1 (1−(1−x)(1−q)) sdFS (s). β 0
Similarly to the preceding section, the maximal transmission rate is determined at x = 0 and
1 dΨ(Ω(x, q))
−1 = ˙ nt β log γ + βFS . (23)
dx γD x=0
The following lemma provides an upper bound on the diversity gain D. Lemma 4.1: For any fading distribution, D ≤ nt nr (Θf + 1). Proof: Suppose D > nt nr (Θf + 1). Then nt 1 1 ˙ < γD γ nr (Θf +1) nt 1 = ˙ P g˜n < , γ n=1 where the second
asymptotic equation is due to (3). ˙ − nt log γ, then If FS−1 γ1D ≥ 1 1 −1 = P {˜ g } |S (˜ g ) ≤ F n n S γD γD
1 ˙ P {˜ ≥ gn }
g˜n ≤ , ∀n γ nt 1 = P g˜n < γ n=1 ˙ >
1 , γD
which results in a contradiction. Therefore D >
˙ t log γ, which results in nt nr (Θf + 1) implies Fs−1 γ1D 0 such that hX (x) feXmx is 4 continuous and satisfies the following conditions: • there exist nX > 0 and cX > 0 so that nX ˙ X (x) 0 and aX > 0 so that (x) ˙ fxXm−1 ˙ 0 so that xm−1 <x x → ∞. 2) X + Y is m + n − Γ − like distributed. 2 We can show that the power gains, namely |gkmn | , of Rayleigh, Ricean and Nakagami-d fading channels are all Θf + 1 − Γ − like distributed. For the case of nr receive antennas, the channel power gains of these fading channels are nr (Θf + 1) − Γ − like distributed. Based on the above definitions and lemmas, we can obtain a relationship between D and r. Proposition 4.6: If the channel power gain is m − Γ − like distributed, then we have D r = 1. (25) + nt nt nr m Proof: When D = nt nr m, we have nt 1 1 = ˙ P g˜n < . γ D n=1 γ −1 (x)= ˙ FX
Thus, FS−1
1 γD
= ˙ − nt log γ,
which results in r = 0. 4 The
subscript X in nX , cX and mX denotes the dependence on X.
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When D < nt nr m, there exists c > 0 so that, ∀γ > c, nt 1 1 > P g ˜ ≤ . n γD γ n=1
25
Then FS−1
1 γD
−1
= ˙
F
= ˙
D log γ, nr m
nt n=1
log g ˜n
1 γD
15
10
5
which results in r = nt − nD and then (25). rm For Rayleigh, Ricean and Nakagami-d fading channels, (25) can be written in a form similar to (14): r D = 1, + nt nt nr (Θf + 1)
20 Spectral Efficiency
Thus, ∀γ > c,
1 1
gn } < D = φ. gn } S {˜ {˜ gn } g˜n ≥ , n = 1, ..., nt ∩ {˜ γ γ
nt=1 nt=2 nt=4 nt=8
(26)
0 −20
0
20
40 average E /N (dB) b
60
80
100
0
Eb Fig. 2. Spectral efficiency versus N with various numbers of transmit 0 antennas in full-multiplexing systems.
16
and the corresponding spectral slope in the high energy region is given by D S∞ = β n t − . (27) (Θf + 1) nr
V. N UMERICAL R ESULTS We now summarize some numerical results that illustrate the analytical results of this paper. We set the outage probability q = 0.05, the number of receive antennas nr = 4, and consider Rayleigh fading in all these results. It should be noted that all numerical results are computed with asymptotic expressions; thus being independent of K and N .
12
Spectral Efficiency
For systems with βnt > 1, we can obtain results similar to (16). We can see that optimal systems achieve a better tradeoff than full-multiplexing systems (for the same diversity gain, the optimal systems achieve higher multiplexing gain). On D , the defining the transmit diversity gain as Dt = (Θf +1)n r transmit diversity-multiplexing tradeoff equation is given by r + Dt = nt , which is intuitive and identical to that of single-user narrow band systems in Rayleigh fading channels [15] with large numbers of receive antennas. The differences between CDMA and narrow-band systems are as follows • The diversity-multiplexing tradeoff is linear for CDMA systems and piecewise-linear for narrow-band systems. • CDMA systems can still achieve multiplexing gain when nr = 1 while this is impossible for narrow band systems. Intuitively, both differences are due to the use of spreading codes in CDMA systems since we can consider the different chips in the spreading codes as being equivalent to different receive antennas. Therefore, the number of equivalent receive antennas is nr N , which is much larger than nt for large systems. Also, the tradeoff equation for narrow-band systems given in [15], namely D = (nr −r)(nt −r), when nr , nt ∈ N, is equivalent to nD + r = nt in CDMA systems with rN sufficiently large N , thus resulting in the linear equation D t + r = nt .
14
10
8
6
4 β=0.4 β=0.8 β=1.2 β=1.6
2
0 −10
−5
0
5
10
15
20
average E b/N0(dB)
Fig. 3. Spectral efficiency versus multiplexing systems.
Eb N0
with various system loads in full-
Eb Figure 2 shows the spectral efficiency versus N for various 0 values of nt in full-multiplexing systems, where β = 0.5. We can see that, in the low energy region, the spectral efficiency decreases as nt increases, which demonstrates the conclusion in Section III-A. In the moderate energy region, adding more transmit antennas can improve the spectral efficiency. However, when the equivalent system is overloaded (nt ≥ 3), the corresponding performance gain is marginal. Also, we notice that for overloaded systems (nt = 4, 8) in the high Eb energy region ( N ≥ 90dB), the spectral efficiency decreases 0 as nt increases. Figure 3 shows the spectral efficiency for various values of β in full-multiplexing systems, when nt = 2. For the case of sum capacity, the spectral efficiency increases with β and saturates for large β [7]. However, as can be observed from Fig. 3, for equal-rate systems in the moderate energy region, the spectral efficiency saturates rapidly at β = 1, which implies that the maximal transmission rate is inversely
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12
VI. C ONCLUSIONS
10
8 Spectral Efficiency
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6
4
sum criterion product criterion full multiplexing
2
0 −10
−5
0
5
10
15
average E /N (dB) b
0
Fig. 4. Comparison between optimal systems and full-multiplexing systems.
14
12
In this paper, we have analyzed the spectral efficiency of uplink DS-CDMA systems with multiple transmit antennas. A key assumption is that different spreading codes are assigned to different antennas of a given user. The conclusions of this analysis are given as follows. • For full multiplexing systems, adding more transmit antennas will decrease the spectral efficiency for any systems in the low energy region and for overloaded systems in the high energy region. For other situations, multiple transmit antennas always benefit the system performance. • For optimal systems, multiple transmit antennas always attain higher spectral efficiency, except for overloaded systems in the high energy region. Compared with fullmultiplexing systems, cooperation across transmit antennas results in considerably higher spectral efficiency in the low energy region and better diversity-multiplexing tradeoff in the high energy region. For overloaded systems in the high energy region, it may be better to consider using the same spreading codes for different antennas of a given user in order to exploit the gain of multiple transmit antennas.
Spectral Efficiency (bps/Hz)
10
A PPENDIX I P ROOF OF L EMMA IV.3
8
Proof: (1) Taking logarithms on both sides of x FX (x) = emt hX (t)dt,
6
4
−∞
0 −10
we obtain
n t=1 n t=2 n t=3
2
0
10
20
30
40
50
average E b/N0(dB)
Fig. 5. Spectral efficiency versus of antennas in optimal systems.
Eb N0
with various number of transmit
proportional to β for overloaded systems. Eb The spectral efficiency versus N for optimal systems and 0 full-multiplexing systems is given in Fig. 4, where β = 0.4, nt = 2 and nr = 4. In Fig. 4, ‘sum criterion’ represents sorting the users by the sum of channel power gains and ‘product criterion’ represents sorting them by the product of channel power gains, which are optimal in the low energy region and the high energy region, respectively, as explained in Section IV. We observe that these two criteria achieve almost the same performance. The full-multiplexing scheme attains considerably lower spectral efficiency than optimal systems in the low energy region. Eb for optimal Figure 5 shows the spectral efficiency versus N 0 systems when β = 2. We can see that, in the high energy region, the spectral efficiency decreases with the number of transmit antennas. This coincides with our conclusion in Section IV-B.
log log FX (x) =1+ mx
x m(t−x) hX (t)dt −∞ e
mx
.
Applying the upper and lower bounds on hX (x) as x → −∞ in Definition IV.2, it is easy to check that the second term in the right hand side of the above equation converges to 0 as x → 0. (2)The PDF of Z = X + Y is given by ∞ fZ (y) = fX (x)fY (y − x)dx −∞ ∞ = emy hX (x)hY (y − x)dx, −∞
which results in hZ (y) =
∞
−∞
hX (x)hY (y − x)dx.
The two conditions of hX in Definition IV.2 are equivalent to the following two conditions: • There exist nX > 0, cX > 0 and Δ1X < 0 so that cX < hX (x) < (−x)nX , ∀x < Δ1X ; • There exist lX > m and Δ2X > 0 so that hX (x) < e−lX x , ∀x > Δ2X . Similarly, we can also define nY , cY , lY , Δ1Y and Δ2Y for hY such that the corresponding inequalities hold.
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Therefore, when y < Δ1X + Δ1Y , hZ (y) can be obtained by integrating over five intervals, namely y−Δ2Y y−Δ1Y hZ (y) = + −∞ Δ1X
+
y−Δ2Y Δ2X
+
∞
+ Δ2X
Δ1X
y−Δ1Y
(hX (x)hY (y − x)) dx.
It is easy to check that, as y → −∞, ∀ > 0, the terms in the above integral have the following properties: n + • the first two terms are upper bounded by (−y) X ; nX +nY +1+ • the third term is upper bounded by (−y) ; n + • the last two terms are upper bounded by (−y) Y . nX +nY +1+ ˙ Thus, as y → −∞, hZ (y) 0. Also, we can show that the third term is lower bounded by cX cY (Δ1X + Δ1Y − Δ) when y < Δ1X + Δ1Y . When y > Δ2X + Δ2Y , we have Δ1X Δ2X y−Δ2Y hZ (y) = + + −∞ y−Δ1Y
+
Δ1X ∞
+ y−Δ2Y
[7] S. Shamai and S. Verd´u, “The impact of frequency-flat fading on the spectral efficiency of CDMA,” IEEE Trans. Inf. Theory, vol. 47, pp. 1302-1327, May 2001. [8] S. Shamai and S. Verd´u, ”Decoding only the strongest CDMA users,” Codes, Graphs, and Systems, R. Blahut and R. Koetter, Eds., pp. 217228. Norwell, MA: Kluwer Academic Publishers, 2002. [9] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE Trans. Inf. Theory, vol. 45, pp. 1456-1467, July 1999. [10] E. Telatar, “Capacity of multi-antenna Gaussian channels”, European Trans. Telecommun., vol. 10, no.6, pp. 585–596, Nov. 1999. [11] D. Tse and S. Hanly, “Linear multiuser receivers: effective interference, effective bandwidth and user capacity,” IEEE Trans. Inf. Theory, vol. 45, pp. 641-657, Mar. 1999. [12] D. N. C. Tse, P. Viswanath, and L. Zheng, “Diversity-multiplexing tradeoff in multiple access channels,” IEEE Trans. Inf. Theory, Vol. 50, pp. 1859-1874, Sept. 2004. [13] S. Verd´u, Multiuser Detection. Cambridge University Press, Cambridge, UK, 1998. [14] S. Verd´u, “Spectral efficiency in the wideband regime,” IEEE Trans. Inf. Theory, vol. 48, pp. 1319-1343, June 2002. [15] L. Zheng and D. N. C. Tse, “Diversity and multiplexing: a fundamental tradeoff in multiple-antenna channels,” IEEE Trans. Inf. Theory, vol. 49, pp. 1073-1096, May 2003.
Δ2X
(hX (x)hY (y − x)) dx.
y−Δ1Y
It is easy to show that, as y → ∞, for any 0 < < min (lX , lY ), the terms in this integral have the following properties: −(lY −)y • the first two terms are upper bounded by e ; −(min(nX ,nY )−) • the third term is upper bounded by e ; −(lX −)y • the last two terms are upper bounded by e ; ˙ −(min(nX ,nY )−) , for some Thus, as y → ∞, hZ (y)<e sufficiently small . Applying Definition IV.2, Z is therefore m − log Γ − like distributed. R EFERENCES [1] T. Cover and J. A. Thomas, Elements of Information Theory. New York: John Wiley and Sons Inc., 1991. [2] G. J. Foschini,“Layered space-time architechture for wireless communication in fading environments when using multi-element antennnas,” Bell Labs Tech. J., pp. 41-59, 1996. [3] S. V. Hanly and D. N. C. Tse, “Resource pooling and effective bandwidths in CDMA networks with multiuser receivers and spatial diverisity,” IEEE Trans. Inf. Theory, vol. 47, pp. 1328-1351, May 2001. [4] H. Li and H. V. Poor, “Power allocaiton and spectral efficiency of DSCDMA systems in fading channels with fixed QoS - part 1: singlerate case,” IEEE Trans. Wireless Commun., vol. 5, pp. 2516-2528, Sept. 2006. [5] A. Mantrovadi, V. V. Veeravalli, and H. Viswanathan, “Spectral efficiency of MIMO multiaccess systems with single-user decoding,” IEEE J. Sel. Areas Commun., vol. 21, pp. 382-394, Apr. 2003. [6] C. Papadias, “On the spectral efficiency of space-time spreading schemes for multiple antenna CDMA systems,” in Proc. 33rd Asilomar Conference on Signals, Systems, and Computers 1999.
Husheng Li (S’00-M’05) received the B.S. and M.S. degrees in electronics engineering from Tsinghua University, Beijing, China, in 1998 and 2000, respectively, and the Ph.D. degrees in electrical engineering from Princeton University, NJ, in 2005. In 2005, he joined Qualcomm Inc. CA. His research interests include statistical signal processing, wireless communication and information theory.
H. Vincent Poor (S’72-M’77-SM’82-F’87) received the Ph.D. degree in EECS from Princeton University in 1977. From 1977 until 1990, he was on the faculty of the University of Illinois at Urbana-Champaign. Since 1990 he has been on the faculty at Princeton, where he is the Dean of Engineering and Applied Science, and the Michael Henry Strater University Professor of Electrical Engineering. He has also held visiting appointments at a number of universities, including recently Imperial College, Stanford and Harvard. Dr. Poor’s research interests are in the areas of wireless networks, advanced signal processing and related fields. Among his publications in these areas is the forthcoming book MIMO Wireless Communications (Cambridge, 2007). Dr. Poor is a member of the National Academy of Engineering, and is a Fellow of the American Academy of Arts and Sciences, the Institute of Mathematical Statistics, the Optical Society of America, and other organizations. He is a past President of the IEEE Information Theory Society, and is the current Editor-in-Chief of the IEEE Transactions on Information Theory. Recent recognition of his work includes the Joint Paper Award of the IEEE Communications and Information Theory Societies (2001), the NSF Director’s Award for Distinguished Teaching Scholars (2002), a Guggenheim Fellowship (2002-03), and the IEEE Education Medal (2005).
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 12, DECEMBER 2006
Spectral Efficiency of Equal-Rate DS-CDMA Systems with Multiple Transmit Antennas Husheng Li, Member, IEEE, and H. Vincent Poor, Fellow, IEEE
Abstract— The spectral efficiency of equal-rate DS-CDMA systems with multiple transmit antennas is analyzed. Explicit expressions for the spectral efficiency are obtained in the low and high energy regions and numerical results are presented for the moderate energy region. The performance gains resulting from the use of multiple transmit antennas are discussed. An equation characterizing the diversity-multiplexing tradeoff in this situation is derived. A conclusion that adding more transmit antennas does not necessarily improve the performance with the constraint of equal rate is drawn. Index Terms— Direct sequence code division multiple access (DS-CDMA), multiple-input multiple-output (MIMO), spectral efficiency.
I. I NTRODUCTION
T
HE use of multiple transmit antennas plays a major role in high data rate wireless communication systems due to the resulting two-fold improvement in the system performance arising from multiplexing and diversity. For multiplexing, it was shown in [10] that the channel capacity in the high energy region is scaled by the number of transmit antennas, provided that the number of receive antennas is sufficiently large. In this context, the well known BLAST receiver [2] has been shown to achieve high data transmission rate for practical applications. Space-time codes were proposed in [9] as a means to achieve full transmit diversity, which enhances the transmission reliability. The tradeoff between multiplexing gain and diversity gain is studied for single-user systems in [15] and for multiple access channels in [12]. However, the analysis in the above papers is based on narrow band systems. For direct sequence code division multiple access (DS-CDMA) systems with multiple transmit antennas, the sum capacity is studied in [5] and the interesting conclusion is drawn that the performance gains in equal power uplink CDMA systems are more sensitive to the number of receive antennas than to the number of transmit antennas. This observation motivates the study in this paper of the performance gain obtained by adding multiple antennas to the transmitting terminals in uplink CDMA systems. Our discussion is focused on equal-rate systems without channel state information at transmitters (CSIT), which is well-suited for practical systems and can be extended to multi-rate systems.
Manuscript received January 26, 2005; revised January 14, 2006; accepted March 29, 2006. The associate editor coordinating the review of this paper and approving it for publication was M. Sawahashi. This research was supported in part by the National Science Foundation under Grant No. ANI-03-38807, and in part by the New Jersey Center for Wireless Telecommunications. H. Li is with Qualcomm Inc., CA (email: [email protected]). H. V. Poor is with the Department of Electrical Engineering, Princeton University, Princeton, NJ (email: [email protected]). Digital Object Identifier 10.1109/TWC.2006.05057.
We consider both full-multiplexing and optimal systems. We will examine the performance of such systems as measured by the corresponding spectral efficiency, particularly in the low and high energy regions. In each case, we develop analytical expressions for the spectral efficiency parameters, namely the minimum energy per bit and spectral slopes in the low and high energy regions [14], and discuss the performance gain provided by multiple transmit antennas. Similarly to the diversity-multiplexing tradeoff discussed in [15] and [12], we also develop equations that describe the diversity-multiplexing tradeoff for full-multiplexing systems and optimal systems, respectively. It should be noted that our discussion is based on asymptotic analysis, which means that the system size tends to infinity while the system load is fixed. In summary, we tackle the following two aspects of multiple transmit antenna CDMA systems: • The spectral efficiency of equal-rate systems without CSIT and the performance gain due to the use of multiple transmit antennas • The diversity-multiplexing tradeoff in equal-rate systems The remainder of this paper is organized as follows. The signal model and spectral efficiency parameters are introduced in Section II. Full-multiplexing systems and optimal systems are discussed in Section III and Section IV, respectively. Simulation results and conclusions are given in Section V and Section VI, respectively. II. S IGNAL M ODEL AND S PECTRAL E FFICIENCY A. Signal Model In this paper, we consider synchronous single-cell uplink DS-CDMA systems with nt transmit antennas per user, nr receive antennas, K active users, spreading gain N , and system load β = K N . The system is illustrated in Fig. 1. We denote the binary spreading code used for the n-th transmit antenna of the k-th user by skn , which satisfies skn 2 = 1. We assume that signals are transmitted through frequency flat fading channels and the complex channel amplitude gain between the n-th transmit antenna of user k and the m-th receive antenna is denoted by gkmn . These gains are assumed to be identically distributed and mutually independent with unit variance. The probability density function (PDF) and cumulative distribution function (CDF) of channel power gain |gkmn |2 are denoted by f and F , respectively. To simplify the analysis, we can convert the multiple receive antenna system into an equivalent system with a single-antenna receiver, by merely replacing f and F with fnr and Fnr , which denote the PDF and CDF of the sum of nr independent random variables distributed with PDF f . This equivalence can be
c 2006 IEEE 1536-1276/06$20.00
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Single cell. Usually, there are multiple cells in cellular systems. However, the inter-cell interference can be incorporated into noise on assuming that soft handoff is not used. Using the above notation and simplifications, the outputs of a chip matched filter at the equivalent single-antenna receiver during a symbol period can be sampled at the chip rate to give an N -vector r of observations: •
User 1
...
1
nt ...
User K
...
1
...
1
nr
r =
nt
=
nt K
gkn skn xkn + n
k=1 n=1 K
Hk xk + n,
(1)
k=1
Fig. 1.
Illustration of uplink MIMO CDMA systems.
justified by the resource pooling effect in [3]. It should be noted that we do not confine our discussion to any specific fading distribution. We assume that CSIT is not available; thus all users transmit with the same power, regardless of the channel conditions. In this paper, we consider only the case of assigning independent random spreading codes to the different transmit antennas of a given user. Thus, adding more transmit antennas has a two-fold impact, namely increasing the number of degrees of freedom in the fading channels and decreasing the number of degrees of freedom in the spreading code space. Thus, as we will see, adding more transmit antennas does not necessarily imply an improvement of system performance, as there is a tradeoff between these two effects. It should be noted that we adopt some assumptions which may not be exact in practical systems. However, these assumptions simplify the analysis and provide insight for practical systems. Some comments on the validity of these assumptions are given as follows. • Fading channels. A typical CDMA system usually experience frequency selective fading channels, instead of frequency flat fading channels. However, it is difficult to obtain explicit expressions for frequency selective fading channels. Therefore, we adopt the assumption of frequency flat channels simplifying the analysis, similar to [5] and [7]. • Synchronicity. For many CDMA systems, such as CDMA2000 and W-CDMA, the transmissions of different users in the uplink are asynchronous. Again, we assume the synchrony in order to simplify the analysis, similar to [3][5][7][9]. It should be noted that the assumption of synchrony is valid for time division duplexing (TDD) systems, such as TDS-CDMA systems. • Random spreading code. This assumption is also used in [5][6]. An alternative scheme is to use the same spreading code for different transmit antennas of the same user. Then, fewer spreading codes are used at the cost of causing more interference (the same as narrow band systems) across the signals transmitted from different antennas of the same user. The discussion of the same code scheme is beyond the scope of this paper.
where gkn is the channel amplitude gain of the equivalent model, n is additive complex white Gaussian noise with variance σn2 , {xkn } are the transmitted channel symbols and1 Hk xk
= =
(gk1 sk1 , ..., gknt sknt ) , T
(xk1 , ..., xknt ) ,
k = 1, ..., K,
k = 1, ..., K,
1 where the constraint on xk is given by E{xk xH k } = nt Int ×nt . Thus, the transmitting signal-to-noise ratio (SNR) is given by γ = σ12 . Throughout this paper, we assume that the receiver n has perfect channel state information, thus enabling coherent detection and decoding. Note that the supports of f and F are in [0, ∞). To analyze the performance with small outage probability in the high energy region, it is of interest to study the behavior of fm (x) and Fm (x) in the vicinity of x = 0. Supposing that f (x) is infinitely differentiable at x = 0, we define Θf min n : f [n] (0) = 0 ,
where f [n] denotes the n-th derivative of f . It is easy to verify that ΘFm = Θfm + 1. Then we have the following lemma Lemma 2.1: For any m > 0, Θfm = m (Θf + 1) − 1 . (2) ΘFm = mΘF Proof: We can actually show a more generous conclusion. Let X and Y be two independent nonnegative random variables distributed according to the PDFs f and h, respectively. Denoting the PDF of random variable Z = X +Y by g, we can show Θg = Θf + Θh + 1. The proof is straightforward by applying the fact that s g(s) = f (t)h(s − t)dt. 0
Taking the n-th derivative on both sides, we have s n−1 [n] [n−1−k] [k] g (s) = f (s)h (0) + f (t)h[n] (s − t)dt. 0
k=0
Thus, we have g [n] (0) =
n−1
f [n−1−k] (0)h[k] (0),
k=0 1 The superscript T denotes transposition and the superscript H denotes conjugate transposition.
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which implies Θg = Θf + Θh + 1. So, by induction on m, we have Θfm = m (Θf + 1) − 1. The conclusion for ΘFm follows immediately since ΘFm = Θfm + 1. Thus when x is sufficiently small, we have the following approximations due to Taylor’s expansion:
fm (x) = C1 xm(Θf +1)−1 +o xm(Θf +1)−1 , Fm (x) = C2 xm(Θf +1) + o xm(Θf +1)
(3)
where C1 and C2 are positive constants. It is easy to show that for both Rayleigh and Ricean fading channels, Θf = 0, and for Nakagami-d fading channels, Θf = d−1. In the following sections, we denote |g|2 by g˜ for notational simplicity, since we are only concerned with the the channel power gain, and the corresponding PDF and CDF are given by fnr and Fnr .
B. Spectral Efficiency Spectral efficiency is defined as C=
K R = βR, N
(4)
where R is the information transmission rate (in nats per symbol). We denote the spectral efficiency (measured in nats/s/Hz) as a function of the transmit SNR γ as C(γ). The relationship among the energy per information bit, the spectral efficiency and SNR is given by βγ log 2 Eb , = N0 C
(5)
where N0 is the one-sided noise spectral density which equals σn2 . An explicit expression for C as a function of β and Eb N0 is difficult to obtain. However, the following important parameters can be computed with the tools developed in [14]2 :
Eb Eb • = β˙log 2 , which represents the minimal N N0 0 C(0) min required for reliable communications. 2 C˙ (0) , which quantifies the increasing slope of • S0 = −2 ¨ C(0) the spectral efficiency in the low energy region. ˙ ), which quantifies the increasing • S∞ = limP →∞ P C(P slope of the spectral efficiency in the high energy region. Since our discussion on spectral efficiency is based on asymptotic analysis, we adopt the following notations of asymptotic equality and inequality. • •
•
log f (x) log h(x)
→ 1 as x → ∞;
(x)) ˙ ˙ f (x)>h(x) and f (x)1 log(h(x))
log(f (x)) and lim log(h(x)) < 1, respectively;
(x)) ˙ ˙ ≥1 f (x)≥h(x) and f (x)≤h(x) means lim log(f log(h(x))
log(f (x)) and lim log(h(x)) ≤ 1, respectively. f (x)=h(x) ˙ means
2 Throughout
this paper, logarithms are base e.
III. F ULL - MULTIPLEXING S YSTEMS In this section, we assume that the information symbols are multiplexed, channel coded and transmitted independently by different transmit antennas of a given user, which is similar to BLAST systems; thus the multiple transmit antennas are used exclusively for multiplexing. We call this a fullmultiplexing system. Since the signals from some transmit antennas experience deep fading, this system demands fairly high transmitter power to achieve reliable communications for all users. Thus, it is reasonable that the receiver decodes only the users with good channel conditions, and announces the rest of the users as experiencing an outage [8]. We denote this outage probability by q. In the single-antenna systems, a user experiences outage when its channel power gain is lower than a cutoff channel gain 3 . However, as we will see, determining an outage is more complicated for multiple-antenna systems. Since each transmit antenna operates independently, the multi-antenna system is equivalent to a single-antenna system with system load nt β, transmitting SNR nγt , outage probability gcut ), q and cutoff channel gain g˜cut , which satisfies q = Fnr (˜ except for the outage probability. A user experiences outage if any of its antennas falls in the outage set of the equivalent single-antenna system, which implies that q = 1 − (1 − q )nt . Thus we consider single-antenna systems first by applying the conclusions in [4], and then extend the results to the multiantenna case using these equivalences. Applying Theorem 2 in [8], the maximum transmission rate of each user (instead of each transmit antenna) with the constraint of equal rate is given by Ψ ((1 − q )(1 − x)) − Ψ(1 − q ) , 0<x≤1 (1 − q )βx
Rmax (q) = inf
(6)
where Ψ ((1 − Pout )(1 − x)) denotes the sum capacity (nor )(1 − x), which is malized by N ) of users 0 to 1 − (1 − Pout given by
= −
Ψ ((1 − q )(1 − x)) ψ(x,q ) g˜γη(γ, x) log 1 + g) nt β dFnr (˜ nt 0 log η(γ, x) + η(γ, x) − 1,
(7)
(1 − (1 − q )(1 − x)) and the multiuser where ψ(x, q ) Fn−1 r efficiency η(γ, x) is given by solving the following Tse-Hanly equation [11] ψ(x,q ) g˜η(γ, x)γ dFnr (˜ η(γ, x) + nt β g ) = 1. (8) n ˜η(γ, x)γ t+g 0 A. Low Energy Region In the low energy region, η(γ, x) → 1 as γ → 0. It is shown in [4] that the infimum of (6) is achieved at x = 0, and is given by
˙ Ψ((1 − q )(1 − x)) x=0 Rmax (q) = (1 − q )β η(γ, 0)˜ gcut γ = nt log 1 + , (9) nt 3 It should be noted that the outage here is defined for individual users, not for the whole system.
LI and POOR: SPECTRAL EFFICIENCY OF EQUAL-RATE DS-CDMA SYSTEMS WITH MULTIPLE TRANSMIT ANTENNAS
where η(γ, 0) is also a function of γ, which is determined by (8). ˙ Then we have C(0) = (1 − q)β˜ gcut , which results in Eb log 2 = , N0 min (1 − q)˜ gcut log 2
. (10) = 1 (1 − q)Fn−1 1 − (1 − q) nt r Due to
(9),
dη 2 dγ
we
can obtain
¨ C(0)
=
(1 −
− g˜cut . Taking the derivative on
g˜ dη both sides of (8), we have dγ = −β 0 cut g˜dFnr (˜ g ),
q)β˜ gcut
γ=0,x=0
γ=0,x=0
thus resulting in S0 =
2β(1 − q)˜ gcut . g˜cut 2β 0 g˜dFnr (˜ g ) + g˜cut
(11)
Eb An interesting conclusion drawn from (10) is that N 0 min increases monotonically with nt since g˜cut decreases with nt , which means that adding more transmit antennas decreases the system performance in the low energy region. Thus the negative impact of multiple transmit antennas on the spreading code space dominates, and the single-antenna system is optimal for the full-multiplexing transmitter in the low energy region. Also, the result implies the necessity of joint channel coding across different transmit antennas of a given user. An intuitive explanation for this phenomenon is based on the fact that, in full-multiplexing systems, the transmission reliability of a given user is determined by the transmit antenna with the weakest channel power gain, when the channel encoding is independent across different antennas. Thus, as nt increases, the expectation of the weakest channel power gain of a given user decreases, demanding more transmitting power to sustain reliable transmission and thereby decreasing the spectral efficiency. B. High Energy Region For the high energy region, we discuss the following two cases for the outage probability q, namely fixed positive q and arbitrarily small q which tends to 0 as γ → ∞. In the former case, the reliability of communication is not enhanced with the increase of transmit SNR γ, thus achieving no diversity gain; in the latter case, part of the gain of multiple transmit antennas is spent on the diversity gain. 1) Fixed Positive Outage Probability: We discuss the equivalent single-antenna systems first, and we assume that nt βq < 1. As γ → ∞, we can obtain that •
•
•
if nt β(1 − (1 − q )(1 − x)) < 1, then η → 1 − nt β(1 − (1 − q )(1 − x)) and Ψ((1 − q )(1 − x))=n ˙ t β(1 − (1 − q )(1 − x)) log γ;
if nt β(1 − (1 − q )(1 − x)) = 1, then η = O √1γ and Ψ((1 − q )(1 − x))= ˙ log γ;
if nt β(1 − (1 − q )(1 − x)) > 1, then η = O γ1 and also Ψ((1 − q )(1 − x))= ˙ log γ.
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Due to the assumption that nt βq < 1, we have Ψ(1 − q )=n ˙ t q β log γ. Substituting these expressions into (6), we can obtain that If β < n1t , then the infimum of (6)is attained at any 0 < x ≤ 1 and we have ˙ t log γ Rmax =n ; (12) S∞ = (1 − q)βnt
•
•
If β > n1t , then the infimum of (6) is achieved at x = 1. Thus, we have ⎧ t βq ⎨ Rmax = ˙ 1−n (1−q )β log γ . (13) ⎩ S∞ = (1−q)(1−n t βq ) (1−q ) When q is small enough, we have q ≈ nqt . Thus S∞ ≈ (1−q)(1−βq) for small outage probabilities. 1− q nt
An interesting observation on these results is that the spectral slope S∞ does not increase monotonically with respect to the number of transmit antennas. When nt > β1 , adding more transmit antennas undermines the system performance. This is due to the fact that S∞ for sum capacity increases linearly with respect to β and saturates at S∞ = 1 for overloaded systems (β > 1), as observed in [14]. Essentially, this phenomenon is due to our use of independent spreading codes for different transmit antennas. For overloaded systems, the negative impact on the spreading code space dominates, thus undermining the system performance. A conclusion of interest is that a single-antenna transmitter attains optimal spectral efficiency for overloaded equal-rate systems in the high energy region. 2) Arbitrarily Small Outage Probability: For the case of arbitrarily small outage probability, we define the diversity gain D (only receive diversity and no transmit diversity due to full multiplexing) and the multiplexing gain r by q = ˙ γ1D and ˙ log γ. The relationship between D and r is revealed Rmax =r by the equation in the following proposition. Proposition 3.1: For full-multiplexing systems with nt β < 1, the diversity gain D and the multiplexing gain r satisfy r D + = 1. nt (Θf + 1) nr Proof: From the definition of g˜cut , we have log g˜cut
= = ˙ = ˙ = ˙
(14)
log Fn−1 (q ) r
(q) log Fn−1 r 1 log Fn−1 r γD D log γ, − nr (Θf + 1)
where the last equation is due to equation (3). Since nt β < 1, we have η ≥ 1 − nt β > 0. When x > 0, with γ large enough, the first term in (7) dominates the quantity of Ψ((1 − q )(1 − x)), thus resulting in Ψ ((1 − q )(1 − x)) =n ˙ t β(1 − (1 − q )(1 − x)) log γ. Thus the increasing slope of Ψ (1 − (1 − q )(1 − x)) with respect to x is equal to nt , which means r = nt . Therefore, the multiplexing gain r is determined at x = 0. ˙ 1 , Ψ((1 − When D ≥ nr (Θf + 1), or equivalently g˜cut ≤ γ q )(1 − x)) = o(log γ) in the vicinity of x = 0, which
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implies no multiplexing gain. Thus, we consider only D < nr (Θf + 1), which means g˜cut γ → ∞ as γ → ∞. When γ is sufficiently large, (7) can be rewritten as Ψ ((1 − q )(1 − x)) = ˙ nt β(1 − (1 − q )(1 − x)) log γ ψ(x,q) 1 + nt β + g˜ Fnr (˜ log g )d˜ g. γ 0
−
D
In the moderate energy region, it is intractable to find an analytical expression for Ω(x, q). Thus, we discuss only the low and high energy regions. A. Low Energy Region
1 ) dψ (x, Pout = . ) f (˜ dx (1 − Pout nr gcut )
Eqn. (14) follows immediately. Using the above results, the spectral slope in the high energy region is given by D S∞ = n t β 1 − . (15) (Θf + 1) nr When nt β > 1, we can combine the above results and obtain 1 D , nt 1 − r = min . (16) β (Θf + 1) From (14), we can see that the maximum diversity gain in the full-multiplexing system is (Θf + 1) nr , which means that the multiple transmit antennas achieve no transmit gain for diversity (all diversity gain is from the multiple receive antennas), since the transmit antennas of a given user operate independently.
In the low energy region, η(γ, x) → 1 as γ → 0, which imnt log 1 + g˜n γη(γ,x) by plies that we can approximate n=1 nt γ nt ˜n . Thus the multi-antenna transmitter is equivalent n=1 g nt to a single-antenna one, whose channel power gain is equal nt to n1t n=1 g˜n . Hence, on denoting the CDF of equivalent channel power gain by Fˆ , where Fˆ (x) = Fnt nr (nt x), we have
1 nt −1 Ω(x, q) = {˜ gn }
gn < Fˆ (1 − (1 − x)(1 − q)) , nt n=1
nt and the users with n1t n=1 gn < Fˆ −1 (q) are announced as experiencing outage. We call this condition the sum criterion in this paper since the outage is determined by the sum of channel power gains of a given user. We can obtain results similar to (10) and (11) as follows. log 2 Eb , (19) = N0 min (1 − q)Fˆ −1 (q) and S0 =
IV. O PTIMAL S YSTEMS In the preceding section, we assume that each antenna of a given user operates independently, and thus all transmit antennas are used for multiplexing. However, for optimal systems without CSIT, cooperation, namely joint channel encoding, across different transmit antennas is indispensable; thus the multi-antenna transmitter cannot be converted into an equivalent single-antenna one. The sum capacity of users 1 to K is given by γ Ψsum = log det I + HHH , nt where H = (H1 , ..., HK ). Thus, the sum capacity is equivalent to that of a single-antenna system with nt K users. Similar to (6), the maximum transmission rate is given by Ψ (Ω(x, q)) − Ψ (Ω(0, q)) , (1 − q)βx
(18)
and the Tse-Hanly equation nt nt g˜n η(γ, x)γ (dFnr (˜ gn )) = 1. η(γ, x) + β ˜n η(γ, x)γ n=1 Ω(x,q) n=1 nt + g
where the second asymptotic equation is due to
0<x≤1
log η(γ, x) + η(γ, x) − 1,
Ω(x, q) = arg min Ψ(D) P (D) = 1 − (1 − x)(1 − q) ∈ Rnt ,
Consequently, the maximum rate is determined by
d
dx (Ψ((1 − q )(1 − x))) x=0 Rmax = β(1 − q ) = ˙ nt log γ + nt log g˜cut Dnt = ˙ nt log γ − log γ, (Θf + 1) nr
Rmax (q) = inf
where for any Borel set D ∈ Rnt , Ψ(D) is the sum capacity (normalized by N1 ) of users in D, which is given by nt nt g˜n γη(γ, x) log 1 + (dFnr (˜ gn )) Ψ(D) = β nt D n=1 n=1
(17)
2β(1 − q)Fˆ −1 (q) . Fˆ −1 (q) 2β 0 gdFˆ (g) + Fˆ −1 (q)
(20)
It is obvious that the spectral efficiency increases in the low energy region as nt increases. This means that optimal systems fully make use of the positive effect of multiple transmit antennas by alleviating the fluctuations in channel power gain, whereas the full-multiplexing systems cannot do this. B. High Energy Region 1) Fixed positive outage probability: For any fixed positive q, using the same approach as in Section III, we can obtain the following results: 1 • If β < n , we obtain the same result as in (12). t 1 • If β > n , we have t t βq ˙ 1−n Rmax = (1−q)β log γ . (21) S∞ = 1 − nt βq
LI and POOR: SPECTRAL EFFICIENCY OF EQUAL-RATE DS-CDMA SYSTEMS WITH MULTIPLE TRANSMIT ANTENNAS
2) Arbitrarily small outage probability: For an arbitrarily small q, we assume that nt β < 1. Since η(x) > 1 − nt β, the last two terms in (18) can be neglected in the high energy region; thus we can obtain the optimal outage user set Ω(x, q) in the high energy region as follows:
gn }) < FS−1 (1 − (1 − x)(1 − q)) , (22) Ω(x, q) = {˜ gn } S({˜
t log γ1 + g˜n and FS is the CDF of where S({˜ gn }) = nn=1 the random variable S({˜ gn }). When γ is sufficiently large, the optimal outage set comprises the users with small product of channel power gains and we call this the product criterion. In this case, Ψ(Ω(x, q)) is given by Ψ(Ω(x, q))
= ˙ +
nt β(1 − (1 − q)(1 − x)) log γ FS−1 (1−(1−x)(1−q)) sdFS (s). β 0
Similarly to the preceding section, the maximal transmission rate is determined at x = 0 and
1 dΨ(Ω(x, q))
−1 = ˙ nt β log γ + βFS . (23)
dx γD x=0
The following lemma provides an upper bound on the diversity gain D. Lemma 4.1: For any fading distribution, D ≤ nt nr (Θf + 1). Proof: Suppose D > nt nr (Θf + 1). Then nt 1 1 ˙ < γD γ nr (Θf +1) nt 1 = ˙ P g˜n < , γ n=1 where the second
asymptotic equation is due to (3). ˙ − nt log γ, then If FS−1 γ1D ≥ 1 1 −1 = P {˜ g } |S (˜ g ) ≤ F n n S γD γD
1 ˙ P {˜ ≥ gn }
g˜n ≤ , ∀n γ nt 1 = P g˜n < γ n=1 ˙ >
1 , γD
which results in a contradiction. Therefore D >
˙ t log γ, which results in nt nr (Θf + 1) implies Fs−1 γ1D 0 such that hX (x) feXmx is 4 continuous and satisfies the following conditions: • there exist nX > 0 and cX > 0 so that nX ˙ X (x) 0 and aX > 0 so that (x) ˙ fxXm−1 ˙ 0 so that xm−1 <x x → ∞. 2) X + Y is m + n − Γ − like distributed. 2 We can show that the power gains, namely |gkmn | , of Rayleigh, Ricean and Nakagami-d fading channels are all Θf + 1 − Γ − like distributed. For the case of nr receive antennas, the channel power gains of these fading channels are nr (Θf + 1) − Γ − like distributed. Based on the above definitions and lemmas, we can obtain a relationship between D and r. Proposition 4.6: If the channel power gain is m − Γ − like distributed, then we have D r = 1. (25) + nt nt nr m Proof: When D = nt nr m, we have nt 1 1 = ˙ P g˜n < . γ D n=1 γ −1 (x)= ˙ FX
Thus, FS−1
1 γD
= ˙ − nt log γ,
which results in r = 0. 4 The
subscript X in nX , cX and mX denotes the dependence on X.
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When D < nt nr m, there exists c > 0 so that, ∀γ > c, nt 1 1 > P g ˜ ≤ . n γD γ n=1
25
Then FS−1
1 γD
−1
= ˙
F
= ˙
D log γ, nr m
nt n=1
log g ˜n
1 γD
15
10
5
which results in r = nt − nD and then (25). rm For Rayleigh, Ricean and Nakagami-d fading channels, (25) can be written in a form similar to (14): r D = 1, + nt nt nr (Θf + 1)
20 Spectral Efficiency
Thus, ∀γ > c,
1 1
gn } < D = φ. gn } S {˜ {˜ gn } g˜n ≥ , n = 1, ..., nt ∩ {˜ γ γ
nt=1 nt=2 nt=4 nt=8
(26)
0 −20
0
20
40 average E /N (dB) b
60
80
100
0
Eb Fig. 2. Spectral efficiency versus N with various numbers of transmit 0 antennas in full-multiplexing systems.
16
and the corresponding spectral slope in the high energy region is given by D S∞ = β n t − . (27) (Θf + 1) nr
V. N UMERICAL R ESULTS We now summarize some numerical results that illustrate the analytical results of this paper. We set the outage probability q = 0.05, the number of receive antennas nr = 4, and consider Rayleigh fading in all these results. It should be noted that all numerical results are computed with asymptotic expressions; thus being independent of K and N .
12
Spectral Efficiency
For systems with βnt > 1, we can obtain results similar to (16). We can see that optimal systems achieve a better tradeoff than full-multiplexing systems (for the same diversity gain, the optimal systems achieve higher multiplexing gain). On D , the defining the transmit diversity gain as Dt = (Θf +1)n r transmit diversity-multiplexing tradeoff equation is given by r + Dt = nt , which is intuitive and identical to that of single-user narrow band systems in Rayleigh fading channels [15] with large numbers of receive antennas. The differences between CDMA and narrow-band systems are as follows • The diversity-multiplexing tradeoff is linear for CDMA systems and piecewise-linear for narrow-band systems. • CDMA systems can still achieve multiplexing gain when nr = 1 while this is impossible for narrow band systems. Intuitively, both differences are due to the use of spreading codes in CDMA systems since we can consider the different chips in the spreading codes as being equivalent to different receive antennas. Therefore, the number of equivalent receive antennas is nr N , which is much larger than nt for large systems. Also, the tradeoff equation for narrow-band systems given in [15], namely D = (nr −r)(nt −r), when nr , nt ∈ N, is equivalent to nD + r = nt in CDMA systems with rN sufficiently large N , thus resulting in the linear equation D t + r = nt .
14
10
8
6
4 β=0.4 β=0.8 β=1.2 β=1.6
2
0 −10
−5
0
5
10
15
20
average E b/N0(dB)
Fig. 3. Spectral efficiency versus multiplexing systems.
Eb N0
with various system loads in full-
Eb Figure 2 shows the spectral efficiency versus N for various 0 values of nt in full-multiplexing systems, where β = 0.5. We can see that, in the low energy region, the spectral efficiency decreases as nt increases, which demonstrates the conclusion in Section III-A. In the moderate energy region, adding more transmit antennas can improve the spectral efficiency. However, when the equivalent system is overloaded (nt ≥ 3), the corresponding performance gain is marginal. Also, we notice that for overloaded systems (nt = 4, 8) in the high Eb energy region ( N ≥ 90dB), the spectral efficiency decreases 0 as nt increases. Figure 3 shows the spectral efficiency for various values of β in full-multiplexing systems, when nt = 2. For the case of sum capacity, the spectral efficiency increases with β and saturates for large β [7]. However, as can be observed from Fig. 3, for equal-rate systems in the moderate energy region, the spectral efficiency saturates rapidly at β = 1, which implies that the maximal transmission rate is inversely
LI and POOR: SPECTRAL EFFICIENCY OF EQUAL-RATE DS-CDMA SYSTEMS WITH MULTIPLE TRANSMIT ANTENNAS
12
VI. C ONCLUSIONS
10
8 Spectral Efficiency
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6
4
sum criterion product criterion full multiplexing
2
0 −10
−5
0
5
10
15
average E /N (dB) b
0
Fig. 4. Comparison between optimal systems and full-multiplexing systems.
14
12
In this paper, we have analyzed the spectral efficiency of uplink DS-CDMA systems with multiple transmit antennas. A key assumption is that different spreading codes are assigned to different antennas of a given user. The conclusions of this analysis are given as follows. • For full multiplexing systems, adding more transmit antennas will decrease the spectral efficiency for any systems in the low energy region and for overloaded systems in the high energy region. For other situations, multiple transmit antennas always benefit the system performance. • For optimal systems, multiple transmit antennas always attain higher spectral efficiency, except for overloaded systems in the high energy region. Compared with fullmultiplexing systems, cooperation across transmit antennas results in considerably higher spectral efficiency in the low energy region and better diversity-multiplexing tradeoff in the high energy region. For overloaded systems in the high energy region, it may be better to consider using the same spreading codes for different antennas of a given user in order to exploit the gain of multiple transmit antennas.
Spectral Efficiency (bps/Hz)
10
A PPENDIX I P ROOF OF L EMMA IV.3
8
Proof: (1) Taking logarithms on both sides of x FX (x) = emt hX (t)dt,
6
4
−∞
0 −10
we obtain
n t=1 n t=2 n t=3
2
0
10
20
30
40
50
average E b/N0(dB)
Fig. 5. Spectral efficiency versus of antennas in optimal systems.
Eb N0
with various number of transmit
proportional to β for overloaded systems. Eb The spectral efficiency versus N for optimal systems and 0 full-multiplexing systems is given in Fig. 4, where β = 0.4, nt = 2 and nr = 4. In Fig. 4, ‘sum criterion’ represents sorting the users by the sum of channel power gains and ‘product criterion’ represents sorting them by the product of channel power gains, which are optimal in the low energy region and the high energy region, respectively, as explained in Section IV. We observe that these two criteria achieve almost the same performance. The full-multiplexing scheme attains considerably lower spectral efficiency than optimal systems in the low energy region. Eb for optimal Figure 5 shows the spectral efficiency versus N 0 systems when β = 2. We can see that, in the high energy region, the spectral efficiency decreases with the number of transmit antennas. This coincides with our conclusion in Section IV-B.
log log FX (x) =1+ mx
x m(t−x) hX (t)dt −∞ e
mx
.
Applying the upper and lower bounds on hX (x) as x → −∞ in Definition IV.2, it is easy to check that the second term in the right hand side of the above equation converges to 0 as x → 0. (2)The PDF of Z = X + Y is given by ∞ fZ (y) = fX (x)fY (y − x)dx −∞ ∞ = emy hX (x)hY (y − x)dx, −∞
which results in hZ (y) =
∞
−∞
hX (x)hY (y − x)dx.
The two conditions of hX in Definition IV.2 are equivalent to the following two conditions: • There exist nX > 0, cX > 0 and Δ1X < 0 so that cX < hX (x) < (−x)nX , ∀x < Δ1X ; • There exist lX > m and Δ2X > 0 so that hX (x) < e−lX x , ∀x > Δ2X . Similarly, we can also define nY , cY , lY , Δ1Y and Δ2Y for hY such that the corresponding inequalities hold.
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Therefore, when y < Δ1X + Δ1Y , hZ (y) can be obtained by integrating over five intervals, namely y−Δ2Y y−Δ1Y hZ (y) = + −∞ Δ1X
+
y−Δ2Y Δ2X
+
∞
+ Δ2X
Δ1X
y−Δ1Y
(hX (x)hY (y − x)) dx.
It is easy to check that, as y → −∞, ∀ > 0, the terms in the above integral have the following properties: n + • the first two terms are upper bounded by (−y) X ; nX +nY +1+ • the third term is upper bounded by (−y) ; n + • the last two terms are upper bounded by (−y) Y . nX +nY +1+ ˙ Thus, as y → −∞, hZ (y) 0. Also, we can show that the third term is lower bounded by cX cY (Δ1X + Δ1Y − Δ) when y < Δ1X + Δ1Y . When y > Δ2X + Δ2Y , we have Δ1X Δ2X y−Δ2Y hZ (y) = + + −∞ y−Δ1Y
+
Δ1X ∞
+ y−Δ2Y
[7] S. Shamai and S. Verd´u, “The impact of frequency-flat fading on the spectral efficiency of CDMA,” IEEE Trans. Inf. Theory, vol. 47, pp. 1302-1327, May 2001. [8] S. Shamai and S. Verd´u, ”Decoding only the strongest CDMA users,” Codes, Graphs, and Systems, R. Blahut and R. Koetter, Eds., pp. 217228. Norwell, MA: Kluwer Academic Publishers, 2002. [9] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE Trans. Inf. Theory, vol. 45, pp. 1456-1467, July 1999. [10] E. Telatar, “Capacity of multi-antenna Gaussian channels”, European Trans. Telecommun., vol. 10, no.6, pp. 585–596, Nov. 1999. [11] D. Tse and S. Hanly, “Linear multiuser receivers: effective interference, effective bandwidth and user capacity,” IEEE Trans. Inf. Theory, vol. 45, pp. 641-657, Mar. 1999. [12] D. N. C. Tse, P. Viswanath, and L. Zheng, “Diversity-multiplexing tradeoff in multiple access channels,” IEEE Trans. Inf. Theory, Vol. 50, pp. 1859-1874, Sept. 2004. [13] S. Verd´u, Multiuser Detection. Cambridge University Press, Cambridge, UK, 1998. [14] S. Verd´u, “Spectral efficiency in the wideband regime,” IEEE Trans. Inf. Theory, vol. 48, pp. 1319-1343, June 2002. [15] L. Zheng and D. N. C. Tse, “Diversity and multiplexing: a fundamental tradeoff in multiple-antenna channels,” IEEE Trans. Inf. Theory, vol. 49, pp. 1073-1096, May 2003.
Δ2X
(hX (x)hY (y − x)) dx.
y−Δ1Y
It is easy to show that, as y → ∞, for any 0 < < min (lX , lY ), the terms in this integral have the following properties: −(lY −)y • the first two terms are upper bounded by e ; −(min(nX ,nY )−) • the third term is upper bounded by e ; −(lX −)y • the last two terms are upper bounded by e ; ˙ −(min(nX ,nY )−) , for some Thus, as y → ∞, hZ (y)<e sufficiently small . Applying Definition IV.2, Z is therefore m − log Γ − like distributed. R EFERENCES [1] T. Cover and J. A. Thomas, Elements of Information Theory. New York: John Wiley and Sons Inc., 1991. [2] G. J. Foschini,“Layered space-time architechture for wireless communication in fading environments when using multi-element antennnas,” Bell Labs Tech. J., pp. 41-59, 1996. [3] S. V. Hanly and D. N. C. Tse, “Resource pooling and effective bandwidths in CDMA networks with multiuser receivers and spatial diverisity,” IEEE Trans. Inf. Theory, vol. 47, pp. 1328-1351, May 2001. [4] H. Li and H. V. Poor, “Power allocaiton and spectral efficiency of DSCDMA systems in fading channels with fixed QoS - part 1: singlerate case,” IEEE Trans. Wireless Commun., vol. 5, pp. 2516-2528, Sept. 2006. [5] A. Mantrovadi, V. V. Veeravalli, and H. Viswanathan, “Spectral efficiency of MIMO multiaccess systems with single-user decoding,” IEEE J. Sel. Areas Commun., vol. 21, pp. 382-394, Apr. 2003. [6] C. Papadias, “On the spectral efficiency of space-time spreading schemes for multiple antenna CDMA systems,” in Proc. 33rd Asilomar Conference on Signals, Systems, and Computers 1999.
Husheng Li (S’00-M’05) received the B.S. and M.S. degrees in electronics engineering from Tsinghua University, Beijing, China, in 1998 and 2000, respectively, and the Ph.D. degrees in electrical engineering from Princeton University, NJ, in 2005. In 2005, he joined Qualcomm Inc. CA. His research interests include statistical signal processing, wireless communication and information theory.
H. Vincent Poor (S’72-M’77-SM’82-F’87) received the Ph.D. degree in EECS from Princeton University in 1977. From 1977 until 1990, he was on the faculty of the University of Illinois at Urbana-Champaign. Since 1990 he has been on the faculty at Princeton, where he is the Dean of Engineering and Applied Science, and the Michael Henry Strater University Professor of Electrical Engineering. He has also held visiting appointments at a number of universities, including recently Imperial College, Stanford and Harvard. Dr. Poor’s research interests are in the areas of wireless networks, advanced signal processing and related fields. Among his publications in these areas is the forthcoming book MIMO Wireless Communications (Cambridge, 2007). Dr. Poor is a member of the National Academy of Engineering, and is a Fellow of the American Academy of Arts and Sciences, the Institute of Mathematical Statistics, the Optical Society of America, and other organizations. He is a past President of the IEEE Information Theory Society, and is the current Editor-in-Chief of the IEEE Transactions on Information Theory. Recent recognition of his work includes the Joint Paper Award of the IEEE Communications and Information Theory Societies (2001), the NSF Director’s Award for Distinguished Teaching Scholars (2002), a Guggenheim Fellowship (2002-03), and the IEEE Education Medal (2005).
