IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 4, APRIL 2007
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Error Probability for MIMO Zero-Forcing Receiver with Adaptive Power Allocation in the Presence of Imperfect Channel State Information Edward K. S. Au, Student Member, IEEE, Cheng Wang, Student Member, IEEE, Sana Sfar, Member, IEEE, Ross D. Murch, Senior Member, IEEE, Wai Ho Mow, Senior Member, IEEE, Vincent K. N. Lau, Senior Member, IEEE, Roger S. Cheng, Member, IEEE, and Khaled Ben Letaief, Fellow, IEEE
Abstract— Link adaptation allows the transmitter to adapt to changing channel conditions. Critical to the design of link adaptation is the accuracy of channel state information at the transmitter. In this paper, we investigate the effect of imperfect channel state information and feedback delay on the performance of a multiple-input multiple-output zero-forcing receiver with adaptive power allocation. A closed-form approximate upper bound on bit error rate is derived for M-ary phase shift keying and M-ary quadrature amplitude modulation and it is valid for arbitrary numbers of transmit and receive antennas. Comparison with Monte Carlo simulations is also provided, showing that the results derived from this analytical bound are useful for system design. Index Terms— Bit error rate, channel estimation error, feedback, MIMO, power allocation, zero-forcing.
I. I NTRODUCTION
M
ULTIPLE-input multiple-output (MIMO) technology is considered as a promising candidate for nextgeneration wireless communications systems [1]. Compared with conventional single-input single-output technology, MIMO has comparative advantages in improving spectral efficiency, enhancing link throughput and capacity, and providing a simple but effective way to increase degrees of freedom of the channels [2]. In addition, it has been well understood that by means of feedback, the transmitter can be provided with channel state information (CSI), estimated at the receiver, and various link adaptation techniques can be applied to accommodate the transmitter to the changed channel condition [3], [4]. Critical to the link adaptation techniques is the accuracy of the estimated CSI. In previous literature [5], adaptation algorithms have been designed based on an ideal Manuscript received August 12, 2005; revised July 7, 2006; accepted November 21, 2006. The associate editor coordinating the review of this paper and approving it for publication was M. Uysal. This work was supported in part by the Hong Kong Research Grants Council under Grants HKUST6113/04E and HKUST6151/05E. E. K. S. Au, C. Wang, R. D. Murch, W. H. Mow, V. K. N. Lau, R. S. Cheng, and K. B. Letaief are with the Department of Electronic and Computer Engineering, the Hong Kong University of Science and Technology, Hong Kong S.A.R., China (e-mail: {eeedward, eeelva, eermurch, eewhmow, eeknlau, eecheng, eekhaled}@ust.hk). S. Sfar was with the Department of Electronic and Computer Engineering, the Hong Kong University of Science and Technology, Hong Kong S.A.R., China. She is currently with Lehigh University, Lehigh, PA (e-mail:
[email protected]). Digital Object Identifier 10.1109/TWC.2007.05621.
assumption that perfect CSI is available at both the transmitter and the receiver. However, because there are errors in practical estimation processes, the estimated CSI is usually imperfect and therefore, performance degradation will occur. In order to understand the effect of imperfect CSI on the performance of MIMO systems with link adaptation, one can resort to time-consuming Monte Carlo simulations but it can only provide limited insight to the problem. Alternatively, one challenging but attractive approach is to derive analytical bounds in closed-form expression. In [6], Zhou and Giannakis analyzed the performance of transmit beamforming with adaptive modulation over Rayleigh MIMO channels and derived a closed-form expression of bit error rate (BER) that requires an integration. In [7], Chen and Tellambura derived cumulative density function, probability density function, and moment generating function of the signal-to-noise ratio (SNR) under a MIMO maximal ratio transmission system over an independent and identical distributed (i.i.d.) Rayleigh fading channel. In this paper, we investigate the effect, on BER, of imperfect CSI and feedback delay at the transmitter with adaptive power allocation when a MIMO zero-forcing receiver is utilized. A closed-form approximate upper bound is derived for both M-ary phase shift keying (MPSK) and M-ary quadrature amplitude modulation (MQAM) and it is valid for arbitrary numbers of transmit and receive antennas. The zero-forcing receiver is focused on because of its simple structure and its ease for providing first-order insight to the problem. However, it should be emphasized that, in spite of its simplicity, the analysis is complicated due to the exploitation of power adaptation at the transmitter and the feedback delay. The rest of this paper is organized as follows. The MIMO system model with practical power allocation strategy is introduced in Section II. Section III details the derivation of the closed-form approximate upper bound and provides analysis in terms of channel estimation error and feedback delay. Finally, numerical results are presented in Section IV and conclusions are given in Section V. Throughout the paper, bold uppercase (lowercase) letters denote matrices (vectors). For a matrix M, MH , M† and M−1 are its Hermitian transpose, pseudo-inverse and inverse, respectively, and [M]i,j refers to its (i, j)-th element. In addition, IN is an identity matrix of size N , diag{.} is a
c 2007 IEEE 1536-1276/07$25.00
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 4, APRIL 2007
n1
……
^ H
H
……
Power Adaptation Matrix F
s
+
Nt transmit antennas
z1
nNr
+
zNr
Nr receive antennas
feedback path
Fig. 1.
Zero-Forcing Detector
^ s
^ H
By substituting (6) into (5), the received signal vector is expressed as ˆsn = ˜sn + n ˜n (7)
Channel Estimation
The transceiver diagram.
diagonal matrix and tr{.} is the trace of the matrix. Lastly, E{.} is an expectation operator, N and N are the ceiling and floor operations, respectively, of the integer N . II. S YSTEM M ODEL We consider an uncoded MIMO system with Nt transmit and Nr receive antennas, where Nt ≤ Nr , as shown in Fig. 1. The received data at time instant n, namely zn ∈ C Nr ×1 , is expressed as (1) zn = Hn Fn sn + nn . Here, Fn is a Nr ×Nt precoding matrix that is used for power adaptation, sn ∈ C Nt ×1 denotes the transmitted data vector whose elements are drawn from a unit energy constellation, Nr ×1 i.e., E{sn sH is an additive white n } = INt , and nn ∈ C Gaussian noise (AWGN) vector with zero-mean and variance σn2 . In addition, Hn ∈ C Nr ×Nt denotes an uncorrelated Rayleigh flat-fading channel and each of the channel elements is assumed to be i.i.d. with complex Gaussian distribution CN (0, 1). In practice, the effect of channel estimation error ˆ n as the imperfect should be taken into account. Denote H estimate of Hn at the receiver, it can be modelled as [8] ˆ n = H n + Ωn H
(2)
Nr ×Nt
where Ωn ∈ C is the channel estimation error matrix that is uncorrelated with Hn and each of its elements is also i.i.d. with complex Gaussian distribution CN (0, σe2 ). In realistic MIMO channels, there exists feedback delay which may also incur additional estimation error. In the following, we extend our channel model (2) to incorporate the effect of feedback delay [9] ˆ n = ρHn + 1 − ρ2 Hn−d + Ωn (3) H where Hn−d is the channel matrix at time instant n − d and ρ is the correlation coefficient between the channel matrices at time instants n and n − d. ρ= 2 σn,n−d
E{Hn HH n−d } 2 σn,n−d
ˆ n can be approximated by in which the pseudo-inverse of H using the linear part of the Taylor’s series as follows [10], [11]. ˆ †n = ρHn + 1 − ρ2 Hn−d + Ωn † H 1 1 INt − H†n 1 − ρ2 Hn−d + Ωn H†n . (6) ρ ρ
˜ n denote the desired signal and the where ˜sn and n interference-plus-noise signal, respectively, and they are written as 1 ˜sn = Fn sn ρ 1 † 1 † † † ˜n = n H nn − 2 Hn Ωn Fn sn + Hn Ωn Hn nn ρ n ρ 2 1−ρ † † † − Hn Hn−d Fn sn + Hn Hn−d Hn nn . ρ2 Then, the post-processing SNR of the k-th data sub-stream is expressed as shown in (8) at the top of the next page. Before proceeding further, we need to determine the precoder structure and the power allocation strategy. Here, we consider the maximization of the sum of all post-processing SNR’s expressed in decibel (dB) at time instant n, which is equivalent to maximize the weighted geometric mean of all sub-streams’ SNR’s [12]. Mathematically, it is max f0 ({γk,n }) =
Nt w γk,n k,n
where wk,n ≥ 0 is the weight of the k-th sub-stream at time instant n. The objective function (9) can equivalently be expressed in terms of mean square error (MSE) as follows. Nt −1 wk,n . M SEk,n max f0 {M SEk,n } =
(10)
k=1
By [12] and [13], it is shown that the objective function is Schur-concave on each sub-stream and therefore, the optimal structure of the precoder Fn is given by [12] Fn = Un Dn
(11)
where Un is an Nt × Nt unitary matrix and Dn = diag{d1,n , . . . , dNt ,n } is an Nt × Nt diagonal matrix. Finally, the maximization problem can be written in the following convex form subject to a total transmit power constraint PT .
(4)
with begin the variance of the channel coefficient. The justification of using the model is that it allows easy tracking of the channel state given the knowledge of the past [9]. Given the equations above, the zero-forcing estimate of the transmitted data vector is given by ˆ †n Hn Fn sn + nn ˆsn = H (5)
(9)
k=1
max{dk,n } subject to
w
f0 ({γk,n }) =
Nt
(γk,n )w˜k,n
k=1 NT
tr{Fn FH n} =
d2k,n = PT
(12)
k=1
where w ˜k,n = Ntk,n is the normalized weight of the k-th i=1 wi,n sub-stream at time instant n. By using Lagrangian multipliers
AU et al.: ERROR PROBABILITY FOR MIMO ZERO-FORCING RECEIVER WITH ADAPTIVE POWER ALLOCATION
γk,n
ρ2 Fn FH 1 n k,k × 2 2 2 tr{(HH H )−1 } H H )−1 ρ σn + (σe2 + 1 − ρ2 ) tr{Fn FH } + σ (H n n n n n n k,k
and solving the KKT conditions, the optimal power allocation strategy is given by ˜k,n PT . (13) dk,n = w By substituting (12) and (13) into (8), the approximate postprocessing SNR of the k-th sub-stream is expressed as follows. γk,n
H ρ 2 Un D n D H n Un
ρ2 σn2
+
(σe2
+1−
1 × −1 (HH H n) n
ρ2 )
PT +
k,k
−1 } σn2 tr{(HH n Hn )
.
f (γk,n ) =
r 1 Δ−1 − k,n e 2 , rk,n − 1)!
2Δ (Δ
PeP SK (M |γk,n ) ≈
In the next section, we shall use (14) to derive a closed-form approximate upper bound on the BER for both MPSK and MQAM.
A. PDF of the post-processing SNR The post-processing SNR of the k-th sub-stream can be simplified by considering the following approximation and upper bound. H ρ2 Un Dn DH n Un k,k γk,n (15) −1 ρ2 σn2 + (σe2 + 1 − ρ2 )PT (HH n Hn )
BER
P SK
. (16)
Here, the approximation in (15) is very accurate for small channel estimation error and the detailed justification can be found in [11], [14]. Note that when σe2 = 0 and ρ = 1, no approximation is required. For the upper bound in (16), the justification is detailed in Appendix. −1 which is a chi-squared Denote rk,n = 1/ (HH n Hn ) k,k random variable with 2(Nr −Nt +1) = 2Δ degrees of freedom [15]. the post-processing SNR of the k-th sub-stream is a weighted chi-squared random variable given by
= ≡
ρ2 PT rk,n ρ2 σn2 + (σe2 + 1 − ρ2 )PT 1+
rk,n
γ˜ rk,n .
(17)
B. Approximate Upper Bound on the BER The BER expression of a modulation scheme for the MIMO zero-forcing receiver with adaptive power allocation in the presence of channel estimation error and feedback delay is given by
BER = Eγk,n Pe (M |γk,n ) ∞ = Pe (M |γk,n )f (γk,n )dγk,n (18) o
(20)
min(2,M/4) Δ−1
2 max(log2 M, 2) c=1 j=0 Δ j 1 − μc Δ − 1 + j 1 + μc j 2 2
=
k,k
PT 2 σn σe2 +1−ρ2 PT 2 ρ2 σn
min(2,M/4)
2 max(log2 M, 2) c=1 (2c − 1)π Q 2˜ γ rk,n sin . M
∞ min(2,M/4)
2 max(log2 M, 2) 0 c=1 (2c − 1)π Q 2˜ γ rk,n sin f (γk,n )dγk,n M
k,k
−1 ρ2 σn2 + (σe2 + 1 − ρ2 )PT (HH n Hn )
(19)
By averaging (20) over (19), the approximate upper bound on the BER in the presence of channel estimation error and feedback delay is
III. P ERFORMANCE A NALYSIS
γk,n
rk,n > 0.
1) MPSK: The BER expression over an AWGN channel has the following tight approximation [17]
(14)
ρ2 PT
(8)
where M is the constellation size, Pe (M |γk,n ) is the BER conditioned on γk,n [16], and f (γk,n ) is the PDF of γk,n which is given by
k,k
1525
(21) where μc =
(2c−1)π M sin2 (2c−1)π M
γ ˜ sin2 1+˜ γ
and γ˜ =
PT 2 σn 2 σe +1−ρ2 PT 1+ 2 ρ2 σn
is
defined in (17). From (21), it can be observed that the BER is not only a function of both the channel estimation error and the feedback delay, but also a function of the constellation size and the receive diversity order. More specifically, (1) Given that the other parameters remain unchanged, the larger the value of σe2 , the smaller the value of γ˜ . As a result, μc decreases and hence the BER increases. (2) The longer the feedback delay, the smaller the value of ρ. Given that the other parameters remain unchanged, a decrease in ρ will lead to a decrease in γ˜ and hence an increase in the BER. (3) Increase the modulation level M results in a loss in the BER performance, due to the fact that the minimum distance decreases. (4) Receive diversity order Δ is always beneficial whether CSI is accurate or not. Lastly, if σn2 → 0, γ˜ asym ≡ lim γ˜ → 2 σn →0
ρ2
= ∞ σe2 + 1 − ρ2
(22)
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when σe2 = 0 and ρ = 1. The asymptotic approximate upper bound on the BER is then given by
QPSK, Nt = 4, Nr = 4
0
10
ρ = 0.9, NMSE = 10%
min(2,M/4) Δ−1
2 max(log2 M, 2) c=1 j=0 j asym Δ 1 − μc Δ − 1 + j 1 + μasym c j 2 2 (23)
ρ2 sin2
−1
10
BER
BER
P SK
2) MQAM: The exact BER expression over an AWGN channel is as follows [18].
10
−k
Analytical bound Monte Carlo simulations 10
0
5
10
15 SNR (dB)
√
BERQAM
where νc =
−k
=
NMSE = 10%
−1
10
NMSE = 5%
νcasym
= →
σn →0
lim νc
Analytical bound Monte Carlo simulations −2
10
0
5
10
15 SNR (dB)
20
25
30
Fig. 3. Impact of σe2 on BER with SNR ranging from 0 to 30dB; Nt = Nr = 6, ρ = 0.98, and 16QAM.
√
ρ2
= ∞ σe2 + 1 − ρ2
(26)
2 →0 σn
3ρ2 (2c
NMSE = 0%
MQAM is given by
3˜ γ (2c+1)2 2(M−1)+3˜ γ (2c+1)2 .
lim γ˜ → 2
30
16QAM, Nt = 6, Nr = 6, ρ = 0.98
0
√
In common with the BER expression for MPSK, (25) is a function of channel estimation error, feedback delay, constellation size and receive diversity order. For the asymptotic analysis, consider σn2 → 0, σe2 = 0, and ρ = 1, γ˜ asym
25
10
Similarly, by averaging (24) over (19), the approximate upper bound on the BER is given by log2 M (1−2 ) M −1 Δ−1
2 √ √ M log2 M k=1 c=1 j=0 c · 2k−1 1 c·2k−1 (−1) M 2k−1 − √ + 2 M Δ j Δ − 1 + j 1 + νc 1 − νc (25) j 2 2
20
Fig. 2. Impact of σe2 and ρ on BER with SNR ranging from 0 to 30dB; Nt = Nr = 4, and QPSK.
√
log2 M (1−2 ) M −1
2 √ = √ M log2 M k=1 c=1 c · 2k−1 k−1 1 c·2M k−1 2 + (−1) − √ 2 M 3˜ γ rk,n . (24) Q (2c + 1) M −1
ρ = 0, NMSE = 0%
−4
BER
PeQAM (M |γk,n )
ρ = 0, NMSE = 10%
−2
10
−3
(2c−1)π
M ≡ limσn2 →0 μc → = 1. where μasym c 2 +1−ρ2 )+ρ2 sin2 (2c−1)π (σe M In other words, in the presence of channel estimation error and/or feedback delay, the BER will not be equal to zero when the noise variance is very small. Instead, an undesirable error floor exists.
√
ρ = 0.9, NMSE = 0%
1)2
+
= 1. 2(M − 1)(σe2 + 1 − ρ2 ) + 3ρ2 (2c + 1)2 (27)
The asymptotic approximate upper bound on the BER for
BERQAM
−k
√
log2 M (1−2 ) M−1 Δ−1
2 √ √ M log2 M k=1 c=1 j=0 k−1 k−1 c·2 1 c·2 + (−1) M 2k−1 − √ 2 M j asym Δ Δ − 1 + j 1 + νcasym 1 − νc . j 2 2 (28)
Hence, it is in common with MPSK that an error floor exists at high SNR in the presence of channel estimation error and/or feedback delay. IV. P ERFORMANCE E VALUATION To assess the validity of the analytical upper bound derived in the previous section and illustrate the effects of imper-
AU et al.: ERROR PROBABILITY FOR MIMO ZERO-FORCING RECEIVER WITH ADAPTIVE POWER ALLOCATION
QPSK, Nt = 2, SNR = 10dB
0
MQAM, Nt = 4, Nr = 4, ρ = 0.98
0
10
1527
10 NMSE = 0% NMSE = 5% NMSE = 10%
Nr = 2
M = 256
−1
10
N =3
BER
BER
r
−2
10
−1
10
M=4 N =4 r
−3
10
SNR = 5dB SNR = 10dB Asymptotic SNR −4
10
0.9
−2
0.92
0.94
ρ
0.96
0.98
1
Fig. 4. Impact of the receive diversity order on BER with ρ ranging from 0.9 to 1 and NMSE = 0%, 5% and 10%; Nt = 2, Nr ∈ {2, 3, 4}, SN R = 10dB, and QPSK.
fect CSI and feedback delay, Monte Carlo simulations are conducted with different numbers of transmit and receive antennas. Note that in all numerical examples presented in the following, we use normalized mean square error (NMSE) to characterize the estimation accuracy and it is a function of e as follows. ˆ n ]i,j 2 E [Hn ]i,j − [H N M SE = 2 E [Hn ]i,j (29) = 2 1 − 1 − e2 . As the first example, we consider the impact of σe2 and ρ on the BER with SNR ranging from 0 to 30dB. Fig. 2 illustrates the performance of the MIMO zero-forcing receiver using QPSK, Nt = 4 transmit and Nr = 4 receive antennas while Fig. 3 illustrates the performance of the system using 16QAM, Nt = 6 transmit and Nr = 6 receive antennas. Referring to Fig. 2, we consider 4 difference scenarios, namely (1) no estimation error and no feedback delay (σe2 = 0 and ρ = 1); (2) no estimation error but with feedback delay (σe2 = 0 and ρ = 0.9); (3) no feedback delay but with estimation error (σe2 = 0.1 and ρ = 1); and (4) with estimation error and feedback delay (σe2 = 0.1 and ρ = 0.9). It can be observed that when there are channel estimation errors and/or feedback delay, undesirable error floors exist, and the result is consistent with our analysis as detailed in Section III.2. In Fig. 3, the feedback delay is fixed at ρ = 0.98. We observe that the BER increases with the channel estimation error, as expected. Lastly, the comparison with Monte Carlo simulations shows that the approximate upper bound works well and it is about 1dB away from the actual value for various values of σe2 and ρ. Because of the tightness, we shall only use these analytical bounds to gain insight into the system performance in the rest of this section. Next, we look at the impact of the receive diversity order on the system performance. In Fig. 4, the performance of the system with QPSK and Nt = 2 transmit antennas is
10
0
0.1
0.2
0.3
0.4
0.5
NMSE
Fig. 5. Impact of M on BER with N M SE ranging from 0 to 50%; Nt = Nr = 4, ρ = 0.98, and MQAM.
illustrated at SN R = 10dB. It can be observed that for various values of ρ and N M SE, the system with additional receive antennas always outperforms that with only Nr = 2 receive antennas. For example, at ρ = 0.93 and N M SE = 5%, BER = 10−1 , 3 × 10−2 and 10−2 when Nr = 2, 3, and 4, respectively. Finally, the impact of the constellation level M is illustrated in Fig. 5. For the system under consideration, there are Nt = 4 transmit and Nr = 4 receive antennas, and ρ = 0.98. It can be observed that, for various values of SN R, the BER increases as M increases. As a remark, note that the asymptotic bound (28) is the best BER that the MIMO zero-forcing receiver can achieve in the presence of channel estimation error and feedback delay.
V. C ONCLUSION In this paper, performance analysis is considered for a MIMO zero-forcing receiver with power adaptation at the transmitter in the presence of both imperfect CSI and feedback delay over an i.i.d. Rayleigh flat-fading channel. For the power allocation strategy, it is determined by the maximization of a weighted sum of SNR’s of all sub-streams subject to a total transmit power constraint. A closed-form approximate upper bound is derived for both M-ary phase shift keying (MPSK) and M-ary quadrature amplitude modulation (MQAM) and it is valid for arbitrary numbers of transmit and receive antennas. From the results derived herein, it is observed that the BER is not only a function of both the channel estimation error and the feedback delay, but also a function of the constellation size and the receive diversity order. In addition, comparison with Monte Carlo simulations confirmed the validity of the analytical bound. Hence, instead of relying on time-consuming simulations, the results derived herein can be used as a design guideline and allow a detailed evaluation for the system performance.
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A PPENDIX A U PPER BOUND IN (16) H First of all, we expand the term Un Dn DH n Un as follows.
Un Dn DH UH ⎛ n n u1,1 u1,2 . . . ⎜ u2,1 u2,2 . . . ⎜ = ⎜ . .. .. ⎝ .. . . uNt ,1 uNt ,2 . . . ⎛ 2 d1,n 0 ... 2 ⎜ 0 d ... 2,n ⎜ ⎜ .. .. .. ⎝ . . . ⎛ ⎜ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ = ⎜ ⎝ ⎛
0
uH 2,1 uH 2,2
.. .
.. .
uH 1,Nt
uH 2,Nt
uH 1,1 ⎜ uH ⎜ 1,2 ⎜ . ⎝ .. uH 1,Nt
. . . uH Nt ,1 . . . uH Nt ,2 .. .. . . . . . uH Nt ,Nt
d22,n u1,2 d22,n u2,2 .. . d22,n uNt ,2
uH 2,1 uH 2,2 .. . uH 2,Nt
⎞ ⎟ ⎟ ⎟ ⎠
uNt ,Nt ⎞ 0 0 ⎟ ⎟ ⎟ .. ⎠ .
. . . d2Nt ,n
0
uH 1,1 uH 1,2
d21,n u1,1 d21,n u2,1 .. . d21,n uNt ,1
u1,Nt u2,Nt .. .
⎞ ⎟ ⎟ ⎟ ⎠
. . . d2Nt ,n u1,Nt . . . d2Nt ,n u2,Nt .. .. . . . . . d2Nt ,n uNt ,Nt ⎞ H
. . . uNt ,1 . . . uH Nt ,2 .. .. . . . . . uH Nt ,Nt
⎞ ⎟ ⎟ ⎟ ⎠
⎟ ⎟ ⎟. ⎠
For each of the diagonal elements, it is given by H Un Dn DH = d21,n u2k,1 + · · · + d2Nt ,n u2k,Nt n Un k,k =
Nt
dj,n u2k,j .
(30)
j=1
Denote uk,max = max ({uk,j }) for all j, (30) can be upperbounded as H Un Dn DH n Un k,k
≤ u2k,max
Nt
d2j,n
(31)
j=1
= u2k,max
Nt
w ˜j,n PT
[5] H. Sampath, P. Stoica, and A. Paulraj, “Generalized linear precoder and decoder design for MIMO channels using the weighted MMSE criterion,” IEEE Trans. Commun., vol. 49, pp. 2198-2206, Dec. 2001. [6] S. Zhou and G. B. Giannakis, “How accurate channel prediction needs to be for transmit-beamforming with adaptive modulation over Rayleigh MIMO channels?” IEEE Trans. Wireless Commun., vol. 3, pp. 1285-1294, July 2004. [7] Y. Chen and C. Tellambura, “Performance analysis of maximal ratio transmission with imperfect channel estimation,” IEEE Commun. Lett., vol. 9, pp. 322-324, April 2005. [8] A. P. Iserte, M. A. Lagunas Hernandez, and A. I. Perez-Neira, “Robust power allocation for minimum BER in a SISO-OFDM system,” in Proc. 2003 IEEE ICC, vol. 2 , pp. 1263-1267. [9] H. T. Nguyen, J. B. Andersen, and G. F. Pedersen, “Capacity and performance of MIMO systems under the impact of feedback delay,” in Proc. 15th IEEE PIMRC 2004, vol. 1, pp. 53-57. [10] T. Haustein et al., “Bit error rates for a MIMO system in Rayleigh and Rician channels,” in Proceedings of IEEE 54th VTC Fall, October 2001, vol. 4, pp. 1984-1987. [11] C. Wang, E. K. S. Au, R. D. Murch, W. H. Mow, R. S. Cheng, and V. K. N. Lau, “On performance of the MIMO zero-forcing receiver in the presence of channel estimation error,” accepted to IEEE Trans. Wireless Commun., April 2006. [12] D. P. Palomar, “A unified framework for communications through MIMO channels,” Ph.D. dissertation, Technical University of Catalonia (UPC), Barcelona, Spain, May 2003. [13] D. P. Palomar, J. M. Cioffi, and M. A. Lagunas, “Joint Tx-Rx beamforming design for multicarrier MIMO channels: a unified framework for convex optimization,” IEEE Trans. Signal Processing, vol. 51, pp. 2381-2401, Sept. 2003. [14] C. Wang, E. K. S. Au, R. D. Murch, and V. K. N. Lau, “Closedform outage probability and BER of MIMO zero-forcing receiver in the presence of imperfect CSI,” in Proc. 7th IEEE SPAWC 2006. [15] D. A. Gore, R. W. Heath Jr., and A. J. Paulraj, “Transmit selection in spatial multiplexing systems,” IEEE Commun. Lett., vol. 6, pp. 491-493, Nov. 2002. [16] J. G. Proakis, Digital Communications, 4th edition. New York: McGrawHill, 2001. [17] J. Lu, K. B. Letaief, J. C. I. Chuang, and M. L. Liou, “M-PSK and M-QAM BER computation using signal-space concepts,” IEEE Trans. Commun., vol. 47, pp. 181-184, Feb. 1999. [18] K. Cho and D. Yoon, “On the general BER expression of one- and twodimensional amplitude modulations,” IEEE Trans. Commun., vol. 50, pp. 1074-1080, July 2002. Edward K. S. Au (S’02) received his Bachelor of Engineering degree with First Class Honors in Computer Engineering from the Hong Kong University of Science and Technology in 2002. Since August 2002, he has been pursuing the Ph.D. degree from the Department of Electronic and Computer Engineering at the same university. His research interests are in the field of communication and information theory, multiple-input multipleoutput (MIMO) wireless systems, and data storage systems. Edward is a recipient of the Teaching Appreciation Award for Teaching Assistant for his outstanding achievement in Teaching Excellence for Academic Year 2003 - 2004. He is a member of the Signal Processing for Storage (SPS) technical committee of IEEE Communications Society. He served as Technical Program Assistant of 2003 IEEE Information Theory Workshop.
j=1
= u2k,max PT ≤ PT
(32)
in which the equality in (32) is achieved when Un = INt . R EFERENCES [1] R. D. Murch and K. B. Letaief, “Antenna systems for broadband wireless access,” IEEE Commun. Mag., vol. 40, pp. 76-83, April 2002. [2] D. N. C. Tse and P. Viswanath, Fundamentals of Wireless Communications. Cambridge University Press, 2005. [3] S. Catreux, V. Erceg, D. Gesbert, and R. W. Heath Jr., “Adaptive modulation and MIMO coding for broadband wireless data networks,” IEEE Commun. Mag., vol. 40, pp. 108-115, June 2002. [4] D. J. Love, R. W. Heath Jr., W. Santipach, and M. L. Hoing, “What is the value of limited feedback for MIMO channels?” IEEE Commun. Mag., vol. 42, pp. 54-59, Oct. 2004.
Cheng Wang (S’03) received the B.S. degree in Electronic Science and Engineering from the Nanjing University, Nanjing, Jiangsu, P. R. China, where she graduated in 2002 and ranked first among 174 students in the department. She is currently working toward the Ph.D. degree in the Department of Electronic and Computer Engineering, the Hong Kong University of Science and Technology, Hong Kong, P. R. China. Her current research interests include multi-user MIMO wireless communication systems, adaptive resource allocation, cross-layer design and optimization, partial CSI and limited feedback. She is the recipient of Schmidt Electronics Asia Award of Excellence, 2006; HKTIIT Post-Graduate Excellence Scholarships, 2006; CenWIT Scholarship: R&D Excellence, 2006; IEEE Communications Society Student Travel Grant Award, 2006; Excellent Student Award of Nanjing University, 1998-2000; First Class Scholarships of Nanjing University, 1998-2001.
AU et al.: ERROR PROBABILITY FOR MIMO ZERO-FORCING RECEIVER WITH ADAPTIVE POWER ALLOCATION
Sana Sfar (S’01-M’05) is currently a visiting researcher at Lehigh University, PA in collaboration with Bell-labs, Lucent, NJ. She received her Ph. D. and MPhil. degrees in 2005 and 2001, respectively, in Electrical and Electronic Engineering from The Hong Kong University of Science and Technology (HKUST) with focus on wireless communications. Prior to that, she received her Engineering Diploma in Telecommunications with Excellency in 1999 from cole Suprieure des Communications de Tunis, Tunisia, (SupCom) with focus on multimedia and networking. Her current research interests are in the area of wireless communication systems and communication theory. These include wireless and mobile networks, Ad-Hoc networks, Broadband wireless access, MIMO techniques, space-time coding, channel coding and modulation, OFDM, CDMA, and After Next Wireless systems. Ross D. Murch (S’85-M’87-SM’98) is a Professor of Electronic and Computer Engineering at the Hong Kong University of Science and Technology. His current research interests include MIMO antenna design, MIMO and cooperative systems, WLAN, B3G and Ultra-Wide-Band (UWB) systems for wireless communications. He has several US patents related to wireless communication, over 150 published papers and acts as a consultant for industry and government. In addition he is an editor for the IEEE Transactions on Wireless Communications, Technical Program Chair for the IEEE Wireless Communication and Networking Conference in 2007 and was the Chair of the Advanced Wireless Communications Systems Symposium at ICC 2002. He is also the founding Director of the Center for Wireless Information Technology at Hong Kong University of Science and Technology which was begun in August 1997. He is the program Director for the MSc in Telecommunications at Hong Kong University of Science and Technology. From August-December 1998 he was on sabbatical leave at Allgon Mobile Communications (which manufactured 1 million antennas per week in 1998), Sweden and AT&T Research Labs, NJ, USA. Wai Ho Mow (S’89-M’93-SM’99) received his M.Phil. and Ph.D. degrees in information engineering from the Chinese University of Hong Kong in 1991 and 1993, respectively. From 1997 to 1999, he was with the Nanyang Technological University, Singapore. He has been with the Hong Kong University of Science and Technology (HKUST) since March 2000. In spite of the relatively short PhD study period, he received the university-wide Young Scholar Dissertation Award and the Best PhD Thesis in Engineering Award. He was also the recipient of the Croucher Research Fellowship (Hong Kong), the Humboldt Research Fellowship (Germany), the TAO Research Fellowship (Japan), the Tan Chin Tuan Fellowship (Singapore), and the Foreign Expert Bureau Fellowship (China). His research interests are in the areas of wireless communications, coding and information theory. He pioneered the lattice approach to signal detection problems (such as sphere decoding and complex lattice reductionaided detection) and unified all known constructions of perfect (or CAZAC) root-of-unity sequences (widely used as preambles and sounding sequences). Dr. Mow published one book, and co-authored six filed patent applications and over 100 technical publications, among which he is the sole author of over 40. He also co-authored a paper that received the ISITA2002 Paper Award for Young Researchers and supervised two undergraduate projects that won the first-runner up prizes in the IEE Hong Kong YMS Project Competitions. Since Jun 2002, he has been the principal investigator of eight funded research projects. In 2005, he chaired the Hong Kong Chapter of the IEEE Information Theory Society that won the Chapter of the Year Award at ISIT’06. He was the technical program chair/co-chair of five conferences, and served the technical program committees of numerous conferences, such as ICC, Globecom, ITW, ISITA and VTC. He was a guest editor for a special section on sequence design in the IEICE Transactions on Fundamentals in 2006. He is currently an associate director of the Wireless IC System Design Center at HKUST Nansha Graduate School. He has been a member of the Radio Spectrum Advisory Committee, Office of the Telecommunications Authority, Hong Kong S.A.R. Government since 2003.
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Vincent K. N. Lau (M’98-SM’01) obtained B.Eng (Distinction 1st Hons) from the University of Hong Kong (1989-1992) and Ph.D. from Cambridge University (1995-1997). He was with HK Telecom (PCCW) as system engineer from 1992-1995 and Bell Labs - Lucent Technologies as member of technical staff from 1997-2003. He joined the Department of ECE, Hong Kong University of Science and Technology (HKUST) as Associate Professor. At the same time, he is a technology advisor of HKASTRI, leading the Advanced Technology Team on Wireless Access Systems. His current research focus is on the robust cross layer scheduling for MIMO/OFDM wireless systems with imperfect channel state information, communication theory with limited feedback as well as cross layer scheduling for users with heterogeneous delay requirements.
Roger S. Cheng (S’85-M’91) received his Ph.D. degree from Princeton University. In 1995, he joined HKUST where he is currently an Associate Professor. He was an Assistant Professor at University of Colorado at Boulder in 1991-1995, and was visiting senior engineer at Qualcomm, San Diego. He has also consulted with Huawei technologies, ZTE, and ASTRI, and had served as Guest Editor for IEEE Journal on Selected Areas in Communications and Associate Editor for IEEE Transactions on Signal Processing and IEEE Transactions on Communications. His current research interests include MIMO, OFDM and wireless communications.
Khaled Ben Letaief (S’85-M’86-SM’97-F’03) received the BS degree with distinction in Electrical Engineering from Purdue University at West Lafayette, Indiana, USA, in December 1984. He received the MS and Ph.D. Degrees in Electrical Engineering from Purdue University, in August 1986, and May 1990, respectively. From January 1985 and as a Graduate Instructor in the School of Electrical Engineering at Purdue University, he has taught courses in communications and electronics. From 1990 to 1993, he was a faculty member at the University of Melbourne, Australia. Since 1993, he has been with the Hong Kong University of Science and Technology where he is currently a Chair Professor of Electronic and Computer Engineering (ECE) and Head of the ECE Department. He is also the Director of the Hong Kong Telecom Institute of Information Technology as well as the Director of the Center for Wireless Information Technology. His current research interests include wireless and mobile networks, Broadband wireless access, OFDM, CDMA, and Beyond 3G systems. In these areas, he has published over 280 journal and conference papers and given invited keynote talks as well as courses all over the world. Dr. Letaief served as consultants for different organizations and is currently the founding Editor-inChief of the IEEE Transactions on Wireless Communications. He has served on the editorial board of other prestigious journals including the IEEE Journal on Selected Areas in Communications - Wireless Series (as Editor-in-Chief) and the IEEE Transactions on Communications. He has been involved in organizing a number of major international conferences and events. These include serving as the Technical Program Chair of the 1998 IEEE Globecom Mini-Conference on Communications Theory, held in Sydney, Australia as well as the Co-Chair of the 2001 IEEE ICC Communications Theory Symposium, held in Helsinki, Finland. In 2004, he served as the Co-Chair of the IEEE Wireless Communications, Networks and Systems Symposium, held in Dallas, USA as well as the Co-Technical Program Chair of the 2004 IEEE International Conference on Communications, Circuits and Systems, held in Chengdu, China. He is also the General Co-Chair of the 2007 IEEE Wireless Communications and Networking Conference, WCNC’07, held in Hong Kong as well as the Technical Program Co-Chair of the 2008 IEEE International Conference on Communication, ICC’08 held in Beijing. He served as the Chair of the IEEE Communications Society Technical Committee on Personal Communications as well as a member of the IEEE ComSoc Technical Activity Council. He is a Fellow of IEEE and an IEEE Distinguished lecturer of the IEEE Communications Society, and an elected member of the IEEE Communications Society Board of Governors.
