IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 5, SEPTEMBER 2007
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Capacity-Approaching Multiple Coding for MIMO Rayleigh Fading Systems With Transmit Channel State Information Jianhan Liu, Member, IEEE, Jinghu Chen, Member, IEEE, Anders Høst-Madsen, Senior Member, IEEE, and Marc P. C. Fossorier, Fellow, IEEE
Abstract—We study transmit power adaptation and capacityapproaching coding/decoding in multiple-input–multiple-output (MIMO) Rayleigh fading channels under the assumption that perfect channel state information (CSI) is known at both the transmitter and the receiver. The capacity in MIMO systems with transmitter CSI can be achieved via two schemes: 1) a multiple-coding scheme with temporal and spatial water filling and 2) a singlecoding scheme with temporal and spatial water filling. The former requires an infinite number of different codes, and the latter one requires interblock coding and, therefore, a very long code. We propose three simple but powerful methods for transforming the MIMO fading channel into a set of additive white Gaussian noise channels to which standard codes for the Gaussian channel can then be applied. We show through a number of examples that these methods can closely approach channel capacity. As a by-product, the closed-form expressions of the marginal probability density function of the ordered and unordered eigenvalues of Wishart matrices are derived. Index Terms—Channel state information (CSI), multiple coding, multiple-input–multiple-output (MIMO), partitioned channel inversion.
I. I NTRODUCTION
M
ULTIPLE-INPUT–multiple-output (MIMO) systems have attracted significant attention since Foschini and Telatar analyzed its potential to attain very high spectral efficiencies [1], [2]. One factor influencing the performance of MIMO systems is the amount of channel state information (CSI) that transmitters and/or receivers have. When the CSI is known neither at the transmitter nor at the receiver, the capacity-achieving signaling matrix has been derived by Marzetta and Hochwald in [3] for MIMO fading channels. In this case, it is shown that the capacity-achieving signals are
Manuscript received April 4, 2006; revised August 14, 2006 and October 25, 2006. The review of this paper was coordinated by Dr. S. Vishwanath. This work was supported in part by the National Science Foundation under Grant CCR 03-29908. J. Liu is with SiBEAM Inc., Sunnyvale, CA 94085 USA (e-mail: jliu@ sibeam.com). J. Chen is with Qualcomm Inc., San Diego, CA 92121 USA (e-mail:
[email protected]). A. Høst-Madsen and M. P. C. Fossorier are with the Department of Electrical Engineering, University of Hawaii at Manoa, Honolulu, HI 96822 USA (e-mail:
[email protected];
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2007.900489
temporally orthogonal among the transmit antennas within the coherence interval, and there is no need to make the number of transmit antennas greater than that of symbols in the coherence time interval. In [2], the capacity of MIMO Rayleigh fading channels is calculated by assuming CSI at the receiver but none at the transmitter, and it is shown that the capacity increases linearly with the minimum between the numbers of transmit and receive antennas. When CSI is known at both the transmitter and the receiver, higher capacity can of course be expected, and in [4]–[6], it is shown that the gain can be considerable when the number of transmit antennas is larger than the number of receive antennas, whereas the gain is limited when there are fewer transmit than receive antennas.1 However, even in the latter case, there can be large advantages to transmitter CSI, in that it can potentially reduce coding complexity considerably (cf., channel feedback in [7]). The aim of this paper is to develop coding schemes that can both exploit the increase in capacity and reduce coding complexity. If the ergodic fading channel that is considered is changing reasonably slowly, the instantaneous CSI, which is typically obtained by estimating unknown channel parameters at the receiver, can be obtained at the transmitter through a dedicated feedback channel. When the instantaneous CSI is available to both the transmitter and the receiver, the optimal power adaptation is known as the “water-filling” scheme if the MIMO system is subject to an average power constraint [2], [6], [7]. It can be seen that a multiplexed coding scheme with “waterfilling” power adaptation [9] can be applied to achieve the capacity. However, the multiplexed coding has high complexity since it requires in principle an infinite number of codes, or at least a very large number of codes, that are adapted to different signal-to-noise ratio (SNR) values to implement the multiplexed coding and decoding. This is still true for constant power water filling, although it simplifies the power adaptation [8]. It has been shown that, in a single-input–single-output (SISO) system [10] and in a MIMO system [11], capacity can in fact be achieved using a single code with a kind of “waterfilling” power adaptation. However, this code in principle needs to span a whole ensemble of fading states, and the code length 1 After correspondence with the authors of [5] and [6], it has been agreed that the numerical results that were presented in [5] and [6] are too optimistic with respect to the effect of CSI. The numerical results in this paper are therefore different from those in [5] and [6].
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and decoding delay therefore could be very long, particularly, in slow fading. In other words, to overcome deep fades, the single-coding scheme seems to require a very long code length to average fades over a sufficiently large number of channel realizations. A scheme to reduce complexity is the truncated channel inversion that is proposed in [9] (for SISO systems) [12] (for single-input–multiple-output (SIMO) systems). With channel inversion, the fading channel is transformed into an additive white Gaussian noise (AWGN) channel, and any code that is designed for the AWGN channel can be used. In addition, in slow fading, the length of the code and coding delay can be chosen independently of the fading dynamics, only taking into account the AWGN properties. However, this simplification comes at the cost of capacity loss, which can be significant, particularly, when the scheme is generalized to MIMO systems. In [13], a channel partition method is proposed to approach the capacity in a SISO Rayleigh fading channel, allowing a tradeoff between complexity and performance. However, the channel inversion is not introduced. Therefore, the codes still operate on fading channels, and a relatively large number of partitions are required to closely approach the capacity. In [22], channel inversion and multiple eigenbeamforming are studied. However, the performance analysis and the general tradeoff between complexity and capacity loss are not elaborated. In this paper, we investigate truncated channel inversion and channel partition for MIMO fading channels and propose simple schemes that achieve good tradeoff between coding/ decoding complexity and capacity loss. The main idea is to transform the MIMO fading system into a set of parallel independent AWGN channels by exploiting the CSI at both the transmitter and the receiver. Therefore, the code length and delay is determined solely by the length of codes that were chosen for the AWGN channel(s) and not by the fading dynamics. The focus of this paper is to develop simple but powerful methods for system transformation and not to find optimum coding. Transforming a channel into a few subchannels allows using a code having a lower rate than that of the single code that is better adapted to deep fades. The justification for the methods is that they can closely approach the channel capacity. The rest of this paper is organized as follows. Section II describes the discrete-time system model for MIMO systems. The marginal probability density functions (pdfs) of the eigenvalues of the channel matrix are presented in Section III. Section IV describes a truncated partitioned channel inversion scheme for beamforming systems. Section V extends the new scheme to multiple eigenbeamforming systems. Conclusions are drawn in Section VI. The notation that is used in this paper is given as follows: Bold letters denote matrices or column vectors. The symbol (·)† denotes the operation of complex conjugate transpose. The matrix I P stands for a P × P identity matrix, E(·) stands for expectation, and tr(·) stands for the trace of a matrix. II. S YSTEM M ODEL We consider a wireless link comprising Nt transmitter antennas and Nr receiver antennas that operates in a frequencyflat Rayleigh fading environment. The fading channel gains
can be described by an Nr × Nt channel matrix H, which is denoted by r ,Nt H = [Hi,j ]N i,j=1
(1)
where Hi,j is the complex channel gain factor between the jth transmit antenna and the ith receive antenna, and it is modeled as an independent identically distributed complex circular Gaussian random variable with zero mean and unit variance. Furthermore, the channel is assumed to be block static, i.e., channel matrix H remains constant within L symbol intervals, then changes to a new independent value for L other symbols, and so on. The channels are also assumed to be ergodic. The received signal can be expressed as y = Hx + z
(2)
where y is an Nr × 1 vector of received signals, x is an Nt × 1 vector of transmitted signals, and z is an Nr × 1 vector of additive receiver noise values, which are independent zeromean circular complex Gaussian random variables with unit variance. For future reference, we define m = min(Nt , Nr ) and n = max(Nt , Nr ), and we refer to such a system as an (n, m) MIMO system. Let the singular value decomposition (SVD) of channel matrix H be H = U Λ1/2 V † , where U and V are unitary matrices, and Λ is a diagonal matrix with nonnegative diagonal elements {λ1 , λ2 , . . . , λm }. Since the CSI is known at both the transmitter and the receiver, the transmitter can transmit x = V Q1/2 s, where s is the coded source data with unit normalized power, and Q determines the transmit power of each symbol. By multiplying the received signal vector y of (2) with the Hermitian of U and ignoring entries beyond m (only m eigenvalues are nonzero), we obtain y˜ = U † y = U †√ HV Q 2 s + U † z λ1 s1 1 . .. Q 2 .. + z ˜ = . √ sm λm 1
(3)
˜ = U † z is still a complex additive Gaussian noise where z vector with zero mean and covariance matrix I m . It is well known that {λ1 , λ2 , . . . , λm } are the eigenvalues of a Wishart matrix W , which is defined by
W =
HH † , for Nr < Nt . H † H, for Nr ≥ Nt
(4)
The joint pdf of the unordered eigenvalues λ1 , λ2 , . . . , λm is given in [14] fΛ (λ1 , λ2 , . . . , λm ) m m = Kn,m e(− i=1 λi ) λn−m i i=1
(λi − λj )2
1≤i λ1 , λk+1 , . . . , λm < λ1 ) . Notice that, because of the symmetry of the unordered eigenvalues, Pr(ηk ≤ η, π(Λ, i) = k) = Pr(ηk ≤ η, π(Λ, 1) = k), i = 2, . . . , m, and that these cases are mutually exclusive. The cumulative density function of ηk , therefore, is Pr(ηk ≤ η) m−1 =m k−1 × Pr (λ1 ≤ η, λ2 , . . . , λk > λ1 , λk+1 , . . . , λm < λ1 , ) . By applying the joint pdf of the unordered eigenvalues (5) and differentiating with respect to η, the marginal pdf of ηk can be obtained as m−1 k (η) = m fn,m k−1 ∞ ∞ η η × ··· · · · fΛ (η, λ2 , . . . , λm ) η
u (λ) = Kn,2 e−λ λn−2 Φ(λ, n, 2) fn,2
(9)
u fn,3 (λ) = Kn,3 e−λ λn−3 Φ(λ, n, 3)
(10)
0
k−1
∞ (11)
(12)
m xi i=0
(13)
(14)
i!
e−t tm dt = m! 1 − e−x
m xi i=0
0
i!
.
(15)
A closed-form expression of (13) can straightforwardly be obtained for any (n, m) MIMO system. This is used to obtain the results in the following sections. However, for a large m, the expressions get very complex, so we will only list the case where m = 2 here, i.e., η
B. Marginal PDF of the Ordered Eigenvalues We sort all the eigenvalues in nonincreasing order and denote the kth-order statistic of λ1 , λ2 , . . . , λm by ηk . In the following, we find the marginal pdf of ηk . Notice that the probability that two eigenvalues are equal is zero, and we can therefore discard this case. Let π be defined by π(Λ, i) = k if ηk = λi , i.e., if λi is the kth largest eigenvalue. Consider all Λs, so that π(Λ, 1) = k. This means that, among the remaining m − 1 unordered eigenvalues, k − 1 is larger than λ1 , and m − k is smaller than λ1 . The k − 1 larger eigenvalues can be selected
× dλ2 · · · dλm .
e−t tm dt = m!e−x
x
x
+ 4λ3 [(n − 1)!(n − 2)! − n!]
+ 2λ2 (n + 1)! + 2n!(n − 2)! − 3((n − 1)!)2
0
m−k
Notice that the fΛ given in (5) is a linear combination of terms of the form λi11 e−λ1 λi22 e−λ2 · · · λimm e−λm . The multidimensional integration in (13) can therefore be written as a sum of products of integrals of the following form:
where
+ 4λ [n!(n − 1)! − (n + 1)!(n − 2)!]
+ 2 (n + 1)!(n − 1)! − (n!)2 .
η
We state explicit formulas for the (n, 2) and (n, 3) MIMO systems (i.e., where the smaller of Nt and Nr is 2 or 3) as important special cases, i.e.,
Φ(λ, n, 2) = (n − 2)!λ2 − 2(n − 1)!λ + n!
Φ(λ, n, 3) = 2λ4 (n − 1)! − ((n − 2)!)2
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1 (η) fn,2
=2
fΛ (η, λ2 )dλ2 0
= 2Kn,2 [ϕ1 (n, n − 2, η1 ) − 2ϕ1 (n − 1, n − 1, η1 ) (16) + ϕ1 (n − 2, n, η1 )] ∞ 2 fn,2 (η) = 2 fΛ (η, λ2 )dλ2 η
= 2Kn,2 [ϕ2 (n, n − 2, η2 ) − 2ϕ2 (n − 1, n − 1, η2 ) (17) + ϕ2 (n − 2, n, η2 )]
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where
−η l
ϕ1 (l, k, η) = e
−η
η k! 1 − e
k ηi i=0
ϕ2 (l, k, η) = e−2η η l k!
k ηi i=0
i!
.
i!
(18)
(19)
IV. B EAMFORMING C ASE
Fig. 1.
Beamforming, which always transmits the signal along the direction corresponding to the largest eigenvalue, has recently received considerable attention since it can maximize the received SNR and exploit the diversity gain of MIMO channels [15]. In [17], it is shown that beamforming with proper power control can achieve significant average transmit power savings compared to the systems with constant transmit power. Therefore, we first consider power adaptation and coding/decoding for beamforming. Based on [10], splitting the problem into power adaptation and coding does not introduce any capacity loss. With the transmit power constraint Ω, the system model for beamforming with power adaptation can therefore be written as (20) y˜ = q(η1 , Ω)η1 s + z˜ where η1 is the largest eigenvalue that is defined in Section III-B, z˜ is a zero-mean additive complex Gaussian noise with unit variance, s is the transmitted symbols satisfying E[|s|2 ] = 1, and q(η1 , Ω) denotes the power control function. The optimal power control scheme can be obtained by solving the following optimization problem: 1 log(1 + q(η1 , Ω)η1 )fn,m (η1 )dη1 C = max q(η1 ,Ω)
η1 1 q(η1 , Ω)fn,m (η1 )dη1 = Ω,
s.t.
q(η1 , Ω) ≥ 0
η1
(21) where f1 (η1 ) is the marginal pdf of the largest eigenvalue η1 . It can be shown that the optimal power adaptation is the “waterfilling” method, i.e., 1 1 η1 ≥ η0 − η , 1 (22) q(η1 , Ω) = η0 0, η1 < η0 where η0 is some cutoff value and can be obtained by solving the average power constraint (21). In [9], a truncated channel inversion scheme was proposed as a suboptimal power adaptation scheme. Since the truncated channel inversion scheme converts the fading channel into one AWGN channel, it has a much lower coding complexity. However, the truncated channel inversion scheme suffers a large capacity penalty relative to the optimal power adaptation in some cases. In order to obtain a tradeoff between coding/decoding complexity and capacity loss, we applied a partitioned channel inversion scheme to convert the beamforming fading channel into a few exclusive AWGN
Truncated partitioned channel inversion.
channels. It uses a finite number of codes to approximate the channel capacity instead of using only one code. The general partitioned channel inversion scheme is illustrated in Fig. 1, where we first partition the value of channel gain into two regions. The first region, from 0 to η1,0 , is called the truncated region because no signal will be transmitted if η1 falls into this region; the second region, which begins at η1,0 , is partitioned into N subregions, each of which can be regarded as a fading subchannel; it is referred to as the untruncated region. Let ρi be the average power that is allocated to the ith subchannel. By applying channel inversion at the transmitter for each fading subchannel, we obtain the following power adaptation scheme in the ith subchannel: ρi (23) qi (η1 , Ω) = ai η1 where ai is a normalizing factor that is given by
η1,i
ai = η1,i−1
1 1 f (η1 )dη1 . η1 n,m
(24)
Substituting the power adaptation scheme into (20), we obtain the following channel model when η1 falls into the ith subregion: y˜ =
ρi s + z˜. ai
(25)
Obviously, the scheme converts the fading subchannels into N AWGN subchannels with different SNR values. It is readily seen that these N AWGN subchannels are independent, and therefore, only N codes are required to approach the capacity of the beamforming. The problem left is to find the channel partition points ηi , i = 0, . . . , N , and the power allocation ρi , i = 1, . . . , N . As will be seen here, the values ρi can be found explicitly, given the values η1,i . However, searching for the values η1,i results in a multidimensional optimization problem that can only be solved numerically. Instead, we suggest a heuristic solution. The simplest possible solution is to divide the probability space (for η1 above cutoff) into N equal partitions. Although this is a simple solution, it can be readily seen that, for N → ∞, it converges toward optimum water filling, and as we will see later, even for small N , it gives a performance that is close to optimum. We also numerically calculated the rates that were achieved by optimal partition for a few cases and observed that the optimal partition only has negligible gain
LIU et al.: CAPACITY-APPROACHING MULTIPLE CODING FOR MIMO RAYLEIGH FADING SYSTEM WITH CSI
TABLE I ALGORITHM FOR NUMERICALLY CALCULATING Ce
over the heuristic method [16]. Let Pi be the probability that η1 falls into the ith subregion, which is given by
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η1,i 1 fn,m (η1 )dη1
Pi = η1,i−1 ∞
1 fn,m (η1 )dη1
=
η1,0
= Pa ,
N
i = 1, . . . , N
(26)
where we denote η1,N = ∞ for convenience. It is clear that, if the cutoff value η1,0 is fixed, all the other partition positions η1,i , for i = 1, 2, . . . , N − 1, are also fixed. Therefore, the multidimensional optimization problem can be avoided. The achievable capacity Ce of the beamforming with the proposed scheme can be obtained by solving the following optimization problem: N ρi Pa log 1 + (27) Ce = max η1,0 ,ρi ai i=1 s.t.
N
ρi ≥ 0.
ρi = Ω,
∞
e−η η −1 dη = E1 (x)
x
where E1 (x) is the exponential integral of order 1, it can be shown that Pa and ai , for any (n, m) MIMO system, can be expressed in closed form. We demonstrate the results explicitly for the (n, 2) system. The integral of ϕ1 (l, k, η) defined in (18), for l ≥ −1, k ≥ 0, can be found to be (33), shown at the bottom of the page. Then, Pa and ai can be expressed as
(28)
P(η1,0 ) N ai = A(η1,i−1 ) − A(η1,i )
Pa =
i=1
For a given set of η1,i , i = 0, 1, . . . , N − 1, this is a standard optimization problem and can be solved by applying the Lagrange multiplier and Kuhn–Tucker conditions [7]. It is shown that the best power adaptation scheme in this case can be acquired from the following equations: ρi,o = [Pi µ − ai ]+ ,
N
(29)
where [x]+ = max{x, 0}, and µ is some constant (“waterfilling level”) that is chosen so that the power constraint (29) is satisfied. Solving the preceding equations and substituting the solution of ρi,o into (27), the achievable capacity can be expressed as N ρi,o log 1 + . Ce = max Pa η1,0 ai i=1
∞
P(x) =
1 fn,m (η1 )dη1
= 2Kn,2 [ϕ˜1 (n, n − 2, x) − 2ϕ˜1 (n − 1, n − 1, x) (36) + ϕ˜1 (n − 2, n, x)] ∞ 1 1 A(x) = f (η1 )dη1 η1 n,m x
= 2Kn,2 [ϕ˜1 (n − 1, n − 2, x) − 2ϕ˜1 (n − 2, n − 1, x) (37) + ϕ˜1 (n − 3, n, x)] respectively. When η1,0 is given, η1,i , for i = 1, 2, . . . , N − 1, can be obtained by recursively solving
(30)
1 (η1 ) or From Section III-B, we notice that each term in fn,m 1 −c2 η1 l η1 , where c1 is some 1/η1 fn,m (η1 ) can be written as c1 e constant coefficient, c2 is a positive scalar, and l is an integer, which is no less than −1. By applying the following identities:
x
l c1 l! −c2 x (c2 x)i , e i! cl+1 2 i=0
ϕ1 (l, k, η)dη = x
l≥0
∞
ϕ˜1 (l, k, x) =
(35)
respectively, where P(x) and A(x) are given by
η1,i 1 fn,m (η1 )dη1
Pa =
c1 e−c2 η1 η1l dη1 =
(34)
x
ρi,o = Ω
i=1
∞
(32)
k!l!e−x
(31)
l
xj j=0 j!
η1,i−1
= P(η1,i−1 ) − P(η1,i )
(38)
i.e., given η1,0 , η1,1 is found; then, with η1,1 found, η1,2 can be found, and so on. The achievable capacity Ce now can be numerically calculated by following the algorithm in Table I. We illustrate the achievable capacity Ce with a few examples in Figs. 2 and 3 and discuss the results next.
− k!
(l+i)! −2x l+i (2x)j i=0 i!2l+i+1 e j=0 j! , k (i−1)! −2x i−1 (2x)j k! i=1 i!2i e j=0 j! ,
k
k!E1 (x) − k!E1 (2x) −
l≥0 l = −1
(33)
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converge toward the optimal water-filling scheme asymptotically. It is more interesting that, for even a small N , the achievable capacity is very close to the optimum water filling. Fig. 3 depicts the achievable capacity of beamforming with channel partitioning in (2, 2) and (4, 4) MIMO Rayleigh fading channels. It is worth mentioning that the capacity penalty of truncated channel inversion (channel partitioning with N = 1) diminishes remarkably as the number of antennas increases. This observation extends the conclusion that was made on SIMO fading channels with diversity combining [12] to MIMO fading channels with beamforming. V. M ULTIPLE E IGENBEAMFORMING C ASE
Fig. 2. Capacity of beamforming in SISO and SIMO Rayleigh fading channels.
If CSI is available at both the transmitter and the receiver, (3) indicates that an (n, m) MIMO fading channel can be decomposed into m parallel channels, and these parallel channels are characterized by an eigenvalue matrix Λ. When the SNR is high, instead of just using the best channel among the m channels as in beamforming, multiple parallel channels could be utilized to support high transmission rates. With an average transmit power constraint, the transmit power also should be dynamically allocated according to the instantaneous CSI. Consequently, the transmit power should be a function of Λ and the average power constraint Ω. Again, the power adaptation and coding can be separated without introducing any capacity loss [6], [11]. Therefore, the multiple eigenbeamforming model with power control can be modeled by ˜ ˜ = Λ1/2 Q(Λ, Ω)1/2 s + z y
(39)
where s is the transmitted symbol satisfying E[s† s] = 1, and Q(Λ, Ω) features the transmit power control scheme. It is readily seen that Q(Λ, Ω) should be a diagonal matrix and satisfy the following average power constraint: E {tr [Q(Λ, Ω)]} = Ω.
Fig. 3. Capacity of beamforming in MIMO Rayleigh fading channels.
Fig. 2 depicts the achievable capacity of beamforming with the proposed scheme in SISO (1, 1) and SIMO (2, 1) Rayleigh fading channels. Truncated channel inversion [9], which corresponds to our scheme with N = 1, has the smallest coding/decoding complexity as it requires only one code that is designed for an AWGN channel. However, from the figure, it can be seen that it suffers a consequent capacity penalty, particularly, at high SNR. By using two or three codes that are designed for AWGN channels, i.e., channel partitioning with N = 2 or N = 3, the capacity penalty can be decreased. Since both the transmitter and the receiver know which code is used and all the subchannels are exclusive, only one pair of encoder/decoder is active at each coherence interval. It can be seen that the suboptimality that is introduced by equal partitioning of the untruncated region is not significant, and as N → ∞, this suboptimality vanishes since it can be readily argued that the capacity with equally truncated channel partitioning will
(40)
It is shown in [5] that optimal power adaptation in the multiple eigenbeamforming technique is still the “water-filling” method with 1 1 λi ≥ λ0 − λ , i (41) Qi,i = λ0 0, λi < λ0 where λ0 is a unique cutoff value for all eigenvalues and can be obtained by solving the following average power constraint equation: ∞ λ0
1 1 − λ0 λ
u (λ)dλ = fn,m
Ω m
(42)
u where fn,m (λ) is the marginal pdf of the unordered eigenvalue for an (n, m) MIMO system. The capacity with the optimal power control scheme is ∞ λ u C = m log (λ)dλ. (43) fn,m λ0 λ0
LIU et al.: CAPACITY-APPROACHING MULTIPLE CODING FOR MIMO RAYLEIGH FADING SYSTEM WITH CSI
Again, there are two methods to achieve the capacity: 1) multiple coding with optimal power adaptation [9] and 2) single coding with optimal power adaptation [11]. The former method requires an infinite number of codes in principle and joint multiplexed coding/decoding among m parallel channels. The latter method requires interblock coding and, therefore, might need quite long codes to compensate the fading, particularly when deep fading appears. To obtain a better tradeoff, we propose a capacity-approaching scheme that converts the dependent fading parallel channels into independent AWGN parallel channels. It turns out that the truncated channel inversion method is a straightforward method to achieve this goal. A. Unordered Multiple Eigenbeamforming
y˜k =
ρi sk + z˜k ai
(44)
where ρi is the average power allocated to the ith subchannel, and ai is defined as λi
ai =
1 u f (λ)dλ λ n,m
(45)
λi−1
where λi , for i = 1, 2, . . . , N − 1, stands for the ith partition
position, λ0 is the truncated value, and λN is defined as λN = ∞. From (44), we can see that the N subchannels of the kth channel are exclusive, and any subchannel of the kth channel is independent of any subchannel of the k th channel if k = k . Thus, we convert the m dependent parallel channels into independent parallel AWGN channels. The achievable capacity Ce in this case can now be found by solving the following optimization problem: N ρi Ce = max mPa log 1 + λ0 ,ρi ai i=1 s.t.
N
ρi =
i=1
Ω , m
ρi ≥ 0
(46)
where we define
Pa =
∞ λ0
u fn,m (λ)dλ
N
.
As in Section IV, the optimal values of ρi can be found, given λ0 , and we get the achievable capacity as N ρi,o Ce = max mPa log 1 + (48) λ0 ai i=1 m Ω (49) ρi,o = ρi,o = [mPa µ − ai ]+ , m i=1 where µ is a “water-filling level” that is chosen so that (49) is satisfied. With a similar argument as in Section IV, we can show that Pa and ai can be expressed in closed form. We will give the expressions for the (n, 2) system. Define ˜ x) = ψ(l,
Let us first consider applying the truncated channel partition among the m unordered parallel channels that are defined by SVD. Notice that, in the unordered case, all channels have the same fading characteristic. Therefore, each channel should be truncated and partitioned in the same way, and equal power should be allocated to each channel, namely, Ω/m. We partition each truncated channel into N subchannels, as in Section IV. After channel inversion on each subchannel, if the value of the kth channel falls into the ith subregion, the input–output relation can be expressed as
(47)
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∞
−η l
e
η dη =
x
l!e−x li=0 E1 (x),
xi i! ,
l≥0 l = −1
(50)
u and using fn,m (λ) derived in Section III-A, we can write Pa and ai in closed form as
Pu (λ0 ) N ai = B(λi−1 )B(λi )
Pa =
(51) (52)
respectively, where Pu (x) and B(x) are given by ! ˜ λ0 ) − 2(n − 1)!ψ(n ˜ − 1, λ0 ) Pu (x) = Kn,2 (n − 2)!ψ(n, " ˜ − 2, λ0 ) (53) + n!ψ(n ! ˜ − 1, x) − 2(n − 1)!ψ(n ˜ − 2, x) B(x) = Kn,2 (n − 2)!ψ(n " ˜ − 3, x) . + n!ψ(n (54) The achievable capacity of the unordered multiple eigenbeamforming with the proposed scheme can therefore easily be found numerically by following the same procedure as in Table I. When N → ∞, it can be easily seen that truncated channel partitioning approaches channel capacity. What is more interesting is that we can come close to the channel capacity with even moderate N , as the following numerical examples will show. In Figs. 4 and 5, we present the achievable capacities of the unordered multiple eigenbeamforming with channel partioning in MIMO Rayleigh fading channels. It can be seen that there exists a considerable capacity penalty for simple truncated channel inversion, particularly, at high SNR. Unlike the conclusion for beamforming, where the capacity penalty diminishes as the number of antennas increases, we notice that the capacity penalty for (4, 4) MIMO channels is actually bigger than the capacity penalties for (3, 3) and (2, 2) MIMO channels. However, from Fig. 4, we observe that the capacity penalty in (n, 2) MIMO channels decreases as n increases. Therefore, in the unordered multiple eigenbeamforming system, we can only say that the capacity penalty diminishes as the maximum between the number of transmit and receive antennas increases while the minimum between the number of transmit and receive antennas is fixed. It can be seen that the capacity penalty
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channel falls into the ith subregion, the input–output relation can be expressed as ρk,i sk + z˜k (55) y˜k = ak,i where we define ηk,i
ak,i = ηk,i−1
1 k f (ηk )dηk ηk n,m
(56)
k (ηk ) is the marginal pdf of the kth largest eigenvalue where fn,m that was defined in Section III-B; ηk,i , for i = 1, 2, . . . , N − 1, stands for the ith partition position of the kth channel; ηk,0 is the truncated value of the kth channel, and λk,N is defined as λk,N = ∞. The subchannels resulting from the truncated channel partition are still independent AWGN channels. We therefore get the following achievable capacity:
Fig. 4. Capacity of unordered multiple eigenbeamforming for (n, 2) MIMO Rayleigh fading channels.
Cg =
K N ρk,i Pk,i log 1 + ηk,i−1 ,ρk,i ak,i i=1 max s.t.
k=1 K N
ρk,i ≥ 0
ρk,i = Ω,
(57) (58)
k=1 i=1
where Pk,i denotes the probability that ηk falls into the subregion between ηk,i−1 and ηk,i , which is given by ηk,i
k fn,m (ηk )dηk .
Pk,i =
(59)
ηk,i−1
Unfortunately, the optimization of (57) is a KN -dimensional optimization problem, which can only be solved numerically. For simplicity, we consider a special case in which we use the same cutoff value for all the channels and then apply channel inversion (N = 1) to the region of each channel above this cutoff value. Let η0 denote the universal cutoff value. The achievable capacity by ordered multiple eigenbeamforming is Fig. 5. Capacity of unordered multiple eigenbeamforming for (3, 3) and (4, 4) MIMO Rayleigh fading channels.
can be greatly decreased by applying channel partitioning. For instance, when N = 4, the capacity penalty becomes very small in all our examples.
Ce = max η0
K ρk,o Pk,a log 1 + ak
where ak and Pk,a are given by ∞ 1 k ak = f (ηk )dηk ηk n,m η0 ∞
B. Ordered Multiple Eigenbeamforming The unordered multiple eigenbeamforming scheme emphasizes power control and coding/decoding in time, and it always needs m pairs of coder/decoders. In this section, we consider the ordered channels, which can provide more efficient power allocation and coding/decoding in both time and space. We assume that the eigenvalues have been ordered and that the K (1 ≤ K ≤ m) best eigenmodes are used in the communication system. In this case, only K pairs of coder/decoders are required. We still apply the truncated channel partition on each ordered channel. Let ρk,i be the average power that is allocated to the ith subchannel of the kth channel; if the value of the kth
(60)
k=1
k fn,m (ηk )dηk
Pk,a =
(61)
η0
and ρk,o can be obtained by solving the following equations: +
ρk,o = [Pk,a µ − ak ] ,
K
ρk,o = Ω.
(62)
k=1
Again, closed-form expressions for Pk,a and ak are obtainable for any MIMO system. For the (n, 2) system, it is readily seen that P1,a = P(η0 ) and a1 = A(η0 ), where P(x) and A(x) are
LIU et al.: CAPACITY-APPROACHING MULTIPLE CODING FOR MIMO RAYLEIGH FADING SYSTEM WITH CSI
Fig. 6. Capacity of ordered multiple eigenbeamforming for (n, 2) MIMO Rayleigh fading channels.
given in (36) and (37), respectively. To calculate P2,a and a2 , we define (63), shown at the bottom of the page. Noticing the 1 2 (η1 ) and fn,2 (η2 ), we can similarity between the forms of fn,2 still obtain P2,a = P(η0 ) and a2 = A(η0 ) through (36) and (37), respectively, except that we need to replace ϕ˜1 (l, k, x) with ϕ˜2 (l, k, x). The achievable capacity Ce of ordered multiple eigenbeamforming can be evaluated from (60) through a similar numerical search algorithm as in Table I. Notice that only η0 has to be optimized numerically, and it is therefore a 1-D numerical optimization problem. Figs. 6 and 7 illustrate the capacities that were achieved by ordered multiple eigenbeamforming in MIMO Rayleigh fading channels. As expected, the more eigenmodes used, the higher the achievable capacities, with the differences increasing with SNR values. It is worth pointing out that the achievable capacity by the ordered multiple eigenbeamforming (K = m) is clearly superior to that of beamforming, which corresponds to the ordered multiple eigenbeamforming with K = 1. It is shown that, if K < m, increasing K provides significant improvement, and if K = m, the choice (N = 1) is sufficient to closely approach the channel capacity. In that case, the influence of the suboptimality that was introduced by simplified optimization is not noticeable. In Fig. 6, it is shown that the capacity penalty diminishes as the maximum of the number of antenna increases while the minimum of the number of antennas is fixed. C. More Numerical Results In Fig. 8, we compare the achievable capacity of the proposed schemes in ordered and unordered multiple eigenbeamforming
∞
ϕ˜2 (l, k, x) =
ϕ2 (l, k, η)dη = x
Fig. 7. Capacity of ordered multiple eigenbeamforming for (3, 3) and (4, 4) MIMO Rayleigh fading channels.
Fig. 8. Comparison between ordered and unordered multiple eigenbeamforming for (3, 3) and (4, 4) MIMO Rayleigh fading channels.
for (3, 3) and (4, 4) MIMO systems. We first compare the achievable capacity by truncated channel inversion. For the (3, 3) MIMO system, we observe that the ordered multiple eigenbeamforming (K = 2) attains higher capacity than the unordered multiple eigenbeamforming (N = 1) if the SNR is below 20 dB. For the (4, 4) MIMO system, the ordered multiple eigenbeamforming (K = 3) attains higher capacity than the unordered multiple eigenbeamforming (N = 1) in the considered SNR region. Remember that unordered
(l+i)! −2x l+i (2x)j i=0 i!2l+i+1 e j=0 j! , k (i−1)! −2x i−1 (2x)j k!E1 (2x) + k! i=1 i!2i e j=0 j! ,
k!
k
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l≥0 l = −1
(63)
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eigenbeamforming. Based on our numerical results, we showed that the capacities that were achieved by using the proposed multiple-coding/decoding scheme can closely approach channel capacity and that the proposed schemes achieve a good tradeoff between coding/decoding complexity and capacity penalty. We also extended the conclusion [12] for the SIMO system into the MIMO system and restated it in a more general claim, i.e., the capacity penalty by using the truncated channel inversion scheme diminishes as the maximum of the number of antenna increases while the minimum of the number of antennas is fixed. The main advantage of the proposed schemes is that they can approach channel capacity for MIMO systems with moderate coding/decoding complexity. To the end, capacityapproaching iteratively decodable codes seem to be particularly promising [18]–[21]. R EFERENCES Fig. 9. Comparison with the capacity of CSI at the receiver only in MIMO Rayleigh fading channels.
eigenbeamforming always uses m parallel coders/decoders, while ordered eigenbeamforming uses K. Therefore, for the truncated channel inversion, the ordered case can use less number of coder/decoders than the unordered case but still achieve higher capacity in a certain SNR region. In Fig. 8, we also observe that both the unordered and ordered cases closely approach the capacity when the same number (K = N = m) codes are used. However, the usage of the codes is quite different for the unordered and ordered cases. For the unordered case, there are N candidate codes can be used by each eigenchannel. Based on the channel realization, each eigenchannel selects one code among these N candidate codes. For the ordered case, K ordered eigenchannel are used, and each ordered eigenchannel uses one fixed code. In Fig. 9, we compare the capacities that were achieved by knowing the CSI at the transmitter with the MIMO capacities in the case when the CSI is available only at the receiver. It is shown that, for the case where the CSI is only available at receiver and Nr ≤ Nt , only increasing the number of transmit antennas Nt cannot provide major capacity gain. However, when the CSI is also known at the transmitter, only increasing the number of transmit antennas Nt can provide considerable capacity gain. (Please note that although the gain is considerable, it is much less than the gain claimed in [5] and [6].) It is also shown that our proposed schemes allow to closely approach the capacity values. VI. C ONCLUSION We investigated power adaptation and coding/decoding schemes in MIMO Rayleigh flat fading channels under the assumption that CSI is available at both the transmitter and the receiver. We suggested coding methods based on channel partition and inversion that provide a tradeoff between coding/ decoding complexity and performance for beamforming, unordered multiple eigenbeamforming, and ordered multiple
[1] G. J. Foschini, “Layered space-time architecture for wireless communication in fading environments when using multi-element antennas,” Bell Lab. Tech. J., vol. 1, no. 2, pp. 41–59, 1996. [2] E. Telatar, “Capacity of multi-antenna Gaussian channels,” Eur. Trans. Telecommun., vol. 10, no. 6, pp. 585–596, Nov. 1999. [3] T. L. Marzetta and B. M. Hochwald, “Capacity of a mobile multipleantenna communication link in Rayleigh flat fading,” IEEE Trans. Inf. Theory, vol. 45, no. 1, pp. 139–157, Jan. 1999. [4] A. Narula, M. J. Lopez, M. D. Trott, and G. W. Wornell, “Efficient use of side information in multiple-antenna data transmission over fading channels,” IEEE J. Sel. Areas Commun., vol. 16, no. 8, pp. 1423–1436, Oct. 1998. [5] S. K. Jayaweera and H. V. Poor, “Capacity of multi-antenna systems with adaptive transmission techniques,” in Proc. 5th Int. Symp. WPMC, Honolulu, HI, Oct. 2002, vol. 2, pp. 392–396. [6] S. K. Jayaweera and H. V. Poor, “Capacity of multiple-antenna systems with both receiver and transmitter channel state information,” IEEE Trans. Inf. Theory, vol. 49, no. 10, pp. 2697–2709, Oct. 2003. [7] T. M. Cover and J. A. Thomas, Elements of Information Theory. Hoboken, NJ: Wiley, 1991. [8] W. Yu and J. Cioffi, “On constant-power waterfilling,” in Proc. IEEE ICC, Jun. 2001, vol. 6, pp. 1665–1669. [9] A. J. Goldsmith and P. P. Varaiya, “Capacity of fading channels with channel side information,” IEEE Trans. Inf. Theory, vol. 43, no. 6, pp. 1986– 1992, Nov. 1997. [10] G. Caire and S. Shamai, “On the capacity of some channels with channel state information,” IEEE Trans. Inf. Theory, vol. 45, no. 6, pp. 2007–2019, Sep. 1999. [11] E. Biglieri, G. Caire, and G. Taricco, “Limiting performance of blockfading channels with multiple antennas,” IEEE Trans. Inf. Theory, vol. 47, no. 4, pp. 1273–1289, May 2001. [12] M. S. Alouini and A. J. Goldsmith, “Capacity of Rayleigh fading channels under different adaptive transmission and diversity-combining techniques,” IEEE Trans. Veh. Technol., vol. 48, no. 4, pp. 1165–1181, Jul. 1999. [13] L. Lin, R. D. Yates, and P. Spasojevic, “Adaptive transmission with discrete code rates,” in Proc. IEEE ICC, May 2002, vol. 3, pp. 1424–1428. [14] A. T. James, “Distributions of matrix variates and latent roots derived from normal samples,” Ann. Math. Statist., vol. 35, no. 2, pp. 475–501, Jun. 1964. [15] S. A. Jafar and A. J. Goldsmith, “Beamforming capacity and SNR maximization for multiple antenna systems,” in Proc. IEEE 53rd VTC—Spring, May 2001, vol. 1, pp. 43–47. [16] J. Liu, “Multiple coding and space-time multi-user detection in multiple antenna systems,” Ph.D. dissertation, Univ. Hawaii Manoa, Honolulu, 2005. [17] R. Knopp and G. Caire, “Power control and beamforming for systems with multiple transmit and receive antennas,” IEEE Trans. Wireless Commun., vol. 1, no. 4, pp. 638–648, Oct. 2002. [18] B. J. Frey, R. Koetter, G. D. Forney, F. R. Kschischang, R. J. McEliece, and D. A. Spielman, “Special issue on codes on graphs and iterative algorithms,” IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 493–497, Feb. 2001.
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[19] P. H. Siegel, D. Divsalar, E. Eleftheriou, J. Hagenauer, D. Rowitch, and W. H. Tranter, “The turbo principle: From theory to practice, part I,” IEEE J. Sel. Areas Commun., vol. 19, no. 5, pp. 793–799, May 2001. [20] P. H. Siegel, D. Divsalar, E. Elepftheriou, J. Hagenauer, and D. Rowitch, “The turbo principle: From theory to practice, part II,” IEEE J. Sel. Areas Commun., vol. 19, no. 9, pp. 1657–1661, Sep. 2001. [21] M. Fossorier and S. Olcer, “Capacity approaching codes, iterative decoding algorithms and their applications,” IEEE Commun. Mag., vol. 41, no. 8, pp. 100–140, Aug. 2003. [22] C. M. James and B. L. Hughes, “Adaptive transmission using channel inversion for multiple antenna systems,” in Proc. CISS, Mar. 2004.
Jianhan Liu (S’04–M’06) received the B.E. and M.E. degrees from Xi’an Jiaotong University, Xi’an, China, in 1996 and 1999, respectively, and the Ph.D. degree from the University of Hawaii, Honolulu, in 2005, all in electrical engineering. He is currently a Senior Algorithm Engineer with SiBEAM Inc., Sunnyvale, CA. His research interests include discrete-time signal processing, wireless communications, and channel coding.
Jinghu Chen (S’01–M’04) received the B.E. and M.E. degrees from the Nanjing University of Posts and Telecommunications, Nanjing, China, in 1995 and 1998, respectively, and the Ph.D. degree in electrical engineering from the University of Hawaii, Honolulu, in 2003. From November 2003 to March 2005, he was a Research Associate with the Department of Computer Science, University of Illinois, Chicago. In April 2005, he joined Qualcomm Inc., San Diego, CA, where he is currently a Senior Engineer. His research interests include wireless communications, error-correcting codes, and algorithms for decoding and detections.
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Anders Høst-Madsen (SM’02) was born in Denmark in 1966. He received the M.Sc. degree in electrical engineering and the Ph.D. degree in mathematics from the Technical University of Denmark, Lyngby, Denmark, in 1990 and 1993, respectively. From 1993 to 1996, he was with Dantec Measurement Technology A/S, Copenhagen, Denmark; from 1996 to 1998, he was an Assistant Professor with Kwangju Institute of Science and Technology, Kwangju, Korea; and from 1998 to 2000, he was an Assistant Professor with the Department of Electrical and Computer Engineering, University of Calgary, Calgary, AB, Canada, and a Staff Scientist with TRLabs, Calgary. Since 2001, he has been with the Department of Electrical Engineering, University of Hawaii at Manoa, Honolulu, where he is currently as an Associate Professor. He was also a Visitor at the Department of Mathematics, University of California, Berkeley, throughout 1992. His research interests are statistical signal processing, information theory, and wireless communications, including ad hoc networks, cooperative diversity, wireless sensor networks, heart monitoring, and life detection. Prof. Høst-Madsen currently serves as Editor for Multiuser Communications for the IEEE TRANSACTIONS ON COMMUNICATIONS and as Associate Editor for Detection and Estimation for the IEEE TRANSACTIONS ON INFORMATION THEORY.
Marc P. C. Fossorier (S’89–M’90–SM’00–F’06) received the B.E. degree from the National Institute of Applied Sciences (INSA), Lyon, France, in 1987 and the M.S. and Ph.D. degrees from the University of Hawaii at Manoa, Honolulu, in 1991 and 1994, all in electrical engineering. In 1996, he joined the faculty of the Department of Electrical Engineering, University of Hawaii at Manoa, as an Assistant Professor of electrical engineering. He was promoted to Associate Professor in 1999 and to Professor in 2004. His research interests include decoding techniques for linear codes, communication algorithms, and statistics. Dr. Fossorier has served as Editor of the IEEE COMMUNICATIONS LETTERS since 1999, served as Editor of the IEEE TRANSACTIONS ON INFORMATION THEORY from 2003 to 2006, served as Editor of the IEEE TRANSACTIONS ON C OMMUNICATIONS from 1996 to 2003 and as Treasurer of the IEEE Information Theory Society from 1999 to 2003. Since 2002, he has also been an elected member of the Board of Governors of the IEEE Information Theory Society, which he is currently serving as First Vice-President. He was the Program Co-Chairman for the 2000 International Symposium on Information Theory and Its Applications and Editor for the Proceedings of the 2006, 2003, and 1999 Symposia on Applied Algebra, Algebraic Algorithms, and Error Correcting Codes. He was a recipient of a 1998 National Science Foundation Career Development Award.
