Space-Time Water-Filling for Composite MIMO Fading Channels

We analyze the ergodic capacity and channel outage probability for a composite MIMO channel model, which includes both fast fading and shadowing e ﬀ ects. The ergodic capacity and exact channel outage probability with space-time water-ﬁlling can be evaluated through numerical integrations, which can be further simpliﬁed by using approximated empirical eigenvalue and maximal eigenvalue distribution of MIMO fading channels. We also compare the performance of space-time water-ﬁlling with spatial water-ﬁlling. For MIMO channels with small shadowing e ﬀ ects, spatial water-ﬁlling performs very close to space-time water-ﬁlling in terms of ergodic capacity. For MIMO channels with large shadowing e ﬀ ects, however, space-time water-ﬁlling achieves signiﬁcantly higher capacity per antenna than spatial water-ﬁlling at low to moderate SNR regimes, but with a much higher channel outage probability. We show that the analytical capacity and outage probability results agree very well with those obtained from Monte Carlo simulations.


INTRODUCTION
Multiple-input multiple-output (MIMO) communication systems exploit the degrees of freedom introduced by multiple transmit and receive antennas to offer high spectral efficiency.In narrowband channels, when channel state information is available at the transmitter and instantaneous adaptation is possible, the capacity achieving distribution is found by using the well-known water-filling algorithm [1,2].With only average power constraints, a two-dimensional water-filling in both the temporal and spatial domains has recently been shown to be optimal [3,4].By studying the empirical distribution of the eigenvalues of Gaussian random matrices [1], two-dimensional water-filling for Rayleigh MIMO channels [3,4] can be transformed into one-dimensional water-filling for a time-varying SISO channel [5].With the freedom to optimize the transmit power in both time and spatial domains, two-dimensional spacetime water-filling disables data transmission when all of the effective channel gains are not high enough to utilize transmit power efficiently, thereby resulting in a larger ergodic capacity when compared to spatial-only water-filling.In [3], a MIMO channel outage probability is defined to quantify how often the transmission is blocked, and upper bounds in Rayleigh fading channels on this outage probability have been developed.Although the ergodic capacity in i.i.d.MIMO Rayleigh fading channels is well understood, the capacity in MIMO Rayleigh fading channels with shadowing effects has not been evaluated, and the exact channel outage probability calculation has not been discussed.Furthermore, while [1][2][3][4] have studied either spatial or space-time waterfilling, the capacity gain of space-time water-filling over spatial water-filling has not been quantified.
In this paper, we perform space-time water-filling for a mixed MIMO channel model that includes both Rayleigh fading and shadowing effects.We show that the ergodic capacity and the exact channel outage probability can both be evaluated through numerical integrations.Hence, the timeconsuming Monte Carlo simulations, that is, generating a large number of channel realizations and then performing averaging, can be avoided.We also show that for Rayleigh channels without shadowing, space-time water-filling gains little in capacity over spatial water-filling.For Rayleigh channels with shadowing, space-time water-filling achieves higher spectral efficiency per antenna over spatial water-filling, with a tradeoff of higher channel outage probability.In either case, space-time water-filling actually has lower computational complexity than spatial water-filling.

SYSTEM MODEL
A point-to-point MIMO system is shown in Figure 1.Let N t and N r denote the number of transmit and receive antennas,  respectively.The symbolwise discrete-time input-output relationship of a narrowband point-to-point MIMO system can be simplified as where H is the N r × N t MIMO channel matrix, x is the N t × 1 transmitted symbol vector, y is the N r × 1 received symbol vector, and v is the N r × 1 additive white Gaussian noise vector, with variance E[vv † ] = σ 2 I, where (•) † denotes the operation of matrix complex conjugate transpose.In this paper, the MIMO channel H is modeled as where H w is an N r × N t Rayleigh fast fading MIMO channel whose entries are i.i.d.complex Gaussian random variables [1], and s is a scalar log-normal random variable, that is, 10 log 10 s ∼ N (0, ρ 2 ), representing the shadowing effect.Notice that log-normal shadowing models the channel power variation from objects on large spatial scales; hence, the square root of s is used in (2).Further, shadowing can be modeled as a multiplicative factor to fast fading [6,7].Since shadowing occurs on large spatial scales, it is assumed that the shadowing value s equally effects all elements of H w .Furthermore, s is assumed to be independent of H w .
As the shadowing effect varies slower relative to fast fading, the channel model discussed in this paper is suitable for transmissions over a long time period.Throughout this paper, we assume perfect channel state information is known at the transmitter.The MIMO channel capacity with imperfect channel state information can be found in [8].Further, we consider MIMO systems with equal numbers of transmit and receive antennas, that is, N t = N r = M, since expressing the channel eigenvalue distribution is simpler than for unequal numbers of transmit and receive antennas [1].The same technique discussed in this paper, however, can be applied to MIMO systems with unequal numbers of transmit and receive antennas.

Spatial water-filling
The problem of spatial water-filling for MIMO Rayleigh fading channels was presented in [1].Channel state information is assumed to be available at the transmitter and power adaption is performed with a total power constraint for each channel realization.The capacity maximization problem can be represented as max where H is the MIMO channel, Q is the autocorrelation matrix of the input vector x, defined as Q = E[xx † ], P is the instantaneous power limit, |A| denotes the determinant of A, and tr(A) denotes the trace of matrix A.
Notice that H † H can be diagonalized as H † H = U † ΛU, where U is a unitary matrix, Λ = diag{λ 1 , . . ., λ M }, and [1] that the optimization in (3) can be carried out over Q = UQU † and the capacity-achieving Q is a diagonal matrix.Let Q = diag{q 1 , q 2 , . . ., q M }, then the optimal value for q i is q , where σ 2 is the noise variance, a + denotes max{0, a}, and Γ (σ 2 ,M) 0 is solved to satisfy M i=1 q i = P.

Space-time water-filling
The problem of two-dimensional space-time water-filling can be formulated as max where P is the average power constraint; H and Q have the same meaning as in (3), that is, is the covariance matrix of the transmitted signal for a particular channel realization H. Hence, Q is a function of H.The expectation in E[tr(Q)] is carried over all MIMO channel realizations.This notation can be understood as the symbol rate is much faster than the MIMO channel variation and Q is evaluated from all symbols within one channel realization.
Notice that where λ k is the kth unordered eigenvalue of H † H, λ denotes any of them, and p(λ) denotes the power adaption as a function of λ.Hence, (4) can be rewritten as max where f (λ) is the empirical eigenvalue probability density function.The problem in ( 6) is essentially the same as in [5].
To find , where λ k is the kth largest eigenvalue of H † H.The ordered joint eigenvalue distribution of Gaussian random matrices H † w H w has been given in [1,9] as where K M is a normalizing factor.In this paper, the empirical eigenvalue distribution for H † w H w is defined to be the probability density function for an eigenvalue t smaller than a certain threshold z.Telatar derived its pdf g(t) by integrating out all other eigenvalues in the unordered joint eigenvalue distribution of Gaussian random matrices [1] to obtain where Since 10 log 10 s ∼ N (0, ρ 2 ), by a simple change of variables, the pdf of s can be written as Furthermore, s is independent of H w , hence s is independent of t.The cdf of λ is Differentiating F(λ) with respect to λ generates the pdf of λ: With f (λ) available, the optimal cutoff value Γ (σ 2 , M) 0 can be found by numerically solving and the ergodic capacity can be expressed as (13)

CHANNEL OUTAGE PROBABILITY
The capacity achieving power distribution from space-time water-filling blocks transmission when all eigenvalues of H † H are not high enough to utilize transmit power efficiently.The channel outage probability defined in [3] is equivalent to the probability that the largest eigenvalue of of H † H are in descending order, the channel outage probability can be expressed as Although the channel outage probability is defined in [3], only upper bounds in MIMO Rayleigh fading channels on this outage probability are derived.In this paper, the exact channel outage probability is expressed in terms of the maximal eigenvalue distribution, denoted as f max (λ 1 ).
Recall that λ 1 = st 1 , where s is the shadowing random variable and t 1 is the maximal eigenvalue of H † w H w .The distribution of t 1 is denoted g max (t 1 ) and can be obtained from ( 7) by integrating out t M , t M−1 , . . ., t 2 , that is, Mathematica's built-in function Integrate can be used to perform the symbolic integration in (15).For example, when ).With g max (t 1 ) available, the same procedure in ( 9)-( 11) can be used to calculate f max (λ 1 ), with t and g(t) replaced by t 1 and g max (t 1 ), respectively.The channel outage probability becomes

APPROXIMATED CAPACITY AND CHANNEL OUTAGE ANALYSIS
Even for medium-sized MIMO systems, for example, M = 4 or 6, the calculation of the empirical eigenvalue distribution g(t) in ( 8) for H † w H w is computationally intensive, and the resultant g(t) is too complicated to be handled in closed form.Therefore, an approximation to g(t) will be utilized to simplify the calculation of Γ (σ 2 , M) 0 .An interesting property of Gaussian random matrices is that the distribution of t/M has a limit as the number of antennas increases [1].Hence, as M → ∞.Simulations show that this approximation holds well even for medium-sized MIMO systems, for example, M = 4 or 6.With (17), for Rayleigh fading channel with shadowing variance ρ, the cutoff value Γ (σ 2 , M) 0 can be found by numerically solving 10M (2π) (3/2) ρ log 10 s 2 e −(10 log 10 s) 2 /2ρ 2 ds dλ = P.

(18)
Although the lengthy calculation of g(t) can be avoided with the approximation in (17), the method in (15) to find the maximal eigenvalue distribution g max (t 1 ) for channel Table 1: Cutoff value Γ (σ 2 , M) 0 for 2 × 2 MIMO fading channels.The average power constraint is P = 1.The exact empirical eigenvalue distribution [8] is used in finding Γ (σ 2 , M) outage probability analysis still requires a certain amount of computation.In [10], Wong showed that the distribution of the largest singular value of H w , that is, √ t 1 , can be well approximated with a Nakagami-m distribution.In other words, g max (t 1 ) can be approximated with where m and Ω are coefficients dependent on the MIMO system size M; Γ(m) is the Gamma function, which is implemented in Mathematica as Gamma [m].Wong also showed the values of m and Ω for different transmit and receive antenna numbers, up to the 6 × 6 MIMO case [10].For example, for M = 4, (m, Ω) = (12.5216,9.7758); for M = 6, (m, Ω) = (24.0821,16.5881).Substituting (19) into ( 16), the outage probability can be calculated as (20)

NUMERICAL RESULTS AND DISCUSSION
In this section, the achievable spectral efficiencies per antenna of the following three cases are compared by Monte Carlo simulations: (1) space-time water-filling, (2) waterfilling in space only, and (3) equal power distribution.We also compare the results from numerical integrations with those obtained from Monte Carlo simulations.In all simulations, the Rayleigh MIMO channel H w has variance of 1/2 for both real and imaginary components.The shadowing effect has a log-normal distribution with standard deviation of ρ [11].For the pure Rayleigh fading channel, s is a constant of 1.For notational simplicity, we denote the pure Rayleigh fading case as ρ = 0. We also study the cases where ρ = 8 and 16.Table 1 shows the cutoff values for a 2×2 MIMO system with different SNRs and log-normal shadowing variances.These cutoff values are obtained from the numerical method NIntegrate in Mathematica 5.0.The average power constraint is P = 1.In Table 1, the columns P sim show the average power obtained in Monte Carlo simulations.If the cutoff value Γ (σ 2 , M) 0 is calculated exactly, then P sim will equal P. Table 1 shows that for ρ = 0 and 8, the cutoff values are very accurate.For ρ = 16, P sim has 1-2% relative error compared to P, which is primarily caused by the limited accuracy in the process of numerically finding Γ (σ 2 , M) 0 for high shadowing variances.
Figure 2 shows the capacity per antenna versus SNR under different shadowing variances.For Rayleigh channels without shadowing, spatial water-filling achieves almost the same capacity as space-time water-filling.However, for Rayleigh channels with shadowing variance ρ = 8, the space-time water-filling algorithm achieves approximately 0.15 bps/Hz/antenna over spatial water-filling at low SNRs, and has a 1.7 dB SNR gain over equal power distribution at a spectral efficiency of 2 bps/Hz/antenna.For Rayleigh fading with shadowing variance ρ = 16, space-time water-filling achieves 0.3 bps/Hz/antenna over spatial water-filling.Notice that compared to the pure Rayleigh fading case, the average channel power is increased with the introduction of shadowing, but this does not affect the comparison between 2D and 1D water-fillings.Further, Figure 2 shows that the numerical results evaluated from (13) with Mathematica 5.0 agree with the Monte Carlo results.
Figure 3 shows the channel outage probability for a 2 × 2 MIMO system.With the increase of the shadowing variance, higher channel outage probability is observed.Figure 3   presents the channel outages evaluated from (16) with Mathematica 5.0, and the results again agree very well with those obtained from Monte Carlo simulations.
Table 2 shows the cutoff values Γ (σ 2 , M) 0 and P sim for 4 × 4 and 6 × 6 MIMO systems.The cutoff values are evaluated with the approximation in (17).Even with the approximated empirical eigenvalue distribution, the cutoff values are still very accurate, which is partially shown by the fact that P sim has a relative error not exceeding 2.5% compared to P.
Figure 4 shows the capacity per antenna for a 4×4 MIMO system.The capacity per antenna for the 6 × 6 case is very close to the 4 × 4 case.From Figures 2 and 4, the capacity per antenna is insensitive to the number of antennas in the system.Numerical results from (13) are also shown in Figure 4.
Figure 5 shows the channel outage probability for the 4 × 4 and 6 × 6 MIMO systems, with shadowing variance ρ = 8.The outage probability is evaluated through (20).For the same shadowing variance, the outage probabilities for the 4 × 4 and 6 × 6 MIMO systems are very close, since the shadowing variable equally effects all eigenvalues of H † w H w and therefore dominates the channel outage probability.Figure 5 shows that even with the approximated maximal eigenvalue distribution, the results from (20) still agree with the Monte Carlo simulations very well.
We also compare the main advantages and disadvantages of space-time water-filling versus spatial water-filling in Table 3.For space-time water-filling, only the cutoff threshold needs to be precomputed, while for spatial waterfilling, the optimal power distribution needs to be computed for each channel realization to achieve capacity.On the other hand, the two-dimensional algorithm requires a priori knowledge of the channel eigenvalue distribution in order for 4 × 4 and 6 × 6 MIMO fading channels.The average power constraint is P = 1.The approximated empirical eigenvalue distribution [8] is used in finding Γ (σ 2 , M)   to calculate the optimal cutoff threshold.Furthermore, the higher capacity achieved by two-dimensional water-filling comes with a larger channel outage probability.Since shadowing changes much slower than fast fading, the transmission of space-time water-filling is subject to long periods of  outage and hence is similar to block transmission.For spatial water-filling, the transmission mode is continuous since for every channel realization, the transmitter always has power to transmit.Further, the capacity gap between space-time and spatial water-filling depends on the distributions of the fast fading and shadowing gains.An analytical expression for the gap, however, is difficult to obtain.

CONCLUSION
In this paper, the ergodic capacity and channel outage probability in a composite MIMO channel model both fading and shadowing have analyzed.With the eigenvalue distribution of MIMO fading channels, both the capacity and the channel outage probability have been evaluated through numerical integration, which avoids timeconsuming Monte Carlo simulations and provides more direct insight into the system.Furthermore, approximations to the empirical eigenvalue distribution and the maximal eigenvalue distribution can greatly simplify the capacity and outage probability analysis.Numerical results illustrate that while the capacity difference is negligible for Rayleigh fading channels, space-time water-filling has an advantage when large-scale fading is taken into account.In all cases, it is simpler to compute the solution for space-time water-filling because it avoids the cutoff value calculation for each channel realization, but it requires knowledge of the channel distribution.The spectral efficiency gain of space-time water-filling over spatial water-filling is also shown to be associated with a higher channel outage probability.Hence, space-time waterfilling is more suitable for burst mode transmission when the channel gain distribution has a heavy tail, and spatial water-filling is preferred for continuous transmission when the channel gain distribution is close to Rayleigh or is unknown.

Figure 2 :
Figure 2: Capacity of 2 × 2 MIMO fading channels.The variance of the log-normal random variable is denoted by ρ.The numerical results are obtained from (13) with Mathematica 5.0.

Figure 3 :
Figure 3: Channel outage probability for 2 × 2 MIMO fading channels.The numerical results are obtained from (16) with Mathematica 5.0.The variance of the log-normal random variable is denoted by ρ.

Figure 4 :
Figure 4: Capacity of 4 × 4 MIMO fading channels.The variance of the log-normal random variable is denoted by ρ.The numerical results are obtained from (13) with Mathematica 5.0.

Table 3 :
Comparison of space-time and spatial water-fillings.