MIMO Broadcast Channels with Spatial ... - Semantic Scholar
1
MIMO Broadcast Channels with Spatial Heterogeneity Illsoo Sohn, Jeffrey G. Andrews, and Kwang Bok Lee
Abstract We develop a realistic model for multiple-input multiple-output (MIMO) broadcast channels, where each randomly located user’s average SNR depends on its distance from the transmitter. With perfect channel state information at the transmitter (CSIT), the average sum capacity is proven to scale for many users like
αM 2
log K
instead of M log log K, where α, M , and K denote the path loss exponent, the number of transmit antennas, and the number of users in a cell. With only partial CSIT, the sum capacity at high SNR eventually saturates due to interference, and the saturation value scales for large B like
MB M −1 ,
where B denotes the quantization
resolution for channel feedback.
I. I NTRODUCTION With perfect channel state information at transmitter (CSIT), dirty paper coding (DPC) is the wellknown optimal transmit strategy for MIMO broadcast channels [1]–[3], and achieves sum capacity. This sum capacity scales for large K like M log log K, where M is the number of transmit antennas and K is the total number of (single-antenna) users, assuming all users have the same average signal-to-noise ratio (SNR). The same capacity scaling has also been derived for linear beamforming schemes like zeroforcing (ZFBF), under the same assumptions and assuming interference orthogonality [4]. Although random orthogonal beamforming does not orthogonalize the interference amongst different users in contrast to ZFBF, surprisingly, it also achieves the aforementioned capacity scaling of M log log K for large K [5], [6]. The key contribution of this work is to revisit these well-known results without the assumption of average SNR, which is clearly invalid for the emerging cellular systems. Whereas all the above work models only the Rayleigh fading as a random variable, cellular users are typically randomly located within a cell and therefore experience different path loss. Therefore, the random geometry of the users as well as the Rayleigh fading should be captured in the system model to determine more accurate scaling laws. Our key findings are: •
With CSIT, the sum capacity of MIMO broadcast channels scales like
αM 2
log K instead of
M log log K, where α denotes the path loss exponent. This means that the additional randomness
2
due to the spatial heterogeneity of users results in much higher growth rate with the number of users than the existing scaling laws. •
With partial CSIT, the system eventually becomes interference-limited at high SNR regardless of the spatial heterogeneity, and the sum capacity at high SNR scales like
MB M −1
for large B, where
B denotes the quantization resolution of channel feedback. The consideration of the spatial heterogeneity of cellular users has also appeared in a recent independent work. A distributed resource allocation for multi-cell single-antenna systems is developed in [7]. As we will show later, mathematical analogies between single-cell MIMO broadcast channels with spatial heterogeneity and multi-cell single-antenna channels lead to an identical scaling law on capacity although the result comes from different mathematical tools. Investigating these analogies may also be interesting to future cellular capacity research. II. S YSTEM M ODEL Fig. 1 illustrates the system model. It is assumed that the transmitter has M transmit antennas and each user has a single receive antenna. Equal power allocation over M selected users is considered. The received signal at the k-th user is given by yk =
√
ρk (hH k wk )xk +
√
ρk
X
(hH k wj )xj + zk ,
j, k ∈ S,
(1)
j6=k
where ρk is the average SNR, xk is the independent data symbol transmitted through M transmit antennas satisfying E[|xk |2 ] = 1, hH k is the independent and identically distributed (i.i.d.) flat Rayleigh fading channel vector (hk ∈ CM ×1 ), wk is the unit-norm linear precoding vector (wk ∈ CM ×1 ), zk is the complex Gaussian noise with unit variance of the k-th user, and S is a set of the simultaneously scheduled users among the entire user set U (|U| = K). The channel is assumed to be block fading where the channel remains static in each block. In most previous works, the average SNR of the user, ρk , is constant for all users. Whereas we consider different average SNRs of users reflecting the spatial heterogeneity of users as ρk =
P , M dα k
where dk , α, and P denote the distance of the k-the user from
the transmitter, path loss exponent, and transmitter power, respectively.
1
One channel direction information (CDI) and one channel quality information (CQI) are considered for partial channel feedback of each user. The k-th user quantizes its CDI using the predefined 1
We have considered a simplified path loss model for tractable analysis. However, when the number of users is extremely large,
an ideal max sum-rate scheduler tends to select users who may be located too close to the transmitter increasing the received power to infinity. The validity of the simplified model with a moderate number of users is verified through simulations, in which we set an exclusion area around the transmitter within which no user can be located. It is observed that the scaling law derived from the simplified path loss model clearly matches the simulation results that have an exclusion area.
3
codebook, C = {c1 , c2 , · · · , cN } with the size of N = 2B , and feeds the quantized CDI index back to the transmitter at every beginning of blocks via feedback channel [8]. The quantized CDI index for feedback is determined as ˜H mk = arg max h c k l ,
(2)
1≤l≤N
˜k = where h
hk khk k
is the unit channel direction vector of the k-th user. Then, the reported CDI of the ˆ k = cm . The CQI of the user-k is quantized using quantization function k-th user at the transmitter is h k
g(·) as √ 2 nk = g( ρk hH k ) = ρk khk k .
(3)
Here, it is assumed that CQI is reported to BS without quantization as in [4], [6] to focus on the effects of quantization of CDI. Based on user feedback, the set of simulataneously scheduled users are determined at the transmitter. The semi-orthogonal user selection (SUS) algorithm in [4], [9] is used to keep reasonable computational complexity. Here, we modify the SUS algorithm as in Table I to capture individual average SNR of users, ρk , since the original algorithm is designed assuming the same average SNR of all users. In Table I, Ti and denote the set of the candidate users for the i-th user selection and semi-orthogonal parameter, respectively. The sum capacity of the MIMO broadcast channels is computed as 2 2 ˜H ρk khk k hk wk X 1 + R = E log 2 P ˜ H 2 . 2 k∈S 1 + ρk khk k hk wj
(4)
j∈S,j6=k
ZFBF [4], [9], [10] is used in this work considering its asymptotic optimality. Thus, the precoding ˆH vector of the k-th user, wk , is determined to satisfy h j wk = 0 for ∀j 6= k (j, k ∈ S). III. S UM C APACITY A NALYSIS WITH CSIT With CSIT, ZFBF completely eliminates inter-user interference terms in (4). In the limit of large K, the sum capacity reduces to " X
R = E
log2
# 2 ˜H 1 + ρk khk k2 h k wk
k∈S
"
(a)
# X
≈ E
log2 1 + ρk khk k
2
k∈S
= E
"M X
# log2 1 + max ρk khk k2 k∈Ti
i=1 (b)
≈
M X i=1
2 E log2 max ρk khk k , k∈Ti
(5)
4
where (a) follows from the fact that the channel gain reduction due to ZFBF becomes negligible ˜ H 2 with proper selection of the semi-orthogonal parameter, i.e., lim→0 hk wk = 1 [4, Lemma 2], and (b) follows from max ρk khk k2 1 with large K. For further analysis of (5), the behavior of k∈Ti max ρk khk k2 needs to be investigated. k∈Ti
Lemma 1. Consider two independent random variables X and Y . Let X denotes the random variable of the average SNR, ρk , and Y denotes the random variable of the norm of Rayleigh fading channel vector, khk k2 , which follows chi-square distribution with 2M degree of freedom. Then, the cumulative distribution function (CDF) of Z is computed as α2 2 M −1 2 z −α X 1 P 2 2 M rα z FZ (z) = 1 − 2 Γ u+ −Γ u+ , , αr M 2 u! α α 2P u=0
(6)
where Γ(·) and Γ(·, ·) represent Gamma function and incomplete Gamma function, respectively. Proof: See Appendix A. Theorem 1. With CSIT, the sum capacity of MIMO broadcast channels with the spatial heterogeneity scales for fixed α and M like lim
K→∞
αM 2
R log K
= 1.
(7)
Proof: For propagation scenarios of α = 1, α = 2 (free-space), or large α (high-attenuation), the parameter of Gamma and incomplete Gamma functions, u + α2 , are integer numbers. Thus, the CDF of Z in (6) reduces to a closed-form expression. In this case, the scaling law is analytically derived based on extreme order statistics technique [11]–[13]. See Appendix B for details. Unfortunately for other path loss exponents, it is very difficult to proceed analytically since the CDF cannot be expressed in closed-form. Instead, we will show by simulations that the scaling law holds for general propagation scenarios. Fig. 2 and Fig. 3 show how the sum capacity increases with the number of users, K, when M = 4 and 8, respectively. The sum capacity is plotted for different wireless propagation scenarios. Close comparisons between analytic results and simulation results have verified that our scaling law holds accurately for all practical propagation scenarios. The gap between analysis and simulation comes from loose bound of extreme order statistics for small K and non-trivial semi-orthogonality parameter, , which is unavoidable in finite simulation time. Note that Theorem 1 provides quite different results from previous work. The additional randomness due to the spatial heterogeneity results in the new capacity scaling law,
αM 2
log K, instead of
M log log K. Now, the path loss exponent also contributes to the slope of the sum capacity while only M does in idealized MIMO broadcast channels. When the path loss exponent increases, the
5
sum capacity becomes more sensitive to the number of users, K. This implies that the importance of exploiting multiuser diversity gain grows as the spatial heterogeneity increases the dynamic range of signal power. Perhaps surprisingly, the capacity scaling law in (7) coincides with the recent result in [7, Theorem 6] except the multiplexing gain M . This is interesting because the two works consider different scenarios: [7] is for multi-cell single-antenna channels and the present paper is for single-cell MIMO broadcast channels with spatial heterogeneity. However, both assume an ideal user scheduler which eliminates interference in the large K limit, rendering the underlying system model identical. In this work, we employ extreme order statistics using an exact form of the composite PDF including path loss and Rayleigh fading, while the proofs are shown for specific path loss exponents. On the contrary, heavy tail behavior is used to describe the composite PDF in [7]. The advantage of the [7] is that the proof generally holds for all path loss exponents without any constraints. IV. S UM C APACITY A NALYSIS WITH PARTIAL CSIT ˆk = ˜ k and the interference term of the denominator in (4) exists in this With only partial CSIT, h 6 h case. From [9], [10], the sum capacity in (4) is computed as 2 2 ˜H ρk khk k hk wk X P R=E log2 1 + , 2 2 1 + ρ kh k sin θ β (1, M − 2) k k k j k∈S
(8)
j∈S, j6=k
where βj (1, M − 2) denotes a Beta-distributed random variable with parameters (1, M − 2). Lemma 2. The expectation of the logarithm of the channel gain reduction is lower-bounded by 2 B ξ ˜ E log2 hk wk , ξ = 2− M −1 , k ∈ S. ≥ log2 (1 − ξ) + M log 2
(9)
Proof: See Appendix C. Theorem 2. With partial CSIT, the sum capacity of MIMO broadcast channels at high SNR, P 1, scales for fixed M like lim
B→∞
Rlimited MB M −1
= 1.
(10)
Proof: See Appendix D. As shown in idealized MIMO broadcast channels [9], [10], the sum capacity of MIMO broadcast channels with the spatial heterogeneity also becomes interference-limited at high SNR, and the interference-limited sum capacity linearly increases with the quantization resolution, B. Fig. 4 verifies that the sum capacity eventually saturates due to interference at high SNR as expected. Fig. 5 further
6
examines the relationship between the interference-limited sum capacity and quantization resolution in detail. Except for small B, the linear scaling is clearly observed, which agrees with our analysis. V. C ONCLUSIONS MIMO broadcast channels considering the spatial heterogeneity of users are analyzed for both CSIT and partial CSIT in this paper. The main contribution is that the geometry of users turns out to be important. With CSIT, the capacity scaling is shown to
αM 2
log K, which is a significantly faster growth
rate with the number of users compared to M log log K. For important practical case of partial CSIT, we verify that the sum capacity eventually becomes interference-limited at high SNR, the interferencelimited sum capacity scales like
MB . M −1
A PPENDIX A P ROOF OF L EMMA 1 From [7], when all users are assumed to be uniformly distributed in a cell, the probability density function (PDF) of the average SNR, ρk = MPdα , is computed as k P 2 P α2 x− α+2 α , ≤x αr2 M M rα fρk (x) = , 0, 0 ≤ x ≤ Pα
(11)
Mr
where r denotes the cell radius. The conditional PDF can be used to find the PDF of Z which is a product of two random variables [14]. The PDF of Z can be computed as Z
∞
fZ (z) = −∞ ∞
Z =
Z0 ∞ = 0
fZ (z| x0 ) fx (x0 ) dx0 z 1 0 fY x fx (x0 ) dx0 0 0 x x z 1 0 fX (x ) fY dx0 . 0 0 x x
Using (11) and chi-square distribution with σ = 1, α2 Z ∞ z M −1 z 1 2 P 1 − α+2 α fZ (z) = x e− 2x dx 2 M (M − 1)! x P x αr M 2 M rα α2 Z 1 P z M −1 ∞ −M −1− 2 − z α e 2x dx = x P αr2 (M − 1)! M 2 α Mr
(12)
(13)
7
Now, integrating the PDF gives the desired result as α2 M −1 Z ∞ Z z 1 P t FZ (z) = 2 P M 2 0 αr (M − 1)! M rα ( 2 α Z ∞ M −1 X 2 z P 2 −1− α − 2x x 1−e = 2 P αr M u=0 M rα α2 M −1 X P 2 1 z − α2 =1− 2 Γ u+ αr M u! 2 u=0
2
t
x−M −1− α e− 2x dxdt z u 2x
)
dx u! 2 2 M rα z −Γ u+ , . α α 2P
(14)
A PPENDIX B P ROOF OF T HEOREM 1 FOR S PECIFIC P ROPAGATION S CENARIOS Derivation details for α = 1, α = 2 (free-space), or large α (high-attenuation) are very similar. Here, we only present the proof for the most practical propagation scenario, α = 2 (free-space), due to size limitation. With α = 2, the CDF of Z in (6) reduces to a closed-form expression as M −1 u 1 − M + e−βz P P (βz)t−1 , z > 0 M r2 βz t! FZ (z) = , β= . u=0 t=0 2P 0, z≤0 Type of Limiting Distribution: Let aK =
MK β
(15)
and bK = 0, and check the type of limiting distribution.
When z > 0 and aK z + bK > 0, " lim K (1 − FZ (aK z + bK )) = lim
K→∞
K→∞
M −1 u K X X (M zK)t−1 1 − z eM zK u=0 t=0 t!
#
1 = . z
(16)
When z ≤ 0 and aK z + bK ≤ 0, lim K (1 − FZ (aK z + bK )) = lim K = ∞.
K→∞
K→∞
(17)
Therefore, the CDF of the j-th largest value, z (j) = j th max {z1 , z2 , · · · , zK }, behaves like Type-1 limiting distribution with Λ1 (z) = exp (−z −1 ) [11], [12]. Rate of Convergence: From [13], if F (z) with normalizing sequences aK and bK is in the domain of attraction of Type-l limiting distribution l ∈ {1, 2, 3},
1 2
< Fj:K (aK z + bK ) < 1, and − log [Λl (z)]
E log2 h − E log2 k wk
!#! sin2 θk
k∈S
X
βj (1, M − 2)
j∈S, j6=k
" #!! X (b) X ˜ H 2 ≥ E log2 hk wk − log2 E sin2 θk E βj (1, M − 2)
k∈S (c)
> M
j∈S, j6=k
log2 (1 − ξ) +
ξ B + M log 2 M − 1
,
b
ξ = 2− M −1
(33)
10
where (a) uses the fact that ρk khk k2 grows approximately as O (K) from (28) assuming proper user selection, (b) comes from Jensen’s inequality, and (c) uses Lemma 2 and [10, Lemma 1]. As shown in (32), the sum capacity becomes independent of the average SNRs of users in the interference-limited system. Now, the derivations of the upper-bound is identical to [10, Theorem 2]. Using the previous results, the interference-limited sum capacity is upper-bounded by B + log2 e + log2 (M − 2) + log2 e . Rlimited ≤ M 1 + M −1
(34)
Using the both bounds in (33),(34) and taking B to infinity, the desired scaling law is drawn. R EFERENCES [1] G. Caire and S. Shamai, “On the achievable throughput of a multiantenna Gaussian broadcast channel,” IEEE Trans. Inf. Theory, vol. 49, no. 7, pp. 1691–1706, Jul. 2003. [2] P. Viswanath and D. Tse, “Sum capacity of the vector Gaussian broadcast channel and uplink-downlink duality,” IEEE Trans. Inf. Theory, vol. 49, no. 8, pp. 1912–1921, Aug. 2003. [3] N. Jindal and A. Goldsmith, “Dirty-paper coding versus TDMA for MIMO broadcast channels,” IEEE Trans. Inf. Theory, vol. 51, no. 5, pp. 1783–1794, May 2005. [4] T. Yoo and A. Goldsmith, “On the optimality of multiantenna broadcast scheduling using zero-forcing beamforming,” IEEE J. Sel. Areas Commun., vol. 24, no. 3, pp. 528–541, Mar. 2006. [5] M. Sharif and B. Hassibi, “On the capacity of MIMO broadcast channels with partial side information,” IEEE Trans. Inf. Theory, vol. 51, no. 2, pp. 506–522, Feb. 2005. [6] K. Huang, J. G. Andrews, and R. W. Heath, “Performance of orthogonal beamforming for SDMA with limited feedback,” IEEE Trans. Veh. Technol., vol. 58, no. 1, pp. 152–164, Jan. 2009. [7] D. Gesbert and M. Kountouris, “Joint power control and user scheduling in multicell wireless networks: Capacity scaling laws,” Sep. 2007, submitted to IEEE Trans. Inf. Theory. [Online]. Available: http://arxiv.org/abs/arxiv:0709.2851 [8] D. J. Love, R. W. Heath, W. Santipach, and M. L. Honig, “What is the value of limited feedback for MIMO channels?” IEEE Commun. Mag., vol. 42, no. 10, pp. 54–59, Oct. 2004. [9] T. Yoo, N. Jindal, and A. Goldsmith, “Multi-antenna downlink channels with limited feedback and user selection,” IEEE J. Sel. Areas Commun., vol. 25, no. 7, pp. 1478–1491, Sep. 2007. [10] N. Jindal, “MIMO broadcast channels with finite-rate feedback,” IEEE Trans. Inf. Theory, vol. 52, no. 11, pp. 5045–5060, Nov. 2006. [11] H. David and H. Nagaraja, Order statistics.
Wiley-Interscience, 2003.
[12] N. Smirnov, “Limit distributions for the terms of a variational series,” Trudy Matematicheskogo Instituta im. VA Steklova, vol. 25, pp. 3–60, 1949. [13] W. Dziubdziela, “On convergence rates in the limit laws of extreme order statistics,” Trans. 7th Prague Conf. and 1974 Europ. Meeting of Statisticians, vol. B, pp. 119–127, 1974. [14] A. Leon-Garcia, Probability and random processes for electrical engineering. Addison-Wesley Publishing Company, Inc., 1994. [15] M. A. Maddah-Ali, M. A. Sadrabadi, and A. K. Khandani, “Broadcast in MIMO systems based on a generalized QR decomposition: Signaling and performance analysis,” IEEE Trans. Inf. Theory, vol. 54, no. 3, pp. 1124–1138, Mar. 2008.
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TABLE I M ODIFIED SEMI - ORTHOGONAL USER SELECTION ALGORITHM .
STEP 1:
Initialize T1 = {1, 2, · · · , K}, i = 1, S = Ø.
STEP 2:
For each k ∈ Ti , calculate gk , the component of hk orthogonal to the subspace spanned by {g(1) , · · · , g(i−1) } H ˆ i−1 P √nk g(j) hk √ ˆ gk = nk h g(j) k − kg k2 j=1
(j)
ˆk when i = 1, gk = h
STEP 3:
Find the ith selected user, Π(i), as follows Π(i) = arg max kgk k, k∈Ti
S←S
S
{Π(i)},
ˆ Π(i) . g(i) = h
STEP 4:
If |S| < M , calculate Ti+1 , set of users semi-orthogonal to g(i) Ti+1 = {k ∈ Ti , k 6= Π(i) | i ← i + 1, Go to STEP 2.
√ H ˆ hk | | nk g(i) √ ˆ k kkg k k nk h (i)
< },
12
M antennas
Transmitter
K packets
User 1
MIMO
User scheduler
Zero-forcing beamformer
broadcast
User 2
channels
M streams User K
CQI,CDI
Fig. 1.
System model.
CDI
13
Nt=4, Nr=1, Fixed P=30dBm, ε=0.5 280
Sum Capacity [bps/Hz]
240
200
Homog. assump. (simulation) Homog. assump. (analysis) Free-space, PE=2 (simulation) Free-space, PE=2 (analysis) Urban-area, PE=3.7 (simulation) Urban-area, PE=3.7 (analysis) Building-area, PE=6 (simulation) Building-area, PE=6 (analysis)
160
120
80
40
0 1 10
10
2
10
3
10
4
Number of Users Fig. 2.
Sum capacity with CSIT versus the number of users, where M = 4, P = 30dBm, r = 1000m, and = 0.5. Homogeneous
assumption implies no spatial heterogeneity of users. In this case, all users are located along the r/2 circle. The exclusion distance around the transmitter is set to 1m.
14
Nt=4, Nr=1, Fixed P=30dBm, ε=0.5 280
Sum Capacity [bps/Hz]
240
200
Homog. assump. (simulation) Homog. assump. (analysis) Free-space, PE=2 (simulation) Free-space, PE=2 (analysis) Urban-area, PE=3.7 (simulation) Urban-area, PE=3.7 (analysis) Building-area, PE=6 (simulation) Building-area, PE=6 (analysis)
160
120
80
40
0 1 10
10
2
10
3
10
4
Number of Users Fig. 3.
Sum capacity with CSIT versus the number of users, where M = 8, P = 30dBm, r = 1000m, and = 0.5. Homogeneous
assumption implies no spatial heterogeneity of users. In this case, all users are located along the r/2 circle. The exclusion distance around the transmitter is set to 1m.
15
30 Perfect CSI at TX Partial CSI at TX (B=12) Partial CSI at TX (B=8) Partial CSI at TX (B=4)
Sum Capacity [bps/Hz]
25
20
15
10
5
0 0
10
20
30
40
TX Power [dBm]
Fig. 4.
Sum capacity with partial CSIT versus SNR, where M = 4, K = 20, r = 1000m, α = 3.7, and = 0.5.
50
16
30 simulation analysis
Sum Capacity [bps/Hz]
25
20
15
10
5
0 2
4
6
8
10
12
14
16
18
Quatization Resolution [bits] Fig. 5.
Interference-limited sum capacity versus quantization resolution, where M = 4, K = 20, P = 50dBm, r = 1000m, α = 3.7,
and = 0.5.
MIMO Broadcast Channels with Spatial Heterogeneity Illsoo Sohn, Jeffrey G. Andrews, and Kwang Bok Lee
Abstract We develop a realistic model for multiple-input multiple-output (MIMO) broadcast channels, where each randomly located user’s average SNR depends on its distance from the transmitter. With perfect channel state information at the transmitter (CSIT), the average sum capacity is proven to scale for many users like
αM 2
log K
instead of M log log K, where α, M , and K denote the path loss exponent, the number of transmit antennas, and the number of users in a cell. With only partial CSIT, the sum capacity at high SNR eventually saturates due to interference, and the saturation value scales for large B like
MB M −1 ,
where B denotes the quantization
resolution for channel feedback.
I. I NTRODUCTION With perfect channel state information at transmitter (CSIT), dirty paper coding (DPC) is the wellknown optimal transmit strategy for MIMO broadcast channels [1]–[3], and achieves sum capacity. This sum capacity scales for large K like M log log K, where M is the number of transmit antennas and K is the total number of (single-antenna) users, assuming all users have the same average signal-to-noise ratio (SNR). The same capacity scaling has also been derived for linear beamforming schemes like zeroforcing (ZFBF), under the same assumptions and assuming interference orthogonality [4]. Although random orthogonal beamforming does not orthogonalize the interference amongst different users in contrast to ZFBF, surprisingly, it also achieves the aforementioned capacity scaling of M log log K for large K [5], [6]. The key contribution of this work is to revisit these well-known results without the assumption of average SNR, which is clearly invalid for the emerging cellular systems. Whereas all the above work models only the Rayleigh fading as a random variable, cellular users are typically randomly located within a cell and therefore experience different path loss. Therefore, the random geometry of the users as well as the Rayleigh fading should be captured in the system model to determine more accurate scaling laws. Our key findings are: •
With CSIT, the sum capacity of MIMO broadcast channels scales like
αM 2
log K instead of
M log log K, where α denotes the path loss exponent. This means that the additional randomness
2
due to the spatial heterogeneity of users results in much higher growth rate with the number of users than the existing scaling laws. •
With partial CSIT, the system eventually becomes interference-limited at high SNR regardless of the spatial heterogeneity, and the sum capacity at high SNR scales like
MB M −1
for large B, where
B denotes the quantization resolution of channel feedback. The consideration of the spatial heterogeneity of cellular users has also appeared in a recent independent work. A distributed resource allocation for multi-cell single-antenna systems is developed in [7]. As we will show later, mathematical analogies between single-cell MIMO broadcast channels with spatial heterogeneity and multi-cell single-antenna channels lead to an identical scaling law on capacity although the result comes from different mathematical tools. Investigating these analogies may also be interesting to future cellular capacity research. II. S YSTEM M ODEL Fig. 1 illustrates the system model. It is assumed that the transmitter has M transmit antennas and each user has a single receive antenna. Equal power allocation over M selected users is considered. The received signal at the k-th user is given by yk =
√
ρk (hH k wk )xk +
√
ρk
X
(hH k wj )xj + zk ,
j, k ∈ S,
(1)
j6=k
where ρk is the average SNR, xk is the independent data symbol transmitted through M transmit antennas satisfying E[|xk |2 ] = 1, hH k is the independent and identically distributed (i.i.d.) flat Rayleigh fading channel vector (hk ∈ CM ×1 ), wk is the unit-norm linear precoding vector (wk ∈ CM ×1 ), zk is the complex Gaussian noise with unit variance of the k-th user, and S is a set of the simultaneously scheduled users among the entire user set U (|U| = K). The channel is assumed to be block fading where the channel remains static in each block. In most previous works, the average SNR of the user, ρk , is constant for all users. Whereas we consider different average SNRs of users reflecting the spatial heterogeneity of users as ρk =
P , M dα k
where dk , α, and P denote the distance of the k-the user from
the transmitter, path loss exponent, and transmitter power, respectively.
1
One channel direction information (CDI) and one channel quality information (CQI) are considered for partial channel feedback of each user. The k-th user quantizes its CDI using the predefined 1
We have considered a simplified path loss model for tractable analysis. However, when the number of users is extremely large,
an ideal max sum-rate scheduler tends to select users who may be located too close to the transmitter increasing the received power to infinity. The validity of the simplified model with a moderate number of users is verified through simulations, in which we set an exclusion area around the transmitter within which no user can be located. It is observed that the scaling law derived from the simplified path loss model clearly matches the simulation results that have an exclusion area.
3
codebook, C = {c1 , c2 , · · · , cN } with the size of N = 2B , and feeds the quantized CDI index back to the transmitter at every beginning of blocks via feedback channel [8]. The quantized CDI index for feedback is determined as ˜H mk = arg max h c k l ,
(2)
1≤l≤N
˜k = where h
hk khk k
is the unit channel direction vector of the k-th user. Then, the reported CDI of the ˆ k = cm . The CQI of the user-k is quantized using quantization function k-th user at the transmitter is h k
g(·) as √ 2 nk = g( ρk hH k ) = ρk khk k .
(3)
Here, it is assumed that CQI is reported to BS without quantization as in [4], [6] to focus on the effects of quantization of CDI. Based on user feedback, the set of simulataneously scheduled users are determined at the transmitter. The semi-orthogonal user selection (SUS) algorithm in [4], [9] is used to keep reasonable computational complexity. Here, we modify the SUS algorithm as in Table I to capture individual average SNR of users, ρk , since the original algorithm is designed assuming the same average SNR of all users. In Table I, Ti and denote the set of the candidate users for the i-th user selection and semi-orthogonal parameter, respectively. The sum capacity of the MIMO broadcast channels is computed as 2 2 ˜H ρk khk k hk wk X 1 + R = E log 2 P ˜ H 2 . 2 k∈S 1 + ρk khk k hk wj
(4)
j∈S,j6=k
ZFBF [4], [9], [10] is used in this work considering its asymptotic optimality. Thus, the precoding ˆH vector of the k-th user, wk , is determined to satisfy h j wk = 0 for ∀j 6= k (j, k ∈ S). III. S UM C APACITY A NALYSIS WITH CSIT With CSIT, ZFBF completely eliminates inter-user interference terms in (4). In the limit of large K, the sum capacity reduces to " X
R = E
log2
# 2 ˜H 1 + ρk khk k2 h k wk
k∈S
"
(a)
# X
≈ E
log2 1 + ρk khk k
2
k∈S
= E
"M X
# log2 1 + max ρk khk k2 k∈Ti
i=1 (b)
≈
M X i=1
2 E log2 max ρk khk k , k∈Ti
(5)
4
where (a) follows from the fact that the channel gain reduction due to ZFBF becomes negligible ˜ H 2 with proper selection of the semi-orthogonal parameter, i.e., lim→0 hk wk = 1 [4, Lemma 2], and (b) follows from max ρk khk k2 1 with large K. For further analysis of (5), the behavior of k∈Ti max ρk khk k2 needs to be investigated. k∈Ti
Lemma 1. Consider two independent random variables X and Y . Let X denotes the random variable of the average SNR, ρk , and Y denotes the random variable of the norm of Rayleigh fading channel vector, khk k2 , which follows chi-square distribution with 2M degree of freedom. Then, the cumulative distribution function (CDF) of Z is computed as α2 2 M −1 2 z −α X 1 P 2 2 M rα z FZ (z) = 1 − 2 Γ u+ −Γ u+ , , αr M 2 u! α α 2P u=0
(6)
where Γ(·) and Γ(·, ·) represent Gamma function and incomplete Gamma function, respectively. Proof: See Appendix A. Theorem 1. With CSIT, the sum capacity of MIMO broadcast channels with the spatial heterogeneity scales for fixed α and M like lim
K→∞
αM 2
R log K
= 1.
(7)
Proof: For propagation scenarios of α = 1, α = 2 (free-space), or large α (high-attenuation), the parameter of Gamma and incomplete Gamma functions, u + α2 , are integer numbers. Thus, the CDF of Z in (6) reduces to a closed-form expression. In this case, the scaling law is analytically derived based on extreme order statistics technique [11]–[13]. See Appendix B for details. Unfortunately for other path loss exponents, it is very difficult to proceed analytically since the CDF cannot be expressed in closed-form. Instead, we will show by simulations that the scaling law holds for general propagation scenarios. Fig. 2 and Fig. 3 show how the sum capacity increases with the number of users, K, when M = 4 and 8, respectively. The sum capacity is plotted for different wireless propagation scenarios. Close comparisons between analytic results and simulation results have verified that our scaling law holds accurately for all practical propagation scenarios. The gap between analysis and simulation comes from loose bound of extreme order statistics for small K and non-trivial semi-orthogonality parameter, , which is unavoidable in finite simulation time. Note that Theorem 1 provides quite different results from previous work. The additional randomness due to the spatial heterogeneity results in the new capacity scaling law,
αM 2
log K, instead of
M log log K. Now, the path loss exponent also contributes to the slope of the sum capacity while only M does in idealized MIMO broadcast channels. When the path loss exponent increases, the
5
sum capacity becomes more sensitive to the number of users, K. This implies that the importance of exploiting multiuser diversity gain grows as the spatial heterogeneity increases the dynamic range of signal power. Perhaps surprisingly, the capacity scaling law in (7) coincides with the recent result in [7, Theorem 6] except the multiplexing gain M . This is interesting because the two works consider different scenarios: [7] is for multi-cell single-antenna channels and the present paper is for single-cell MIMO broadcast channels with spatial heterogeneity. However, both assume an ideal user scheduler which eliminates interference in the large K limit, rendering the underlying system model identical. In this work, we employ extreme order statistics using an exact form of the composite PDF including path loss and Rayleigh fading, while the proofs are shown for specific path loss exponents. On the contrary, heavy tail behavior is used to describe the composite PDF in [7]. The advantage of the [7] is that the proof generally holds for all path loss exponents without any constraints. IV. S UM C APACITY A NALYSIS WITH PARTIAL CSIT ˆk = ˜ k and the interference term of the denominator in (4) exists in this With only partial CSIT, h 6 h case. From [9], [10], the sum capacity in (4) is computed as 2 2 ˜H ρk khk k hk wk X P R=E log2 1 + , 2 2 1 + ρ kh k sin θ β (1, M − 2) k k k j k∈S
(8)
j∈S, j6=k
where βj (1, M − 2) denotes a Beta-distributed random variable with parameters (1, M − 2). Lemma 2. The expectation of the logarithm of the channel gain reduction is lower-bounded by 2 B ξ ˜ E log2 hk wk , ξ = 2− M −1 , k ∈ S. ≥ log2 (1 − ξ) + M log 2
(9)
Proof: See Appendix C. Theorem 2. With partial CSIT, the sum capacity of MIMO broadcast channels at high SNR, P 1, scales for fixed M like lim
B→∞
Rlimited MB M −1
= 1.
(10)
Proof: See Appendix D. As shown in idealized MIMO broadcast channels [9], [10], the sum capacity of MIMO broadcast channels with the spatial heterogeneity also becomes interference-limited at high SNR, and the interference-limited sum capacity linearly increases with the quantization resolution, B. Fig. 4 verifies that the sum capacity eventually saturates due to interference at high SNR as expected. Fig. 5 further
6
examines the relationship between the interference-limited sum capacity and quantization resolution in detail. Except for small B, the linear scaling is clearly observed, which agrees with our analysis. V. C ONCLUSIONS MIMO broadcast channels considering the spatial heterogeneity of users are analyzed for both CSIT and partial CSIT in this paper. The main contribution is that the geometry of users turns out to be important. With CSIT, the capacity scaling is shown to
αM 2
log K, which is a significantly faster growth
rate with the number of users compared to M log log K. For important practical case of partial CSIT, we verify that the sum capacity eventually becomes interference-limited at high SNR, the interferencelimited sum capacity scales like
MB . M −1
A PPENDIX A P ROOF OF L EMMA 1 From [7], when all users are assumed to be uniformly distributed in a cell, the probability density function (PDF) of the average SNR, ρk = MPdα , is computed as k P 2 P α2 x− α+2 α , ≤x αr2 M M rα fρk (x) = , 0, 0 ≤ x ≤ Pα
(11)
Mr
where r denotes the cell radius. The conditional PDF can be used to find the PDF of Z which is a product of two random variables [14]. The PDF of Z can be computed as Z
∞
fZ (z) = −∞ ∞
Z =
Z0 ∞ = 0
fZ (z| x0 ) fx (x0 ) dx0 z 1 0 fY x fx (x0 ) dx0 0 0 x x z 1 0 fX (x ) fY dx0 . 0 0 x x
Using (11) and chi-square distribution with σ = 1, α2 Z ∞ z M −1 z 1 2 P 1 − α+2 α fZ (z) = x e− 2x dx 2 M (M − 1)! x P x αr M 2 M rα α2 Z 1 P z M −1 ∞ −M −1− 2 − z α e 2x dx = x P αr2 (M − 1)! M 2 α Mr
(12)
(13)
7
Now, integrating the PDF gives the desired result as α2 M −1 Z ∞ Z z 1 P t FZ (z) = 2 P M 2 0 αr (M − 1)! M rα ( 2 α Z ∞ M −1 X 2 z P 2 −1− α − 2x x 1−e = 2 P αr M u=0 M rα α2 M −1 X P 2 1 z − α2 =1− 2 Γ u+ αr M u! 2 u=0
2
t
x−M −1− α e− 2x dxdt z u 2x
)
dx u! 2 2 M rα z −Γ u+ , . α α 2P
(14)
A PPENDIX B P ROOF OF T HEOREM 1 FOR S PECIFIC P ROPAGATION S CENARIOS Derivation details for α = 1, α = 2 (free-space), or large α (high-attenuation) are very similar. Here, we only present the proof for the most practical propagation scenario, α = 2 (free-space), due to size limitation. With α = 2, the CDF of Z in (6) reduces to a closed-form expression as M −1 u 1 − M + e−βz P P (βz)t−1 , z > 0 M r2 βz t! FZ (z) = , β= . u=0 t=0 2P 0, z≤0 Type of Limiting Distribution: Let aK =
MK β
(15)
and bK = 0, and check the type of limiting distribution.
When z > 0 and aK z + bK > 0, " lim K (1 − FZ (aK z + bK )) = lim
K→∞
K→∞
M −1 u K X X (M zK)t−1 1 − z eM zK u=0 t=0 t!
#
1 = . z
(16)
When z ≤ 0 and aK z + bK ≤ 0, lim K (1 − FZ (aK z + bK )) = lim K = ∞.
K→∞
K→∞
(17)
Therefore, the CDF of the j-th largest value, z (j) = j th max {z1 , z2 , · · · , zK }, behaves like Type-1 limiting distribution with Λ1 (z) = exp (−z −1 ) [11], [12]. Rate of Convergence: From [13], if F (z) with normalizing sequences aK and bK is in the domain of attraction of Type-l limiting distribution l ∈ {1, 2, 3},
1 2
< Fj:K (aK z + bK ) < 1, and − log [Λl (z)]
E log2 h − E log2 k wk
!#! sin2 θk
k∈S
X
βj (1, M − 2)
j∈S, j6=k
" #!! X (b) X ˜ H 2 ≥ E log2 hk wk − log2 E sin2 θk E βj (1, M − 2)
k∈S (c)
> M
j∈S, j6=k
log2 (1 − ξ) +
ξ B + M log 2 M − 1
,
b
ξ = 2− M −1
(33)
10
where (a) uses the fact that ρk khk k2 grows approximately as O (K) from (28) assuming proper user selection, (b) comes from Jensen’s inequality, and (c) uses Lemma 2 and [10, Lemma 1]. As shown in (32), the sum capacity becomes independent of the average SNRs of users in the interference-limited system. Now, the derivations of the upper-bound is identical to [10, Theorem 2]. Using the previous results, the interference-limited sum capacity is upper-bounded by B + log2 e + log2 (M − 2) + log2 e . Rlimited ≤ M 1 + M −1
(34)
Using the both bounds in (33),(34) and taking B to infinity, the desired scaling law is drawn. R EFERENCES [1] G. Caire and S. Shamai, “On the achievable throughput of a multiantenna Gaussian broadcast channel,” IEEE Trans. Inf. Theory, vol. 49, no. 7, pp. 1691–1706, Jul. 2003. [2] P. Viswanath and D. Tse, “Sum capacity of the vector Gaussian broadcast channel and uplink-downlink duality,” IEEE Trans. Inf. Theory, vol. 49, no. 8, pp. 1912–1921, Aug. 2003. [3] N. Jindal and A. Goldsmith, “Dirty-paper coding versus TDMA for MIMO broadcast channels,” IEEE Trans. Inf. Theory, vol. 51, no. 5, pp. 1783–1794, May 2005. [4] T. Yoo and A. Goldsmith, “On the optimality of multiantenna broadcast scheduling using zero-forcing beamforming,” IEEE J. Sel. Areas Commun., vol. 24, no. 3, pp. 528–541, Mar. 2006. [5] M. Sharif and B. Hassibi, “On the capacity of MIMO broadcast channels with partial side information,” IEEE Trans. Inf. Theory, vol. 51, no. 2, pp. 506–522, Feb. 2005. [6] K. Huang, J. G. Andrews, and R. W. Heath, “Performance of orthogonal beamforming for SDMA with limited feedback,” IEEE Trans. Veh. Technol., vol. 58, no. 1, pp. 152–164, Jan. 2009. [7] D. Gesbert and M. Kountouris, “Joint power control and user scheduling in multicell wireless networks: Capacity scaling laws,” Sep. 2007, submitted to IEEE Trans. Inf. Theory. [Online]. Available: http://arxiv.org/abs/arxiv:0709.2851 [8] D. J. Love, R. W. Heath, W. Santipach, and M. L. Honig, “What is the value of limited feedback for MIMO channels?” IEEE Commun. Mag., vol. 42, no. 10, pp. 54–59, Oct. 2004. [9] T. Yoo, N. Jindal, and A. Goldsmith, “Multi-antenna downlink channels with limited feedback and user selection,” IEEE J. Sel. Areas Commun., vol. 25, no. 7, pp. 1478–1491, Sep. 2007. [10] N. Jindal, “MIMO broadcast channels with finite-rate feedback,” IEEE Trans. Inf. Theory, vol. 52, no. 11, pp. 5045–5060, Nov. 2006. [11] H. David and H. Nagaraja, Order statistics.
Wiley-Interscience, 2003.
[12] N. Smirnov, “Limit distributions for the terms of a variational series,” Trudy Matematicheskogo Instituta im. VA Steklova, vol. 25, pp. 3–60, 1949. [13] W. Dziubdziela, “On convergence rates in the limit laws of extreme order statistics,” Trans. 7th Prague Conf. and 1974 Europ. Meeting of Statisticians, vol. B, pp. 119–127, 1974. [14] A. Leon-Garcia, Probability and random processes for electrical engineering. Addison-Wesley Publishing Company, Inc., 1994. [15] M. A. Maddah-Ali, M. A. Sadrabadi, and A. K. Khandani, “Broadcast in MIMO systems based on a generalized QR decomposition: Signaling and performance analysis,” IEEE Trans. Inf. Theory, vol. 54, no. 3, pp. 1124–1138, Mar. 2008.
11
TABLE I M ODIFIED SEMI - ORTHOGONAL USER SELECTION ALGORITHM .
STEP 1:
Initialize T1 = {1, 2, · · · , K}, i = 1, S = Ø.
STEP 2:
For each k ∈ Ti , calculate gk , the component of hk orthogonal to the subspace spanned by {g(1) , · · · , g(i−1) } H ˆ i−1 P √nk g(j) hk √ ˆ gk = nk h g(j) k − kg k2 j=1
(j)
ˆk when i = 1, gk = h
STEP 3:
Find the ith selected user, Π(i), as follows Π(i) = arg max kgk k, k∈Ti
S←S
S
{Π(i)},
ˆ Π(i) . g(i) = h
STEP 4:
If |S| < M , calculate Ti+1 , set of users semi-orthogonal to g(i) Ti+1 = {k ∈ Ti , k 6= Π(i) | i ← i + 1, Go to STEP 2.
√ H ˆ hk | | nk g(i) √ ˆ k kkg k k nk h (i)
< },
12
M antennas
Transmitter
K packets
User 1
MIMO
User scheduler
Zero-forcing beamformer
broadcast
User 2
channels
M streams User K
CQI,CDI
Fig. 1.
System model.
CDI
13
Nt=4, Nr=1, Fixed P=30dBm, ε=0.5 280
Sum Capacity [bps/Hz]
240
200
Homog. assump. (simulation) Homog. assump. (analysis) Free-space, PE=2 (simulation) Free-space, PE=2 (analysis) Urban-area, PE=3.7 (simulation) Urban-area, PE=3.7 (analysis) Building-area, PE=6 (simulation) Building-area, PE=6 (analysis)
160
120
80
40
0 1 10
10
2
10
3
10
4
Number of Users Fig. 2.
Sum capacity with CSIT versus the number of users, where M = 4, P = 30dBm, r = 1000m, and = 0.5. Homogeneous
assumption implies no spatial heterogeneity of users. In this case, all users are located along the r/2 circle. The exclusion distance around the transmitter is set to 1m.
14
Nt=4, Nr=1, Fixed P=30dBm, ε=0.5 280
Sum Capacity [bps/Hz]
240
200
Homog. assump. (simulation) Homog. assump. (analysis) Free-space, PE=2 (simulation) Free-space, PE=2 (analysis) Urban-area, PE=3.7 (simulation) Urban-area, PE=3.7 (analysis) Building-area, PE=6 (simulation) Building-area, PE=6 (analysis)
160
120
80
40
0 1 10
10
2
10
3
10
4
Number of Users Fig. 3.
Sum capacity with CSIT versus the number of users, where M = 8, P = 30dBm, r = 1000m, and = 0.5. Homogeneous
assumption implies no spatial heterogeneity of users. In this case, all users are located along the r/2 circle. The exclusion distance around the transmitter is set to 1m.
15
30 Perfect CSI at TX Partial CSI at TX (B=12) Partial CSI at TX (B=8) Partial CSI at TX (B=4)
Sum Capacity [bps/Hz]
25
20
15
10
5
0 0
10
20
30
40
TX Power [dBm]
Fig. 4.
Sum capacity with partial CSIT versus SNR, where M = 4, K = 20, r = 1000m, α = 3.7, and = 0.5.
50
16
30 simulation analysis
Sum Capacity [bps/Hz]
25
20
15
10
5
0 2
4
6
8
10
12
14
16
18
Quatization Resolution [bits] Fig. 5.
Interference-limited sum capacity versus quantization resolution, where M = 4, K = 20, P = 50dBm, r = 1000m, α = 3.7,
and = 0.5.
