MIMO Broadcast Channels with Spatial Heterogeneity

Transcription

1 I TRANSACTIONS ON WIRLSS COMMUNICATIONS, VOL. 9, NO. 8, AUGUST 449 MIMO Broadcast Channels with Spatial Heterogeneity Illsoo Sohn, Member, I, Jeffrey G. Andrews, Senior Member, I, and wang Bok Lee, Senior Member, I 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 M log instead of M log log, where, M, and 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,whereB denotes the quantization resolution for channel M feedback. Index Terms MIMO systems, broadcast channels, multiuser diversity, limited feedback. I. INTRODUCTION WITH perfect channel state information at transmitter CSIT, dirty paper coding DC is the well-known optimal transmit strategy for MIMO broadcast channels [] [3], and achieves sum capacity. This sum capacity scales for large like M log log, wherem is the number of transmit antennas and 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 zero-forcing 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 for large [5], [6]. The key contribution of this work is to revisit these wellknown 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: Manuscript received November, 9; revised February, ; accepted May 3,. The associate editor coordinating the review of this letter and approving it for publication was A. Sezgin. I. Sohn and J. G. Andrews are with the Wireless Networking and Communications Group WNCG, Department of lectrical and Computer ngineering, The University of Texas at Austin, University Station C83, Austin, TX 787-4, USA isohn@mail.utexas.edu, jandrews@ece.utexas.edu.. B. Lee is with the School of lectrical ngineering and Computer Science, Seoul National University, Seoul, orea klee@snu.ac.kr. This work was supported by the National Research Foundation of orea NRF grant funded by the orea government MST No This paper was presented in part at the I International Symposium on Information Theory, Austin, TX, USA, June. Digital Object Identifier.9/TWC With CSIT, the sum capacity of MIMO broadcast channels scales like M log instead of M log log, where denotes the path loss exponent. This means that the additional randomness 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 for large B, whereb 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. Investigating these analogies may also be interesting to future cellular capacity research. II. SYSTM MODL Fig. illustrates the system model. It is assumed that the transmitter has M transmit antennasand each user has a single receive antenna. qual power allocation over M selected users is considered. The received signal at the k-th user is given by y k ρ k h H k w k x k + ρ k h H k w j x j + z k, j,k S, j k where ρ k is the average SNR, x k is the independent data symbol transmitted through M transmit antennas satisfying [x k ], h H k is the independent and identically distributed i.i.d. flat Rayleigh fading channel vector h k C M, w k is the unit-norm linear precoding vector w k C M, z k 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. 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, ρ k, is constant for all users. Whereas we consider different average SNRs of users reflecting the spatial heterogeneity of users as ρ k Md,whered k,, and k denote the distance of the k-the user from the transmitter, /$5. c I This was first revealed in the context of multi-cell scheduling [7]. The results have been confirmed with different mathematical tools in this work.

2 45 I TRANSACTIONS ON WIRLSS COMMUNICATIONS, VOL. 9, NO. 8, AUGUST packets Transmitter M antennas TABL I MODIFID SMI-ORTHOGONAL USR SLCTION ALGORITHM. Fig.. scheduler CQI,CDI System model. M streams Zero-forcing beamformer CDI MIMO broadcast channels path loss exponent, and transmitter power, respectively. 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 codebook, C c, c,, c N } with the size of N B, 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 k m k argmax l N h H k c l, where h k h k is the unit channel direction vector of the k-th user. Then, the reported CDI of the k-th user at the transmitter is ĥk c mk. The CQI of the user-k is quantized using quantization function g as n k g ρ k h H k ρ k h 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 semiorthogonal 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, ρ k, since the original algorithm is designed assuming the same average SNR of all users. In Table I, T i and ε 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 ρ k h k h H R log + k w k +ρ k h k h H k w j. j S,j k 4 ZFBF [4], [9], [] is used in this work considering its asymptotic optimality. Thus, the precoding vector of the k-th user, w k, is determined to satisfy ĥh j w k for j k j, k S. 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. ST : ST : ST 3: ST 4: Initialize T,,,}, i, S. For each k T i, calculate g k, the component of h k orthogonal to the subspace spanned by g,, g i } g k n k ĥ k i nk g H jĥk g j j g j when i, g k ĥk Find the ith selected user, Πi, as follows Πi arg max g k, S S Πi}, g i ĥπi. If S <M, calculate T i+, set of users whose channels are semi-orthogonal to g i T i+ k T i,k Πi i i +, Go to ST. H n k g iĥk <ε}, n k ĥ k g i III. SUM CAACITY ANALYSIS WITH CSIT With CSIT, ZFBF completely eliminates inter-user interference terms in 4. In the limit of large, the sum capacity reduces to [ R log +ρ k h k ] hh k w k a b [ log +ρ k h k ] [ M log +max ρ k h k }] i M [log max ρ k h k }], 5 i where a follows from the fact that the channel gain reduction due to ZFBF becomes negligible with proper selection of the semi-orthogonal parameter, i.e., lim ε hh k w k [4,Lemma],andb follows from max ρ k h k } u with large. For further analysis of 5, the behavior of max ρ k h k } needs to be investigated. Lemma. Consider two independent random variables X and Y.LetX denotes the random variable of the average SNR, ρ k, and Y denotes the random variable of the norm of Rayleigh fading channel vector, h k, which follows chisquare distribution with M degree of freedom. Then, the cumulative distribution function CDF of Z is computed as F Z z z r M M Γ u + u! Γ u +, Mr z }, 6

3 I TRANSACTIONS ON WIRLSS COMMUNICATIONS, VOL. 9, NO. 8, AUGUST Homog. assump. simulation Homog. assump. analysis Free-space, simulation Free-space, analysis Urban-area, 3.7 simulation Urban-area, 3.7 analysis Building-area, 6 simulation Building-area, 6 analysis Homog. assump. simulation Homog. assump. analysis Free-space, simulation Free-space, analysis Urban-area, 3.7 simulation Urban-area, 3.7 analysis Building-area, 6 simulation Building-area, 6 analysis Number of s Fig.. Sum capacity with CSIT versus the number of users, where M 4, 3dBm, r m, and ε.5. Homogeneous assumption implies no spatial heterogeneity of users. In this case, all users are located along the r/ circle. The exclusion distance around the transmitter is set to m. 3 4 Number of s Fig. 3. Sum capacity with CSIT versus the number of users, where M 8, 3dBm, r m, and ε.5. Homogeneous assumption implies no spatial heterogeneity of users. In this case, all users are located along the r/ circle. The exclusion distance around the transmitter is set to m. where Γ and Γ, represent Gamma function and incomplete Gamma function, respectively. roof: See Appendix A. Theorem. With CSIT, the sum capacity of MIMO broadcast channels with the spatial heterogeneity scales for fixed and M like R lim. 7 M log roof: For propagation scenarios of, freespace, or large high-attenuation, the parameter of Gamma and incomplete Gamma functions, u+, 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 [] [3]. 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. and Fig. 3 show how the sum capacity increases with the number of users,, whenm 4and 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 and non-trivial semiorthogonality parameter, ε, which is unavoidable in finite simulation time. Note that Theorem provides quite different results from previous work. The additional randomness due to the spatial heterogeneity results in the new capacity scaling law, M log, instead of M log log. Now,thepathlossexponent 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 sum capacity becomes more sensitive to the number of users,. Thisimpliesthat the importance of exploiting multiuser diversity gain grows as the spatial heterogeneity increases the dynamic range of signal power. erhaps 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 singleantenna 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 limit, rendering the underlying system model identical. In this work, we employ extreme order statistics using an exact form of the composite DF 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 DF in [7]. The advantage of the [7] is that the proof generally holds for all path loss exponents without any constraints. IV. SUM CAACITY ANALYSIS WITH ARTIAL CSIT With only partial CSIT, ĥk h k and the interference term of the denominator in 4 exists in this case. From [9], [], the sum capacity in 4 is computed as R ρ k h k hh k w k log + +ρ k h k sin, θ k β j j S, j k 8 where β j denotes a Beta-distributed random variable with parameters,m. Lemma. The expectation of the logarithm of the channel gain reduction is lower-bounded by [ ] log hk w k log ξ+ ξ M log, ξ B M, k S. 9

4 45 I TRANSACTIONS ON WIRLSS COMMUNICATIONS, VOL. 9, NO. 8, AUGUST erfect CSI at TX artial CSI at TX B artial CSI at TX B8 artial CSI at TX B TX ower [dbm] Fig. 4. Sum capacity with partial CSIT versus SNR, where M 4,, r m, 3.7, andε Analysis Simulation Quantization Resolution [bits] Fig. 5. Interference-limited sum capacity versus quantization resolution, where M 4,, 5dBm, r m, 3.7, andε.5. roof: See Appendix C. Theorem. With partial CSIT, the sum capacity of MIMO broadcast channels at high SNR,, scalesforfixed M like lim B R limited MB M. roof: See Appendix D. As shown in idealized MIMO broadcast channels [9], [], 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 examines the relationship between the interference-limited sum capacity and quantization resolution in detail. xcept for small B, the linear scaling is clearly observed, which agrees with our analysis. V. CONCLUSIONS 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 M log, which is a significantly faster growth rate with the number of users compared to M log log. For important practical case of partial CSIT, we verify that the sum capacity eventually becomes interferencelimited at high SNR, the interference-limited sum capacity scales like MB M. ANDIX A ROOF OF LMMA From [7], when all users are assumed to be uniformly distributed in a cell, the probability density function DF of the average SNR, ρ k Md, is computed as k f ρk x r M x +, Mr x, x, Mr where r denotes the cell radius. The conditional DF can be used to find the DF of Z which is a product of two random variables [4]. The DF of Z can be computed as f Z z f Z z x f x x dx z x f Y x x f x x dx z x f X x f Y x dx. Using and chi-square distribution with σ, f Z z x Mr r M M r M! x + z M e M x z dx M! x z M x M e x z dx Mr 3 Now, integrating the DF gives the desired result as F Z z r M r M [ Γ u + x Mr M Γ u e z x z u! u +, Mr z M u z u x u! } dx ]. 4 ANDIX B ROOF OF THORM FOR SCIFIC ROAGATION SCNARIOS Derivation details for, free-space, or large high-attenuation are very similar. Here, we only present the proof for the most practical propagation scenario,

5 I TRANSACTIONS ON WIRLSS COMMUNICATIONS, VOL. 9, NO. 8, AUGUST 453 free-space, due to size limitation. With,theCDFof Z in 6 reduces to a closed-form expression as F Z z M βz M + e βz u u t βz t t!, z >, z 5 where β Mr. Type of Limiting Distribution: Let a M β and b, and check the type of limiting distribution. When z> and a z + b >,, computed at the points, z and z, into 8 gives 3 F j: r log Υ j log + o O log, 3 F j: r log Υ j log + o log j O. 4 lim F Z a z + b [ lim z M u e Mz u t When z and a z + b, ] Mz t t! z. 6 lim F Z a z + b lim. 7 Therefore, the CDF of the j-th largest value, z j j th max z,z,,z }, behaves like Type- limiting distribution with Λ z exp z [], []. Rate of Convergence: From [3], if F z with normalizing sequences a and b is in the domain of attraction of Type-l limiting distribution l,, 3}, <F j: a z + b <, and log [Λ l z] <, then for natural number j Fj: az + b Υj l z+ δ g j, δ Θz+ [ π exp δ 4 6 δ3 3 δ + 9 δ3 δ δ + δ exp δ + 4 } ] δ 3 δ, δ 8 where all the relevant functions are defined as j log [Λ l z]} i z Λ l z, 9 i! i Θz exp [ δ j [δ z] i z] Υ j l z i!, i δ z F a z + b,, z exp θ, <z g z,θ μ +<z μ +, exp θ, Υ j l θ μ+ μ+! θμ μ! μ,,, It is shown in the following how the CDF of the j-th largest value, F j: z, is close to the limiting distribution. Asymptotic behaviors at two points, z log and z log, are considered. utting all 9- values Combining 3 and 4, we have F j: r log F j: r log O log, 5 or equivalently, [ r r log zj r log ] O log. 6 Taking log on both ends gives [ r log r log log log z j log r +log log ] O log. 7 Finally, for large number of users, the largest value of signal power based on proper user selection is approximated as log max ρ k h k } Ti log Now, putting the result in 5 gives M [ Ti R log r i i M i log Ti r r +O log log T i. 8 + O log log ] T i 9 From [4, Lemma 3], the size of candidate user pool is bounded by ε M T i, M ε M M log r R log r, 3 i Hence, the sum capacity scales for fixed M like. R lim M log. 3 which is the desired capacity scaling law for the propagation scenario of free-space. 3 Derivation details are omitted here for space limitation. Taylor series expansions of exponential functions followed by Big-O notations are used repeatedly. Similar derivations are also shown in the proof of [5, Lemma 7] and proof of [9, Theorem ].

6 454 I TRANSACTIONS ON WIRLSS COMMUNICATIONS, VOL. 9, NO. 8, AUGUST ANDIX C ROOF OF LMMA From the CDF of sin θ k in [9], the DF of sin θ k is simply derived as M f sin θ x B x M, x ξ, 3, x ξ where ξ B M. Then, the following computation gives the desired result. [ ] [ log hk w k log cos ] θ k ξ log xm B x M dx B ξ log ξ+ x M dx log ξ log ξ+ M log. 33 ANDIX D ROOF OF THORM In the interference-limited system,, the sum capacity is lower-bounded as R limited a h H k w k log + sin θ k β j,m > [log ] hhk w k log sin θ k j S, j k j S, j k b [log ] hhk w k log [ sin ] θ k c >M log ξ+ ξ M log + β j j S, j k B M β j, ξ b M 34 where a uses the fact that ρ k h k grows approximately as O from 8 assuming proper user selection, b comes from Jensen s inequality, and c uses Lemma and [, Lemma ]. As shown in a, the sum capacity becomes independent of the average SNRs of users in the interferencelimited system. Now, the derivations of the upper-bound is identical to [, Theorem ]. Using the previous results, the interference-limited sum capacity is upper-bounded by R limited M + B +log e +log M M + log e. 35 Using the both bounds in 34,35 and taking B to infinity, the desired scaling law is drawn. RFRNCS [] G. Caire and S. Shamai, On the achievable throughput of a multiantenna Gaussian broadcast channel," I Trans. Inf. Theory, vol. 49, no. 7, pp , July 3. []. Viswanath and D. Tse, Sum capacity of the vector Gaussian broadcast channel and uplink-downlink duality," I Trans. Inf. Theory, vol. 49, no. 8, pp. 9-9, Aug. 3. [3] N. Jindal and A. Goldsmith, Dirty-paper coding versus TDMA for MIMO broadcast channels," I Trans. Inf. Theory, vol. 5, no. 5, pp , May 5. [4] T. Yoo and A. Goldsmith, On the optimality of multiantenna broadcast scheduling using zero-forcing beamforming," I J. Sel. Areas Commun., vol. 4, no. 3, pp , Mar. 6. [5] M. Sharif and B. Hassibi, On the capacity of MIMO broadcast channels with partial side information," I Trans. Inf. Theory, vol. 5, no., pp. 56-5, Feb. 5. [6]. Huang, J. G. Andrews, and R. W. Heath, erformance of orthogonal beamforming for SDMA with limited feedback," I Trans. Veh. Technol., vol. 58, no., pp. 5-64, Jan. 9. [7] D. Gesbert and M. ountouris, Joint power control and user scheduling in multicell wireless networks: capacity scaling laws," Sep. 7, submitted to I Trans. Inf. Theory. [Online]. Available: [8] D. J. Love, R. W. Heath, W. Santipach, and M. L. Honig, What is the value of limited feedback for MIMO channels?" I Commun. Mag., vol. 4, no., pp , Oct. 4. [9] T. Yoo, N. Jindal, and A. Goldsmith, Multi-antenna downlink channels with limited feedback and user selection," I J. Sel. Areas Commun., vol. 5, no. 7, pp , Sep. 7. [] N. Jindal, MIMO broadcast channels with finite-rate feedback," I Trans. Inf. Theory, vol. 5, no., pp , Nov. 6. [] H. David and H. Nagaraja, Order Statistics. Wiley-Interscience, 3. [] N. Smirnov, Limit distributions for the terms of a variational series," Trudy Matematicheskogo Instituta im. VA Steklova, vol. 5, pp. 3-6, 949. [3] W. Dziubdziela, On convergence rates in the limit laws of extreme order statistics," Trans. 7th rague Conf. 974 urop. Meeting Statisticians, vol. B, pp. 9-7, 974. [4] A. Leon-Garcia, robability and Random rocesses for lectrical ngineering. Addison-Wesley ublishing Company, Inc., 994. [5] M. A. Maddah-Ali, M. A. Sadrabadi, and A.. handani, Broadcast in MIMO systems based on a generalized QR decomposition: signaling and performance analysis," I Trans. Inf. Theory, vol. 54, no. 3, pp. 4-38, Mar. 8.