Marios Kountouris - Transmission Capacity of Multiuser MIMO in Wireless Ad Hoc Networks

Transcript

1. Transmission Capacity of Multiuser MIMO in Wireless Ad Hoc Networks Marios Kountouris Dept. of Telecommunications SUPELEC, France joint work with Jeff Andrews (UT Austin, USA) SCEE Seminar Series Rennes, France February 11, 2010 SUPELEC - SCEE Seminar – p. 1/23
2. Prolegomena We consider a wireless ad hoc network with multi-antenna (MIMO) nodes SUPELEC - SCEE Seminar – p. 2/23
3. Prolegomena We consider a wireless ad hoc network with multi-antenna (MIMO) nodes Questions: What is the transmission capacity of SDMA in ad hoc networks? How does capacity scale with the number of Tx/Rx antennas? How to efficiently use multiple antennas to enhance network performance? SUPELEC - SCEE Seminar – p. 2/23
4. What do we know about the capacity of multiuser MIMO? In the downlink (broadcast channel), assuming Gaussian noise/interference, high SNR, iid exponential channels, sum capacity can be stated as: N transmit antennas, N receive antennas, U = 1 receiving user [FosGan98,Tel99]: C = O(N log SNR) Can be achieved using BLAST (perfect CSIR) or precoding methods (perfect CSIT). SUPELEC - SCEE Seminar – p. 3/23
5. What do we know about the capacity of multiuser MIMO? In the downlink (broadcast channel), assuming Gaussian noise/interference, high SNR, iid exponential channels, sum capacity can be stated as: N transmit antennas, N receive antennas, U = 1 receiving user [FosGan98,Tel99]: C = O(N log SNR) Can be achieved using BLAST (perfect CSIR) or precoding methods (perfect CSIT). N transmit antennas, 1 receive antenna, U = N receiving users, perfect CSIT [CaiSha03]: C = O(N log SNR) Can be achieved using Dirty Paper Coding (DPC) and user selection. SUPELEC - SCEE Seminar – p. 3/23
6. What do we know about the capacity of multiuser MIMO? In the downlink (broadcast channel), assuming Gaussian noise/interference, high SNR, iid exponential channels, sum capacity can be stated as: N transmit antennas, N receive antennas, U = 1 receiving user [FosGan98,Tel99]: C = O(N log SNR) Can be achieved using BLAST (perfect CSIR) or precoding methods (perfect CSIT). N transmit antennas, 1 receive antenna, U = N receiving users, perfect CSIT [CaiSha03]: C = O(N log SNR) Can be achieved using Dirty Paper Coding (DPC) and user selection. N transmit antennas, N receive antennas, any SNR, U ≫ N receiving users, perfect CSIT [ShaHas07]: C = O(N log log NU) Can be achieved using Dirty Paper Coding and user/antenna selection. SUPELEC - SCEE Seminar – p. 3/23
7. What changes in decentralized networks? Aspect Centralized Decentralized Channel i.i.d. exp(1) ind. exp(d−α i ) Capacity metric sum Capacity (Shannon) transmission capacity (outage-based) Interference generally ignored/Gaussian explicitly considered Interferer locations irrelevant Poisson point process Key differences in an ad hoc network: Interference is principally a product of the network geometry. There is an undeniable spatial aspect to the capacity, e.g. sum-rate per square meter. No agreed upon capacity formulation even for single antenna nodes. Symmetry in Tx/Rx antennas is typical. SUPELEC - SCEE Seminar – p. 4/23
8. Possible Routes Forward Transport Capacity and Scaling Laws [GupKum00, etc] Hierarchical Cooperation, virtual MIMO gives Θ(n1−δ) [OzgLevTse’07]. Electromagnetic limits state instead Θ( √ n) [FraMigMin’07]. Scaling laws are too coarse to give much design insight on multiuser MIMO Capacity Approximations: deterministic model [AveDigTse’07] and DoF [Jafar et al.’07]. Not without promise, but also asymptotic in nature Cut-set bounds, erasure networks [Dana et al’06]. Still result in scaling laws; inherently assume significant cooperation SUPELEC - SCEE Seminar – p. 5/23
9. Transmission Capacity (TC) Framework Baseline Definition: the maximum intensity λ of successful concurrent transmissions per unit area for which each transmission exceeds SINR β with probability 1 − ǫ. C = λ(1 − ǫ) log(1 + β) bps/Hz/m2. Key assumptions homogeneous Poisson field of transmitters. uncoordinated network (ALOHA-type MAC). initially for single-hop networks (extended in [AWKH’09]). Why transmission capacity? Exactly derived or tightly bounded achievable rates. Provides intuitive closed-form expressions, quantitative comparisons. SUPELEC - SCEE Seminar – p. 6/23
10. ‘Single User’ MIMO TC results to date With M transmit antennas, N receive antennas, K = 1 streams Channel-aware beamforming: diversity-oriented techniques result in C = Ω(max{N, M} 2 α ) and C = O((NM) 2 α ). MRC/MRT provides sublinear TC scaling [HunAndWeb’07]. Interference-aware beamforming (M = N): canceling the N − 1 strongest interferers ⇒ C = O(N1− 2 α ) [Huang et al.’08] Partial ZF: perform Rx ZF with k < N streams and MRC with N − k streams ⇒ C = O(N) [Jindal et al.’09] MMSE Receiver: for fixed density, the expected SINR scales as E{SINR} = O(N α 2 ) [GovBliSta’07] Also some recent results on Sp. MUX from Louie et al., Stamatiou et al., Vaze and Heath. SUPELEC - SCEE Seminar – p. 7/23
11. Our Goal Extend TC framework to take into account point-to-multipoint transmissions using MU-MIMO. Quantify the capacity gains (if any) of sending multiple streams. Fill in the following blanks: TC for a MU-MIMO network, i.e. K ≤ N streams to up to N users each with N Rx antennas (best-case) non-linear precoding (DPC) linear precoding (ZFBF) TC for MISO, i.e. K ≤ N streams sent to up to N users each using 1 receive antenna. SUPELEC - SCEE Seminar – p. 8/23
12. Network Model Random access, uncoordinated networks. Transmitters are distributed as 2-D homogeneous Poisson point process (PPP) on the plane. Received SINR at a typical Rx is SINR = ρH0 d−α i∈Π(λ) ρHi |Xi |−α + η . (1) ρ fixed transmit power H0 channel gain between S-D Hi channel gain from i-th interferer d distance between source and destination α pathloss exponent (α > 2) η noise power |Xi | distance from interferer i Π(λ) = {Xi } i.i.d. stationary Poisson point process with intensity λ SUPELEC - SCEE Seminar – p. 9/23
13. MIMO SDMA Channel Model Each Tx with M antennas communicates with |K| = K ≤ M Rx, each with N receive antennas (in practice M = N). The received signal yk at receiver k ∈ K is given by yk = √ ρd− α 2 H0kxk + √ ρ i∈Π(λ) |Xi |− α 2 Hikxi + n (2) where H0k ∈ CN×M : channel between T0 and receiver k, Hik ∈ CN×M : channel between Rx k and interfering Tx Ti, xk: normalized transmit signal vector, n: complex additive Gaussian noise. Perfect channel state information at the transmitter/receiver (CSIT/CSIR). SUPELEC - SCEE Seminar – p. 10/23
14. Key Performance Metrics Outage Probability: Pout = P0 {SINR ≤ β} ≤ ǫ. - depends on node distribution. - continuous monotone increasing in λ. Optimum per-stream contention density: λǫ = max {λ : P {SINRk ≤ βk } ≤ ǫk , ∀k} . maximum intensity of concurrent (single-stream) transmissions λǫ per m2 Multi-stream Transmission Capacity: Assuming transmission at the Shannon target rate b = log2 (1 + β) bps/Hz Cǫ = Kλǫ(1 − ǫ)b bps/Hz/m2 SUPELEC - SCEE Seminar – p. 11/23
15. Case 1: M × N MIMO with DPC and Rx Diversity M-antenna transmitters and N-antenna receivers: receive diversity Theorem 1: The maximum contention density when each Tx serves K Rx each with N antennas using DPC is upper bounded by λub DPC ≤ (4MN) 2 α IK β 2 α d2 − log(1 − ǫ) + ηβdα 4MNρ where IK = 2π α K−1 m=0 K m B(m + 2/α, K − (m + 2/α)). SUPELEC - SCEE Seminar – p. 12/23
16. Case 1: M × N MIMO with DPC and Rx Diversity M-antenna transmitters and N-antenna receivers: receive diversity Theorem 1: The maximum contention density when each Tx serves K Rx each with N antennas using DPC is upper bounded by λub DPC ≤ (4MN) 2 α IK β 2 α d2 − log(1 − ǫ) + ηβdα 4MNρ where IK = 2π α K−1 m=0 K m B(m + 2/α, K − (m + 2/α)). Since IK ∼ πΓ(1 − 2/α)K 2 α for large K, we can show that at high SNR, the multi-stream TC scales as Ω(K1− 2 α [(M − K + 1)N] 2 α ) ≤ CDPC ≤ O(K1− 2 α (MN) 2 α ). SUPELEC - SCEE Seminar – p. 12/23
17. Case 1 - Scaling results M-antenna transmitters and N-antenna receivers using MRC For K = M = N, the multi-stream transmission capacity scales as Ω(N) ≤ CDPC ≤ O(N1+ 2 α ). For K = 1 and M = N, i.e. single-stream eigenbeamforming C = Θ(N 4 α ) Which implies that the upper bound of [HunAndWeb’09] is tight for MRT/MRC. TC lower bound gives the optimal number of streams K∗ = 1 − 2 α (M + 1). Intuitively, upper bound is from O(N) streams from DPC, with O(N 2 α ) rate per stream from MRC. SUPELEC - SCEE Seminar – p. 13/23
18. Case 1 - Accurate results M-antenna transmitters and N-antenna receivers using MRC For small outage constraints ǫ: Lemma 1: The optimal contention density λmimo DPC employing DPC is given by λmimo DPC = FMN ǫ IM β2/αD2 e− ηβDα ρ (3) where FMN =   MN−1 k=0 k j=0 k j η ρ k−j 1 j! j−1 m=0 (m − 2/α)   −1 . (4) For large number of antennas, FMN scales as O((MN) 2 α ) ⇒ CDPC = O(N1+ 2 α ). SUPELEC - SCEE Seminar – p. 14/23
19. Case 2: M × N MIMO with DPC and Rx MRC/IC M-antenna Tx’s and N-antenna Rx’s with DPC at the Tx and both MRC and interference cancellation (IC) at the Rx Each Rx employs K DoF to for diversity reception, and the remaining N − K antennas to cancel the L = ⌊M K ⌋ − 1 closest interferers. SUPELEC - SCEE Seminar – p. 15/23
20. Case 2: M × N MIMO with DPC and Rx MRC/IC M-antenna Tx’s and N-antenna Rx’s with DPC at the Tx and both MRC and interference cancellation (IC) at the Rx Each Rx employs K DoF to for diversity reception, and the remaining N − K antennas to cancel the L = ⌊M K ⌋ − 1 closest interferers. Theorem 2: The multi-stream TC employing DPC with receiver interference cancellation scales with the number of antennas/streams at high SNR as Ω((KL)1− 2 α ∆ 2 α ) ≤ CDPC ≤ O(K1− 2 α ∆ 2 α L) with ∆ = (M − K + 1)(N − K + 1). SUPELEC - SCEE Seminar – p. 15/23
21. Case 2: M × N MIMO with DPC and Rx MRC/IC M-antenna Tx’s and N-antenna Rx’s with DPC at the Tx and both MRC and interference cancellation (IC) at the Rx Each Rx employs K DoF to for diversity reception, and the remaining N − K antennas to cancel the L = ⌊M K ⌋ − 1 closest interferers. Theorem 2: The multi-stream TC employing DPC with receiver interference cancellation scales with the number of antennas/streams at high SNR as Ω((KL)1− 2 α ∆ 2 α ) ≤ CDPC ≤ O(K1− 2 α ∆ 2 α L) with ∆ = (M − K + 1)(N − K + 1). For K = κM (0 < κ < 1) and M = N: Θ((1 − κ) 4 α N1+ 2 α ) ≤ CDPC ≤ Θ((κ−1(1 − κ)2) 2 α N1+ 2 α ) TC unambiguously scales superlinearly with N. SUPELEC - SCEE Seminar – p. 15/23
22. Case 3: M × 1 MISO with Dirty Paper Coding Theorem 3: The transmission capacity employing DPC to K ≤ M single-antenna receivers scales at high SNR as Ω(K1− 2 α (M − K + 1) 2 α ) ≤ Cmiso DPC ≤ O(K1− 2 α M 2 α ). For fully loaded SDMA network (K = M), we have Ω(M1− 2 α ) ≤ Cmiso DPC ≤ O(M) For K = κM (0 < κ < 1), TC scales as Θ(κ1− 2 α (1 − κ) 2 α M) ≤ Cmiso DPC ≤ Θ(κ1− 2 α M). κ∗ lb = 1 − 2 α ⇒ Cmiso DPC = Θ(M). Not using multiple receive antennas limits the throughput (unlike in centralized MIMO BC), but in ad hoc network likely to have M = N. SUPELEC - SCEE Seminar – p. 16/23
23. Case 4: ZF Transmit Beamforming With M = KN and receive combining (MRC): C = Θ(M1− 2 α ) or alternatively C = Θ(κ1− 2 α N2− 4 α ) for K = κN. With M = N and receive antenna selection: C = O(M), so selection provides an M 2 α -fold increase in Rx signal power vs. combining without an interference penalty as in MRC. With just N = 1 receive antenna: For large M, the C = Θ(M1− 2 α ); as expected from [Huang et al.’09]. Conclusion: Possession and judicious use of Rx antennas seems very important. SUPELEC - SCEE Seminar – p. 17/23
24. DPC Performance Evaluation 2 3 4 5 6 7 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 Number of Tx antennas Transmission capacity C ε DPC UB 1 DPC UB 2 DPC closed−form DPC sim. perf. DPC capacity scaling CDPC vs. nb. of Tx antennas (M = N) for α = 4. Superlinear scaling behavior of DPC with the number of antennas. Tightness of the upper bound depends on the pathloss exponent α and M (being tighter for α decreasing). Substantial gains appear even when only a few streams are transmitted. SUPELEC - SCEE Seminar – p. 18/23
25. ZF Precoding Performance Evaluation 1 2 3 4 5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 x 10−3 Number of Tx antennas Transmission capacity C ε DPC with single−antenna Rx ZFBF with Rx Antenna Selection ZFBF with single−antenna Rx ZFBF with multi−antenna Rx Transmission capacity vs. nb. of antennas for different SDMA precoding techniques and α = 4. Linear scaling of MISO DPC and sublinear capacity behavior of linear precoding. Diversity-oriented receive processing combined with linear transmit processing is not sufficient to achieve linear scaling. SUPELEC - SCEE Seminar – p. 19/23
26. Summary Standard MU-MIMO precoding provides at best superlinear TC growth but with DPC (theoretical) + perfect CSIT Partial ZF is key technique for achieving superlinear scaling in MU-MIMO It is often beneficial to send K < M streams (≈ (1 − 2 α )M) SDMA ZFBF achieves at best linear TC scaling with perfect CSIT. MU-MISO ad hoc communication is not very appealing. SUPELEC - SCEE Seminar – p. 20/23
27. Summary Standard MU-MIMO precoding provides at best superlinear TC growth but with DPC (theoretical) + perfect CSIT Partial ZF is key technique for achieving superlinear scaling in MU-MIMO It is often beneficial to send K < M streams (≈ (1 − 2 α )M) SDMA ZFBF achieves at best linear TC scaling with perfect CSIT. MU-MISO ad hoc communication is not very appealing. Orderwise: Cmiso ZF ≤ Cmrc ZF < CAS ZF ≈ CPZF miso = Cmiso DPC < Cmrc DPC ≤ CPZF DPC MU-MIMO vs. SIMO Comparison Per-stream basis: SIMO-PZF without CSIT > MU-MIMO with CSIT. Orderwise: MU-MISO with CSIT ≡ SIMO without CSIT. SUPELEC - SCEE Seminar – p. 20/23
28. Conclusions MIMO allows for denser packing of simultaneous transmissions. Superlinear capacity growth can in theory be achieved with multiple antennas (+ perfect CSIT). Receive antenna processing is of prime importance in MIMO ad hoc networks. MU-MIMO requires careful deliberation in ad hoc networks, since goal of multiple independent streams in an area can also be achieved simply with greater spatial reuse. Need for novel, MANET-tailored SDMA techniques (interference alignment with local CSIT?). MAC layer coordination, stream control, and scheduling to the rescue? SUPELEC - SCEE Seminar – p. 21/23
29. Merci! SUPELEC - SCEE Seminar – p. 22/23
30. References Results Given Here: SDMA: M. Kountouris and J. G. Andrews, “Capacity Scaling of SDMA in Wireless Ad Hoc Networks", in Proc. ITW’09 and journal pre-submission. Most-Related Prior Work: [WebAndJin’07] S. Weber, J. G. Andrews, and N. Jindal, “The effect of fading, channel inversion, and threshold scheduling on ad hoc networks," IEEE Trans. Inform. Theory, vol. 53, no. 11, pp. 4127-4149, Nov. 2007. [WebAndJin’08] S. Weber, J. G. Andrews, and N. Jindal, “Transmission capacity: applying stochastic geometry to uncoordinated ad hoc networks," arxiv.org/abs/0809.0016. [Huang et al.’08] K. Huang, J. G. Andrews, R. W. Heath, Jr., D. Guo, and R. A. Berry, “Spatial Interference Cancellation for Multi-Antenna Mobile Ad Hoc Networks", submitted to IEEE Trans. on Information Theory, July 2008. (on Arxiv). [HunAndWeb ’08] A. M. Hunter, J. G. Andrews, and S. P. Weber, “Transmission capacity of ad hoc networks with spatial diversity," IEEE Trans. on Wireless Communications, Vol. 7, No. 12, pp. 5058-71, Dec. 2008. SUPELEC - SCEE Seminar – p. 23/23