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
(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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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