Applications of Random Matrices to Small Cell Networks Jakob Hoydis, Mari Kobayashi and M´ erouane Debbah Department of Telecommunications Alcatel-Lucent Chair on Flexible Radio Sup´ elec, Gif-Sur-Yvette, France jakob.hoydis@supelec.fr Random Matrix Symposium October 11–13, 2010 T´ el´ ecom Paristech Jakob Hoydis (Sup´ elec) RMT in Small Cells October 13, T´ el´ ecom Paristech 1 / 22
Outline What are Small Cell Networks (SCNs)? 1 A general channel model for SCNs 2 Applications of RMT for the performance analysis of SCNs 3 Jakob Hoydis (Sup´ elec) RMT in Small Cells October 13, T´ el´ ecom Paristech 2 / 22
“Globally, mobile data traffic will double every year through 2014, increasing 39 times between 2009 and 2014.” (Cisco, 2009) Jakob Hoydis (Sup´ elec) RMT in Small Cells October 13, T´ el´ ecom Paristech 3 / 22
How to increase the capacity of current cellular networks? Options More spectrum → hardly available New modulation/coding schemes → not to be expected Interference avoidance/cancellation → less interference, less spectral efficiency Cooperation (network MIMO) → less interference, benefits only at cell edge Cognitive radio → exploit spectrum holes, does it work? Will this be enough? Jakob Hoydis (Sup´ elec) RMT in Small Cells October 13, T´ el´ ecom Paristech 4 / 22
Will this be enough? Efficiency Gains 1950-2000 Voice coding MAC & Modulation 10 3 More Spectrum Spatial Reuse 10 2 2700 × increase 10 1 Source: ArrayComm & William Webb (Ofcom, 2007) Network densification seems only option to carry the forecasted traffic Increasing the macro cell density does not scale: too expensive to plan/deploy (CAPEX) and to maintain (OPEX) Radical network design change required. Jakob Hoydis (Sup´ elec) RMT in Small Cells October 13, T´ el´ ecom Paristech 5 / 22
Small Cell Networks Dense deployment of low-cost low-power BSs as additional capacity layer Advantages → Collocate with existing street furniture no cell site acquisition Use existing backhaul infrastructure → reduced deployment cost Self-organizing/optimization → no planning or maintenance needed SCNs can provide high network capacity and reduce CAPEX & OPEX. Jakob Hoydis (Sup´ elec) RMT in Small Cells October 13, T´ el´ ecom Paristech 6 / 22
Challenges Line-of-sight channels to several base stations User mobility requires base station cooperation Path loss Imperfect channel knowledge Limited backhaul capacity New tools for the performance analysis of SCNs are required. Jakob Hoydis (Sup´ elec) RMT in Small Cells October 13, T´ el´ ecom Paristech 7 / 22
The Role of RMT for Small Cell Networks Jakob Hoydis (Sup´ elec) RMT in Small Cells October 13, T´ el´ ecom Paristech 8 / 22
A general uplink channel model x 1 BS 1 UT 1 N × K MIMO Channel: C y = √ ρ H x + z y x 2 C BS 2 UT 2 CS H Gaussian Signaling x ∼ CN (0 , I K ) C ρ ≥ 0 x K BS B UT K Rician fading channel with a variance profile: H = W + A σ 2 [ W ] ij ∼ CN (0 , K ) ij σ 2 ij ≥ 0, variance profile → proportional to path loss A , deterministic → LOS components Noise, quantization error and correlated interference � zz H � △ = E = + Q ( C ) + z ∼ CN (0 , Z ) Z I N ∆ , ���� ���� � �� � correlated interference thermal noise quantization errors Jakob Hoydis (Sup´ elec) RMT in Small Cells October 13, T´ el´ ecom Paristech 9 / 22
Why RMT for small cells? Mutual Information � � 1 I N + ρ Z − 1 2 HH H Z − 1 I ( ρ ) = N log det 2 Ergodic Mutual Information I ( ρ ) = E H [ I ( ρ )] Outage Probability P out = Pr [ N I ( ρ ) ≤ R ] SINR at the output of the MMSE detector � 1 � − 1 γ k = h H ρ Z + H [ k ] H H h k k [ k ] RMT can provide close approximations of all these quantities. Jakob Hoydis (Sup´ elec) RMT in Small Cells October 13, T´ el´ ecom Paristech 10 / 22
Applications Jakob Hoydis (Sup´ elec) RMT in Small Cells October 13, T´ el´ ecom Paristech 11 / 22
Approximation of the ergodic mutual information � 1 �� � I N + ρ Z − 1 2 HH H Z − 1 I ( ρ ) = E N log det 2 � 1 I N + Q ( C ) + ∆ + ρ HH H �� � − 1 = E N log det N log det Z Note that Q ( C ) + ∆ + ρ HH H = ( Γ + Φ ) ( Γ + Φ ) H where Γ = [ √ ρ W 0 N × N ] � √ ρ A ( Q ( C ) + ∆ ) � 1 Φ = . 2 This is a Rician fading channel with a variance profile. Theorem 4.1 (Hachem, Loubaton, Najim: AAP’2007) Under some mild technical assumptions, the following limit holds true: I ( ρ ) − I ( ρ ) − N , K →∞ 0 . − − − − → Jakob Hoydis (Sup´ elec) RMT in Small Cells October 13, T´ el´ ecom Paristech 12 / 22
Approximation of the ergodic mutual information: Numerical results 8 N = 2 , K = 2 N = 6 , K = 3 N = 3 , K = 6 6 I ( ρ ) [ bits/s/Hz ] 4 2 0 − 10 0 10 20 30 SNR [ dB ] Jakob Hoydis (Sup´ elec) RMT in Small Cells October 13, T´ el´ ecom Paristech 13 / 22
Approximation of the outage probability Claim 1 (Hoydis, Kammoun, Najim, Debbah: ICC’11) Let Z = I N . Under some mild technical assumptions, the mutual information I ( ρ ) satisfies N � � D I ( ρ ) − I ( ρ ) − N , K →∞ N (0 , 1) − − − − → Θ N , K where Θ 2 N , K = − log det( J N , K ) . Remark J N , K is the Jacobian matrix of the fundamental equations of the random matrix model. This result can be used to approximate the outage probability: � � △ R − NI ( ρ ) P out ( R ) = Pr( N I ( ρ ) < R ) ≈ 1 − Q Θ N , K Jakob Hoydis (Sup´ elec) RMT in Small Cells October 13, T´ el´ ecom Paristech 14 / 22
Approximation of the outage probability: Numerical Results 10 0 Histogram 0 . 4 N (0 , 1) 10 − 1 0 . 3 K = 3 K = 6 K = 9 Frequency P out ( R ) 10 − 2 0 . 2 10 − 3 0 . 1 det. approx. simulation 0 10 − 4 − 4 − 3 − 2 − 1 0 1 2 3 4 6 8 10 12 14 16 18 20 N Θ N,K ( I ( ρ ) − V ( ρ )) SNR [ dB ] N = 6 , K = 9 Jakob Hoydis (Sup´ elec) RMT in Small Cells October 13, T´ el´ ecom Paristech 15 / 22
Performance of linear receivers with imperfect CSI Partial channel knowledge H = ˆ H + ˜ H σ 2 σ 2 ˆ ˜ ˆ K ), ˜ H ij ∼ CN (0 , ij H ij ∼ CN (0 , K ) ij H ij and ˜ ˆ H ij mutually independent SINR at the MMSE detector � 2 � � − 1 ˆ � ˆ H [ k ] ˆ ˆ h H H H [ k ] + 1 ρ I N h k k γ k ˆ = � − 1 ˜ 2 � − 1 ˆ � � � − 1 � � � � � � ˆ ˆ H [ k ] ˆ + ˆ H [ k ] ˆ ˆ H [ k ] ˆ ˆ � h H H H [ k ] + 1 � h H H H [ k ] + 1 H [ k ] H H [ k ] + 1 H H [ k ] + 1 ρ I N h k ρ I N ρ I N ρ I N h k � � k k � Corollary 1 (Hoydis, Kobayashi, Debbah: Asilomar’10) Under some mild assumptions, the following limit holds true: a.s. γ k − γ k ˆ − N , K →∞ 0 . − − − − → Jakob Hoydis (Sup´ elec) RMT in Small Cells October 13, T´ el´ ecom Paristech 16 / 22
Performance of linear receivers with imperfect CSI: Numerical results MMSE 20 ZF MF perfect CSI 10 imperfect CSI γ k [ dB ] 0 − 10 − 20 − 10 − 5 0 5 10 15 20 SNR [ dB ] N = 12 , K = 16 Jakob Hoydis (Sup´ elec) RMT in Small Cells October 13, T´ el´ ecom Paristech 17 / 22
Optimal channel training Partial channel knowledge through channel training The channel H remains constant during T channel uses of which τ are used for channel estimation: σ 2 ˆ ij ( τ ), concave increasing σ 2 ˜ ij ( τ ), convex decreasing Goal: Maximization of the net ergodic achievable rate � 1 �� � � � 1 − τ τ ∗ = arg max I N + ρ Z − 1 H H Z − 1 2 ( τ )ˆ H ˆ 2 ( τ ) N log det E T τ Solution: Maximize the deterministic equivalent approximation instead � � ˆ 1 − τ τ ∗ = arg max I ( ρ, τ ) T τ Theorem 4 (Hoydis, Kobayashi, Debbah: IEEE Trans. SP’10) τ ∗ − τ ∗ − − − − − → N , K →∞ 0 . Jakob Hoydis (Sup´ elec) RMT in Small Cells October 13, T´ el´ ecom Paristech 18 / 22
Optimal channel training: Numerical Results 1 . 2 τ ∗ deterministic equivalent τ ∗ exhaustive search 1 R net ( τ ) [ bits/channel use ] Optimal training length 25 0 . 8 C = 1 0 . 6 C = 5 C = 10 20 0 . 4 0 . 2 15 0 0 20 40 60 80 100 − 10 − 5 0 5 10 15 20 Training length τ SNR [ dB ] N = 6 , K = 3 Jakob Hoydis (Sup´ elec) RMT in Small Cells October 13, T´ el´ ecom Paristech 19 / 22
Current work Polynomial expansion detectors Closed-form expression of the asymptotic moments of the matrix HH H These results can be used to compute an approximation of the matrix � � − 1 L � HH H � l HH H + 1 � ≈ ρ I N λ l l =0 where the λ l are only related to the asymptotic moments and can be precomputed. Distributed Downlink Beamforming (Lakshminarayana, Hoydis, Debbah, Assad: PIMRC’10) Optimal downlink beamforming vectors can be computed by a distributed algorithm which requires the exchange of full CSI (Dahrouj, Yu: CISS’08). Proposed algorithm requires only exchange of statistical CSI. Significant reduction of message exchange over the backhaul network. Jakob Hoydis (Sup´ elec) RMT in Small Cells October 13, T´ el´ ecom Paristech 20 / 22
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