Applications of Random Matrices to Small Cell Networks Jakob Hoydis, - - PowerPoint PPT Presentation

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

1

What are Small Cell Networks (SCNs)?

2

A general channel model for SCNs

3

Applications of RMT for the performance analysis of SCNs

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?

101 102 103 2700× increase Efficiency Gains 1950-2000 Voice coding MAC & Modulation More Spectrum Spatial Reuse

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

CS BS1 BS2 BSB UT1 x1 x2 xK UT2 UTK C C C y H

N × K MIMO Channel: y = √ρ H x + z

Gaussian Signaling

x ∼ CN(0, IK) ρ ≥ 0 Rician fading channel with a variance profile: H = W + A [W]ij ∼ CN(0,

σ2

ij

K )

σ2

ij ≥ 0, variance profile →

proportional to path loss A, deterministic → LOS components Noise, quantization error and correlated interference Z

△

= E

• zzH

= IN

• thermal noise

+ Q(C)

quantization errors

+ ∆

• correlated interference

, z ∼ CN(0, Z)

Jakob Hoydis (Sup´ elec) RMT in Small Cells October 13, T´ el´ ecom Paristech 9 / 22

Why RMT for small cells?

Mutual Information I(ρ) = 1 N log det

• IN + ρZ− 1

2 HHHZ− 1 2

• Ergodic Mutual Information

I(ρ) = EH [I(ρ)] Outage Probability Pout = Pr [NI(ρ) ≤ R] SINR at the output of the MMSE detector γk = hH

k

1 ρZ + H[k]HH

[k]

−1 hk 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

I(ρ) = E 1 N log det

• IN + ρZ− 1

2 HHHZ− 1 2

• = E

1 N log det

• IN + Q(C) + ∆ + ρHHH

− 1 N log det Z Note that Q(C) + ∆ + ρHHH = (Γ + Φ) (Γ + Φ)H where Γ = [√ρW 0N×N] Φ = √ρA (Q(C) + ∆)

1 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

−10 10 20 30 2 4 6 8 SNR [dB] I(ρ) [bits/s/Hz] N = 2, K = 2 N = 6, K = 3 N = 3, K = 6

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 = IN. Under some mild technical assumptions, the mutual information I(ρ) satisfies N ΘN,K

• I(ρ) − I(ρ)
• D

− − − − − →

N,K→∞ N(0, 1)

where Θ2

N,K = − log det(JN,K) .

Remark JN,K is the Jacobian matrix of the fundamental equations of the random matrix model. This result can be used to approximate the outage probability: Pout(R)

△

= Pr(NI(ρ) < R) ≈ 1 − Q

• R−NI(ρ)

ΘN,K

• Jakob Hoydis (Sup´

elec) RMT in Small Cells October 13, T´ el´ ecom Paristech 14 / 22

Approximation of the outage probability: Numerical Results

−4 −3 −2 −1 1 2 3 4 0.1 0.2 0.3 0.4

N ΘN,K (I(ρ) − V (ρ))

Frequency Histogram N(0, 1) 6 8 10 12 14 16 18 20 10−4 10−3 10−2 10−1 100 K = 3 K = 6 K = 9 SNR [dB] Pout(R)

• det. approx.

simulation

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 ˆ Hij ∼ CN(0,

ˆ σ2

ij

K ), ˜

Hij ∼ CN(0,

˜ σ2

ij

K )

ˆ Hij and ˜ Hij mutually independent SINR at the MMSE detector

ˆ γk =

• ˆ

hH

k

• ˆ

H[k] ˆ HH

[k] + 1 ρ IN

−1 ˆ hk 2

• ˆ

hH

k

• ˆ

H[k] ˆ HH

[k] + 1 ρ IN

−1 ˜ hk

• 2

+ ˆ hH

k

• ˆ

H[k] ˆ HH

[k] + 1 ρ IN

−1 H[k]HH

[k] + 1 ρ IN

ˆ H[k] ˆ HH

[k] + 1 ρ IN

−1 ˆ hk

Corollary 1 (Hoydis, Kobayashi, Debbah: Asilomar’10) Under some mild assumptions, the following limit holds true: ˆ γk − γk

a.s.

− − − − − →

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

−10 −5 5 10 15 20 −20 −10 10 20 SNR [dB] γk [dB] MMSE ZF MF perfect CSI imperfect CSI

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: ˆ σ2

ij(τ), concave increasing

˜ σ2

ij(τ), convex decreasing

Goal: Maximization of the net ergodic achievable rate τ ∗ = arg max

τ

• 1 − τ

T

• E

1 N log det

• IN + ρZ− 1

2 (τ)ˆ

Hˆ HHZ− 1

2 (τ)

• Solution: Maximize the deterministic equivalent approximation instead

τ ∗ = arg max

τ

• 1 − τ

T ˆ I(ρ, 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

20 40 60 80 100 0.2 0.4 0.6 0.8 1 1.2 Training length τ Rnet(τ) [bits/channel use] C = 1 C = 5 C = 10 −10 −5 5 10 15 20 15 20 25 SNR [dB] Optimal training length τ ∗ deterministic equivalent τ ∗ exhaustive search

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 HHH These results can be used to compute an approximation of the matrix

• HHH + 1

ρIN −1 ≈

L

• l=0

λl

• HHHl

where the λl 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

Conclusions

Small cells are a promising network architecture to provide high capacity coverage. Smaller cell sizes pose many new challenges:

◮ LOS channels ◮ How to deal with imperfect CSI? ◮ Cooperation of BSs necessary to handle user mobility ◮ Limited backhaul capacity

Asymptotic results of information-theoretic quantities for involved matrix models Close approximations for realistic system dimensions Simplify optimization problems Develop distributed algorithms

Jakob Hoydis (Sup´ elec) RMT in Small Cells October 13, T´ el´ ecom Paristech 21 / 22

Thank you !

Jakob Hoydis (Sup´ elec) RMT in Small Cells October 13, T´ el´ ecom Paristech 22 / 22