David versus Goliath:Small Cells versus Massive MIMO. Jakob Hoydis and Mérouane Debbah

Transcription

1 David versus Goliath:Small Cells versus Massive MIMO Jakob Hoydis and Mérouane Debbah

2 The total worldwide mobile traffic is expected to increase 33 from The average 3G smart phone user consumed 375 MB/month. The average 3G broadband (HSPA/+) user consumed 5 GB/month. The average LTE consumer used GB/month of data. 2 1 Source: IDATE for UMTS Forum 2 Press release of a Scandinavian operator (Nov. 2010) 2 / 25

3 Network densification Efficiency Gains Voice coding MAC & Modulation More Spectrum Spatial Reuse 2700 increase Spectral efficiency antenna 2 antennas 8 antennas 4 antennas SNR (db) The exploding demand for wireless data traffic requires a massive network densification: Densification: Increasing the number of antennas per unit area 3 / 25

4 David vs Goliath or Small Cells vs Massive MIMO How to densify: More antennas or more BSs? Questions: Should we install more base stations or simply more antennas per base? How can massively many antennas be efficiently used? Can massive MIMO simplify the signal processing? 4 / 25

5 Vision Bell Labs lightradio antenna module the next generation small cell (picture from

6 A thought experiment Consider an infinite large network of randomly uniformly distributed base stations and user terminals. What would be better? A 2 more base stations B 2 more antennas per base station 5 / 25

7 A thought experiment Consider an infinite large network of randomly uniformly distributed base stations and user terminals. What would be better? A 2 more base stations B 2 more antennas per base station Stochastic geometry can provide an answer. 5 / 25

8 System model: Downlink Received signal at a tagged UT at the origin: y = 1 r α/2 0 h H 0 x 0 + }{{} desired signal 1 i=1 r α/2 i h H i x i } {{ } interference + n h i CN (0, I N ): fast fading channel vectors r i : distance to ith closest BS P = E [ x H i x i ] : average transmit power constraint per BS 6 / 25

9 System model: Downlink Received signal at a tagged UT at the origin: y = 1 r α/2 0 h H 0 x 0 + }{{} desired signal 1 i=1 r α/2 i h H i x i } {{ } interference + n h i CN (0, I N ): fast fading channel vectors r i : distance to ith closest BS P = E [ x H i x i ] : average transmit power constraint per BS Assumptions: infinitely large network of uniformly randomly distributed BSs and UTs with densities λ BS and λ UT, respectively single-antenna UTs, N antennas per BS each UT is served by its closest BS distance-based path loss model with path loss exponent α > 2 total bandwidth W, re-used in each cell 6 / 25

10 Transmission strategy: Zero-forcing Assumptions: K = λ UT λ BS UTs need to be served by each BS on average total bandwidth W divided into L 1 sub-bands K = K/L N UTs are simultaneously served on each sub-band 7 / 25

11 Transmission strategy: Zero-forcing Assumptions: K = λ UT λ BS UTs need to be served by each BS on average total bandwidth W divided into L 1 sub-bands K = K/L N UTs are simultaneously served on each sub-band Transmit vector of BS i: x i = P K K w i,k s i,k k=1 s i,k CN (0, 1): message determined for UT k from BS i w i,k C N 1 : ZF-beamforming vectors 7 / 25

12 Performance metric: Average throughput Received SINR at tagged UT: γ = i=1 r α i r α 0 K k=1 h H 0 w 0,1 2 h H i w i,k 2 + K P = r α 0 S i=1 r α i g i + K P Coverage probability: P cov(t ) = P (γ T ) Average throughput per UT: C = W L E [log(1 + γ)] = W L P cov (e z 1) dz 0 8 / 25

13 Performance metric: Average throughput Received SINR at tagged UT: γ = i=1 r α i r α 0 K k=1 h H 0 w 0,1 2 h H i w i,k 2 + K P = r α 0 S i=1 r α i g i + K P Coverage probability: P cov(t ) = P (γ T ) Average throughput per UT: C = W L E [log(1 + γ)] = W L P cov (e z 1) dz 0 Remarks: expectation with respect to fading and BSs locations S = h H 0 w 0,1 2 Γ(N K + 1, 1), gi = K h H i w i,k 2 Γ(K, 1) K impacts the interference distribution, N impacts the desired signal k=1 for P, the SINR becomes independent of λ BS 8 / 25

14 A closed-form result Theorem (Combination of Baccelli 09, Andrews 10) ( P cov(t ) = L Ir0 (i2πr0 α Ts) exp r 0 >0 i2πr 0 αtk P ) LS ( i2πs) 1 s f r0 (r 0 )dsdr 0 i2πs where ( ( ) ) 1 L Ir0 (s) = exp 2πλ BS 1 r 0 (1 + sv α ) K vdv ( ) 1 N K+1 L S (s) = 1 + s f r0 (r 0 ) = 2πλ BS r 0 e λ BSπr 2 0 The computation of P cov(t ) requires in general three numerical integrals. J. G. Andrews, F. Baccelli, R. K. Ganti, A Tractable Approach to Coverage and Rate in Cellular Networks IEEE Trans. Wireless Commun., submitted F. Baccelli, B. B laszczyszyn, P. Mühlethaler, Stochastic Analysis of Spatial and Opportunistic Aloha Journal on Selected Areas in Communications, / 25

15 Example Density of UTs: λ UT = 16 Constant transmit power density: P λ BS = 10 Number of BS-antennas: N = λ UT /λ BS Path loss exponent: α = 4 UT simultaneously served on each band: K = λ UT /(λ BS L) Only two parameters: λ BS and L 10 / 25

16 Example Density of UTs: λ UT = 16 Constant transmit power density: P λ BS = 10 Number of BS-antennas: N = λ UT /λ BS Path loss exponent: α = 4 UT simultaneously served on each band: K = λ UT /(λ BS L) Only two parameters: λ BS and L Table: Average spectral efficiency C/W in (bits/s/hz) sub-bands L λ BS = 1 λ BS = 2 λ BS = 4 λ BS = 8 λ BS = x x x x x x x x x Fully distributing the antennas gives highest throughput gains! 10 / 25

17 First conclusions Distributed network densification is preferable over massive MIMO if the average throughput per UT should be increased. More antennas increase the coverage probability, but more BSs lead to a linear increase in area spectral efficiency (with constant total transmit power). If we use other metrics such as coverage probability or goodput, the picture might change. 11 / 25

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21 Infinitely Many Antennas: Forward-Link Capacity For 20 MHz Bandwidth, 42 Terminals per Cell, 500 sec Slot Interference-limited: energy-per-bit can be made arbitrarily small! Frequency Reuse.95-Likely SIR (db).95-likely Capacity per Terminal (Mbits/s) Mean Capacity per Terminal (Mbits/s) Mean Capacity per Cell (Mbits/s) Mean Capacity per Cell (Mbits/s) LTE Advanced (>= Release 10) 74

22 Motivation of massive MIMO Consider a N K MIMO MAC: K y = h k x k + n k=1 where h k, n are i.i.d. with zero mean and unit variance. Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 4 / 30

23 Motivation of massive MIMO Consider a N K MIMO MAC: y = K h k x k + n k=1 where h k, n are i.i.d. with zero mean and unit variance. By the strong law of large numbers: 1 N hmh a.s. y x m N, K=const. Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 4 / 30

24 Motivation of massive MIMO Consider a N K MIMO MAC: y = K h k x k + n k=1 where h k, n are i.i.d. with zero mean and unit variance. By the strong law of large numbers: 1 N hmh a.s. y x m N, K=const. With an unlimited number of antennas, uncorrelated interference and noise vanish, the matched filter is optimal, the transmit power can be made arbitrarily small. T. Marzetta, Noncooperative cellular wireless with unlimited numbers of base station antennas IEEE Trans. Wireless Commun., vol. 9, no. 11, pp , Nov Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 4 / 30

25 About some fundamental assumptions The receiver has perfect channel state information (CSI). What happens if the channel must be estimated? Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 5 / 30

26 About some fundamental assumptions The receiver has perfect channel state information (CSI). What happens if the channel must be estimated? The number of interferers K is small compared to N. What does small mean? Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 5 / 30

27 About some fundamental assumptions The receiver has perfect channel state information (CSI). What happens if the channel must be estimated? The number of interferers K is small compared to N. What does small mean? The channel provides infinite diversity, i.e., each antenna gives an independent look on the transmitted signal. What if the degrees of freedom are limited? Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 5 / 30

28 About some fundamental assumptions The receiver has perfect channel state information (CSI). What happens if the channel must be estimated? The number of interferers K is small compared to N. What does small mean? The channel provides infinite diversity, i.e., each antenna gives an independent look on the transmitted signal. What if the degrees of freedom are limited? The received energy grows without bounds as N. Clearly wrong, but might hold up to very large antenna arrays if the aperture scales with N. Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 5 / 30

29 On channel estimation and pilot contamination 1 The receiver estimates the channels based on pilot sequences. 2 The number of orthogonal sequences is limited by the coherence time. 3 Thus, the pilot sequences must be reused. Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 6 / 30

30 On channel estimation and pilot contamination 1 The receiver estimates the channels based on pilot sequences. 2 The number of orthogonal sequences is limited by the coherence time. 3 Thus, the pilot sequences must be reused. Assume that transmitter m and j use the same pilot sequence: ĥ m = h m + h j }{{} + n }{{} m pilot contamination estimation noise Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 6 / 30

31 On channel estimation and pilot contamination 1 The receiver estimates the channels based on pilot sequences. 2 The number of orthogonal sequences is limited by the coherence time. 3 Thus, the pilot sequences must be reused. Assume that transmitter m and j use the same pilot sequence: ĥ m = h m + h j }{{} + n }{{} m pilot contamination estimation noise Thus, 1 N ĥmh a.s y x m + x j N,K=const. Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 6 / 30

32 On channel estimation and pilot contamination 1 The receiver estimates the channels based on pilot sequences. 2 The number of orthogonal sequences is limited by the coherence time. 3 Thus, the pilot sequences must be reused. Assume that transmitter m and j use the same pilot sequence: ĥ m = h m + h j }{{} + n }{{} m pilot contamination estimation noise Thus, 1 N ĥmh a.s y x m + x j N,K=const. With an unlimited number of antennas, uncorrelated interference, noise and estimation errors vanish, the matched filter is optimal, the transmit power can be made arbitrarily small ( 1/ N [Ngo 11]), but the performance is limited by pilot contamination. T. Marzetta, Noncooperative cellular wireless with unlimited numbers of base station antennas IEEE Trans. Wireless Commun., vol. 9, no. 11, pp , Nov Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 6 / 30

33 Uplink Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 7 / 30

34 System model and channel estimation Uplink: L BSs with N antennas, K UTs per cell. Received signal at BS j: y j = L ρ H jl x l + n j l=1 Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 8 / 30

35 System model and channel estimation Uplink: L BSs with N antennas, K UTs per cell. Received signal at BS j: y j = ρ The columns of H jl (N K) are modeled as L H jl x l + n j l=1 h jlk = R 1 2 jlk w jlk, w jlk CN (0, I N ) Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 8 / 30

36 System model and channel estimation Uplink: L BSs with N antennas, K UTs per cell. Received signal at BS j: y j = ρ The columns of H jl (N K) are modeled as L H jl x l + n j l=1 h jlk = R 1 2 jlk w jlk, w jlk CN (0, I N ) Channel estimation: y τ jk = h jjk + l j h jlk + 1 ρτ n jk Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 8 / 30

37 System model and channel estimation Uplink: L BSs with N antennas, K UTs per cell. Received signal at BS j: y j = ρ The columns of H jl (N K) are modeled as L H jl x l + n j l=1 h jlk = R 1 2 jlk w jlk, w jlk CN (0, I N ) Channel estimation: y τ jk = h jjk + l j h jlk + 1 ρτ n jk MMSE estimate: h jjk = ĥjjk + h jjk Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 8 / 30

38 System model and channel estimation Uplink: L BSs with N antennas, K UTs per cell. Received signal at BS j: y j = ρ The columns of H jl (N K) are modeled as L H jl x l + n j l=1 h jlk = R 1 2 jlk w jlk, w jlk CN (0, I N ) Channel estimation: y τ jk = h jjk + l j h jlk + 1 ρτ n jk MMSE estimate: h jjk = ĥjjk + h jjk ĥ jjk CN (0, Φ jjk ), h jjk CN (0, R jjk Φ jjk ) Φ jlk = R jjk Q jk R jlk, ( 1 Q jk = I N + ) 1 R jlk ρ τ l Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 8 / 30

39 Achievable rates with linear detectors Ergodic achievable rate of UT m in cell j: γ jm = [ E rjm H with an arbitrary receive filter r jm. R jm = EĤjj [log 2 (1 + γ jm )] rjmĥjjm H 2 ( 1 ρ I N + h jjm h H jjm h jjmh H jjm + l H jlh H jl ) r jm Ĥ jj ] B. Hassibi and B. M. Hochwald, How much training is needed in multiple-antenna wireless links? IEEE Trans. Inf. Theory., vol. 49, no. 4, pp , Nov Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 9 / 30

40 Achievable rates with linear detectors Ergodic achievable rate of UT m in cell j: γ jm = [ E rjm H with an arbitrary receive filter r jm. Two specific linear detectors r jm : R jm = EĤjj [log 2 (1 + γ jm )] rjmĥjjm H 2 ( 1 ρ I N + h jjm h H jjm h jjmh H jjm + l H jlh H jl r MF jm = ĥjjm r MMSE jm = (Ĥjj Ĥ H jj + Z j + NλI N ) 1 ĥjjm ) r jm Ĥ jj ] where λ > 0 is a design parameter and Z j = E H jj H H jj + H jl H jl = l j k (R jjk Φ jjk ) + l j R jlk. k B. Hassibi and B. M. Hochwald, How much training is needed in multiple-antenna wireless links? IEEE Trans. Inf. Theory., vol. 49, no. 4, pp , Nov Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 9 / 30

41 Large system analysis based on random matrix theory Assume N, K at the same speed. Then, γ jm γ jm a.s. 0 R jm log 2 (1 + γ jm ) a.s. 0 Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 10 / 30

42 Large system analysis based on random matrix theory Assume N, K at the same speed. Then, γ jm γ jm a.s. 0 where R jm log 2 (1 + γ jm ) a.s. 0 γ MF jm = 1 tr Φ ρn 2 jjm + 1 N l,k ( 1 N tr Φ jjm) 2 1 N tr R jlkφ jjm + l j 1 tr Φ N jlm 2 γ MMSE jm = 1 ρn 2 tr Φ jjm T j + 1 N δ 2 jm l,k µ jlkm + l j ϑ jlm 2 and δ jm, µ jlkm, θ jlm, T j can be calculated numerically. Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 10 / 30

43 A simple multi-cell scenario Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 11 / 30

44 A simple multi-cell scenario intercell interference factor α [0, 1] transmit power per UT: ρ H jl = [h jl1 h jlk ] = N/PAW jl A C N P composed of P N columns of a unitary matrix W ij C P K have i.i.d. elements with zero mean and unit variance Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 11 / 30

45 A simple multi-cell scenario intercell interference factor α [0, 1] transmit power per UT: ρ H jl = [h jl1 h jlk ] = N/PAW jl A C N P composed of P N columns of a unitary matrix W ij C P K have i.i.d. elements with zero mean and unit variance Assumptions: P channel degrees of freedom, i.e., rank (H jl ) = min(p, K) [Ngo 11] energy scales linearly with N, i.e., E [ tr H jl H H jl ] = KN only pilot contamination, i.e., no estimation noise: ĥ jjk = h jjk + α l j h jlk Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 11 / 30

46 Asymptotic performance of the matched filter Assume that N, K and P grow infinitely large at the same speed: SINR MF L ρn }{{} noise 1 K + P L 2 + α( L 1) }{{}}{{} pilot contamination multi-user interference where L = 1 + α(l 1). Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 12 / 30

47 Asymptotic performance of the matched filter Assume that N, K and P grow infinitely large at the same speed: SINR MF L ρn }{{} noise 1 K + P L 2 + α( L 1) }{{}}{{} pilot contamination multi-user interference where L = 1 + α(l 1). Observations: The effective SNR ρn increases linearly with N. The multiuser interference depends on P/K and not on N. Ultimate performance limit: SINR MF a.s SINR 1 = N,P, K=const. α( L 1) J. Hoydis, S. ten Brink, and M. Debbah, Massive MIMO: How many antennas do we need, Allerton Conference, Urbana-Champaing, Illinois, US, Sep [Online] Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 12 / 30

48 Asymptotic performance of the MMSE detector Assume that N, K and P grow infinitely large at the same speed: SINR MMSE L ρn X }{{} noise 1 K + P L 2 Y + α( L 1) }{{}}{{} pilot contamination multi-user interference where L = 1 + α(l 1) and X, Y are given in closed-form. Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 13 / 30

49 Asymptotic performance of the MMSE detector Assume that N, K and P grow infinitely large at the same speed: SINR MMSE L ρn X }{{} noise 1 K + P L 2 Y + α( L 1) }{{}}{{} pilot contamination multi-user interference where L = 1 + α(l 1) and X, Y are given in closed-form. Observations: As for the MF, the performance depends only on ρn and P/K. The ultimate performance of MMSE and MF coincide: SINR MMSE a.s SINR 1 = N,P, K=const. α( L 1) J. Hoydis, S. ten Brink, and M. Debbah, Massive MIMO: How many antennas do we need, Allerton Conference, Urbana-Champaing, Illinois, US, Sep [Online] Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 13 / 30

50 Numerical results Ergodic achievable rate (b/s/hz) R P = N P = N/3 MF approx. MMSE approx. ρ = 1, K = 10, α = 0.1, L = 4 Simulations Number of antennas N Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 14 / 30

51 Conclusions (I) - Uplink Massive MIMO can be seen as a particular operating condition where noise + interference pilot contamination. Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 18 / 30

52 Conclusions (I) - Uplink Massive MIMO can be seen as a particular operating condition where noise + interference pilot contamination. If this condition is satisfied depends on: P/K : degrees of freedom per UT ρn : effective SNR (transmit power number of antennas) α : path loss (or intercell interference) Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 18 / 30

53 Conclusions (I) - Uplink Massive MIMO can be seen as a particular operating condition where noise + interference pilot contamination. If this condition is satisfied depends on: P/K : degrees of freedom per UT ρn : effective SNR (transmit power number of antennas) α : path loss (or intercell interference) Connection between N and P is crucial, but unclear for real channels. Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 18 / 30

54 Conclusions (I) - Uplink Massive MIMO can be seen as a particular operating condition where noise + interference pilot contamination. If this condition is satisfied depends on: P/K : degrees of freedom per UT ρn : effective SNR (transmit power number of antennas) α : path loss (or intercell interference) Connection between N and P is crucial, but unclear for real channels. As N, MF and MMSE detector achieve identical performance. For finite N, the MMSE detector largely outperforms the MF. Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 18 / 30

55 Conclusions (I) - Uplink Massive MIMO can be seen as a particular operating condition where noise + interference pilot contamination. If this condition is satisfied depends on: P/K : degrees of freedom per UT ρn : effective SNR (transmit power number of antennas) α : path loss (or intercell interference) Connection between N and P is crucial, but unclear for real channels. As N, MF and MMSE detector achieve identical performance. For finite N, the MMSE detector largely outperforms the MF. The number of antennas needed for massive MIMO depends on all these parameters! Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 18 / 30

56 Downlink Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 19 / 30

57 System model: Downlink L BSs with N antennas, K UTs per cell. Received signal at mth UT in cell j: y jm = L ρ h H ljms l + q jm l=1 Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 20 / 30

58 System model: Downlink L BSs with N antennas, K UTs per cell. Received signal at mth UT in cell j: where y jm = ρ s l = λ l K m=1 1 λ l = tr W l Wl H L h H ljms l + q jm l=1 w lm x lm = λ l W l x l ] = E [ρs H l s l = ρ Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 20 / 30

59 System model: Downlink L BSs with N antennas, K UTs per cell. Received signal at mth UT in cell j: where y jm = ρ s l = λ l K m=1 1 λ l = tr W l Wl H L h H ljms l + q jm l=1 w lm x lm = λ l W l x l ] = E [ρs H l s l = ρ Channel estimation through uplink pilots (as before): h jjk = ĥjjk + h jjk ĥ jjk CN (0, Φ jjk ), hjjk CN (0, R jjk Φ jjk ) Φ jlk = R jjk Q jk R jlk, ( 1 Q jk = I N + ) 1 R jlk ρ τ l Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 20 / 30

60 Achievable rates with linear precoders Ergodic achievable rate of UT m in cell j: γ jm = R jm = log 2 (1 + γ jm ) E [ λj h H jjmw jm ] var [ [ λ ρ j h H jjm w ] jm + (l,k) (j,m) E λl h H ljm w lk 2]. J. Jose, A. Ashikhmin, T. Marzetta, and S. Vishwanath, Pilot contamination and precoding in multi-cell TDD systems, IEEE Trans. Wireless Commun., no. 99, pp. 1 12, Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 21 / 30

61 Achievable rates with linear precoders Ergodic achievable rate of UT m in cell j: γ jm = R jm = log 2 (1 + γ jm ) E [ λj h H jjmw jm ] var [ [ λ ρ j h H jjm w ] jm + (l,k) (j,m) E λl h H ljm w lk 2]. Two specific precoders W j : W BF j W RZF j = Ĥjj = where α > 0 and F j are design parameters. (Ĥjj Ĥ H jj + F j + NαI N ) 1 Ĥjj J. Jose, A. Ashikhmin, T. Marzetta, and S. Vishwanath, Pilot contamination and precoding in multi-cell TDD systems, IEEE Trans. Wireless Commun., no. 99, pp. 1 12, Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 21 / 30

62 Large system analysis based on random matrix theory Assume N, K at the same speed. Then, γ jm γ jm a.s. 0 R jm log 2 (1 + γ jm ) a.s. 0 J. Hoydis, S. ten Brink, M. Debbah, Comparison of linear precoding schemes for downlink Massive MIMO, ICC 12, Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 22 / 30

63 Large system analysis based on random matrix theory Assume N, K at the same speed. Then, γ jm γ jm a.s. 0 where R jm log 2 (1 + γ jm ) a.s. 0 γ BF jm = γ RZF jm = λ j ( 1 N tr Φ jjm) 2 K + 1 Nρ N l,k λ 1 l tr R N ljmφ llk + λ l j j 1 λ j δjm 2 tr Φ N ljm K (1 + δ Nρ jm) N l,k λ ( ) 1+δjm 2 l µljmk 1+δ lk + ( ) 1+δjm 2 l j λ l ϑljm 1+δ lm 2 2 and λ j, δ jm, µ jlkm and ϑ jlm can be calculated numerically. J. Hoydis, S. ten Brink, M. Debbah, Comparison of linear precoding schemes for downlink Massive MIMO, ICC 12, Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 22 / 30

64 Downlink: Numerical results 3 2 Base stations User terminals 1 2 UT k djlk cell l 0 cell j cells, K = 10 UTs distributed on a circle of radius 3/4 Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 23 / 30

65 Downlink: Numerical results 3 2 Base stations User terminals 1 2 UT k djlk cell l 0 cell j cells, K = 10 UTs distributed on a circle of radius 3/4 path loss exponent β = 3.7, ρ τ = 6 db, ρ = 10 db Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 23 / 30

66 Downlink: Numerical results 3 2 Base stations User terminals 1 2 UT k djlk cell l 0 cell j cells, K = 10 UTs distributed on a circle of radius 3/4 path loss exponent β = 3.7, ρ τ = 6 db, ρ = 10 db Two channel models: No correlation R jlk = d β/2 jlk [A 0 N N P ], where A = [a(φ 1 ) a(φ P )] C N P with a(φ p) = 1 [ 1, e i2πc sin(φ),..., e i2πc(n 1) sin(φ)] T P Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 23 / 30

67 Downlink: Numerical results Average rate per UT (bits/s/hz) No Correlation Physical Model Simulations R = bits/s/hz RZF BF Number of antennas N P = N/2, α = 1/ρ, F j = 0 Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 24 / 30

68 Conclusions (II) - Downlink For finite N, RZF is largely superior to BF: A matrix inversion can reduce the number of antennas by one order of magnitude! Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 25 / 30

69 Conclusions (II) - Downlink For finite N, RZF is largely superior to BF: A matrix inversion can reduce the number of antennas by one order of magnitude! Whether or not massive MIMO will show its theoretical gains in practice depends on the validity of our channel models. Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 25 / 30

70 Conclusions (II) - Downlink For finite N, RZF is largely superior to BF: A matrix inversion can reduce the number of antennas by one order of magnitude! Whether or not massive MIMO will show its theoretical gains in practice depends on the validity of our channel models. Reducing signal processing complexity by adding more antennas seems a bad idea. Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 25 / 30

71 Conclusions (II) - Downlink For finite N, RZF is largely superior to BF: A matrix inversion can reduce the number of antennas by one order of magnitude! Whether or not massive MIMO will show its theoretical gains in practice depends on the validity of our channel models. Reducing signal processing complexity by adding more antennas seems a bad idea. Many antennas at the BS require TDD (FDD: overhead scales linearly with N) Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 25 / 30

72 Conclusions (II) - Downlink For finite N, RZF is largely superior to BF: A matrix inversion can reduce the number of antennas by one order of magnitude! Whether or not massive MIMO will show its theoretical gains in practice depends on the validity of our channel models. Reducing signal processing complexity by adding more antennas seems a bad idea. Many antennas at the BS require TDD (FDD: overhead scales linearly with N) Related work: Overview paper: Rusek, et al., Scaling up MIMO: Opportunities and Challenges with Very Large Arrays, IEEE Signal Processing Magazine, to appear. Constant-envelope precoding: S. Mohammed, E. Larsson, Single-User Beamforming in Large-Scale MISO Systems with Per-Antenna Constant-Envelope Constraints: The Doughnut Channel, Network MIMO TDD systems: Huh, Caire, et al., Achieving Massive MIMO Spectral Efficiency with a Not-so-Large Number of Antennas, Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 25 / 30

73 Conclusions (III) - Interesting topics for future work Is large-scale MIMO the smartest option? How else could additional antennas at the BSs be used? (Example: Interference reduction in heterogeneous networks) Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 26 / 30

74 Conclusions (III) - Interesting topics for future work Is large-scale MIMO the smartest option? How else could additional antennas at the BSs be used? (Example: Interference reduction in heterogeneous networks) If we could double the number of antennas in a network, what should we do? More BSs or more antennas per BS? Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 26 / 30

75 Conclusions (III) - Interesting topics for future work Is large-scale MIMO the smartest option? How else could additional antennas at the BSs be used? (Example: Interference reduction in heterogeneous networks) If we could double the number of antennas in a network, what should we do? More BSs or more antennas per BS? How much can be gained through cooperation? Can we compensate for densification by cooperation? Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 26 / 30

76 Conclusions (III) - Interesting topics for future work Is large-scale MIMO the smartest option? How else could additional antennas at the BSs be used? (Example: Interference reduction in heterogeneous networks) If we could double the number of antennas in a network, what should we do? More BSs or more antennas per BS? How much can be gained through cooperation? Can we compensate for densification by cooperation? Optimal placement of N antennas to cover a given area Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 26 / 30

77 Conclusions (III) - Interesting topics for future work Is large-scale MIMO the smartest option? How else could additional antennas at the BSs be used? (Example: Interference reduction in heterogeneous networks) If we could double the number of antennas in a network, what should we do? More BSs or more antennas per BS? How much can be gained through cooperation? Can we compensate for densification by cooperation? Optimal placement of N antennas to cover a given area 3D-beamforming in dense networks Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 26 / 30

78 Conclusions (III) - Interesting topics for future work Is large-scale MIMO the smartest option? How else could additional antennas at the BSs be used? (Example: Interference reduction in heterogeneous networks) If we could double the number of antennas in a network, what should we do? More BSs or more antennas per BS? How much can be gained through cooperation? Can we compensate for densification by cooperation? Optimal placement of N antennas to cover a given area 3D-beamforming in dense networks Massive MIMO becomes a necessity for communications at millimeter waves... Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 26 / 30

79 Conclusions (III) - Interesting topics for future work Is large-scale MIMO the smartest option? How else could additional antennas at the BSs be used? (Example: Interference reduction in heterogeneous networks) If we could double the number of antennas in a network, what should we do? More BSs or more antennas per BS? How much can be gained through cooperation? Can we compensate for densification by cooperation? Optimal placement of N antennas to cover a given area 3D-beamforming in dense networks Massive MIMO becomes a necessity for communications at millimeter waves... Full-duplex radios for cellular communications? Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 26 / 30

80 Related publications T. L. Marzetta Noncooperative cellular wireless with unlimited numbers of base station antennas IEEE Trans. Wireless Commun., vol. 9, no. 11, pp , Nov H. Q. Ngo, E. G. Larsson, T. L. Marzetta Analysis of the pilot contamination effect in very large multicell multiuser MIMO systems for physical channel models Proc. IEEE SPAWC 11, Prague, Czech Repulic, May H. Q. Ngo, E. G. Larsson, T. L. Marzetta Uplink power efficiency of multiuser MIMO with very large antenna arrays Allerton Conference, Urbana-Champaing, Illinois, US, Sep J. Hoydis, S. ten Brink, M. Debbah Massive MIMO: How many antennas do we need? Allerton Conference, Urbana-Champaing, Illinois, US, Sep. 2011, [Online] J. Hoydis, S. ten Brink, M. Debbah Comparison of linear precoding schemes for downlink Massive MIMO ICC 12, submitted, Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 29 / 30

81 Thank you! Jakob Hoydis (Supélec) Massive MIMO: How many antennas do we need? 30 / 30