Cells Operate Independently, Each Serving Single- Antenna Terminals via Multi-User MIMO: TDD Only! Maximum number of terminals limited by the time that it takes to send reverse pilots: pilot-interval divided by the channel delay- spread Coherence interval: 500 sec (7 LTE OFDM symbols) – TGV speeds! 3 symbols for reverse-link pilots 3 symbols for data 1 symbol for computations and dead time 42 terminals per cell served simultaneously
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 .95-Likely Mean Capacity Mean Capacity (dB) Capacity per per Terminal per Cell (Mbits/s) Terminal (Mbits/s) (Mbits/s) 1 -29 .016 44 1800 3 -5.8 .89 28 1200 7 8.9 3.6 17 730 Mean Capacity per Cell (Mbits/s) LTE Advanced 74 (>= Release 10)
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 .95-Likely Mean Capacity Mean Capacity (dB) Capacity per per Terminal per Cell (Mbits/s) Terminal (Mbits/s) (Mbits/s) 1 -29 .016 44 1800 3 -5.8 .89 28 1200 7 8.9 3.6 17 730 Mean Capacity per Cell (Mbits/s) LTE Advanced 74 (>= Release 10)
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 (Sup´ elec) Massive MIMO: How many antennas do we need? 4 / 30
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. By the strong law of large numbers: 1 a.s. H y − − − − − − − − − − → N h m N →∞ , K =const. x m Jakob Hoydis (Sup´ elec) Massive MIMO: How many antennas do we need? 4 / 30
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. By the strong law of large numbers: 1 a.s. H y − − − − − − − − − − → N h m N →∞ , K =const. x m 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. 35903600, Nov. 2010. Jakob Hoydis (Sup´ elec) Massive MIMO: How many antennas do we need? 4 / 30
About some fundamental assumptions The receiver has perfect channel state information (CSI). What happens if the channel must be estimated? Jakob Hoydis (Sup´ elec) Massive MIMO: How many antennas do we need? 5 / 30
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 (Sup´ elec) Massive MIMO: How many antennas do we need? 5 / 30
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 (Sup´ elec) Massive MIMO: How many antennas do we need? 5 / 30
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 (Sup´ elec) Massive MIMO: How many antennas do we need? 5 / 30
On channel estimation and pilot contamination The receiver estimates the channels based on pilot sequences. 1 The number of orthogonal sequences is limited by the coherence time. 2 Thus, the pilot sequences must be reused. 3 Jakob Hoydis (Sup´ elec) Massive MIMO: How many antennas do we need? 6 / 30
On channel estimation and pilot contamination The receiver estimates the channels based on pilot sequences. 1 The number of orthogonal sequences is limited by the coherence time. 2 Thus, the pilot sequences must be reused. 3 Assume that transmitter m and j use the same pilot sequence: ˆ h m = h m + h j + n m ���� ���� estimation noise pilot contamination Jakob Hoydis (Sup´ elec) Massive MIMO: How many antennas do we need? 6 / 30
On channel estimation and pilot contamination The receiver estimates the channels based on pilot sequences. 1 The number of orthogonal sequences is limited by the coherence time. 2 Thus, the pilot sequences must be reused. 3 Assume that transmitter m and j use the same pilot sequence: ˆ h m = h m + h j + n m ���� ���� estimation noise pilot contamination Thus, 1 ˆ H y a.s − N →∞ , K =const. x m + x j − − − − − − − − → h m N Jakob Hoydis (Sup´ elec) Massive MIMO: How many antennas do we need? 6 / 30
On channel estimation and pilot contamination The receiver estimates the channels based on pilot sequences. 1 The number of orthogonal sequences is limited by the coherence time. 2 Thus, the pilot sequences must be reused. 3 Assume that transmitter m and j use the same pilot sequence: ˆ h m = h m + h j + n m ���� ���� estimation noise pilot contamination Thus, 1 ˆ H y a.s − N →∞ , K =const. x m + x j − − − − − − − − → h m N 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. 35903600, Nov. 2010. Jakob Hoydis (Sup´ elec) Massive MIMO: How many antennas do we need? 6 / 30
Uplink Jakob Hoydis (Sup´ elec) Massive MIMO: How many antennas do we need? 7 / 30
System model and channel estimation Uplink: L BSs with N antennas, K UTs per cell. Received signal at BS j : L y j = √ ρ � H jl x l + n j l =1 Jakob Hoydis (Sup´ elec) Massive MIMO: How many antennas do we need? 8 / 30
System model and channel estimation Uplink: L BSs with N antennas, K UTs per cell. Received signal at BS j : L y j = √ ρ � H jl x l + n j l =1 The columns of H jl ( N × K ) are modeled as 1 h jlk = R jlk w jlk , 2 w jlk ∼ CN ( 0 , I N ) Jakob Hoydis (Sup´ elec) Massive MIMO: How many antennas do we need? 8 / 30
System model and channel estimation Uplink: L BSs with N antennas, K UTs per cell. Received signal at BS j : L y j = √ ρ � H jl x l + n j l =1 The columns of H jl ( N × K ) are modeled as 1 h jlk = R jlk w jlk , 2 w jlk ∼ CN ( 0 , I N ) Channel estimation: 1 � y τ jk = h jjk + h jlk + √ ρ τ n jk l � = j Jakob Hoydis (Sup´ elec) Massive MIMO: How many antennas do we need? 8 / 30
System model and channel estimation Uplink: L BSs with N antennas, K UTs per cell. Received signal at BS j : L y j = √ ρ � H jl x l + n j l =1 The columns of H jl ( N × K ) are modeled as 1 h jlk = R jlk w jlk , 2 w jlk ∼ CN ( 0 , I N ) Channel estimation: 1 � y τ jk = h jjk + h jlk + √ ρ τ n jk l � = j h jjk = ˆ h jjk + ˜ MMSE estimate: h jjk Jakob Hoydis (Sup´ elec) Massive MIMO: How many antennas do we need? 8 / 30
System model and channel estimation Uplink: L BSs with N antennas, K UTs per cell. Received signal at BS j : L y j = √ ρ � H jl x l + n j l =1 The columns of H jl ( N × K ) are modeled as 1 h jlk = R jlk w jlk , 2 w jlk ∼ CN ( 0 , I N ) Channel estimation: 1 � y τ jk = h jjk + h jlk + √ ρ τ n jk l � = j h jjk = ˆ h jjk + ˜ MMSE estimate: h jjk ˆ ˜ h jjk ∼ CN ( 0 , Φ jjk ) , h jjk ∼ CN ( 0 , R jjk − Φ jjk ) � � − 1 1 � Φ jlk = R jjk Q jk R jlk , Q jk = ρ τ I N + R jlk l Jakob Hoydis (Sup´ elec) Massive MIMO: How many antennas do we need? 8 / 30
Achievable rates with linear detectors Ergodic achievable rate of UT m in cell j : R jm = E ˆ H jj [log 2 (1 + γ jm )] � � 2 jm ˆ � � r H � h jjm � γ jm = � � � � � jjm + � ρ I N + ˜ h jjm ˜ � ˆ 1 � r H h H jjm − h jjm h H l H jl H H E r jm H jj jm jl with an arbitrary receive filter r jm . 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. 951–963, Nov. 2003. Jakob Hoydis (Sup´ elec) Massive MIMO: How many antennas do we need? 9 / 30
Achievable rates with linear detectors Ergodic achievable rate of UT m in cell j : R jm = E ˆ H jj [log 2 (1 + γ jm )] � � 2 jm ˆ � � r H � h jjm � γ jm = � � � � � jjm + � ρ I N + ˜ h jjm ˜ � ˆ 1 � r H h H jjm − h jjm h H l H jl H H E r jm H jj jm jl with an arbitrary receive filter r jm . Two specific linear detectors r jm : r MF jm = ˆ h jjm � − 1 ˆ � r MMSE H jj ˆ ˆ H H = jj + Z j + N λ I N h jjm jm where λ > 0 is a design parameter and � � � � ˜ H jj ˜ H H = ( R jjk − Φ jjk ) + Z j = E jj + H jl H jl R jlk . l � = j l � = j k 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. 951–963, Nov. 2003. Jakob Hoydis (Sup´ elec) Massive MIMO: How many antennas do we need? 9 / 30
Large system analysis based on random matrix theory Assume N , K → ∞ at the same speed. Then, a.s. γ jm − ¯ γ jm − − → 0 a.s. R jm − log 2 (1 + ¯ γ jm ) − − → 0 Jakob Hoydis (Sup´ elec) Massive MIMO: How many antennas do we need? 10 / 30
Large system analysis based on random matrix theory Assume N , K → ∞ at the same speed. Then, a.s. γ jm − ¯ γ jm − − → 0 a.s. R jm − log 2 (1 + ¯ γ jm ) − − → 0 where � 1 � 2 N tr Φ jjm γ MF ¯ jm = � � � N tr R jlk Φ jjm + � � 2 ρ N 2 tr Φ jjm + 1 1 1 � 1 N tr Φ jlm N l , k l � = j δ 2 γ MMSE jm ¯ = jm � l , k µ jlkm + � ρ N 2 tr Φ jjm ¯ 1 j + 1 l � = j | ϑ jlm | 2 T ′ N and δ jm , µ jlkm , θ jlm , ¯ T ′ j can be calculated numerically. Jakob Hoydis (Sup´ elec) Massive MIMO: How many antennas do we need? 10 / 30
A simple multi-cell scenario Jakob Hoydis (Sup´ elec) Massive MIMO: How many antennas do we need? 11 / 30
A simple multi-cell scenario intercell interference factor α ∈ [0 , 1] transmit power per UT: ρ � H jl = [ h jl 1 · · · h jlK ] = N / P AW jl N × P composed of P ≤ N columns of a unitary matrix A ∈ C P × K have i.i.d. elements with zero mean and unit variance W ij ∈ C Jakob Hoydis (Sup´ elec) Massive MIMO: How many antennas do we need? 11 / 30
A simple multi-cell scenario intercell interference factor α ∈ [0 , 1] transmit power per UT: ρ � H jl = [ h jl 1 · · · h jlK ] = N / P AW jl N × P composed of P ≤ N columns of a unitary matrix A ∈ C P × K have i.i.d. elements with zero mean and unit variance W ij ∈ C Assumptions: P channel degrees of freedom, i.e., rank ( H jl ) = min( P , K ) [Ngo’11] � � tr H jl H H energy scales linearly with N , i.e., E = KN jl only pilot contamination, i.e., no estimation noise: h jjk = h jjk + √ α � ˆ h jlk l � = j Jakob Hoydis (Sup´ elec) Massive MIMO: How many antennas do we need? 11 / 30
Asymptotic performance of the matched filter Assume that N , K and P grow infinitely large at the same speed: 1 SINR MF ≈ ¯ L K L 2 ¯ α (¯ L − 1) + + ρ N P � �� � ���� ���� pilot contamination multi-user interference noise where ¯ L = 1 + α ( L − 1). Jakob Hoydis (Sup´ elec) Massive MIMO: How many antennas do we need? 12 / 30
Asymptotic performance of the matched filter Assume that N , K and P grow infinitely large at the same speed: 1 SINR MF ≈ ¯ L K L 2 ¯ α (¯ L − 1) + + ρ N P � �� � ���� ���� pilot contamination multi-user interference noise where ¯ L = 1 + α ( L − 1). Observations: The effective SNR ρ N increases linearly with N . The multiuser interference depends on P / K and not on N . Ultimate performance limit: 1 N , P →∞ , K =const. SINR ∞ = a.s SINR MF − − − − − − − − − − − → 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. 2011. [Online] http://arxiv.org/abs/1107.1709 Jakob Hoydis (Sup´ elec) Massive MIMO: How many antennas do we need? 12 / 30
Asymptotic performance of the MMSE detector Assume that N , K and P grow infinitely large at the same speed: 1 SINR MMSE ≈ ¯ L K L 2 Y ¯ α (¯ + + L − 1) ρ N X P � �� � � �� � � �� � pilot contamination multi-user interference noise where ¯ L = 1 + α ( L − 1) and X , Y are given in closed-form. Jakob Hoydis (Sup´ elec) Massive MIMO: How many antennas do we need? 13 / 30
Asymptotic performance of the MMSE detector Assume that N , K and P grow infinitely large at the same speed: 1 SINR MMSE ≈ ¯ L K ¯ L 2 Y α (¯ + + L − 1) ρ N X P � �� � � �� � � �� � pilot contamination multi-user interference noise where ¯ L = 1 + α ( L − 1) and X , Y are given in closed-form. Observations: As for the MF, the performance depends only on ρ N and P / K . The ultimate performance of MMSE and MF coincide: 1 N , P →∞ , K =const. SINR ∞ = a.s SINR MMSE − − − − − − − − − − − → 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. 2011. [Online] http://arxiv.org/abs/1107.1709 Jakob Hoydis (Sup´ elec) Massive MIMO: How many antennas do we need? 13 / 30
Numerical results 5 P = N Ergodic achievable rate (b/s/Hz) R ∞ 4 3 2 P = N/ 3 1 MF approx. MMSE approx. Simulations ρ = 1 , K = 10 , α = 0 . 1 , L = 4 0 0 100 200 300 400 Number of antennas N Jakob Hoydis (Sup´ elec) Massive MIMO: How many antennas do we need? 14 / 30
Conclusions (I) - Uplink Massive MIMO can be seen as a particular operating condition where noise + interference ≪ pilot contamination . Jakob Hoydis (Sup´ elec) Massive MIMO: How many antennas do we need? 18 / 30
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 ρ N : effective SNR (transmit power × number of antennas) α : path loss (or intercell interference) Jakob Hoydis (Sup´ elec) Massive MIMO: How many antennas do we need? 18 / 30
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 ρ N : 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 (Sup´ elec) Massive MIMO: How many antennas do we need? 18 / 30
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 ρ N : 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 (Sup´ elec) Massive MIMO: How many antennas do we need? 18 / 30
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 ρ N : 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 (Sup´ elec) Massive MIMO: How many antennas do we need? 18 / 30
Downlink Jakob Hoydis (Sup´ elec) Massive MIMO: How many antennas do we need? 19 / 30
System model: Downlink L BSs with N antennas, K UTs per cell. Received signal at m th UT in cell j : L y jm = √ ρ � h H ljm s l + q jm l =1 Jakob Hoydis (Sup´ elec) Massive MIMO: How many antennas do we need? 20 / 30
System model: Downlink L BSs with N antennas, K UTs per cell. Received signal at m th UT in cell j : L y jm = √ ρ � h H ljm s l + q jm l =1 where K � � � s l = λ l w lm x lm = λ l W l x l m =1 1 � � ρ s H λ l = = ⇒ E l s l = ρ tr W l W H l Jakob Hoydis (Sup´ elec) Massive MIMO: How many antennas do we need? 20 / 30
System model: Downlink L BSs with N antennas, K UTs per cell. Received signal at m th UT in cell j : L y jm = √ ρ � h H ljm s l + q jm l =1 where K � � � s l = λ l w lm x lm = λ l W l x l m =1 1 � � ρ s H λ l = = ⇒ E l s l = ρ tr W l W H l Channel estimation through uplink pilots (as before): h jjk = ˆ h jjk + ˜ h jjk ˆ ˜ h jjk ∼ CN ( 0 , Φ jjk ) , h jjk ∼ CN ( 0 , R jjk − Φ jjk ) � � − 1 1 � Φ jlk = R jjk Q jk R jlk , Q jk = ρ τ I N + R jlk l Jakob Hoydis (Sup´ elec) Massive MIMO: How many antennas do we need? 20 / 30
Achievable rates with linear precoders Ergodic achievable rate of UT m in cell j : R jm = log 2 (1 + γ jm ) � �� �� � 2 λ j h H � E jjm w jm γ jm = 2 � . �� √ λ l h H � �� � + � 1 λ j h H � � ρ + var jjm w jm ( l , k ) � =( j , m ) E ljm w lk � � 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, 2011. Jakob Hoydis (Sup´ elec) Massive MIMO: How many antennas do we need? 21 / 30
Achievable rates with linear precoders Ergodic achievable rate of UT m in cell j : R jm = log 2 (1 + γ jm ) � �� �� � 2 λ j h H � E jjm w jm γ jm = 2 � . �� √ λ l h H � �� � + � 1 λ j h H � � ρ + var jjm w jm ( l , k ) � =( j , m ) E ljm w lk � � Two specific precoders W j : △ = ˆ W BF H jj j � − 1 ˆ � H jj ˆ ˆ W RZF △ H H = jj + F j + N α I N H jj j where α > 0 and F j are design parameters. 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, 2011. Jakob Hoydis (Sup´ elec) Massive MIMO: How many antennas do we need? 21 / 30
Large system analysis based on random matrix theory Assume N , K → ∞ at the same speed. Then, a.s. γ jm − ¯ − − → 0 γ jm a.s. R jm − log 2 (1 + ¯ − − → 0 γ jm ) J. Hoydis, S. ten Brink, M. Debbah, “Comparison of linear precoding schemes for downlink Massive MIMO”, ICC’12, 2011. Jakob Hoydis (Sup´ elec) Massive MIMO: How many antennas do we need? 22 / 30
Large system analysis based on random matrix theory Assume N , K → ∞ at the same speed. Then, a.s. γ jm − ¯ − − → 0 γ jm a.s. R jm − log 2 (1 + ¯ − − → 0 γ jm ) where � 1 � 2 ¯ λ j N tr Φ jjm γ BF ¯ jm = � � � N tr R ljm Φ llk + � � 2 l , k ¯ l � = j ¯ N ρ + 1 K λ l 1 � 1 λ j N tr Φ ljm N ¯ λ j δ 2 γ RZF jm ¯ = jm � � 2 � � 2 N ρ (1 + δ jm ) 2 + 1 � l , k ¯ 1+ δ jm µ ljmk + � l � = j ¯ 1+ δ jm | ϑ ljm | 2 K λ l λ l N 1+ δ lk 1+ δ lm 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, 2011. Jakob Hoydis (Sup´ elec) Massive MIMO: How many antennas do we need? 22 / 30
Downlink: Numerical results 3 Base stations User terminals 2 UT k 1 d jlk 2 cell l 0 cell j −1 3 4 −2 −3 −3 −2 −1 0 1 2 3 7 cells, K = 10 UTs distributed on a circle of radius 3 / 4 Jakob Hoydis (Sup´ elec) Massive MIMO: How many antennas do we need? 23 / 30
Downlink: Numerical results 3 Base stations User terminals 2 UT k 1 d jlk 2 cell l 0 cell j −1 3 4 −2 −3 −3 −2 −1 0 1 2 3 7 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 (Sup´ elec) Massive MIMO: How many antennas do we need? 23 / 30
Downlink: Numerical results 3 Base stations User terminals 2 UT k 1 d jlk 2 cell l 0 cell j −1 3 4 −2 −3 −3 −2 −1 0 1 2 3 7 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 N × P with R jlk = d − β/ 2 ◮ ˜ [ A 0 N × N − P ], where A = [ a ( φ 1 ) · · · a ( φ P )] ∈ C jlk 1 1 , e − i 2 π c sin( φ ) , . . . , e − i 2 π c ( N − 1) sin( φ ) � T � a ( φ p ) = √ P Jakob Hoydis (Sup´ elec) Massive MIMO: How many antennas do we need? 23 / 30
Downlink: Numerical results 12 R ∞ = 15 . 75 bits/s/Hz No Correlation Physical Model Average rate per UT (bits/s/Hz) 10 Simulations 8 RZF 6 4 2 BF 0 0 100 200 300 400 Number of antennas N P = N / 2 , α = 1 /ρ, F j = 0 Jakob Hoydis (Sup´ elec) Massive MIMO: How many antennas do we need? 24 / 30
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 (Sup´ elec) Massive MIMO: How many antennas do we need? 25 / 30
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 (Sup´ elec) Massive MIMO: How many antennas do we need? 25 / 30
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 (Sup´ elec) Massive MIMO: How many antennas do we need? 25 / 30
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 (Sup´ elec) Massive MIMO: How many antennas do we need? 25 / 30
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. http://liu.diva-portal.org/smash/record.jsf?pid=diva2:450781 Constant-envelope precoding: S. Mohammed, E. Larsson, “Single-User Beamforming in Large-Scale MISO Systems with Per-Antenna Constant-Envelope Constraints: The Doughnut Channel”, http://arxiv.org/abs/1111.3752v1 Network MIMO TDD systems: Huh, Caire, et al., “Achieving “Massive MIMO” Spectral Efficiency with a Not-so-Large Number of Antennas”, http://arxiv.org/abs/1107.3862 Jakob Hoydis (Sup´ elec) Massive MIMO: How many antennas do we need? 25 / 30
Related publications ◮ T. L. Marzetta Noncooperative cellular wireless with unlimited numbers of base station antennas IEEE Trans. Wireless Commun., vol. 9, no. 11, pp. 3590–3600 , Nov. 2010. ◮ 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 2011. ◮ 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. 2011. ◮ J. Hoydis, S. ten Brink, M. Debbah Massive MIMO: How many antennas do we need? Allerton Conference, Urbana-Champaing, Illinois, US , Sep. 2011, [Online] http://arxiv.org/abs/1107.1709. ◮ J. Hoydis, S. ten Brink, M. Debbah Comparison of linear precoding schemes for downlink Massive MIMO ICC’12, submitted , 2010. Jakob Hoydis (Sup´ elec) Massive MIMO: How many antennas do we need? 29 / 30
A two-tier network architecture Massive MIMO base stations (BS) overlaid with many small cells (SCs) BSs ensure coverage and serve highly mobile UEs SCs drive the capacity (hot spots, indoor coverage) Jakob Hoydis (Bell Labs) Massive MIMO, Small Cells, and TDD CTW 2013 3 / 23
A two-tier network architecture Massive MIMO base stations (BS) overlaid with many small cells (SCs) BSs ensure coverage and serve highly mobile UEs SCs drive the capacity (hot spots, indoor coverage) Intra- and inter-tier interference is the main performance bottleneck. Jakob Hoydis (Bell Labs) Massive MIMO, Small Cells, and TDD CTW 2013 3 / 23
A two-tier network architecture Massive MIMO base stations (BS) overlaid with many small cells (SCs) BSs ensure coverage and serve highly mobile UEs SCs drive the capacity (hot spots, indoor coverage) Intra- and inter-tier interference is the main performance bottleneck. There are many excess antennas in the network which should be exploited! Jakob Hoydis (Bell Labs) Massive MIMO, Small Cells, and TDD CTW 2013 3 / 23
The essential role of TDD A network-wide synchronized TDD protocol and the resulting channel reciprocity have the following advantages: The downlink channels can be estimated from uplink pilots. → Necessary for massive MIMO Channel reciprocity holds for the desired and the interfering channels. → Knowledge about the interfering channels can be acquired for free. TDD enables the use of excess antennas to reduce intra-/inter-tier interference. Jakob Hoydis (Bell Labs) Massive MIMO, Small Cells, and TDD CTW 2013 4 / 23
An idea from cognitive radio The secondary BS listens to the transmission from the primary UE: 1 y = h x + n ...and computes the covariance matrix of the received signal: 2 = hh H + SNR − 1 I � yy H � E Jakob Hoydis (Bell Labs) Massive MIMO, Small Cells, and TDD CTW 2013 5 / 23
An idea from cognitive radio With the knowledge of the SNR, the secondary BS designs a precoder w which is 3 orthogonal to the sub-space spanned by hh H . Jakob Hoydis (Bell Labs) Massive MIMO, Small Cells, and TDD CTW 2013 6 / 23
An idea from cognitive radio With the knowledge of the SNR, the secondary BS designs a precoder w which is 3 orthogonal to the sub-space spanned by hh H . The interference to the primary UE can be entirely eliminated without explicit 4 knowledge of h . Jakob Hoydis (Bell Labs) Massive MIMO, Small Cells, and TDD CTW 2013 6 / 23
Translating this idea to HetNets Every device estimates its received interference covariance matrix and precodes (partially) orthogonally to the dominating interference subspace. Jakob Hoydis (Bell Labs) Massive MIMO, Small Cells, and TDD CTW 2013 7 / 23
Translating this idea to HetNets Every device estimates its received interference covariance matrix and precodes (partially) orthogonally to the dominating interference subspace. Advantages Reduces interference towards the directions from which most interference is received. No feedback or data exchange between the devices is needed. Every device relies only on locally available information. The scheme is fully distributed and, thus, scalable. Jakob Hoydis (Bell Labs) Massive MIMO, Small Cells, and TDD CTW 2013 7 / 23
About the literature Cognitive radio ◮ R. Zhang, F. Gao, and Y. C. Liang, “Cognitive Beamforming Made Practical: Effective Interference Channel and Learning-Throughput Tradeoff,” IEEE Trans. Commun., 2010. ◮ F. Gao, R. Zhang, Y.-C. Liang, X. Wang, “Design of Learning-Based MIMO Cognitive Radio Systems,” IEEE Trans. Veh. Tech., 2010. ◮ H. Yi, “Nullspace-Based Secondary Joint Transceiver Scheme for Cognitive Radio MIMO Networks Using Second-Order Statistics,” ICC, 2010. TDD Cellular systems ◮ S. Lei and S. Roy, “Downlink multicell MIMO-OFDM: an architecture for next generation wireless networks,” WCNC, 2005. ◮ B. O. Lee, H. W. Je, I. Sohn, O. S. Shin, and K. B. Lee, “Interference-aware Decentralized Precoding for Multicell MIMO TDD Systems,” Globecom. 2008. Blind nullspace learning ◮ Y. Noam and A. J. Goldsmith, “Exploiting spatial degrees of freedom in MIMO cognitive radio systems,” ICC, 2012. and many more... Jakob Hoydis (Bell Labs) Massive MIMO, Small Cells, and TDD CTW 2013 8 / 23
System model and signaling Each BS has N antennas and serves K single-antenna MUEs. S SCs per BS with F antennas serving 1 single-antenna SUE each The BSs and SCs have perfect CSI for the UEs they want to serve. Every device knows perfectly its interference covariance matrix and the noise power. Linear MMSE detection at all devices Jakob Hoydis (Bell Labs) Massive MIMO, Small Cells, and TDD CTW 2013 9 / 23
System model and signaling Each BS has N antennas and serves K single-antenna MUEs. S SCs per BS with F antennas serving 1 single-antenna SUE each The BSs and SCs have perfect CSI for the UEs they want to serve. Every device knows perfectly its interference covariance matrix and the noise power. Linear MMSE detection at all devices The BSs and SCs use precoding vectors of the structure: � − 1 � P HH H + κ Q + σ 2 I w ∼ h ◮ h channel vector to the targeted UE ◮ H channel matrix to other UEs in the same cell ◮ P , σ 2 : transmit and noise powers ◮ Q interference covariance matrix ◮ κ : regularization parameter ( α for BSs, β for SCs) About the regularization parameters For α, β = 0, the BSs and SCs transmit as if they were in an isolated cell, i.e., MMSE precoding (BSs) and maximum-ratio transmissions (SCs). By increasing α, β , the precoding vectors become increasingly orthogonal to the interference subspace. Jakob Hoydis (Bell Labs) Massive MIMO, Small Cells, and TDD CTW 2013 9 / 23
Comparison of duplexing schemes and co-channel deployment FDD TDD SC DL SC UL SC DL frequency frequency SC UL BS DL BS UL BS DL BS UL time time co-channel TDD co-channel reverse TDD SC UL SC DL SC DL SC UL frequency frequency BS UL BS DL BS UL BS DL time time FDD: Channel reciprocity does not hold TDD: Only intra-tier interference can be reduced co-channel (reverse) TDD: Inter and intra-tier interference can be reduced Jakob Hoydis (Bell Labs) Massive MIMO, Small Cells, and TDD CTW 2013 10 / 23
TDD versus reverse TDD (RTDD) Order of UL/DL periods decides which devices interfere with each other. The BS-SC channels change very slowly. Thus, the estimation of the covariance matrix becomes easier for RTDD. Jakob Hoydis (Bell Labs) Massive MIMO, Small Cells, and TDD CTW 2013 11 / 23
Numerical results 1000 m 3 × 3 grid of BSs with wrap around S = 81 SCs per cells on a regular grid K = 20 MUEs randomly distributed 40 m 1 SUE per SC randomly distributed on a disc around each SC 3GPP channel model with path loss, 111 m shadowing and fast fading, N/LOS links TX powers: 46 dBm (BS), 24 dBm (SC), 23 dBm (MUE/SUE) 20 MHz bandwidth @ 2 GHz No user scheduling, power control Averages over channel realizations and UE locations TDD UL/DL cycles of equal length SC MUE BS SUE Jakob Hoydis (Bell Labs) Massive MIMO, Small Cells, and TDD CTW 2013 12 / 23
Downlink spectral area efficiency regions FDD ( N = 20 , F = 1) b/s/Hz/km 2 � 400 � SC DL area spectral efficiency 300 200 100 0 0 20 40 60 80 � b/s/Hz/km 2 � Macro DL area spectral efficiency Jakob Hoydis (Bell Labs) Massive MIMO, Small Cells, and TDD CTW 2013 13 / 23
Downlink spectral area efficiency regions FDD ( N = 20 , F = 1) b/s/Hz/km 2 � FDD/TDD ( N = 100 , F = 4) 400 � SC DL area spectral efficiency 300 F = 1 → 4 200 FDD region 100 more antennas N = 20 → 100 0 0 20 40 60 80 � b/s/Hz/km 2 � Macro DL area spectral efficiency Jakob Hoydis (Bell Labs) Massive MIMO, Small Cells, and TDD CTW 2013 13 / 23
Downlink spectral area efficiency regions FDD ( N = 20 , F = 1) b/s/Hz/km 2 � FDD/TDD ( N = 100 , F = 4) 400 TDD ( N = 100 , F = 4 , α = 1 , β = 1) β = 0 → 1 � SC DL area spectral efficiency 300 F = 1 → 4 200 TDD region FDD region 100 more antennas less intra-tier interf. N = 20 → 100 α = 0 → 1 0 0 20 40 60 80 � b/s/Hz/km 2 � Macro DL area spectral efficiency Jakob Hoydis (Bell Labs) Massive MIMO, Small Cells, and TDD CTW 2013 13 / 23
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