Lecture 11 Multiuser MIMO Capacity • General model • SIMO uplink: 10.1 • MIMO uplink: 10.2 Mikael Skoglund, Theoretical Foundations of Wireless 1/21 Multiuser MIMO • Generic transmission model, K ul � y k [ n ] = H ℓ [ n ] x ℓ [ n ] + w k [ n ] ℓ =1 • H ℓ [ n ] ∈ C N × M • x ℓ [ n ] ∈ C M , ℓ = 1 , . . . , K ul • y k [ n ] ∈ C N , k = 1 , . . . , K dl • w k [ n ] ∈ C N , i.i.d zero-mean Gaussian, E | w k [ n ] | 2 = σ 2 Mikael Skoglund, Theoretical Foundations of Wireless 2/21
• K -user SIMO uplink, • N = N r receive antennas, M = 1 transmit antenna, K ul = K , K dl = 1 • K -user MIMO uplink, • N = N r receive antennas, M = N t transmit antennas, K ul = K , K dl = 1 • K -user SIMO downlink, • N = N r receive antennas, M = 1 transmit antenna, K ul = 1 , K dl = K • K -user MIMO downlink, • N = N r receive antennas, M = N t transmit antennas, K ul = 1 , K dl = K (equal number of t and r antennas for different users) Mikael Skoglund, Theoretical Foundations of Wireless 3/21 K -user SIMO Uplink K � y [ n ] = h ℓ [ n ] x ℓ [ n ] + w [ n ] ℓ =1 • x ℓ [ n ] ∈ C , ℓ = 1 , . . . , K ; y [ n ] ∈ C N r ; w [ n ] ∈ C N r ; h ℓ ∈ C N r • Powers: P k , k = 1 , . . . , K • Rates: R k , k = 1 , . . . , K • Block-length: N c Mikael Skoglund, Theoretical Foundations of Wireless 4/21
Capacities , K = 2 • Fixed and deterministic ; h ℓ [ n ] = h ℓ , n = 1 , 2 , . . . , N c • Achievable rates, 1 + � h 1 � 2 P 1 � � R 1 < log σ 2 1 + � h 2 � 2 P 2 � � R 2 < log σ 2 � I + 1 � σ 2 ( P 1 h 1 h ∗ 1 + P 2 h 2 h ∗ R 1 + R 2 < log det 2 ) • Extension to K > 2 similar as in the SISO case • The sum-rate constraint ↔ K = 1 , N t = 2 MIMO with independent coding on the transmit antennas • SDMA multiplexing, for high SNR the maximum sum-rate is ≈ K log SNR Mikael Skoglund, Theoretical Foundations of Wireless 5/21 • Orthogonal MA (dotted line) is strictly suboptimal! Fig. 10.3 in the textbook Mikael Skoglund, Theoretical Foundations of Wireless 6/21
• To achieve the point ’ A ’ use MMSE + interference cancellation, • The point ’ A ’ 1 + � h 2 � 2 P 2 � � R 2 = log σ 2 1 + P 1 h ∗ 1 ( σ 2 I + P 2 h 2 h ∗ 2 ) − 1 h 1 � � R 1 = log (etc.) Fig. 10.4 in the textbook Mikael Skoglund, Theoretical Foundations of Wireless 7/21 • Slow fading, perfect CSIR, no CSIT ; h ℓ [ n ] = h ℓ , n = 1 , . . . , N c • ( R 1 , R 2 ) ∈ C ( h 1 , h 2 ) iff 1 + � h 1 � 2 P 1 � � R 1 ≤ log σ 2 1 + � h 2 � 2 P 2 � � R 2 ≤ log σ 2 � � I + 1 σ 2 ( P 1 h 1 h ∗ 1 + P 2 h 2 h ∗ R 1 + R 2 ≤ log det 2 ) • Outage probability , p ul-SIMO � � = Pr ( R 1 , R 2 ) / ∈ C ( h 1 , h 2 ) out Mikael Skoglund, Theoretical Foundations of Wireless 8/21
• Outage probability, symmetric case ( R 1 = R 2 = R ) , p ul-SIMO � � ( R ) = Pr ( R, R ) / ∈ C ( h 1 , h 2 ) out � � � � I + 1 σ 2 ( P 1 h 1 h ∗ 1 + P 2 h 2 h ∗ = Pr log det 2 ) < 2 R • ε - outage symmetric capacity , � � C sym p ul-SIMO = arg sup Pr ( R ) < ε ε out R Mikael Skoglund, Theoretical Foundations of Wireless 9/21 • Diversity–multiplexing tradeoff , symmetric case, K users • N r ≥ K assumed (see textbook for N r < K ) • Upper curve: optimal (ML) receiver • Lower curve: decorrelation receiver Fig. 10.5 in the textbook Mikael Skoglund, Theoretical Foundations of Wireless 10/21
• Fast fading, perfect CSIR, no CSIT ; { h ℓ [ n ] } i.i.d (stationary and ergodic) in n • K = 2 , achievable rates 1 + � h 1 � 2 P 1 � � R 1 < E log σ 2 1 + � h 2 � 2 P 2 � � R 2 < E log σ 2 � � I + 1 σ 2 ( P 1 h 1 h ∗ 1 + P 2 h 2 h ∗ R 1 + R 2 < E log det 2 ) with expectation over the stationary distribution Mikael Skoglund, Theoretical Foundations of Wireless 11/21 • Fast fading, perfect CSIR, perfect CSIT • P k [ n ] = π k ( h 1 [ n ] , h 2 [ n ]) = power allocated to user k • C ( π 1 , π 2 ) = ( R 1 , R 2 ) ’s satisfying 1 + � h 1 � 2 π k ( h 1 , h 2 ) � � R 1 < E log σ 2 1 + � h 2 � 2 π k ( h 1 , h 2 ) � � R 2 < E log σ 2 � I + 1 � σ 2 ( π k ( h 1 , h 2 ) h 1 h ∗ 1 + π k ( h 1 , h 2 ) h 2 h ∗ R 1 + R 2 < E log det 2 ) (over the stationary distribution) Mikael Skoglund, Theoretical Foundations of Wireless 12/21
• The capacity region C is the convex hull of the set � C ( π 1 , π 2 ) π 1 ,π 2 over all ( π 1 , π 2 ) such that E [ π i ( h 1 , h 2 )] ≤ P i , i = 1 , 2 Mikael Skoglund, Theoretical Foundations of Wireless 13/21 • Multiuser Diversity • Consider the sum capacity ( K users), � C sum = sup { R = R k : ( R 1 , . . . , R K ) achievable } k • In the SISO case: only one user transmits optimal, no natural generalization to SIMO • Force one user at a time ⇒ the sum rate π k ∗ ( h k ∗ ) � h k ∗ � 2 � � 1 + ¯ E log ≤ C sum σ 2 (assuming i.i.d h k ’s) is achievable, where ¯ π is the “waterfilling over time” policy and k ∗ = arg max k {� h k � 2 } • Multiuser diversity gain lower than in the SISO case, since the tail probability for � h k � 2 decreases with N r Mikael Skoglund, Theoretical Foundations of Wireless 14/21
• Optimal power allocation • No simple and general conclusions can be drawn • In the case N r ≈ K → ∞ , the policy that all users transmit with waterfilling over their own states is close to optimal; � 1 � + I 0 π k ( h 1 , . . . , h K ) = λ − � h k � 2 where � h ℓ [ n ] x ℓ [ n ] + w [ n ] � 2 I 0 = E � ℓ � = k is the noise plus interference power • Reduced gain compared to the SISO case. . . Mikael Skoglund, Theoretical Foundations of Wireless 15/21 K -user MIMO Uplink K � y [ n ] = H ℓ [ n ] x ℓ [ n ] + w [ n ] ℓ =1 Mikael Skoglund, Theoretical Foundations of Wireless 16/21
Capacities , K = 2 • Transmit, Fig. 10.11 in the textbook Mikael Skoglund, Theoretical Foundations of Wireless 17/21 • Receive, Fig. 10.11 in the textbook Mikael Skoglund, Theoretical Foundations of Wireless 18/21
• Fixed and deterministic ; H ℓ [ n ] = H ℓ , n = 1 , 2 , . . . , N c • Let N m = min( N t , N r ) (assuming equal number of t antennas for different users) • Let L k = diag ( p 1 k , . . . , p N t k ) with, if N m < N t , p nk = 0 for n > N m , and subject to � n p nk = P k • Let U 1 and U 2 be unitary • Achievable rates, � I + 1 � σ 2 H k K k H ∗ R k < log det k � � I + 1 σ 2 ( H 1 K 1 H ∗ 1 + H 2 K 2 H ∗ R 1 + R 2 < log det 2 ) for any K k = U k L k U ∗ k • Capacity region = closure of the convex hull over all power allocations and rotations Mikael Skoglund, Theoretical Foundations of Wireless 19/21 • Fast fading, perfect CSIR, no CSIT ; { H ℓ [ n ] } i.i.d (stationary and ergodic) in n • Achievable rates, � � I + 1 σ 2 H k K k H ∗ R k < E log det k � � I + 1 σ 2 ( H 1 K 1 H ∗ 1 + H 2 K 2 H ∗ R 1 + R 2 < E log det 2 ) for any K k = U k L k U ∗ k • Capacity region = closure of the convex hull over all power allocations and rotations • Special case: uncorrelated Rayleigh fading, K k = P k N t I N t is optimal ⇒ the capacity region is a pentagon Mikael Skoglund, Theoretical Foundations of Wireless 20/21
• Fast fading, perfect CSIR, perfect CSIT ; { H ℓ [ n ] } i.i.d (stationary and ergodic) in n , • similar conclusions as for SIMO. . . Mikael Skoglund, Theoretical Foundations of Wireless 21/21
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