Lecture 12 Multiuser MIMO Capacity • MISO downlink: 10.3 • Precoding: 10.3.3–4 • MIMO downlink: 10.4 Mikael Skoglund, Theoretical Foundations of Wireless 1/17 K -user MISO Downlink y k [ n ] = h ∗ k [ n ] x [ n ] + w k [ n ] • Number of transmit antennas: N t • x [ n ] ∈ C N t ; h k [ n ] ∈ C N t , k = 1 , . . . , K • Block-length: N c • Power constraint, N c 1 � x [ n ] � 2 ≤ P � N c n =1 • Rates: R k , k = 1 , . . . , K Mikael Skoglund, Theoretical Foundations of Wireless 2/17
Duality • Uplink–downlink duality , linear processing Fig. 10.16 in the textbook Mikael Skoglund, Theoretical Foundations of Wireless 3/17 • Fixed and deterministic channels, perfect CSIR/T • Downlink, linear superposition K � x [ n ] = x ℓ [ n ] u ℓ ˜ ℓ =1 • Signature vectors u k ∈ C N t , “transmit filters” • Choose { u k } to minimize the total transmit power, subject to a constraint on each user’s SINR. . . • The dual SIMO uplink , with receive filters u k ∈ C N r • The dual uplink problem can be used to solve the downlink beamforming problem. . . • Linear superposition is not optimal for the MISO/MO downlink! • The problem is non-degraded; there is no natural ordering of users Mikael Skoglund, Theoretical Foundations of Wireless 4/17
Precoding for Known Interference • Symbol-by-symbol, s w x y ˆ ω ω β α • ω ∈ { 0 , . . . , M − 1 } , data • s ∈ C , interference • x = α ( ω, s ) ∈ C , E | x | 2 ≤ P , transmitted symbol • w ∈ C , zero-mean Gaussian noise, E | w | 2 = σ 2 • ˆ ω = β ( y ) ∈ { 0 , . . . , M − 1 } Mikael Skoglund, Theoretical Foundations of Wireless 5/17 • Tomlinson–Harashima • PAM alphabet, • Replicated PAM alphabet, • Transmission, Figs. 10.18–20 in the textbook Mikael Skoglund, Theoretical Foundations of Wireless 6/17
• Quantization: Assume ω = i , let q i ( s ) = i -symbol closest to s • Transmit, x = q i ( s ) − s • Receive, y = q i ( s ) + w • Decode to closest i -symbol • “Almost” removes the interference completely, but a gap to no-interference performance Mikael Skoglund, Theoretical Foundations of Wireless 7/17 • Costa or dirty-paper precoding s n w n x n y n ω ˆ ω β α • ω ∈ { 0 , . . . , M − 1 } , data • s n = ( s 1 , . . . , s n ) ∈ C n , interference sequence • x n = ( x 1 , . . . , x n ) = α ( ω, s n ) ∈ C n , n − 1 � m E | x m | 2 ≤ P , transmitted codeword • w ∈ C , zero-mean Gaussian noise, E | w | 2 = σ 2 ω = β ( y n ) ∈ { 0 , . . . , M − 1 } • ˆ Costa, “Writing on dirty paper,” IEEE Trans. IT , 1983 Mikael Skoglund, Theoretical Foundations of Wireless 8/17
• All rates � � R = log M 1 + P < log σ 2 n can be achieved ⇒ no loss compared to the case of no interference • The textbook discusses a generalization of Tomlinson–Harashima based on, • nested lattices • MMSE estimation + quantization Mikael Skoglund, Theoretical Foundations of Wireless 9/17 Costa Precoding Achieves Capacity for the MISO/MO Downlink 1 • Fixed deterministic channels, perfect CSIR/T • K = 2 SISO users , y k [ n ] = h k x [ n ] + w k [ n ] Let x [ n ] = x 1 [ n ] + x 2 [ n ] , with x k [ n ] (power P k , rate R k ) intended for user k ⇒ y k [ n ] = h k [ n ] x 1 [ n ] + h k x 2 [ n ] + w k [ n ] 1 Weingarten, Steinberg and Shamai, “The capacity region of the Gaussian MIMO broadcast channel,” IEEE ISIT 2004 Mikael Skoglund, Theoretical Foundations of Wireless 10/17
• Treat h 1 x 2 [ n ] as known interference when transmitting x 1 [ n ] to y 1 [ n ] ⇒ Costa precoding achieves 1 + | h 1 | 2 P 1 � � R 1 = log σ 2 | h 2 | 2 P 2 � � R 2 = log 1 + | h 2 | 2 P 1 + σ 2 • This point can only be achieved with superposition coding + interference cancellation at the receiver if | h 1 | 2 ≥ | h 2 | 2 • Order can be reversed 1 ↔ 2 Mikael Skoglund, Theoretical Foundations of Wireless 11/17 • K = 2 MISO users , y k [ n ] = h ∗ k x [ n ] + w k [ n ] • Let x [ n ] = u 1 x 1 [ n ] + u 2 x 2 [ n ] , with x k [ n ] (power P k , rate R k ) intended for user k ⇒ y k [ n ] = h ∗ k u 1 x 1 [ n ] + h ∗ k u 2 x 2 [ n ] + w k [ n ] • Treat h ∗ 1 u 2 x 2 [ n ] as known interference ⇒ 1 u 1 | 2 P 1 � � 1 + | h ∗ R 1 = log σ 2 2 u 2 | 2 P 2 � � | h ∗ R 2 = log 1 + | h ∗ 2 u 1 | 2 P 1 + σ 2 achievable; order can be reversed • These rates may not be achievable with superposition coding Mikael Skoglund, Theoretical Foundations of Wireless 12/17
• K > 2 MISO users • Let π : I K → I K represent a permutation of I K = { 1 , . . . , K } • Fix { u k } , with � u k � 2 ≤ 1 , k = 1 , . . . , K • Let x [ n ] = � k u k x k [ n ] , with x k [ n ] (power P k , rate R k ) intended for user k ⇒ K � y k [ n ] = h ∗ u ℓ x ℓ [ n ] + w k [ n ] k ℓ =1 • For user π ( k ) , k = 1 , . . . , K , treat � h ∗ u π ( ℓ ) x π ( ℓ ) [ n ] π ( k ) ℓ<k as known interference • Achieves the rate � π ( k ) u π ( k ) | 2 P π ( k ) � | h ∗ R π ( k ) = log 1 + � π ( ℓ ) u π ( ℓ ) | 2 P π ( ℓ ) + σ 2 ℓ>k | h ∗ for user π ( k ) , k = 1 , . . . , K Mikael Skoglund, Theoretical Foundations of Wireless 13/17 • Let C π ( u 1 , . . . , u K ; P 1 , . . . , P K ) be the set of ( R 1 , . . . , R K ) ’s described by � π ( k ) u π ( k ) | 2 P π ( k ) � | h ∗ R π ( k ) < log 1 + π ( ℓ ) u π ( ℓ ) | 2 P π ( ℓ ) + σ 2 � ℓ>k | h ∗ Then, the closure of the convex hull of the set � C π ( u 1 , . . . , u K ; P 1 , . . . , P K ) over all permutations π , all u k subject to � u k � 2 < 1 , and all P k subject to � k P k ≤ P , is the capacity region of the MISO downlink Mikael Skoglund, Theoretical Foundations of Wireless 14/17
• Duality : transmit beamforming with Costa precoding ↔ receive filtering with SIC • single-user MIMO: diagonalizing • multi-user uplink MIMO: triangularization at receiver • multi-user downlink MIMO: triangularization at transmitter Mikael Skoglund, Theoretical Foundations of Wireless 15/17 • Slow fading , no CSIT: { h k } are random, outage, ε -achievable rates,. . . • Fast fading • Perfect CSIT/R: Linear beamforming with Costa precoding achieves capacity, • waterfilling over time, adaptive precoding ordering (the permutation π ), adaptive beamforming,. . . • No CSIT: Full capacity region unknown, Costa precoding cannot be used,. . . • Partial CSIT: Opportunistic beamforming with multiple beams. . . Mikael Skoglund, Theoretical Foundations of Wireless 16/17
• SIMO downlink • Receivers do receive beamforming (MRC) ⇒ equivalent to SISO downlink ⇒ • No fading : superposition coding optimal • Fading, no CSIT : general capacity region unknown • Fading, perfect CSIT : opportunistic scheduling optimal (for sum rate), role of multiuser diversity diminished with more antennas. . . • MIMO downlink • No fading : Costa precoding optimal • Fading, no CSIT : general capacity region unknown • Fading, perfect CSIT : adaptive Costa precoding optimal Mikael Skoglund, Theoretical Foundations of Wireless 17/17
Recommend
More recommend
