Lecture 12 Multiuser MIMO Capacity MISO downlink: 10.3 Precoding: - - PDF document

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

yk[n] = h∗

k[n]x[n] + wk[n]

• Number of transmit antennas: Nt
• x[n] ∈ CNt; hk[n] ∈ CNt, k = 1, . . . , K
• Block-length: Nc
• Power constraint,

1 Nc

Nc

• n=1

x[n]2 ≤ P

• Rates: Rk, 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

x[n] =

K

• ℓ=1

˜ xℓ[n]uℓ

• Signature vectors uk ∈ CNt, “transmit filters”
• Choose {uk} to minimize the total transmit power, subject to a

constraint on each user’s SINR. . .

• The dual SIMO uplink, with receive filters uk ∈ CNr
• 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,

ω ˆ ω α β s w x y

• 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 = σ2
• 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 qi(s) = i-symbol closest to s
• Transmit,

x = qi(s) − s

• Receive,

y = qi(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

ω ˆ ω α β sn wn xn yn

• ω ∈ {0, . . . , M − 1}, data
• sn = (s1, . . . , sn) ∈ Cn, interference sequence
• xn = (x1, . . . , xn) = α(ω, sn) ∈ Cn, n−1

m E|xm|2 ≤ P,

transmitted codeword

• w ∈ C, zero-mean Gaussian noise, E|w|2 = σ2
• ˆ

ω = β(yn) ∈ {0, . . . , M − 1}

Costa, “Writing on dirty paper,” IEEE Trans. IT, 1983

Mikael Skoglund, Theoretical Foundations of Wireless 8/17

• All rates

R = log M n < log

• 1 + P

σ2

• 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 Downlink1

• Fixed deterministic channels, perfect CSIR/T
• K = 2 SISO users,

yk[n] = hkx[n] + wk[n] Let x[n] = x1[n] + x2[n], with xk[n] (power Pk, rate Rk) intended for user k ⇒ yk[n] = hk[n]x1[n] + hkx2[n] + wk[n]

1Weingarten, Steinberg and Shamai, “The capacity region of the Gaussian MIMO

broadcast channel,” IEEE ISIT 2004

Mikael Skoglund, Theoretical Foundations of Wireless 10/17

• Treat h1x2[n] as known interference when transmitting x1[n] to

y1[n] ⇒ Costa precoding achieves R1 = log

• 1 + |h1|2P1

σ2

• R2 = log
• 1 +

|h2|2P2 |h2|2P1 + σ2

• This point can only be achieved with superposition coding +

interference cancellation at the receiver if |h1|2 ≥ |h2|2

• Order can be reversed 1 ↔ 2

Mikael Skoglund, Theoretical Foundations of Wireless 11/17

• K = 2 MISO users,

yk[n] = h∗

kx[n] + wk[n]

• Let x[n] = u1x1[n] + u2x2[n], with xk[n] (power Pk, rate Rk)

intended for user k ⇒ yk[n] = h∗

ku1x1[n] + h∗ ku2x2[n] + wk[n]

• Treat h∗

1u2x2[n] as known interference ⇒

R1 = log

• 1 + |h∗

1u1|2P1

σ2

• R2 = log
• 1 +

|h∗

2u2|2P2

|h∗

2u1|2P1 + σ2

• 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 : IK → IK represent a permutation of IK = {1, . . . , K}
• Fix {uk}, with uk2 ≤ 1, k = 1, . . . , K
• Let x[n] =

k ukxk[n], with xk[n] (power Pk, rate Rk)

intended for user k ⇒ yk[n] = h∗

k K

• ℓ=1

uℓxℓ[n] + wk[n]

• For user π(k), k = 1, . . . , K, treat

h∗

π(k)

• ℓ<k

uπ(ℓ)xπ(ℓ)[n] as known interference

• Achieves the rate

Rπ(k) = log

• 1 +

|h∗

π(k)uπ(k)|2Pπ(k)

• ℓ>k |h∗

π(ℓ)uπ(ℓ)|2Pπ(ℓ) + σ2

• for user π(k), k = 1, . . . , K

Mikael Skoglund, Theoretical Foundations of Wireless 13/17

• Let Cπ(u1, . . . , uK; P1, . . . , PK) be the set of (R1, . . . , RK)’s

described by Rπ(k) < log

• 1 +

|h∗

π(k)uπ(k)|2Pπ(k)

• ℓ>k |h∗

π(ℓ)uπ(ℓ)|2Pπ(ℓ) + σ2

• Then, the closure of the convex hull of the set
• Cπ(u1, . . . , uK; P1, . . . , PK)
• ver all permutations π, all uk subject to uk2 < 1, and all Pk

subject to

k Pk ≤ 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: {hk} 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