Lecture 11 Multiuser MIMO Capacity General model SIMO uplink: 10.1 - - PDF document

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Lecture 11 Multiuser MIMO Capacity General model SIMO uplink: 10.1 - - PDF document

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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 [ n ] x [

slide-1
SLIDE 1

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,

yk[n] =

Kul

  • ℓ=1

Hℓ[n]xℓ[n] + wk[n]

  • Hℓ[n] ∈ CN×M
  • xℓ[n] ∈ CM, ℓ = 1, . . . , Kul
  • yk[n] ∈ CN, k = 1, . . . , Kdl
  • wk[n] ∈ CN, i.i.d zero-mean Gaussian, E|wk[n]|2 = σ2

Mikael Skoglund, Theoretical Foundations of Wireless 2/21

slide-2
SLIDE 2
  • K-user SIMO uplink,
  • N = Nr receive antennas, M = 1 transmit antenna,

Kul = K, Kdl = 1

  • K-user MIMO uplink,
  • N = Nr receive antennas, M = Nt transmit antennas,

Kul = K, Kdl = 1

  • K-user SIMO downlink,
  • N = Nr receive antennas, M = 1 transmit antenna,

Kul = 1, Kdl = K

  • K-user MIMO downlink,
  • N = Nr receive antennas, M = Nt transmit antennas,

Kul = 1, Kdl = K (equal number of t and r antennas for different users)

Mikael Skoglund, Theoretical Foundations of Wireless 3/21

K-user SIMO Uplink

y[n] =

K

  • ℓ=1

hℓ[n]xℓ[n] + w[n]

  • xℓ[n] ∈ C, ℓ = 1, . . . , K; y[n] ∈ CNr; w[n] ∈ CNr; hℓ ∈ CNr
  • Powers: Pk, k = 1, . . . , K
  • Rates: Rk, k = 1, . . . , K
  • Block-length: Nc

Mikael Skoglund, Theoretical Foundations of Wireless 4/21

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SLIDE 3

Capacities, K = 2

  • Fixed and deterministic; hℓ[n] = hℓ, n = 1, 2, . . . , Nc
  • Achievable rates,

R1 < log

  • 1 + h12P1

σ2

  • R2 < log
  • 1 + h22P2

σ2

  • R1 + R2 < log det
  • I + 1

σ2 (P1h1h∗

1 + P2h2h∗ 2)

  • Extension to K > 2 similar as in the SISO case
  • The sum-rate constraint ↔ K = 1, Nt = 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

slide-4
SLIDE 4
  • To achieve the point ’A’ use MMSE + interference cancellation,
  • The point ’A’

R2 = log

  • 1 + h22P2

σ2

  • R1 = log
  • 1 + P1h∗

1(σ2I + P2h2h∗ 2)−1h1

  • (etc.)
  • Fig. 10.4 in the textbook

Mikael Skoglund, Theoretical Foundations of Wireless 7/21

  • Slow fading, perfect CSIR, no CSIT; hℓ[n] = hℓ, n = 1, . . . , Nc
  • (R1, R2) ∈ C(h1, h2) iff

R1 ≤ log

  • 1 + h12P1

σ2

  • R2 ≤ log
  • 1 + h22P2

σ2

  • R1 + R2 ≤ log det
  • I + 1

σ2 (P1h1h∗

1 + P2h2h∗ 2)

  • Outage probability,

pul-SIMO

  • ut

= Pr

  • (R1, R2) /

∈ C(h1, h2)

  • Mikael Skoglund,

Theoretical Foundations of Wireless 8/21

slide-5
SLIDE 5
  • Outage probability, symmetric case (R1 = R2 = R),

pul-SIMO

  • ut

(R) = Pr

  • (R, R) /

∈ C(h1, h2)

  • = Pr
  • log det
  • I + 1

σ2 (P1h1h∗

1 + P2h2h∗ 2)

  • < 2R
  • ε-outage symmetric capacity,

Csym

ε

= arg sup

R

Pr

  • pul-SIMO
  • ut

(R) < ε

  • Mikael Skoglund,

Theoretical Foundations of Wireless 9/21

  • Diversity–multiplexing tradeoff, symmetric case, K users
  • Nr ≥ K assumed (see textbook for Nr < K)
  • Upper curve: optimal (ML) receiver
  • Lower curve: decorrelation receiver
  • Fig. 10.5 in the textbook

Mikael Skoglund, Theoretical Foundations of Wireless 10/21

slide-6
SLIDE 6
  • Fast fading, perfect CSIR, no CSIT; {hℓ[n]} i.i.d (stationary and

ergodic) in n

  • K = 2, achievable rates

R1 < E log

  • 1 + h12P1

σ2

  • R2 < E log
  • 1 + h22P2

σ2

  • R1 + R2 < E log det
  • I + 1

σ2 (P1h1h∗

1 + P2h2h∗ 2)

  • with expectation over the stationary distribution

Mikael Skoglund, Theoretical Foundations of Wireless 11/21

  • Fast fading, perfect CSIR, perfect CSIT
  • Pk[n] = πk(h1[n], h2[n]) = power allocated to user k
  • C(π1, π2) = (R1, R2)’s satisfying

R1 < E log

  • 1 + h12πk(h1, h2)

σ2

  • R2 < E log
  • 1 + h22πk(h1, h2)

σ2

  • R1 + R2 < E log det
  • I + 1

σ2 (πk(h1, h2)h1h∗

1 + πk(h1, h2)h2h∗ 2)

  • (over the stationary distribution)

Mikael Skoglund, Theoretical Foundations of Wireless 12/21

slide-7
SLIDE 7
  • The capacity region C is the convex hull of the set
  • π1,π2

C(π1, π2)

  • ver all (π1, π2) such that

E[πi(h1, h2)] ≤ Pi, i = 1, 2

Mikael Skoglund, Theoretical Foundations of Wireless 13/21

  • Multiuser Diversity
  • Consider the sum capacity (K users),

Csum = sup {R =

  • k

Rk : (R1, . . . , RK) achievable}

  • In the SISO case: only one user transmits optimal, no natural

generalization to SIMO

  • Force one user at a time ⇒ the sum rate

E log

  • 1 + ¯

πk∗(hk∗)hk∗2 σ2

  • ≤ Csum

(assuming i.i.d hk’s) is achievable, where ¯ π is the “waterfilling over time” policy and k∗ = arg maxk{hk2}

  • Multiuser diversity gain lower than in the SISO case, since the

tail probability for hk2 decreases with Nr

Mikael Skoglund, Theoretical Foundations of Wireless 14/21

slide-8
SLIDE 8
  • Optimal power allocation
  • No simple and general conclusions can be drawn
  • In the case Nr ≈ K → ∞, the policy that all users transmit with

waterfilling over their own states is close to optimal; πk(h1, . . . , hK) = 1 λ − I0 hk2 + where I0 = E

  • ℓ=k

hℓ[n]xℓ[n] + w[n]2 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

y[n] =

K

  • ℓ=1

Hℓ[n]xℓ[n] + w[n]

Mikael Skoglund, Theoretical Foundations of Wireless 16/21

slide-9
SLIDE 9

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

slide-10
SLIDE 10
  • Fixed and deterministic; Hℓ[n] = Hℓ, n = 1, 2, . . . , Nc
  • Let Nm = min(Nt, Nr) (assuming equal number of t antennas for

different users)

  • Let Lk = diag (p1k, . . . , pNtk) with, if Nm < Nt, pnk = 0 for

n > Nm, and subject to

n pnk = Pk

  • Let U1 and U2 be unitary
  • Achievable rates,

Rk < log det

  • I + 1

σ2 HkKkH∗

k

  • R1 + R2 < log det
  • I + 1

σ2 (H1K1H∗

1 + H2K2H∗ 2)

  • for any Kk = UkLkU∗

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; {Hℓ[n]} i.i.d (stationary and

ergodic) in n

  • Achievable rates,

Rk < E log det

  • I + 1

σ2 HkKkH∗

k

  • R1 + R2 < E log det
  • I + 1

σ2 (H1K1H∗

1 + H2K2H∗ 2)

  • for any Kk = UkLkU∗

k

  • Capacity region = closure of the convex hull over all power

allocations and rotations

  • Special case: uncorrelated Rayleigh fading,

Kk = Pk Nt INt is optimal ⇒ the capacity region is a pentagon

Mikael Skoglund, Theoretical Foundations of Wireless 20/21

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SLIDE 11
  • Fast fading, perfect CSIR, perfect CSIT; {Hℓ[n]} i.i.d (stationary and

ergodic) in n,

  • similar conclusions as for SIMO. . .

Mikael Skoglund, Theoretical Foundations of Wireless 21/21