On the Optimality of Beamforming for Multi-User MISO Interference Channels with Single-User Detection

Interference Channels with Single-User Detection

Xiaohu Shang1, Biao Chen2, and H. Vincent Poor1

1: Department of EE, Princeton University 2: Department of EECS, Syracuse University

Multi-user MISO Interference Channel (IC) Model Y_i = m X j=1,j6=i hhhT_jixxx_j + N_i • m transmitter-receiver pairs.

• xxx_i: t_i × 1 transmitted signal at transmitter i. kxxx_ik2 ≤ P_i.

• Y_i: received scalar signal at receiver i .

• N_i ∼ N(0, 1).

• hhh_ji: t_j × 1 channel vector from transmitter j to receiver i.

Single-User Detection Rate Region max m X i=1 µ_iR_i (µ_i ≥ 0) subject to R_i = 1 2 log 1 + hhhT_iiSihhhii 1 + Pm_j=1,j6=ihhhT_jiSjhhhji ! tr(Si) ≤ Pi, Si 0, i = 1,· · ·m. • xxx_i ∼ N (0,Si).

• Receiver i decodes xxx_i from Y_i by treating interference as noise.

• Single-user detection (SUD) rate region is defined as

[

all µ=[µ1,···,µm]

Problem Reformulation

• Why: Non-convex optimization problem.

µ₁log 1 + h T 11S1h11 1 + hT₂₁S₂h₂₁ + µ₂log 1 + h T 22S2h22 1 + hT₁₂S₁h₁₂

• How: non-convex problem ⇛ convex problem

maximize µ₁log 1 + h T 11S1h11 1 + z₂2 + µ₂log 1 + h T 22S2h22 1 + z₁2 subject to hT₁₂S₁h₁₂ = z₁2, hT₂₁S₂h₂₁ = z₂2, tr (Si) ≤ Pi, Si 0, i = 1,2.

New Convex Optimization Problem

max hhhT_iiSihhhii

subject to hhhT_ijSihhhij ≤ z_ij2,

tr(Si) ≤ Pi, Si 0 i, j = 1, . . . , m, i 6= j,

• Preselect z_ji2 as the interference power caused by the jth trans-mitter to the ith receiver.

• Maximize useful signal power with the interference constraint.

Comparison of the Complexity

• Old problem

m

Y

i=1

(num of µ_i) non-convex problems

• New problem m X i=1 m X j=1,j6=i

(num of z_ij) convex problems

• Advantage of problem reformulation

– Non-convex ⇛ convex

Covariance Matrices

All

S

S

_{for boundary}

points

S

_{by problem}

reformulation

Matrix Optimization Lemma If K =  K11 K T 21 K₂₁ K₂₂   0, tr(K) ≤ P

and K₁₁ 0 is a preselected sub-matrix, then

 xxx yyy   T K  xxx yyy   ≤ pxxxT_K 11xxx + kyyyk p P − tr(K₁₁)2 and there exists K∗ that achieves the equality with

Problem Simplification max  xxx yyy   T K  xxx yyy   subject to h_i (K11) = 0, gj (K11) ≤ 0 tr(K) ≤ P, K 0, ⇔ max pxxxTK₁₁xxx + kyyykpP − tr(K₁₁)2 subject to h_i (K11) = 0, gj (K11) ≤ 0, tr(K₁₁) ≤ P, K 0.

A Revisit of the Problem max hhhT_mmSmhhhmm subject to hhhT_mjSihhhmj ≤ z_mj2 , j = 1, . . . , m − 1 tr(Sm) ≤ Pm, Sm 0. ⇒ max  hhh b hhh   T ˜ S  hhh b h hh   subject to  hhhj 0   T ˜ S  hhhj 0   ≤ z_mj2 , j = 1, · · ·m − 1, tr ˜S ≤ P_m, S˜ 0,

A Revisit of the Problem (continue) ⇒ max hhhTS˜₁₁hhh subject to hhhT_j S˜₁₁hhh_j ≤ z_mj2 , j = 1, . . . , m − 1 tr(˜S₁₁) ≤ P_m, S˜₁₁ 0, where

dim(hhh) = dim(hhh_j) = min{t_m, m − 1} , _m¯

˜ S =  S˜11 S˜ T 21 ˜ S₂₁ S˜₂₂  .

Problem Simplification Procedure

• Step 1:

– Singular value decomposition

hhh_m1 = U₁  khhhm1k 0_(tm−1)×1  , where UT₁U₁ = I – then update h(1)_mj = UT₁hmj, j = 1, · · · , m. S(1)_m = UT₁SmU1.

Problem Simplification Procedure (continue) • Step 2: h(1)_m2 =    h(1)_m2 1 h(1)_m2 2,···,tm    =  1 0 0 U₂         h(1)_m2 1 h(1)_m2 2,···,tm 0_(tm−2)×1       , UT₂U2 = I h(2)_mj =  1 0 0 U₂   T h(1)_mj, j = 1, · · ·m. S(2)_m =  1 0 0 U₂   T S(1)_m  1 0 0 U₂  

Final State max  hhh b h hh   T  S˜11 S˜ T 21 ˜ S₂₁ S˜₂₂    hhh b hhh   subject to  hhhj 0   T  S˜11 S˜ T 21 ˜ S₂₁ S˜₂₂    hhhj 0   ≤ z_mj2 , j = 1, · · ·m,¯ tr    S˜11 S˜ T 21 ˜ S₂₁ S˜₂₂     ≤ P_m,  S˜11 S˜ T 21 ˜ S₂₁ S˜₂₂   0,

Optimality of Beamforming

• Theorem:

There exist an S∗ 0 and rank(S∗) ≤ 1 that is optimal for

max hhhT_mmSmhhhmm

subject to hhhT_mjSihhhmj ≤ z_mj2 , j = 1, . . . , m − 1 tr(Sm) ≤ Pm, Sm 0.

• Proof:

– Karush-Kuhn-Tucker conditions.

Conclusion

For an m-user MISO interference, the boundary points of the single-user detection rate region can be achieved by restricting each transmitter to implementing beamforming.

Numerical Example 0 0.5 1 0 0.5 1 1.5 0 0.5 1 1.5 R1 in nat/Hz/s R2 R3