6.2 DoF Analysis with Static Coefficients and Distributed CSI
6.2.2 DoF Analysis in the 3-user Square MIMO IC
Perfect CSIT Solution We consider now a 3-user IC withMi =M, Ni = N,∀i and di =d,∀i. We also assume for the description of the IA scheme that perfect CSIT is available such that we denote the precoder used at TXj byU⋆j. Since we consider the tightly-feasible case [108], we haveM =N = 2d. In that case, the IA constraints can be written as [54]
span
In particular, this system of equations can be easily seen to be fulfilled if the precoders verify
U⋆1Λ1= ( ˜HH3,1)−1H˜H3,2( ˜HH1,2)−1H˜H1,3( ˜HH2,3)−1H˜H2,1U⋆1 U⋆3= ( ˜HH2,3)−1H˜H2,1U⋆1
U⋆2= ( ˜HH1,2)−1H˜H1,3U⋆3
(6.21)
for some diagonal matrixΛ1. We also define for clarity the matrixY⋆equal to
Y⋆,( ˜HH3,1)−1H˜H3,2( ˜HH1,2)−1( ˜HH1,3)( ˜HH2,3)−1H˜H2,1. (6.22) The system of equations (6.21) is then fulfilled by setting
U⋆1 = 1
√dEVD(Y⋆)
e1, . . . ,ed
U⋆3 = 1
k( ˜HH2,3)−1H˜H2,1U⋆1kF
( ˜HH2,3)−1H˜H2,1U⋆1 U⋆2 = 1
k( ˜HH1,2)−1H˜H1,3U⋆3kF
( ˜HH1,2)−1H˜H1,3U⋆3
(6.23)
where we denote by EVD(A) the eigenvector basis of the Hermitian ma-trix A.
Distributed CSIT Solution With distributed CSIT, TX j computes using its channel estimate ˜H(j) the matrix
Y(j)= (( ˜H(j)3,1)H)−1(( ˜H(j)3,2)H(( ˜H(j)1,2)H)−1( ˜H(j)1,3))H(( ˜H(j)2,3)H)−1( ˜H(j)2,1))H. (6.24) The precoding matrices are then obtained from
U(j)1 = 1
√dEVD(Y(j))
e1, . . . ,ed
U(j)3 = 1
k(( ˜H(j)2,3)H)−1( ˜H(j)2,1)HU(j)1 kF
(( ˜H(j)2,3)H)−1( ˜H(j)2,1)HU(j)1 U(j)2 = 1
k(( ˜H(j)1,2)H)−1( ˜H(j)1,3)HU(j)3 kF
(( ˜H(j)1,2)H)−1( ˜H(j)1,3)HU(j)3 .
(6.25)
Note that only U(j)j is effectively used for the transmission due to the dis-tributed precoding assumption.
In that case, we can give the following result on the DoF achieved.
Theorem 10. Using the 3-User IA scheme described above with distributed CSIT, the DoF achieved at user iis denoted by DoFDCSIi and verifies
DoFDCSIi ≥dmin
j6=i min
k,ℓ,k6=ℓA(j)k,ℓ. (6.26)
Proof. The main idea of the proof is to consider only the rate achieved over the channel realizations which are “well enough” conditioned. Over these channel realizations, the precoding is robust enough to the errors in the CSIT. Due to the continuous distribution of the channel matrices, the probability of the “badly conditioned” channel realizations is small enough such that the loss due to removing these channel realizations can be made arbitrarily small.
We consider hereafter that∀k, ℓ, j, A(j)k,ℓ>0 since the result is otherwise trivial. We also consider without loss of generality the precoding at TX j.
For a givenε >0, we define the following channel subsets:
Xε,{H˜|∀i, k, λmin( ˜Hi,k)≥ε} (6.27) Yε,{H˜|∀i6=j, |λi(Y⋆)−λj(Y⋆)| ≥ε} (6.28) and
Hε,Xε\
Yε. (6.29)
Since we aim at deriving a lower bound for the DoF (and the rate is nonneg-ative), we can consider only the rate achieved for the channel realizations belonging to Hε. From (6.13), we can then write
Ri≥EHεh
It can be easily seen from the continuous distribution of the channel matrices that ∀η >0,∃ε >0,Pr(Hε)≥1−η. Hence, it follows that
We now need to upper bound the second term of (6.30) which we denote by Jiε. We can then proceed similarly to (6.16) to write
Jiε≤di error due to the imperfect CSIT at TXj on one of the matrix inversion re-quired to computeY(j)in (6.24). We start by introducing∆(j)i,k to represent the error done in computing the channel inverse:
∆(j)i,k , 1
Using the resolvent equality recalled in Lemma 2 in Appendix .1, we can write
∆(j)i,k =−(( ˜Hi,k)H)−1(N(j)i,k)H(( ˜Hi,k)H)−1+σ(j)i,kΘ(j)i,k, ∀i, k, (6.35) where we have defined
Θ(j)i,k ,(( ˜H(j)i,k)H)−1(N(j)i,k))H(( ˜Hi,k)H)−1(N(j)i,k)H( ˜Hi,k)H)−1, ∀i, k.
(6.36) We can then use the properties of the Frobenius norm to obtain the upper bound
k∆(j)i,kkF≤ kN(j)i,kkFkH˜−i,k1k2F+σ(j)i,kkΘ(j)i,kkF. (6.37) Taking the expectation, we have then
EHε[k∆(j)i,kk2F]≤EHε
kN(j)i,kkFkH˜−1i,kk2F+σ(j)i,kkΘ(j)i,kkF
2
(6.38) The expectation in (6.38) exists and is finite becauseH∈ Hε such that the channel matrix ˜Hi,k(and ˜H(j)i,k) has its eigenvalues bounded away from zero.
We have therefore obtained
EHε[k∆(j)i,kk2F] ˙≤1. (6.39)
We can then write
Y(j)= ( ˜H−3,11+σ(j)3,1∆(j)3,1)H( ˜H3,2+σ(j)3,2N(j)3,2)H( ˜H−1,21+σ(j)1,2∆(j)1,2)H ( ˜H1,3+σ(j)1,3N(j)1,3)H( ˜H−12,3+σ(j)2,3∆(j)2,3)H( ˜H2,1+σ(j)2,1N(j)2,1)H.
(6.40) The relation (6.39) holds for every matrix inversion in (6.31) such that putting all the errors terms together, we can write from (6.40) that
EHε[kY(j)−Y⋆k2F] ˙≤max
ℓ6=k(σ(j)ℓk)2. (6.41) Since H ∈ Hε, all the eigenvalues of Y⋆ (and Y(j)) are different and the matrices Y⋆ andY(j) are diagonalizable. Let Y⋆=V⋆Λ(V⋆)H, withV⋆ ∈ CM×M andΛ⋆ ∈CM×M, andY(j) =V(j)Λ(j)(V(j))H, with V(j) ∈CM×M and Λ(j) ∈CM×M, be the spectral decomposition of Y⋆ andY(j), respec-tively. Applying Theorem 2.1 from [109] to Y⋆ and Y(j) and taking the expectation we can show that for some constantγ(j)>0 independent of the SNR P,
EHε[kV(j)−V⋆k2F]≤γ(j)EHε[kY(j)−Y⋆k2F] (6.42)
≤˙ max
ℓ6=k(σℓ,k(j))2 (6.43)
=. P−minℓ6=kA(j)ℓ,k. (6.44) Let us denote by ¯V(j) and ¯V⋆ the matrices made of the first d columns of V(j)andV⋆, respectively. The precoding scheme is such thatU⋆1= ¯V⋆ and U(1)1 = ¯V(1). Hence,
EHε[k∆U(1)1 k2F] ˙≤P−minℓ6=kA(1)ℓ,k. (6.45) The relation (6.45) is easily extended to the other precoders U(2)2 andU(3)3 to obtain that
3
X
j=1,j6=i
EHε[k∆U(j)j k2F] ˙≤
3
X
j=1,j6=i
P−minℓ6=kA(j)ℓ,k (6.46)
≤˙ P−minj6=iminℓ6=kA(j)ℓ,k. (6.47)
Coming back to (6.30), this gives Ri ≥˙ di
(1−η) log2(P)−log2(1 +P P−minj6=iminℓ6=kA(j)ℓ,k
(6.48)
≥˙ di
minj6=i min
ℓ6=k A(j)ℓ,k−η
log2(P). (6.49)
Choosing ηarbitrarily small concludes the proof.
We have shown that for the 3-user IA closed-form alignment scheme, the achieved DoF is larger than the worst accuracy of the channel estimates across the TXs. Note that this lower bound is in fact conjectured to be tight.
Interestingly, the lower bound at RXj is limited by the accuracy of the estimates relative to the channels of all the other RXs. This result is in strong contrast with the centralized setting where the DoF of useridepends solely on the accuracy with which the channel matrices from the TXs to RXi are feedback.
6.3 Simulations
In this section, we validate by Monte-Carlo simulations the results in the 3-user square IC channel studied in Subsection 6.2.2. We considerM =N = 4 and d = 2 and we average the performance over 10000 realizations of a Rayleigh fading channel. We consider the distributed CSIT configuration described in Section 6.1. The quantization error is modeled using (6.6) with (σi,k(j))2 = 2−B(j)i,k/(NiMj−1) and N(j)i,k having its elements i.i.d. NC(0,1). We choose the CSIT scaling coefficients as
∀(i, k, j)∈ {1,2,3}3\ {(3,2,2),(3,2,3)}, A(j)i,k = 1, A(2)3,2 = 0.5, A(3)3,2= 0.
(6.50) Following Theorem 10, we have for the CSIT configuration described in (6.50) that DoF1 ≥ 0, DoF2 ≥ 0, and DoF3 ≥ 0.5d = 1. The average rate achieved is shown for each user in Fig. 6.2. For comparison, we have also simulated the average rate per-user achieved based on perfect CSIT and with distributed CSIT when the CSIT scaling coefficients are set equal to 1 for every TX (∀i, k, j, A(j)i,k = 1). It can then be verified that having all CSIT scaling coefficients equal to one allows to achieve the maximal DoF.
With the CSIT configuration described in (6.50), the slope of the rate of user 3 decreases as the SNR increases, revealing a very slow convergence
10 15 20 25 30 35 40 45 50 55 60 0
5 10 15 20 25 30 35 40
Average per−TX SNR [dB]
Average rate per user [bits/Hz/s]
With Perfect CSI
With Distributed CSI for Aik(j)=1 R1 with distributed CSI R2 with distributed CSI R3 with distributed CSI
Figure 6.2: Average rate per user in the square setting M = N = 4 with d= 2 for the CSIT scaling coefficients given in (6.50).
to the DoF. This makes it difficult to accurately observe the DoF achieved.
Yet, it can be seen that having onlyA(3)3,2 equal to zero leads already to the saturation of the rates of RX 1 and RX 2 (i.e., their DoF is equal to 0), which tends to confirm our conjecture.