We have discussed the problem of optimizing the CSIT dissemination in a network MIMO scenario. In particular, following a generalized DoF analysis,
we have exhibited a CSIT allocation which allows to achieve the optimal generalized DoF while restricting the cooperation to a local scale. This behavior is critical for the cooperation of a large number of TXs to be possible in a realistic network where the backhaul links are imperfect and of finite capacity. The proposed CSIT allocation can be seen as an alternative to clustering where the hard boundaries of the cluster are replaced by a smooth decrease of the cooperation strength.
In this chapter, we have then shown how optimizing the CSI dissemina- tion allows to achieve the required performance with a CSIT dissemination strategy being much more parcimonious than the conventional one. Inter- estingly, this new approach allows to overcome limitations of the TX coop- eration which seemed fundamental but were in fact only a consequence of the particular CSIT dissemination strategy chosen. Studying the CSIT dis- semination in other application scenarios will be the topic of future research and is expected to lead to further savings and new interesting behaviours.
Interference Alignment with Incomplete CSIT
In the previous chapter, we have shown how to exploit the pathloss atten- uation to reduce the CSIT requirements. The approach could not easily be extended to IA because of the different kind of coordination required between the TXs (the dimensions where the interference are “aligned” have to be jointly chosen). However, we will now show that there is another dimension that can be exploited in the case of MIMO IA.
Let us consider as toy example a K-user IC with all TXs and all RXs having respectivelyM andNantennas. It is shown in [50] that IA is feasible if and only if M +N ≥K+ 1 for single-stream transmissions. This result is obtained with the assumption of perfect CSIT at all RXs. However, if M = 1 and N = K, it is clear that no CSI is necessary at the TXs since the RXs have enough antennas to ZF theK−1 dimensions of interference.
Similarly, ifM =KandN = 1, each TX can apply conventional ZF to emit no interference to the other RXs. It is then not necessary to provide each TX with the CSI relative to the full multi-user channel but solely with the
“local” CSI relative to this particular TX.
Those particular examples highlight the fact that IA can be achieved in some cases without the requirement for each TX to receive the CSI relative to the full multi-user channel. This is very interesting practically as it means that it is possible to reduce the amount of CSI shared in the backhaul at the cost of no performance reduction. We investigate in the following what is the minimal – in the sense of a metric which will be introduced in the following – CSIT allocation which is necessary in order to achieve IA.
Practically, this means revisiting the so-called “feasibility problem” for
MIMO IA, which consists in determining for given antenna configurations whether IA is feasible or not [49, 50, 118, 119], under the scope of incomplete CSIT at the TXs.
8.1 Incomplete CSIT Configuration and Problem Statement
We consider in this section a particular case of distributed CSIT which we call incomplete CSIT. In this model, a TX has either perfect knowledge of a channel coefficient or no information at all on that element. We represent the CSIT structure at TXjby theCSIT matrix F(j)∈ {0,1}Ntot×Mtot such that {F(j)}ik = 1 if {HH}ik is known at TX j, and 0 otherwise. Denoting by ˆH(j) the available CSI at TXj, we obtain
( ˆH(j))H=F(j)⊙HH (8.1) with ⊙ denoting the element-wise (or Hadamart) product. We define the CSIT allocation F as the set of CSI representations available at all TXs:
F ={F(j)|F(j)∈ {0,1}Ntot×Mtot, j= 1, . . . , K} (8.2) and we define the space Fcontaining all the possible CSIT allocations. We can then define the size of an incomplete CSIT allocation as follows.
Definition 4. The size of a CSIT allocation F, denoted by s(F), is equal to the overall number of complex channel coefficients fed back to the TXs.
Thus,
s(F),
K
X
j=1
kF(j)k2F. (8.3)
Remark 14. This size definition can be linked to the size used in Chapter 7.
Indeed, considering a quantization of the channel elements withlog2(P)bits per-element, the size defined in (8.3)is then equal to the total pre-log factor of the number of CSI bits exchanged in the backhaul. In fact, the size in (8.3) can be seen as the size introduced in Definition 2 for the particular case where the number of quantization bits used for a channel element can only be log2(P) or 0.
To check whether IA feasibility is preserved with a given CSIT allocation, we introduce the functionffeaswhich takes as argument a CSIT allocationF and an antenna configuration QK
k=1(Nk, Mk) and returns 1 if IA is feasible
with these parameters and 0 otherwise. Note that this means that there exists one algorithm achieving IA with this CSIT allocation but it does not precise the algorithm. We also define the set Ffeas containing all the CSIT allocations for which IA is feasible. Hence,
Ffeas,{F|F ∈F, ffeas
F,
K
Y
k=1
(Nk, Mk)
= 1}. (8.4)
Only the interfering channel matrices HHij with i6= j are required to fulfill the IA constraints, and not the direct channel matrices HHjj. Thus, from a DoF point of view, we can always skip the direct channel matrices HHjj in the feedback, which leads to the following definition.
Definition 5. Acomplete CSIT allocation, denoted byFcomp, is defined by the knowledge of all the interfering channel matrices HHij with i 6= j at all TXs. Thus, the size of a complete CSIT allocation is
s (Fcomp) =K NtotMtot−
K
X
i=1
NiMi
!
. (8.5)
A CSIT allocation with a size smaller than s(Fcomp) is said to be strictly incomplete.
At this stage, a natural question is to ask what is the most incomplete CSIT allocation which preserves the feasibility of IA, i.e., to find
Fmin= argmin
F∈Ffeas
s(F). (8.6)
Note that we limit here our study to the IA feasible settings, i.e., such that Fcomp∈Ffeas.