Network Performance with Channel Uncertainty

transmitter (CSIT) is usually utilized to allocate power at the transmitter, and thus channel uncertainty is not important in terms of DoF for SU-MIMO.

While the sum DoF of multiuser uplink MAC is not a↵ected by partial or lack of CSIT, channel uncertainty could severely degrade the performance of downlink BC and IC in the sense of DoF. In what follows, the performance in terms of DoF is described with regard to various channel feedback availability.

Perfect Feedback

When channel knowledge with infinite precision and zero-latency is avail-able at the transmitter (referred to as “perfect CSIT”), the optimal DoF value of K-user M transmit-antenna MISO BC is min{M, K} achieved by linear strategies such as ZF beamforming. When it comes to K-user MIMO BC with M antennas at the transmitter and N_k antennas at kth receiver, the optimal sum DoF min{M,PK

k=1N_k} can also be achieved by ZF precoding at both transmitters and receivers [26]. While the optimal sum DoF of K-user SISO interference channels were shown to be ^K₂ [7], the DoF characterization of the general MIMO case with arbitrary antenna configurations is open and still attracting more attention [11, 27]. Partic-ularly, for the two-user MIMO IC, where the transmitters and receivers are equipped with M₁, M₂, N₁, N₂ antennas, respectively, the sum DoF is min{M1+M2, N1+N2,max{M1, N2},max{M1, N2}}with perfect CSIT [28].

Nevertheless, the performance promised by perfect CSIT is depressed by the harsh constraints of infinite precision and zero-latency feedback, which is impractical.

No Feedback

On the other hand, in the absence of knowledge of channel realizations at the transmitter (referred to as “no CSIT”), a collapse of DoF was predicted by information theoretic studies. Specifically, some previous works with no CSIT settings have observed this collapse of DoF, under the additional assumption of homogeneity, such as i.i.d. isotropic fading, channel degradedness or statistical equivalence of receivers [29–32]. For instance, in a two-user MISO BC with generic (i.e., channel coefficients are drawn from a continuous distribution) and constant or time-varying channels, if the transmitter is blind to both users’ channel states, then the best known sum DoF outer bound is ⁴₃ whereas the best known inner bound is still 1. Remarkably,

a surprising result by Jafar [33] showed that, under some heterogeneous block fading model, the sole knowledge of channel coherence intervals can improve sum DoF. For example, for a two-user MISO BC, given that one user experiences time-selective fading whereas the other one experiences frequency-selective fading, the optimal DoF ⁴₃ can be achieved. Nevertheless, whether the outer or inner bound is tight in general no CSIT setting is still unknown.

It was further claimed in [34] that DoF collapse with fixed imprecision of CSIT whose error does not scale with SNR, and the DoF value of a two-user MISO BC is upper bounded by ⁴₃. This DoF upper bound was proved to be tight by Gou, Jafar and Wang in [35], and by Maddah-Ali in [36] under the finite state compound setting, in which DoF are shown to be robust to channel uncertainty and the optimal ⁴₃ DoF are achievable for the finite state compound MISO BC, regardless of the number of states. On the other hand, it was conjectured that the sum DoF of a two-user MISO BC must collapse to 1 under the general no CSIT setting without any additional assumptions.

To prove or disprove this conjecture has been a longstanding open problem, until a recent breakthrough by Davoodi and Jafar [37], who showed that the inner bound is tight such that DoF collapse to unity as conjectured.

Remarkably, this is the first result to show the total collapse of DoF under channel uncertainty.

Feedback with Finite Precision

It is also well known that the full DoF in BC can be maintained under imperfect CSIT if the error in CSIT decreases as O(P ¹) as P grows [38, 39]. The interpretation is that such a quantization strategy maintains the quantization noise at a level no greater than the thermal noise, hence avoiding to make quantization a bottleneck of transmission as the SNR grows.

Moreover, for the case of the temporally correlated fading channel such that the transmitter can predict the current state with error decaying asO(P ^↵) for some constant↵2[0,1], ZF can only achieve a fraction↵ of the optimal DoF per user [38, 39]. This result somehow reveals the bottleneck of a family of precoding schemes relying only on the feedback precision of instantaneous CSIT as the temporal correlation decreases.

Feedback with finite precision was also considered in interference channels in conjunction with interference alignment. Among many others, in [40], the number of required feedback bits via broadcast feedback links was characterized to maintain full DoF with perfect global CSIT. The minimum number of broadcasting feedback bits should scale as the number of users

and logP, meaning that the channel uncertainty decreases as O(P ¹) as P grows, which agrees with the BC case.

Feedback with Delay

When it comes to the feedback delay, the conventional wisdom suggests to predict the current channel from the delayed feedback by exploiting the channel time correlation. The predicted channel state is then used for precoding as if it is the true channel state. It works (although not necessarily optimal) when the coherence time is larger than the overall feedback delay.

Otherwise, the delayed feedback bears no information on the current true channel, and the precoding built upon this prediction o↵ers no multiplexing gain at all.

When the CSI feedback is fully obsolete (i.e., uncorrelated with the current true channel, referred to as fully “delayed CSIT”), it would seem such channel information is non-exploitable in view of improving multiplexing gains. Recently, this viewpoint was challenged by an interesting information theoretic work [41], in which Maddah- Ali and Tse showed a surprising result that even completely outdated CSIT can be very useful in terms of degrees of freedom, as long as it accurately describes past channel realizations, i.e., the error in describing past channel states should decay at least as fast as O(P ¹). For the two-user MISO BC, the proposed scheme in [41] (referred to as “MAT”) achieves the DoF of ²₃ per user – irrespectively of the temporal correlation – achieving strictly better DoF than what is obtained without any CSIT, even in extreme situations when the delayed channel feedback is made totally obsolete by a feedback delay exceeding the channel coherence time. The role of perfect delayed CSIT can be re-interpreted as a feedback of information characterizing the past signal/interference heard by the receivers.

This side information enables the transmitter to perform “retrospective”

interference alignment in the space and time domain, as demonstrated in di↵erent multiuser network systems [42–47]. Despite its DoF optimality, these MAT-inspired schemes are designed assuming the worst case scenario where the delayed channel feedback provides no information about the current channel state.

The further investigation on channel uncertainty in this avenue includes the so-called “alternating CSIT” [48], in which the users experience time-varying CSI availability, e.g., the channel uncertainty varies among perfect, delayed and no CSIT settings, and many others.

This delayed feedback is the form of channel uncertainty considered in Chapters 3 and 4.

Topological Feedback

In partially connected networks, the knowledge of network connectivity can be fairly easy to be acquired by the transmitters via feedback (referred to as

“topological feedback”), as the long term (statistical) channel connectivity often varies slower than channel realizations, and the feedback overhead of this topological information (with binary value indicating whether the channel is strong or weak) is negligible. Specifically, the sole knowledge of network connectivity at the transmitters via topological feedback was shown to be beneficial to improve network performance in partially connected interference networks, whether the channel is slow fading [49] or fast fading [50, 51].

Remarkably, the interference channels with no CSIT beyond topological information were formulated under the topological interference management framework, in which this topological feedback problem was shown to be equivalent to the well-defined index coding problem [52] under linear solutions.

This topological feedback is the setup envisioned in Chapters 5 and 6.