Broadcast channels and Shannon strategies

A complete characterization of C(P₁, . . . , P_L) is not surprisingly hard to obtain.

One of the main issues is that the characterization of the capacity of the broadcast channel (BC), which is a particular case of the considered setup, is itself a noto-rious and long lasting open problem. However, the best known coding schemes for the BC are known to perform well, or even optimally, in many setups. There-fore, inspired by the results on decentralized transmission towards a single RX, in what follows we revisit these schemes transposed to distributed CSIT by means of Shannon strategies.

Specifically, we readily obtain multi-RX decentralized transmission techniques by letting X_l=f_l(U, S_l) for some auxiliary variable U ∈ U,L distributed precod-ing functionsf_l:U × S_l → X_l, and by applying known coding schemes for the BC without state, input U, and K outputs Yk. The resulting achievable schemes are conceptually identical to their centralized counterparts, and relegate the impact of

the distributed CSIT assumption to the design of the distributed precoding func-tions f_l. Starting from the popular treating interference as noise (TIN) approach, we obtain the following inner bound:

Lemma 3.3 (TIN). A rate tuple (R₁, . . . , R_K) is achievable if

R_k ≤I(U_k;Y_k), ∀k∈ K, (3.34) forK auxiliary variables(U1, . . . , UK)∈QK

k=1Uk=:U, with pmfQK

k=1p(uk), inde-pendent of(S, S₁, . . . , S_L), andLfunctionsf_l:U ×S_l → X_l, x_l=f_l(u₁, . . . , u_K, s_l), satisfying E[η_l(X_l)]≤P_l for l∈ L.

Proof. The proof follows by the physical device argument used in Theorem 3.1 and from the first part of [7, Section 8.3].

In (3.34), each of the auxiliary variables U_k represents the data-bearing signal over which Wk is independently encoded. Although generally suboptimal, TIN offers an excellent performance / complexity compromise in many practical cases;

its main feature is that it keeps both the TX and RX architectures simple, and mostly relies on a good design of the distributed precoding functions fl for miti-gating the interference. Therefore, intuitively, TIN performs well whenever these functions are able to drive the interference down to the noise floor; for example, it is close-to-optimal for centralized MIMO fading channels with sufficiently high quality CSIT [50, 2].

A second more complex approach is to let the RXs decode (some of) the in-terference. Focusing on K = 2 RXs for simplicity, this idea leads to the following inner bound:

Lemma 3.4 (Superposition coding). A rate pair (R₁, R₂) is achievable if

R1 ≤I(U1;Y1|U0) +I(U0;Y1), (3.35) R₂ ≤I(U₂;Y₂|U₀) +I(U₀;Y₂), (3.36) R₁+R₂ ≤I(U₁;Y₁|U₀) +I(U₂;Y₂|U₀) + min{I(U₀;Y₁), I(U₀;Y₂)}, (3.37) for some auxiliary variables (U₀, U₁, U₂) ∈ U₀ × U₁ × U₂ =: U independent of (S, S₁. . . , S_L), with pmf factorizing as p(u₀)p(u₁|u₀)p(u₂|u₀), and L functions f_l : U × S_l→ X_l, x_l =f_l(u₀, u₁, u₂, s_l), satisfying E[η_l(X_l)]≤P_l, ∀l ∈ L.

Proof. The proof follows, e.g., from a particular case of [7, Proposition 8.1].

The superposition coding inner bound improves upon TIN by allowing part of (W₁, W₂) to be encoded over a data-bearing signalU₀ to be decoded by both RXs.

This is known to be capacity achieving for the degraded BC without state [7], and for the degraded BC with causal CSIT [51]. The following theorem proves that this holds also for decentralized transmission.

CHAPTER 3. CODING THEOREMS 25 Theorem 3.3. Consider physically degraded channels, i.e., such that

p(y₁, y₂|x₁, x₂, s) =p(y₁|x₁, x₂, s)p(y₂|y₁), (3.38) or, more in general, stochastically degraded channels, i.e., such that there is a phys-ically degraded channel p⁰(y₁, y₂|x₁, x₂, s) = p(y₁|x₁, x₂, s)p⁰(y₂|y₁) with the same marginal p(y₂|x₁, x₂, s). Then, C(P₁, . . . , P_L) is given by the convex hull of the set of all rate pairs (R₁, R₂) satisfying

R₁ ≤I(U₁;Y₁|U₂), (3.39)

R₂ ≤I(U₂;Y₂), (3.40)

for some auxiliary variables (U₁, U₂)∈ U₁× U₂ =:U independent from (S, S₁, S₂), and L functions f_l : U × S_l → X_l, x_l = f_l(u₁, u₂, s_l), satisfying E[η_l(X_l)] ≤ P_l,

∀l ∈ L.

Proof. The proof is given in Appendix B.1.3.

The above theorem extends the result in [51] to the considered network with distributed CSIT. The setup in [51] is recovered for the centralized and perfect CSIT case S₁ = S₂ = S. The optimal transmission scheme exploits the degrad-edness property which ensures that RX 1 can always decode the full message W₂ encoded over U₀ = U₂. Unfortunately, MIMO fading channels are typically not degraded. However, we remark that superposition coding can still be very useful for MIMO fading channels (where it is often referred to as rate splitting or non-orthogonal multiple access), especially when the available CSIT does not allow to bring the interference down to the noise floor, making TIN highly suboptimal [52].

Remarkably, the scheme in Lemma 3.4 is shown in [20] to achieve the optimal throughput scaling law (the so-called sum DoF) for some decentralized MIMO fading channels with non-trivial distributed CSIT configurations.

A third approach is to interpret the interference to be generated as a known non-casual state, whose impact can be preventively mitigated at the encoder side.

By focusing again on K = 2 RXs for simplicity, this idea leads to the following inner bound:

Lemma 3.5 (Marton’s inner bound). A rate pair (R₁, R₂) is achievable if

R₁ ≤I(U₁;Y₁), (3.41)

R₂ ≤I(U₂;Y₂), (3.42)

R1+R2 ≤I(U1;Y1) +I(U2;Y2)−I(U1, U2), (3.43) for some auxiliary variables(U₁, U₂)∈ U₁×U₂ =:U independent from(S, S₁. . . , S_L), and L functions f_l : U × S_l → X_l, x_l = f_l(u₁, u₂, s_l), satisfying E[η_l(X_l)] ≤ P_l,

∀l ∈ L.

Proof. The proof follows from [7, Theorem 8.3].

The main benefit of Marton’s inner bound w.r.t. the TIN bound is that the former allows for arbitrary dependence between the data-bearing signals U₁ and U₂. This, however, comes at a penality on the achievable sum rate. Remarkably, a particular case of Lemma 3.5, known asdirty papercoding [53], is known to achieve the capacity region of MIMO fading channels with perfect CSIT and CSIR [54].

Its main drawbacks are the considerable increase of complexity at the TX end, and the stringent CSIT quality requirements for the interferenceprecancelation idea to be effective.

Remark 3.6 (Marton’s inner bound with superposition coding). Marton’s coding can be readily combined with superposition coding, leading to the largest known single-letter achievable region for the BC [7, Proposition 8.1]. Therefore, the tech-niques presented in this section give the largest known single-letter achievable re-gion R(P₁, . . . , P_L) ⊆ C(P₁, . . . , P_L) for the considered decentralized network, if we restrict the encoders to neglect the past CSIT sequences (i.e., to use Shannon strategies). As already mentioned, under this restriction the difference between de-centralized and de-centralized transmission is confined to the design of good distributed precoding functions f_l.