On the Degrees-of-Freedom of the MIMO Three-Way Channel with Intermittent Connectivity
Anas Chaaban Aydin Sezgin Mohamed-Slim Alouini




On the Degrees-of-Freedom of the MIMO Three-Way Channel with Intermittent Connectivity
    
Anas Chaaban, Aydin Sezgin, and Mohamed-Slim Alouini


A. Chaaban and M.-S. Alouini are with the Division of Computer, Electrical, and Mathematical Sciences and Engineering (CEMSE) at King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia. Email: {anas.chaaban,slim.alouini}@kaust.edu.sa.

A. Sezgin is with the Institute of Digital Communication Systems at the Ruhr-Universität Bochum, Bochum, Germany. Email: aydin.sezgin@rub.de.


    December 30, 2023
================================================================================================================================================================================================================================================================================================================================================================================================================================================================================





The degrees-of-freedom (DoF) of the multi-antenna three-way channel (3WC) with an intermittent node is studied. Special attention is given to the impact of adaptation. A nonadaptive transmission scheme based on interference alignment, zero-forcing, and erasure-channel treatment is proposed, and its corresponding DoF region is derived. Then, it is shown that this scheme achieves the sum-DoF of the intermittent channel, in addition to the DoF region of the nonintermittent one. Thus, adaptation is not necessary from those perspectives. To the contrary, it is shown that adaptation is necessary for achieving the DoF region of the intermittent case. This is shown by deriving an outer bound for the intermittent channel with nonadaptive encoding, and giving a counterexample of an adaptive scheme which achieves DoF tuples outside this bound. This highlights the importance of cooperation in this intermittent network.






§ INTRODUCTION

Multi-way, full-duplex, and device-to-device (D2D) communications are important techniques that are expected to gain more prominence in future communication systems. Multi-way communication refers to communication between multiple nodes each acting as a source, a destination, and possibly a relay. Full-duplex operation is defined as when these three functionalities take place over the same time/frequency resources, and D2D communication refers to direct communication between users without or with limited base-station intervention. Those techniques attracted and continue to attract research attention <cit.>.

Consider a setup where three nodes (D2D users e.g.) communicate with each other in a multi-way fashion. This setup can be modeled as a three-way channel (3WC), an extension of Shannon's two-way channel <cit.> which has been studied in <cit.>. Therein, it is assumed that the nodes are always connected. This assumption is not always valid in practice since a node might have intermittent connectivity, e.g. due to shadowing, or if a D2D node causes strong interference to a cellular user, in which case it is not permitted to use its band <cit.>.

The impact of intermittency on the performance of various networks was studied in <cit.> for instance. In this paper, we study the impact of intermittency on the multiple-input multiple-output (MIMO) 3WC. We consider a full-duplex MIMO 3WC with full message-exchange, where each node has an independent message to each of the other two nodes. The permanent nodes have only causal knowledge of the availability of the intermittent node, which can be obtained by estimating its activity from the received signals. For this model, we study the degrees-of-freedom (DoF), i.e., the capacity scaling versus signal-to-noise ratio (SNR) in a dB scale. We pay particular attention to the necessity or the lack thereof, of adaptive encoding where the transmit signal of each node is allowed to depend on its previously received signals. This issue has been studied for various channels earlier <cit.>.

First, we devise a nonadaptive scheme based on interference alignment and zero-forcing, where intermittency is treated as an erasure channel, and we derive its achievable DoF region. Then, we derive DoF upper bounds that prove that this scheme achieves the sum-DoF of the channel. It follows that as far as the sum-DoF is concerned, adaptation is not necessary. 
This scheme also achieves the DoF region of the channel without intermittency, and hence, for the nonintermittent channel, adaptation is not necessary for achieving the DoF region. After showing the unnecessity of adaptation in those two cases, we prove that adaptation is necessary to achieve the DoF region of the intermittent channel. To show this, we derive a DoF outer bound that holds under nonadaptive encoding, and provide an adaptive scheme that achieves rates that violate this outer bound. This proves that adaptation enlarges the DoF region in the intermittent 3WC.

Throughout the paper, we use x_i^n for some i to denote (x_i,1,…,x_i,n). The N× N identity matrix is denoted _N. We write ∼𝒞𝒩(0,Q) to indicate that  is a complex Gaussian random variable with zero mean and covariance matrix Q. We write x^+ to denote max{0,x} for some x∈ℝ, _i to denote the ℓ_i-norm of , and ^̋†, ^̋H, and span()̋ to denote the pseudo-inverse, the Hermitian transpose, and the the subspace spanned by the columns of $̋.

















































§ SYSTEM MODEL

Consider a system where three MIMO full-duplex nodes communicate in a multi-way manner using the same medium, with one of the nodes being intermittently available (Fig. <ref>). For some transmission durationn∈ℕ(in symbols), lets^ndenote the intermittency of node 1, where forℓ∈{1,…,n},s_ℓ=1means that node 1 is available, ands_ℓ=0otherwise. The states^nis a sequence of independent and identically distributed (i.i.d) Bernoulli random variablesS_ℓwith probabilityℙ(s_ℓ=1)=τandℙ(s_ℓ=0)=1-τ≜τ̅. This sequence is known at node 1. However, knowledge ofs_ℓis only available causally at nodei∈{2,3}, i.e., nodeidoes not knows_ℓat the beginning of theℓ-th transmission, and can only obtain it after receiving theℓ-th received signal from which the activity of node 1 can be detected with certainty.

Nodei∈{1,2,3}is equipped withM_itransmit and receive antennas. Its transmit signal at time indexℓis represented by_i,ℓ; a realization of a random vector_i,ℓ∈ℂ^M_i×1that satisfies a power constraint[Any power discrepancy is absorbed into the channel gains.]∑_ℓ=1^n𝔼[_i,ℓ_2^2]≤nP. 

Clearly_1,ℓ=0ifs_ℓ=0. The received signals are 

    _1,ℓ   =_̋21_2,ℓ+_̋31_3,ℓ+_1,ℓ,  if s_ℓ=1
    _2,ℓ   =_̋12_1,ℓ+_̋32_3,ℓ+_2,ℓ,
    _3,ℓ   =_̋13_1,ℓ+_̋23_2,ℓ+_3,ℓ,

and_1,ℓ=0ifs_ℓ=0, where_̋ji∈ℂ^M_j×M_iand_̋ki∈ℂ^M_k×M_irepresent the channel matrices from nodesjandkto nodei, respectively, and_i,ℓis a realization of_i,ℓ∼𝒞𝒩(0,σ^2_M_i), i.i.d. with respect toℓ. 

We denoteP/σ^2byρand call it SNR throughout the paper. We assume without loss of generality thatM_2≥M_3. We also assume thatM_1≥M_2. The channel matrices are generated randomly from a continuous distribution, held constant throughout the transmission, and are known globally. The message sets, encoding, and decoding, and achievability are defined in the standard Shannon sense <cit.>. The encoder at nodei,ℰ_i,ℓ, can be either adaptive where dependence of_i,ℓon_i^ℓ-1is allowed, or restricted (nonadaptive) where it is not. 
These possibilities are shown in Table <ref>. 

The DoF region is the set of achievable DoF tuples=̣(d_12,d_13,d_21,d_23,d_31,d_32)∈ℝ_+^6defined as in <cit.>. Roughly speaking, if a rate tuple (function ofρ)

    (ρ)   =(R_12(ρ),R_13(ρ),R_21(ρ),R_23(ρ),R_31(ρ),R_32(ρ))

whereR_ij(ρ)is the rate of the message from nodeito nodej, is achievable, then the DoF tuple$̣ with d_ij=limsup_ρ→∞R_ij(ρ)/log(ρ) is achievable. We denote the DoF region under restricted encoding and adaptive encoding for a given τ by 𝒟_ r,τ and 𝒟_ a,τ, respectively, and we define the sum-DoF as d_ r,τ=max_∈̣𝒟_ r,τ_1 and d_ a,τ=max_∈̣𝒟_ a,τ_1.






Next, we describe a restricted transmission scheme, and we derive its achievable DoF region.







§ RESTRICTED ENCODING TRANSMISSION SCHEME


In this section, we prove the following theorem.



The DoF region of the 3WC satisfies 𝒟_ r,τ^[ in]⊆𝒟_ r,τ⊆𝒟_ a,τ, where the achievable inner bound 𝒟_ r,τ^[ in] is the set of ∈̣ℝ_+^6 satisfying the following for i,j∈{2,3}, i≠ j:

    d_1i+d_1j+τ d_ij   ≤τ M_1,   d_31+τ d_32   ≤τ M_3,
    
    d_i1+d_j1+τ d_ij   ≤τ M_1,   d_13+τ d_23   ≤τ M_3,
    
    d_i1+d_1j+τ d_ij   ≤τ M_2.



The inclusion of 𝒟_ r,τ in 𝒟_ a,τ is obvious. The achievability of 𝒟_ r,τ^[ in] is proved in the rest of this section.


Note that the factor τ in the inequalities above imposes a larger penalty on the streams going through the intermittent links. If we interpret τ M_3 in d_31+τ d_32≤τ M_3 as available resources, then increasing d_32 by 1 `eats' τ units of resources, while increasing d_31 by 1 `eats' 1 unit of resources. Thus, transmission between nodes 2 and 3 is `cheaper' by a factor of τ, no matter how large M_1 is as we shall see later. Next, we prove the achievability of 𝒟_ r,τ^[ in].








  §.§.§ Encoding

Each node splits its message w_ij into w_ij^[1] and w_ij^[2] to be sent using zero-forcing and interference alignment, respectively. Encoding proceeds as follows.

Since node 1 is available for a fraction of time, say m=s^n_0 out of the n transmissions, it encodes w_12^[q] and w_13^[q], q∈{1,2}, into codewords _12^[q]m and _13^[q]m with i.i.d. 𝒞𝒩(0,p_1_a_12^[q]) and 𝒞𝒩(0,p_1_a_13^[q]) symbols, respectively. Here a_ij^[q] is the vector length, and p_1 is the power of each component of _12^[q]m and _13^[q]m. Then, those codewords are extended to length n codewords _12^[q]n and _13^[q]n by inserting zeros where s_ℓ=0 (note that s^n is known at node 1).

Now, nodes 2 and 3 are available all the time, but they do not have apriori knowledge of s_ℓ. Thus, these nodes use standard random Gaussian codebooks to encode their messages, and treat the channel to node 1 as an erasure channel with erasure probability τ̅. Node 2 encodes w_21^[q] and w_23^[q], q∈{1,2}, into codewords _21^[q]n and _23^[q]n with i.i.d. 𝒞𝒩(0,p_2_a_21^[q]) and 𝒞𝒩(0,p_2_a_23^[q]) symbols, respectively. Similarly, node 3 encodes w_31^[q] and w_32^[q], q∈{1,2}, into codewords _31^[q]n and _32^[q]n with i.i.d. 𝒞𝒩(0,p_3_a_31^[q]) and 𝒞𝒩(0,p_3_a_32^[q]) symbols, respectively.

To satisfy the power constraint, the powers are chosen as

    p_1   =(a_12^[1]+a_12^[2]+a_13^[1]+a_13^[2])^-1m^-1nP,
    
    p_i   =(a_i1^[1]+a_i1^[2]+a_ij^[1]+a_ij^[2])^-1P,
 
i,j∈{2,3}, i≠ j. This encoding is restricted since it uses neither s^ℓ-1 at nodes 2 and 3, nor _i^ℓ-1 at nodes 1, 2, and 3.





  §.§.§ Transmission

At time ℓ, node i sends

    _i,ℓ   =∑_q=1^2[_ij^[q]_ij,ℓ^[q]+_ik^[q]_ik,ℓ^[q]],
 
where j,k∈{1,2,3}∖{i}, j≠ k, and _ij^[q]∈ℂ^M_i× a_ij^[q] is a beamforming matrix. Zero-forcing is achieved by choosing the matrices _ij^[1] so that _̋ik_ij^[1]=0 for distinct i,j,k∈{1,2,3}. These matrices exist if

    (M_i-M_k)^+≥ a_ij^[1],

ensuring that node i has enough antennas to send a_ij^[1] streams to node j without interfering with node k. To avoid any overlap of the transmit signals in the transmit signal space, we require

    ∑_q=1^2 (a_ij^[q]+a_ik^[q])≤ M_i.






  §.§.§ Decoding

Node 1 receives _1,ℓ=0 if s_ℓ=0 and 

    _1,ℓ   =∑_j=2^3∑_q=1^2_̋j1_j1^[q]_j1,ℓ^[q]+_23[[  _23ℓ^[2]; _32,ℓ^[2] ]]+_1,ℓ

otherwise, where _23=[_̋21_23^[2], _̋31_32^[2]]. This signal consists of four desired signals plus interference. To decode a desired signals, say _21^[1]n, node 1 zero-forces the remaining signals by multiplying _1^n by a post-coder _21^[1]∈ℂ^a_21^[1]× M_1 satisfying _21^[1]_21^[1]H=_a_21^[1] and

    _21^[1][_̋21_21^[2], _̋31_31^[1], _̋31_31^[2], _23]   =0,
    rank(_21^[1]_̋21_21^[1])   =a_21^[1].
 
After post-coding, node 1 is left with the signal 

_21,ℓ=_21^[1]_̋21_21^[1]_21,ℓ^[1]+_21^[1]_1,ℓ

if s_ℓ=1 and _21,ℓ=0 otherwise. This is an erasure channel over which the rate I(_21,ℓ^[1];_21,ℓ,s_ℓ)= I(_21,ℓ^[1];_21,ℓ|s_ℓ)=τ I(_21,ℓ^[1];_21,ℓ|s_ℓ=1) is achievable from node 2 to node 1 for n large.[Recall that s^n is known at the decoding stage.] Since we used Gaussian i.i.d. codes, this rate is

    τlog|_a_21^[1]+p_2/σ^2_21^[1]_̋21_21^[1]_21^[1]H_̋21^H_21^[1]H|,

leading to a DoF of τ a_21^[1] as long as (<ref>) is satisfied (cf. (<ref>)).

A similar procedure can be applied for decoding _21^[2]n, _31^[1]n, and _31^[2]n, to achieve DoF of τ a_21^[2], τ a_31^[1], τ a_31^[2]. The existence of the post-coders _i1^[q] which allow this procedure is guaranteed as long as the columns of 

    [_̋21_21^[1], _̋21_21^[2], _̋31_31^[1], _̋31_31^[2], _23]

are linearly independent. Let a̅_23^[2] be the dimension of span(_̋21_23^[2])∩ span(_̋31_32^[2]). Then, span(_23) has a_23^[2]+a_32^[2]-a̅_23^[2] dimensions, and the above linear independence is possible if we choose

    ∑_q=1^2(a_21^[q]+a_31^[q])+a_23^[2]+a_32^[2]-a̅_23^[2]≤ M_1.

To minimize the impact of interference, we choose _ij^[q] so that a̅_23^[2] is maximized. This can not be chosen arbitrarily large, as it has to be smaller than each of a_23^[2] and a_32^[2], and also smaller than the dimension of span(_̋21)∩ span(_̋31), which is (M_2+M_3-M_1)^+ almost surely. Thus,

    min{a_23^[2],a_32^[2],(M_2+M_3-M_1)^+}≥a̅_23^[2].



The same arguments can be applied at nodes 2 and 3, for decoding their desired signals. This achieves τ a_12^[1], τ a_12^[2], a_32^[1], and a_32^[2] DoF at node 2, and τ a_13^[1], τ a_13^[2], a_23^[1], and a_23^[2] DoF at node 3, leading to similar constraints as (<ref>) and (<ref>).







  §.§.§ Achievable DoF Region

The constraints can be combined as follows

    (M_i-M_k)^+   ≥ a_ij^[1],
    
    
    ∑_q=1^2 (a_ij^[q]+a_ik^[q])   ≤ M_i,
    min{a_ij^[2],a_ji^[2],(M_i+M_j-M_k)^+}   ≥a̅_ij^[2],
    ∑_q=1^2(a_ji^[q]+a_ki^[q])+a_jk^[2]+a_kj^[q]-a̅_jk^[2]   ≤ M_i.

for distinct i,j,k∈{1,2,3}, where a̅_12^[2] and a̅_13^[2] are the dimensions of span(_̋13_12^[2])∩ span(_̋23_21^[2]) and span(_̋12_13^[2])∩ span(_̋32_31^[2]), respectively. By adding the achievable DoF per stream, we obtain d_ij (e.g. d_21=τ a_21^[1]+τ a_21^[2]). Substituting d_ij in (<ref>)–(<ref>), using M_1≥ M_2≥ M_3 and Fourier Motzkin's elimination leads to the DoF region in Theorem <ref>. Details are omitted due to space limitations.





Next, we study the optimality of this scheme.






§ OPTIMALITY DISCUSSION





 §.§ Sum-DoF

We first consider the sum-DoF of the channel, and start by presenting the following DoF upper bounds.



The following must be satisfied by any DoF tuple ∈̣𝒟_ a,τ (and hence also ∈̣𝒟_ r,τ):

    d_13+d_23+d_21   ≤τ M_2+τ̅M_3,
    
    d_31+d_32+d_12   ≤τ M_2+τ̅M_3.



For brevity, we denote (W_ij,W_ik) by _i, and use ϵ_1n, ϵ_2n, and ϵ_3n to denote quantities that vanish as n→∞. Let _23 be an (M_2-M_3)× M_2 matrix so that _23≜[_̋23^T, _23^T]^T has full rank M_2. Such a matrix exists almost surely. Also, let _3,ℓ be defined as _23_2,ℓ+_3,ℓ if S_ℓ=1 and 0 otherwise, where _3,ℓ∼𝒞𝒩(0,σ_3^2_M_2-M_3), and define _3,ℓ=[_3,ℓ^T, _3,ℓ^T]^T. Now, consider any code for the 3WC, and let us establish a bound on R_13+R_23+R_21.[We write R_ij(ρ) simply as R_ij for brevity.] We give (_3^n,W_12) and (_3^n,_3,W_23) as side information to nodes 3 and 1, respectively. By Fano's inequality, we have

    n(R_13+R_23-ϵ_1n)   ≤ I(W_13,W_23;_3^n,S^n,_3,W_12),
    
    n(R_21-ϵ_2n)   ≤ I(W_21;_1^n,_3^n,S^n,_1,_3,W_23).

Recall that each node can estimate S^n with certainty from the received signals as assumed in the system model. Using the chain rule, the independence of the messages of each other and of S^n, and combining the two bounds yields

    n(R_13+R_23+R_21-ϵ_3n)   ≤ I(_2,W_13;_3^n|S^n,_3,W_12)
        +I(W_21;_1^n|_3^n,S^n,_1,_3,W_23).

The second term in this bound is equal to ∑_ℓ=1^n I(W_21;_1,ℓ|_1^ℓ-1,_3^n,S^n,_1,_3,W_23), which is no(log(ρ)),[lim_ρ→∞o(log(ρ))/log(ρ)=0.] since given _1, _3, _1^ℓ-1, _3^n, and S^n, and using the adaptive encoder, we can construct a noisy version of _1,ℓ for all ℓ with S_ℓ=1 given by _̋21_23^-1[[ _3,ℓ-_̋13_1,ℓ;          _3,ℓ ]]+_̋31_3,ℓ. 












On the other hand, using standard steps

    I(_2,W_13;_3^n|S^n,_3,W_12)
    
    
    
       ≤∑_ℓ=1^n I(_1,ℓ,_2,ℓ;_3,ℓ|S_ℓ)
    
    
    
    
       = ∑_ℓ=1^n τ I(_1,ℓ,_2,ℓ;_3,ℓ|S_ℓ=1)+τ̅ I(_2,ℓ;_3,ℓ|S_ℓ=0)
       ≤ n(τ M_2+τ̅M_3)log(ρ)+no(log(ρ)),

since the first and second terms represent (M_1+M_2)× M_2 and M_2× M_3 MIMO channels with M_2 and M_3 DoF almost surely (M_3≤ M_2), respectively. Combining terms, dividing by n and letting n→∞, this yields the bound

    R_13+R_23+R_21≤ (τ M_2+τ̅M_3)log(ρ)+o(log(ρ)),

which consequently leads to the first DoF bound. The second is obtained similarly by giving (_2^n,_2,W_32) and W_13 as side information to nodes 1 and 2, respectively.




































Based on Lemma <ref>, we can state the following theorem.


The sum-DoF of the intermittent 3WC is given by d_ r,τ=d_ a,τ=2τ M_2+2τ̅M_3.


Achievability follows from Theorem <ref> by using the simplex method <cit.> to maximize the sum-DoF subject to the DoF constraints. In particular, it follows by setting a_12^[1]=a_21^[1]=M_2-M_3, a_23^[2]=a_32^[2]=M_3, and a_12^[2]=a_21^[2]=a_23^[1]=a_32^[1]=d_13=d_31=0 in the scheme described in Sec. <ref>. The converse follows by adding the DoF bounds in Lemma <ref>.


This agrees with intuition. To maximize the sum-DoF, one should capitalize on the stable links between nodes 2 and 3, and use any remaining resources for communicating with the intermittent node 1. This theorem proves that adaptation is not necessary for achieving the sum-DoF of the intermittent 3WC. The same does not hold true from a DoF region perspective as we shall see next.





 §.§ DoF Region

In this section, we show that adaptation is necessary for achieving the DoF region of the intermittent 3WC. This result is particularly interesting in light of the following statement.


The scheme in Sec. <ref> achieves the DoF region of the nonintermittent 3WC (τ=1) given by 𝒟_ a,1=𝒟_ r,1^[ in].


The proof is based on upper bounds in <cit.>, and is omitted for lack of space.


Therefore, from a DoF-region point-of-view, adaptation is not necessary in the nonintermittent case.[Adaptation is still necessary from an achievable rate point-of-view <cit.>, but the gain of adaptation does not scale with ρ.] Interestingly, the same is not true in the intermittent case. To prove this, first we need a DoF outer bound for the restricted intermittent 3WC, and second, we need an adaptive scheme which achieves DoF tuples outside this outer bound. The first step is tackled in the following lemma. 



Under restricted encoding, we have 𝒟_ r,τ⊂𝒟_ r,τ^[ out] defined as the set of ∈̣ℝ_+^6 satisfying d_31+τ d_32≤τ M_3.


Let _2,ℓ=S_ℓ_2,ℓ and let us give (_2^n,_2) to node 1 as side information. From Fano's inequality, we have 

    n(R_31-ϵ_1n)
       ≤ I(W_31;_1^n,_2^n|S^n,_1,_2)
       = I(W_31;_2^n|S^n,_1,_2)
          + I(W_31;_1^n|S^n,_1,_2,_2^n).

Given _2^n, _1, _2, and S^n, we can construct a noisy version of _1^n for S_ℓ=1 given by _̋31_̋32^†(_2,ℓ-_̋12_1,ℓ)+_̋21_2,ℓ, where _̋32^† exists almost surely. Thus, I(W_31;_1^n|S^n,_1,_2,_2^n)=nτ o(log(ρ)), and hence


    n(R_31-ϵ_1n-τ o(log(ρ)))
    ≤ I(W_31;_2^n|S^n,_1,_2)
    
          = τ∑_ℓ=1^n I(W_31;_2,ℓ|S^n,_1,_2,_2^ℓ-1,S_ℓ=1).

On the other hand, giving (_1,W_31) to node 3 as side information and using Fano's inequality, we have

    n(τ R_32-ϵ_2n)
       ≤τ I(W_32;_2^n|S^n,_1,_2,W_31)
    
       = τ∑_ℓ=1^n [h(_2,ℓ|S^n,_1,_2,W_31,_2^ℓ-1)
          -h(_2,ℓ|S^n,_1,_2,_3,_2^ℓ-1)].

Since conditioning does not increase entropy, the first term is upper bounded by h(_2,ℓ|S^n,_1,_2,W_31,_2^ℓ-1). Moreover, since restricted encoding can be used to generate _1^n and _3^n from _1 and _3, the second entropy term is equal to h(_2,ℓ)=h(_2,ℓ|S^n,_1,_2,_3,_2^ℓ-1). Thus,

    n(τ R_32-ϵ_2n)
    
    
    ≤τ∑_ℓ=1^n I(W_32;_2,ℓ|S^n,_1,_2,W_31,_2^ℓ-1)
    
    
    
         = τ∑_ℓ=1^n I(W_32;_2,ℓ|S^n,_1,_2,W_31,_2^ℓ-1,S_ℓ=1),

since for a given ℓ, this mutual information is equal to I(W_32;_̋32_3,ℓ+_2,ℓ|S^n,_1,_2,W_31,_2^ℓ-1) independent of the state S_ℓ. Combining the two bounds yields

    n(R_31+τ R_32-ϵ_3n-τ o(log(ρ)))
       ≤τ∑_ℓ=1^n I(_3;_2,ℓ|S^n,_1,_2,_2^ℓ-1,S_ℓ=1)
    
    
    
    
    
    
       ≤ nτ M_3log(ρ)+nτ o(log(ρ))

which follows using similar steps as in the proof of Lemma <ref>.

This leads to the desired result.















Despite its simplicity, Lemma <ref> suffices for our purpose. Based on this lemma, the following theorem proves the necessity of adaptation in the intermittent case.


For an intermittent 3WC with M_1>M_3, 𝒟_ a,τ⊄𝒟_ r,τ^[ out], and hence adaptation is necessary.


It suffices to show that ∃∉̣𝒟_ r,τ^[ out] which is achievable using an adaptive scheme. To is end, suppose that only node 3 has a message to node 1, while node 2 acts as a relay to support node 3 which employs block-Markov encoding. Consider B transmission blocks, each consisting of n channel uses, and let a_2,a_3∈[0,1], be chosen so that a_2M_2,a_3M_3∈ℕ. In block 1, node 3 encodes a message w_31(1) to a codeword _31^n with _31,ℓ∈ℂ^a_3M_3, and sends it to node 1 using a_3M_3 antennas. Node 1 receives only m symbols corresponding to s_1,ℓ=1 where s_1^n is the state sequence in this block, with m≤ n and m/n≈τ as n grows. However, node 2 receives all symbols, and thus, obtains n-m codeword symbols from node 3 that have not been received by node 1. In block 2, node 3 sends w_31(2) similar to block 1, while nodes 2 cooperates with node 3. It does so by multiplying the received signal in block 1 by _̋32^† to obtain a noisy version of _31^n given by _31^n=_31^n+_2^n where _2,ℓ consists of a_3M_3 components of _̋32^†_2,ℓ, and then constructing _̌31^m out of _31,ℓ with ℓ∈{t∈{1,…,n}|s_1,t=0}, where _̌31,ℓ∈ℂ^a_2M_2. Then, it sends a new symbol of _̌31^m to node 1 in transmission ℓ if s_2,ℓ-1=1, and repeats the previously transmitted symbol otherwise. This construction requires ma_2M_2≤ (n-m)a_3M_3. The signal _̌31^m is sent from node 2 so that it is received linearly independent of _31^n at node 1. Thus, node 1 receives a total of m a_2M_2+m a_3M_3 symbols in this block if a_2M_2+a_3M_3≤ M_1. At the end of this block, node 1 is able to decode w_31(1) by combining its received signals from blocks 1 and 2. The same is repeated over blocks 3,…,B-1. In block B, only node 2 is active and delivers m a_2M_2 symbols to node 1. The achievable DoF is the ratio of the total number of delivered symbols to the total number of transmissions, i.e.,

    d_31=(B-1)(m a_3M_3+m a_2M_2)/nB≈τ a_3M_3+τ a_2M_2,

for large n and B. The constraints from above are

    0≤ a_2,a_3   ≤ 1,   a_2M_2,a_3M_3   ∈ℕ,
    
    a_2M_2+a_3M_3   ≤ M_1,   τ a_2M_2-τ̅a_3M_3   ≤0.

Now, we need to maximize d_31 with respect to a_2 and a_3 subject to these constraints. A feasible solution can be obtained as follows. First, we ignore the second constraint, which leads to a linear program which can be solved using the simplex method <cit.>. Solving the linear program leads to a_3^*=1 and a_2^*=min{M_1-M_3/M_2,1,τ̅M_3/τ M_2}. Then, we round a_2^*M_2 and a_3^*M_3 down to the nearest integer to obtain a_3=1, and a_2M_2=min{M_1-M_3,M_2,⌊τ̅M_3/τ⌋}. This leads to the achievability of min{τ M_1,τ M_2+τ M_3, τ M_3+τ⌊τ̅M_3/τ⌋}≜ d_31, a. Thus, the DoF tuple _̣ a=(0,0,0,0,d_31, a,0)∉𝒟_ r,τ^[ out] is achievable, which proves the desired result.


This theorem proves the necessity of adaptation in the intermittent case, where cooperation between nodes 2 and 3 becomes necessary to achieve higher DoF.










§ CONCLUSION


In this paper, we have investigated the impact of intermittency on the DoF region of the MIMO three-way channel. We have seen that adaptive encoding, can be either necessary or not, depending on the performance criterion. As far as the sum-DoF is concerned, adaptive encoding is not necessary, and the optimal sum-DoF can be achieved with restricted (nonadaptive) encoding. Since the sum-DoF might be unfair in terms of per-user or per-stream DoF, the DoF region is of high importance. In this case, we have shown that adaptation is in fact necessary, and that collaboration between nodes using adaptive encoding enlarges the DoF region beyond what can be achieved using adaptive encoding. 



10

url@samestyle








ChaabanSezgin_FnT
A. Chaaban and A. Sezgin, “Multi-way communications: An information theoretic
  perspective,” Found. Trends Commun. Inf. Theory, vol. 12, no. 3-4,
  pp. 185–371, 2015.

SongDevroyeShaoNgo
Y. Song, N. Devroye, H.-R. Shao, and C. Ngo, “Lattice coding for the two-way
  two-relay channel,” submitted to IEEE Trans. Inf. Theory,
  arXiv:1212.1198 [cs.IT], Dec. 2012.

ChengDevroyeLiu
Z. Cheng, N. Devroye, and T. Liu, “The Degrees of Freedom of full-duplex
  bidirectional interference networks with and without a MIMO relay,”
  IEEE Trans. Wireless Commun., vol. 15, no. 4, pp. 2912–2924, April
  2016.

SabharwalSchniterGuo
A. Sabharwal, P. Schniter, D. Guo, D. W. Bliss, S. Rangarajan, and R. Wichman,
  “In-band full-duplex wireless: Challenges and opportunities,” IEEE
  J. Sel. Areas Commun., vol. 32, no. 9, pp. 1637–1652, Sep. 2014.

TehraniUysalYanikomeroglu
M. N. Tehrani, M. Uysal, and H. Yanikomeroglu, “Device-to-device
  communications in 5G cellular netowrks: Challenges, solutions, and future
  directions,” IEEE Commun. Mag., pp. 86–92, May 2014.

Shannon_TWC
C. E. Shannon, “Two-way communication channels,” in Proc. of 4th
  Berkeley Symp. Math. Stat. Probab., vol. 1, 1961, pp. 611–644.

ChaabanMaierSezginMathar
A. Chaaban, H. Maier, A. Sezgin, and R. Mathar, “Three-way channels with
  multiple unicast sessions: Capacity approximation via network
  transformation,” IEEE Trans. Inf. Theory, vol. 62, no. 12, pp.
  7086–7102, Dec. 2016.

Ong
L. Ong, “Capacity results for two classes of three-way channels,” in
  Proc. IEEE Int. Symp. Commun. Inf. Tech. (ISCIT), Gold Coast, QLD,
  Oct. 2012.

ElmahdyKeyiMohassebElBatt
A. M. Elmahdy, A. El-Keyi, Y. Mohasseb, T. ElBatt, M. Nafie, and K. G. Seddik,
  “Asymmetric degrees of freedom of the full-duplex MIMO 3-way channel,” in
  Proc. IEEE Inf. Theory Workshop, Sep. 2016, pp. 469–473.

mach2015band
P. Mach, Z. Becvar, and T. Vanek, “In-band device-to-device communication in
  OFDMA cellular networks: A survey and challenges,” Commun. Surveys
  Tuts, vol. 17, no. 4, pp. 1885–1922, Nov. 2015.

KarakusWangDiggavi
C. Karakus, I. H. Wang, and S. Diggavi, “Gaussian interference channel with
  intermittent feedback,” IEEE Trans. Inf. Theory, vol. 61, no. 9, pp.
  4663–4699, Sep. 2015.

WangSuhDiggaviViswanath
I. H. Wang, C. Suh, S. Diggavi, and P. Viswanath, “Bursty interference channel
  with feedback,” in Proc. IEEE Int. Symp. Inf. Theory, July 2013, pp.
  21–25.

VahidMaddahAliAvestimehr
A. Vahid, M. A. Maddah-Ali, and A. S. Avestimehr, “Binary fading interference
  channel with no CSIT,” in Proc. IEEE Int. Symp. Inf. Theory, June
  2014, pp. 666–670.

YehWang
S. Y. Yeh and I. H. Wang, “Degrees of freedom of the bursty MIMO X channel
  without feedback,” in Proc. IEEE Int. Symp. Inf. Theory, July 2016,
  pp. 1312–1316.

Varshney
L. R. Varshney, “Two way communication over exponential family type
  channels,” in Proc. IEEE Int. Symp. Inf. Theory, July 2013, pp.
  2795–2799.

ChengDevroye
Z. Cheng and N. Devroye, “Two-way networks: When adaptation is useless,”
  IEEE Trans. Inf. Theory, vol. 60, no. 3, pp. 1793–1813, March 2014.

SongAlajajiLinder
L. Song, F. Alajaji, and T. Linder, “Adaptation is useless for two discrete
  additive-noise two-way channels,” in Proc. IEEE Int. Symp. Inf.
  Theory, July 2016, pp. 1854–1858.

CoverThomas
T. Cover and J. Thomas, Elements of Information Theory (Second
  Edition).1em plus 0.5em minus 0.4emJohn Wiley and Sons, Inc.,
  2006.

JafarShamai_XChannel
S. A. Jafar and S. Shamai, “Degrees of freedom region for the MIMO X
  channel,” IEEE Trans. Inf. Theory, vol. 54, no. 1, pp. 151–170,
  Jan. 2008.

MatousekGartner
J. Matous̆ek and B. Gärtner, Understanding and using linear
  programming.1em plus 0.5em minus 0.4emSpringer, 2006.

MaierChaabanMathar
H. Maier, A. Chaaban, and R. Mathar, “Degrees of Freedom of the MIMO 3-way
  channel,” in Proc. Int. Symp. Inf. Theory Appl., Melbourne,
  Australia, Oct. 2014, pp. 625–629.



