Joint Uplink-Downlink Cell Associations for Interference Networks with Local Connectivity
Manik Singhal Aly El Gamal





Joint Uplink-Downlink Cell Associations for Interference Networks with Local Connectivity
    Manik Singhal and Aly El Gamal
ECE Department, Purdue University
 Email: {msingha,elgamala}@purdue.edu
    Received: date / Accepted: date
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We study information theoretic models of interference networks that consist of K Base Station (BS) - Mobile Terminal (MT) pairs. Each BS is connected to the MT carrying the same index as well as L following MTs, where the connectivity parameter L ≥ 1. We fix the value of L and study large networks as K goes to infinity. We assume that each MT can be associated with N_c BSs, and these associations are determined by a cloud-based controller that has a global view of the network. An MT has to be associated with a BS, in order for the BS to transmit its message in the downlink, or decode its message in the uplink. In previous work, the cell associations that maximize the average uplink-downlink per user degrees of freedom (puDoF) were identified for the case when L=1. Further, when only the downlink is considered, the problem was settled for all values of L when we are restricted to use only zero-forcing interference cancellation schemes. In this work, we first propose puDoF inner bounds for arbitrary values of L when only the uplink is considered, and characterize the uplink puDoF value when only zero-forcing schemes are allowed. We then introduce new achievable average uplink-downlink puDoF values. We show that the new scheme is optimal for the range when N_c ≤L/2 when we restrict our attention to zero forcing schemes. Additionally we conjecture that the having unity puDoF during uplink is optimal when N_c ≥ L.




§ INTRODUCTION



The fifth generation of cellular networks is expected to bring new paradigms to wireless communications, that exploit recent technological advancements like cloud computing and cooperative communication (also known as Coordinated Multi-Point or CoMP). In particular, the rising interest in Cloud Radio Access Networks (C-RAN) (see e.g., <cit.>-<cit.>) holds a promise for such new paradigms. These paradigms require new information theoretic frameworks to identify fundamental limits and suggest insights that are backed by rigorous analysis. The focus of this work is to identify associations between cell edge mobile terminals and base stations, that maximize the average rate across both uplink and downlink sessions, while allowing for associating one mobile terminal with more than one base station and using cooperative transmission and reception schemes between base stations in the downlink and uplink sessions, respectively. With a cloud-based controller, optimal decisions for these associations can take into account the whole network topology, with the goal of maximizing a sum rate function. 



Cloud-based CoMP communication is a promising new technology that could significantly enhance the rates of cell edge users (see <cit.> for an overview of CoMP). In <cit.>, an information theoretic model was studied where cooperation was allowed between transmitters, as well as between receivers (CoMP transmission and reception). CoMP transmission and reception in cellular networks are applicable in the downlink and uplink, respectively. The model in <cit.> assumed that each message can be available at M_t transmitters and can be decoded through M_r received signals. It was shown that full Degrees of Freedom (DoF) can be achieved if M_t+M_r ≥ K+1, where K is the number of transmitter-receiver paris (users) in the network.  

Recently in <cit.>, alternative frameworks for cooperation in both downlink and uplink were introduced. The new frameworks are based on the concept of message passing between base stations. In the downlink, quantized versions of the analog transmit signals are being shared between base station transmitters. The supporting key idea is that information about multiple messages can be shared from one transmitter to another with the cost of sharing only one whole message (of the order of log P, where P is the transmit power), if we only share information needed to cancel the interference caused by the messages at unintended receivers, through dirty paper coding (see <cit.>). In the uplink, decoded messages are shared from one base station receiver to another, where they are used to cancel interference. It was shown in <cit.> that there is a duality in this framework between schemes that are used in the downlink and those that are used for the uplink, with the clear advantage that the same backhaul infrastructure can be used to support both scenarios. 



In this work, we first characterize the puDoF of message passing decoding in the uplink of locally connected interference networks when N_c < L/2. We then consider the problem of jointly optimizing the assignment of messages over the backhaul to maximize the average puDoF across both downlink and uplink sessions. We assume that each base station can be associated with N_c mobile terminals, and that an association is needed whenever a mobile terminal's message is used by a base station in either the downlink or the uplink. This usage of a message could be either for delivering the message in downlink, decoding the message in uplink, or for interference cancellation. This problem was first considered in <cit.>, where the average puDoF was characterized for the case when L=1. Here, we consider general values of L, and first show how our new result for the uplink settles the average puDoF problem when N_c ≤L/2. We then tackle this problem when N_c > L, by fixing the uplink scheme to the optimal uplink-only scheme, that associates each mobile terminal with the L+1 base stations connected to it, and characterize the optimal downlink scheme under this constraint. The intuition behind this step is that full DoF is achieved in the uplink when N_c > L through associating each mobile terminal with all L+1 base stations connected to it: any change in that cell association is expected to decrease the uplink puDoF with a factor greater than the gain achieved for the downlink puDoF.


When considering this work, it is important to note that the assumptions in a theoretical framework need not reflect directly a practical setting, but are rather used to define a tractable problem whose solution can lead to constructive insights. For example, it was shown in <cit.> that imposing a downlink backhaul constraint where each message can be available at a specified maximum number of transmitters (maximum transmit set size constraint), can lead to solutions that are also useful to solve the more difficult and more relevant to practice problem, where an average transmit set size constraint is used instead of the maximum. Also, in <cit.>, it was shown that solutions obtained for the locally connected network models, that are considered in this work, can be used to obtain solutions for the more practical cellular network models, by viewing the cellular network as a set of interfering locally connected subnetworks and designing a fractional reuse scheme that avoids interference across subnetworks.



 §.§ Prior Work

In <cit.>, the considered problem was studied for Wyner's linear interference networks (channel model was introduced in <cit.>). The optimal message assignment and puDoF value were characterized. Linear networks form the special case of our problem when L=1. Here, all our results are for general values of the connectivity parameter L. Also, in <cit.>, the downlink part of our problem was considered, and the optimal message assignment (cell association) and puDoF value were characterized for general values of the connectivity parameter L, when we restrict our attention to zero-forcing (or interference avoidance) scheme.  


 §.§ Document Organization

In Section <ref>, we present the problem setup. In Section <ref>, we discuss previous work on zero-forcing CoMP transmission schemes for the downlink. We then present bounds for the puDoF of the uplink in Section <ref>, and prove the converse in Sections  <ref> and  <ref>. In Section <ref>, we present new achievable puDoF values when the average of the uplink and downlink is considred. We finally present concluding remarks in Section <ref>.



§ SYSTEM MODEL AND NOTATION

For each of the downlink and uplink sessions, we use the standard model for the K-user interference channel with single-antenna transmitters and receivers,

    Y_i(t) = ∑_j=1^K H_i,j(t) X_j(t) + Z_i(t),

where t is the time index, X_j(t) is the transmitted signal of transmitter j, Y_i(t) is the received signal at receiver i, Z_i(t) is the zero mean unit variance Gaussian noise at receiver i, and H_i,j(t) is the channel coefficient from transmitter j to receiver i over time slot t. We remove the time index in the rest of the paper for brevity unless it is needed. The signals Y_i and X_i correspond to the receive and transmit signals at the i^th base station and mobile terminal in the uplink, respectively, and the i^th mobile terminal and base station in the downlink, respectively. For consistency of notation, we will always refer to H_i,j as the channel coefficient between mobile terminal i and base station j.



 §.§ Channel Model

We consider the following locally connected interference network. The mobile terminal with index i is connected to base stations {i,i-1,⋯,i-L}, except the first L mobile terminals, which are connected only to all the base stations with a similar or lower index. More precisely, 


    H_i,j = 0  iff  i ∉{j,j+1,⋯,j+L},∀ i,j ∈ [K],

and all non-zero channel coefficients are drawn from a continuous joint distribution. Finally, we assume that global channel state information is available at all mobile terminals and base stations. 



 §.§ Cell Association

For each i ∈ [K], let C_i ⊆ [K] be the set of base stations, with which mobile terminal i is associated, i.e., those base stations that carry the terminal's message in the downlink and will have its decoded message for the uplink. The transmitters in C_i cooperatively transmit the message (word) W_i to mobile terminal i in the downlink. In the uplink, one of the base station receivers in C_i will decode W_i and pass it to the remaining receivers in the set. We consider a cell association constraint that bounds the cardinality of the set C_i by a number N_c; this constraint is one way to capture a limited backhaul capacity constraint where not all messages can be exchanged over the backhaul. 

    | C_i| ≤ N_c, ∀ i∈ [K].


We would like to stress on the fact that we only allow full messages to be shared over the backhaul. More specifically, splitting messages into parts and sharing them as in <cit.>, or sharing of quantized signals as in <cit.> is not allowed. 



 §.§ Degrees of Freedom

Let P be the average transmit power constraint at each transmitter, and let W_i denote the alphabet for message W_i. Then the rates R_i(P) = log| W_i|/n are achievable if the decoding error probabilities of all messages can be simultaneously made arbitrarily small for a large enough coding block length n, and this holds for almost all channel realizations. The degrees of freedom d_i, i∈[K], are defined as d_i=lim_P →∞R_i(P)/log P. The DoF region D is the closure of the set of all achievable DoF tuples. The total number of degrees of freedom (η) is the maximum value of the sum of the achievable degrees of freedom, η=max_ D∑_i ∈ [K] d_i.

For a K-user locally connected with connectivity parameter L, we define η(K,L,N_c) as the best achievable η on average taken over both downlink and uplink sessions over all choices of transmit sets satisfying the backhaul load constraint in (<ref>). 
In order to simplify our analysis, we define the asymptotic per user DoF (puDoF) τ(L,N_c) to measure how η(K,L,N_c) scales with K while all other parameters are fixed,

    τ(L,N_c) = lim_K→∞η(K,L,N_c)/K.


We further define τ_D (L,N_c) and τ_U (L,N_c) as the puDoF when we optimize only for the downlink and uplink session, respectively.



 §.§ Interference Avoidance Schemes
 
We consider in this work the class of interference avoidance schemes, where every receiver is either active or inactive. An active receiver can observe its desired signal with no interference. In the downlink, we are considering cooperative zero-forcing where a message's interference is cancelled over the air through cooperating transmitters. In the uplink, we are considering message passing decoding where a decoded message is passed through a cooperating receiver to other receivers wishing to remove the message's interference.

We add the superscript zf to the puDoF symbol when we impose the constraint that the coding scheme that can be used has to be a zero-forcing scheme. For example, τ_U^zf(L,N_c) denotes the puDoF value when considering only the uplink and impose the restriction to zero-forcing schemes. 



§ PRIOR WORK: DOWNLINK-ONLY SCHEME

In <cit.>, the considered setting was studied for only downlink transmission. When restricting our choice of coding scheme to zero-forcing schemes, the puDoF value was characterized as,

    τ_D^zf(L,N_c)=2N_c/2N_c+L,

and the achieving cell association was found to be the following. The network is split into subnetworks; each with consecutive 2N_c+L transmitter-receiver pairs. The last L transmitters in each subnetwork are inactive to avoid inter-subnetwork interference. The zero-forcing scheme aims to deliver 2N_c messages free of interference in each subnetwork, so that the acheived puDoF value is as in (<ref>). In order to do that with a cooperation constraint that limits each message to be available at N_c transmitters, we create two Multiple Input Single Output (MISO) Broadcast Channels (BC) within each subnetwork; each with N_c transmitter-receiver pairs, and ensure that interference across these channels is eliminated. We now discuss the cell association in the first subnetwork, noting that the remaining subnetworks follow an analogous pattern. The first MISO BC consists of the first N_c transmitter-receiver pairs. For each i∈{1,2,⋯,N_c}, message W_i is associated with base stations with indices in the following set, C_i={i,i+1,⋯,N_c}. The second MISO BC consists of the N_c transmitters with indices in the set {N_c+1,N_c+2,⋯,2N_c} and the N_c receivers with indices in the set {N_c+L+1,N_c+L+2,⋯,2N_c+L}. Note that the middle L receivers in each subnetwork are deactivated to eliminate interference between the two MISO BCs. For each i∈{N_c+L+1,N_c+L+2,⋯,2N_c+L}, message W_i is associated with transmitters that have indices in the set C_i={i-L,i-L-1,⋯,N_c+1}. It was shown in <cit.> that the puDoF value of (<ref>) achieved by this scheme is that best achievable value in the downlink using the imposed cooperation constraint and zero-forcing schemes.




§ UPLINK-ONLY SCHEME

We discuss in this section backhaul designs that optimize only the uplink rate, and consider only zero-forcing coding schemes. More precisely, we show that the following theorem holds.



The asymptotic puDoF for the uplink when considering message passing schemes is characterised by the following equation: 

    τ_U^zf(L,N_c) =
                
                    1     L + 1 ≤ N_c, 
    N_c + 1/L + 2   L/2≤ N_c ≤ L, 
    2N_c/2N_c + L    1 ≤ N_c ≤L/2 - 1.



The cell association that is used to achieve the above is as follows. When N_c ≥ L+1, each mobile terminal is associated with the L + 1 base stations connected to it. The last base station, with index K, in the network decodes the last message and then passes it on to the L other base stations connected to the K^th mobile terminal, eliminating all interference caused by that mobile terminal. Each preceding base station then decodes its message and passes it on to the other base stations, eliminating the interference caused by the message. Thus, one degree of freedom is achieved for each user.

In the second range L/2≤ N_c ≤ L, the cell association that is used to achieve a puDoF value of N_c + 1/L + 2 is as follows. The network is split into subnetworks, each with consecutive L+2 transmitter-receiver pairs. In each subnetwork, the last N_c + 1 words are decoded. For each i ∈{L+2, L+1, ⋯, L + 2 - N_c + 1}, message W_i is associated with base stations {i,i-1,⋯,L + 2 - N_c + 1}⊆ C_i. Thus the last N_c words are decoded. The base stations with indices in the set {2,3,⋯,L + 2 - N_c} are inactive as there is interference from the last transmitter in the subnetwork which cannot be eliminated. The first base station decodes W_L+2-N_c. To eliminate the interference caused by the transmitters in the set S = {L + 2 - N_c + 1, L + 2 - N_c + 2, ⋯, L + 1} at the first base station of the subnetwork, we add the first base station to each C_i, ∀ i ∈ S. Now for messages with indices in the set S, we have used β_i = 2 + i - (L + 2 - N_c + 1) associations up to this point; the factor of two comes from the base station resolving W_i and the first base station of the subnetwork. But each transmitter with indices in the set S\{L + 1} also interferes with the subnetwork directly preceding this subnetwork. ∀ i ∈ S\{L + 1}, the message W_i interferes with the bottom L + 1 - i base stations of the preceding subnetwork, which is precisely the number of associations left for the respective message i.e. N_c - β_i = L + 1 - i, thus inter-subnetwork interference can be eliminated at those base stations. 

In the third range 1 ≤ N_c ≤L/2 - 1, the cell association that is used to achieve the lower bound of 2N_c/2N_c + L is similar to the one described in Section <ref> for the downlink. The network is split into disjoint subnetworks; each with consecutive 2N_c + L transmitter-receiver pairs. For the uplink, we consider two sets of indices for transmitters A_T = {1,2,⋯, N_c} and B_T = {N_c + L + 1,N_c + L + 2⋯, 2N_c + L}, and corresponding sets of receivers A_R = {1,2,⋯, N_c} and B_R = {N_c + 1,N_c + L + 2⋯, 2N_c}. For each i ∈ A_T, the message W_i is associated with the receivers receiving it in A_R. Receiver i decodes W_i and the other associations in C_i exist for eliminating interference. Similarly For each j ∈ B_T, the message W_j is associated with the receivers receiving it in B_R, but now receiver j - L decodes W_j and the other associations in C_j are for eliminating interference.

We observe that if we were not restricted to the zero-forcing coding scheme then for the third range, we could achieve 1/2 puDoF using asymptotic interference alignment <cit.>, which is higher than the value achieved by zero-forcing. The next sections complete the proof of Theorem <ref>



§ CONVERSE PROOF WHEN L/2≤ N_C ≤ L


In this section, we provide a converse proof for the second range of (<ref>). More precisely, we show that the following holds.

    τ_U^zf (L,N_c) = N_c+1/L+2,     L/2≤ N_c ≤ L.

We start by proving the case when N_c=L, The optimal zero-forcing puDoF for the uplink can be characterised as:


    τ_U^zf(L,L) =           
                    L + 1/L + 2.


We begin by dividing the network into subnetworks of L + 2 consecutive transmitters-receiver pairs. We observe that in any subnetwork, if we have N_c + 1=L+1 consecutive active receivers (base stations), then the transmitter connected to all these receivers must be inactive, because a message's interference cannot be canceled at N_c or more receivers. Let Γ_BS be the set of subnetworks where all N_c + 2 receivers are active, and Φ_BS be the set of subnetworks with at most N_c active receivers. Similarly, let Γ_MT and Φ_MT be the subnetworks with N_c+2 active transmitters and at most N_c active transmitters, with respect to order. To be able to achieve a higher puDoF than (<ref>), it must be true that both conditions hold: | Γ_BS|>| Φ_BS| and | Γ_MT|>| Φ_MT|. Now note that for any subnetwork that belongs to Γ_BS, at most N_c transmitters will be active, because the interference caused by any message cannot be canceled at N_c or more receivers. Hence Γ_BS⊆Φ_MT. Further, the same logic applies to conclude that for any subnetwork with N_c+1 active receivers, the number of active transmitters is at most N_c+1, and hence Γ_MT⊆Φ_BS. It follows that if |Γ_BS|>|Φ_BS|, then |Γ_MT| < |Φ_MT|, and hence the statement in (<ref>) is proved.




To aid in the next step we define MT-BS pairs (m_i, b_j) as decoding pairs if W_i is decoded at base station j.
To prove that τ_U^zf(L,N_c)=N_c + 1/L + 2 when L/2≤ N_c < L, we use the following lemmas :



	For any zero-forcing scheme, one of the following is true for any two decoding pairs (m_i_1, b_j_1) and (m_i_2, b_j_2): j_2 ∉{i_1, i_1 -1, ⋯, i_1 - L} or j_1 ∉{i_2, i_2 -1, ⋯, i_2 - L}.


If the claim were not true, i.e. j_2 ∈{i_1, i_1 -1, ⋯, i_1 - L} and j_1 ∈{i_2, i_2 -1, ⋯, i_2 - L} then W_i_1 and W_i_2 would interfere with one another and could not be decoded using the zero-forcing scheme. This is a consequence of the work done in  <cit.>.




	For any set L ⊆ [K] of L + 1 consecutive indices, a maximum of N_c mobile terminals with indices in L can be decoded at base stations with indices in L for any zero-forcing scheme.


We prove this claim by contradiction. If N_c + 1 or more mobile terminals with indices in L are decoded at base stations with indices in L, then at least one of the mobile terminals would be associated with more than N_c base stations. This violates the constraint in  (<ref>).


From Lemma <ref>, we have the following corollary:


	For any two decoding pairs (m_i_1, b_j_1) and (m_i_2, b_j_2) in a zero-forcing scheme, if i_1 > i_2 then j_1 > j_2 and vice versa.


Immediately from Lemma <ref> we have that subnetwork only decoding, i.e. transmissions from a subnetwork are decoded in the same subnetwork, can only decode at most N_c + 1 words in each subnetwork of L+2 consecutive BS-MT pairs.

 Our proof will be based on the concept that to break the inner bound described in (<ref>), at least one subnetwork of L+2 consecutive MT-BS pairs must have more than N_c + 1 active mobile terminals. And all such subnetworks must borrow base stations from the subnetwork above it to decode words corresponding to its own mobile terminals. 
This happens because for a consecutive set of L+2 mobile terminals, the only base stations that can help decode their transmissions are the corresponding base stations or the other L base stations with preceding indices that are connected to the set. 

To aid in the writing we define : α_k = (L+2) × (k-1), here α_k denotes the first index of each subnetwork L_k. In this sense L_k is topologically below L_k-1, i.e. mobile terminals from L_k are connected to some base stations in L_k-1. Additionally MT_i denotes mobile terminal i, and BS_j denotes base station j


We use the above lemmas and definitions to define a best case scenario for inter-subnetwork interference. A best case scenario is where the interference from one subnetwork's (e.g., L_k) mobile terminals to another subnetwork's (e.g., L_k-1) base stations is focused on the bottom most base stations. This is defined as the best case scenario because from Lemma <ref>, we know that for L_k-1's own mobile terminals to be decoded in L_k-1 , we need base stations that are indexed outside the range of the interference from the mobile terminals of L_k. We also define that if there exists decoding pairs (m_i, b_j) such that i ≥α_k and j < α_k, i.e. the mobile terminal is in L_k and the base station is in L_k-1, then L_k borrows a base station from L_k-1. Similarly, if there exists certain consecutive base stations in L_k-1 indexed by (α_k - μ, α_k - μ + 1, ⋯α_k - 1 ) such that no words can be decoded here in the zero-forcing scheme due to the cooperation constraint being met in L_k, we say that L_k blocks μ base stations in L_k-1.

We introduce two new variables x and δ. Here x defines the number of extra mobile terminals (beyond N_c+1) active in a subnetwork of L+2 consecutive mobile terminals and base stations, and δ defines the number of base stations that L_k borrows from L_k-1 to help decode words from L_k.
When L/2≤ N_c ≤ L it follows from the network topology and the defined cooperation constraint that we have 1 ≤ x,δ≤ N_c. 

We want to show that τ_U^zf(L,N_c) ≤N_c + 1/L + 2 when L > N_c ≥L/2. It follows from the pigeonhole principle that to break this bound, there must be at least one subnetwork (say  L_k) where we have N_c + 1 + x mobile terminals active. Now by Lemma <ref>, we have that L_k must borrow at least x base stations from L_k-1, thus x ≤δ≤ N_c. We now consider possible cases for the value of δ.

When δ = 1, thus x = 1, so  L_k has N_c + 2 active mobile terminals. As L_k  is borrowing one base station, say base station j, N_c + 1 words must have been decoded in  L_k. By Lemma <ref>, there exists at least one decoding pair (m_i,b_n) where i,n ≥α_k, such that b_n is not connected to the highest indexed active mobile terminal in L_k. Due to the size of the subnetwork, this forces n = α_k. Hence, mobile terminal i's transmission is decoded at the first base station of L_k. By Lemma <ref>, this implies that j ∉{i, i-1, ... i-L}. It follows that the best case scenario occurs when i = α_k + (L + 2 - (N_c + 1)), making j ≤α_k - N_c = α_k-1 + L+2-N_c. Let the number of available base stations left in L_k-1 be θ. As N_c ≥L/2, it follows that θ≤ L + 2 - N_c ≤ N_c + 2. Additionally, due to the borrowed base station, the number of associations allowed for MT_α_k-1 + L+1 (the last mobile terminal in L_k-1) has effectively reduced by one. From Lemma <ref>, we have that a maximum of N_c mobile terminals can be decoded in  L_k-1. It follows that either the average number of active mobile terminals over the two subnetworks is still N_c + 1 per subnetwork, or L_k-1 will have to borrow at least one base station from  L_k-2. We do not consider the former case, as we just have to restart our argument from L_k-2 because all subnetworks with higher indexes  will have an average of N_c +1 active mobile terminals per subnetwork. Hence, we only consider the latter case where L_k-1 borrows at least one base station from  L_k-2. As base station α_k-1 is being used in L_k-1, the lowest possible indexed base station that L_k-1 borrows from L_k-2 is base station α_k-2 + (L+2 - N_c). Therefore the argument for L_k-2 borrowing base stations from L_k-3 is exactly the same as the argument shown for L_k-1 borrowing from L_k-2. It follows that this borrowing will continue till either we stop borrowing at some subnetwork L_i, where i < k, or L_1 needs to borrow at least one more base station, which is not possible. If L_i does not borrow from L_i-1, we have that L_i and L_k have at most N_c and N_c + 1 + 1 active mobile terminals, respectively, and all other subnetworks between them have at most N_c + 1 active mobile terminals, resulting in an average of N_c +1 active mobile terminals per subnetwork over these k-i subnetworks. Thus we can discard them as they do not break the inner bound and start the same argument over from L_i-1. If we continue borrowing till L_1, we have that L_1 and L_k have at most N_c and N_c + 1 + 1 active mobile terminals respectively and all other subnetworks have at most N_c + 1 active mobile terminals, resulting that the average number of active mobile terminals over the whole network is N_c + 1  per subnetwork which implies that τ_U^zf(L,N_c) ≤N_c + 1/L + 2. This presents the simplest case for our iterative argument.


When δ > 1, we have a similar argument as described in the previous paragraph. By Lemma <ref>, we have that the borrowed base stations in  L_k-1 will have to send the associations downwards, i.e. the lowest indexed borrowed base station in  L_k-1 will have to be exclusively connected to the lowest indexed active mobile terminal of  L_k. As the index of the lowest active mobile terminal in L_k is at most α_k + (L + 2 - (N_c + 1 + x)) - 1, we have that the index of the lowest borrowed base station in L_k-1 is α_k-1 + (L+3 - N_c - x). Therefore the number of available base stations in L_k-1 can be expressed as L + 3 - N_c - x. These available base stations must at least decode N_c + 1 + (1 - x) mobile terminals' transmissions to have an average greater than N_c + 1 active mobile terminals per subnetwork over L_k and L_k-1 without L_k-1 borrowing base stations from L_k-2. This cannot happen when L + 3 - N_c - x < N_c + 1 + 1 - x, which is only possible when N_c > L+1/2. Hence, the condition N_c > L+1/2 implies that  L_k-1 has to borrow at least one base station from  L_k-2, which presents an iterative argument as the one shown when δ=1.

Now we consider the case when L/2≤ N_c ≤L+1/2. By Lemma <ref>, we also have that the maximum number of mobile terminals from  L_k-1 decoded in L_k-1's available base stations is N_c. As we only need N_c + 2 - x active mobile terminals decoded to break the inner bound defined, L_k-1 will not have to borrow from  L_k-2 when x ≥ 2. At least N_c + 2 - x mobile terminals' transmissions must be decoded in  L_k-1, but MT_α_k-1 + L+1 has its associations reduced by δ≥ x.  Using MT_α_k-1 + L+1, a maximum of N_c - δ + 1 ≤ N_c + 1 - x transmitted words can be decoded within L_k-1, which will lead us to have L_k-1 borrowing at least one base station from  L_k-2. This presents another iterative argument, akin to the one shown above.

In order to achieve a case where L_k-1 does not have to borrow base stations from L_k-2, our best case scenario guides us to find the first mobile terminal in L_k-1, which is connected to at most  x - 2 base stations that are being borrowed by L_k, but still connected to at least N_c + 2 - x available base stations in L_k-1. Assume that the index of that mobile terminal is α_k-1 + ν. Clearly, ν≤ (L + 2 - N_c - x) + (x - 2) = L - N_c. So in L_k-1 we have N_c + 2 - x active mobile terminals without borrowing from L_k-2, but mobile terminal α_k-1 + ν has already used up all its associations and it is connected to some base stations in L_k-2, specifically at least N_c base stations. Hence, L_k-2 has a maximum L + 2 - N_ c ≤ N_c + 2 base stations available to decode more transmissions, and we need at least N_c + 1 words to be decoded here, which can be done, but this would imply that at least two mobile terminals are associated with N_c base stations. These two mobile terminals are indexed higher than κ, where κ = α_k-2 + L + 1 - (N_c + 1). Hence, L_k-2 blocks at least N_c of the bottom L base stations in L_k-3, and one can see that each further subnetwork blocks at least one base station from the preceding subnetwork for the average number of active mobile terminals per subnetwork to remain above N_c+1. If say L_i does not block any base stations in L_i-1, then L_i can have at most N_c active mobile terminals decoded in L_i. It follows that either L_i borrows from L_i-1 or only has N_c active mobile terminals. If L_i borrows from L_i-1 we have a similar iterative argument as shown above. Otherwise, L_i has only N_c active mobile terminals, making the average number of active mobile terminals through the considered k-i subnetworks N_c + 1 per subnetwork. Hence, each subnetwork continues blocking base stations in the preceding subnetwork and the extra active mobile terminals in the whole network does not scale and is fixed by the constant x, which shows that the average number of active mobile terminals asymptotically approaches N_c + 1 for every subnetwork of size L + 2.

We have shown that if any subnetwork has more than N_c + 1 active mobile terminals when L ≥ N_c ≥L/2, either the number of extra active mobile terminals do not scale with size of the network, or the average over the whole network remains bounded by N_c + 1 active mobile terminals per subnetwork. This forces that the average number of decoded words per subnetwork is at most N_c + 1, implying that the asymptotic puDoF during the uplink using zero forcing, τ_U^zf(L,N_c) ≤N_c + 1/L + 2. We have shown in Section <ref> that τ_U^zf(L,N_c) ≥N_c + 1/L + 2, implying that τ_U^zf(L,N_c) = N_c + 1/L + 2 whenever L/2≤ N_c ≤ L. The proof of (<ref>) is thus complete.



§ CONVERSE PROOF FOR UPLINK WHEN N_C < L/2


In this section, we provide a converse proof for the third range of (<ref>). More precisely, we show that the following holds.

    τ_U^zf (L,N_c) = 2N_c/2N_c + L,      N_c < L/2.


Similar to Section <ref> Our proof will be based on the concept that to break the inner bound described in (<ref>), at least one subnetwork of 2N_c + L consecutive MT-BS pairs must have more than 2N_c active mobile terminals. And all such subnetworks must either borrow or block base stations from the subnetwork above it to decode words corresponding to its own mobile terminals. 
 
This happens because for a consecutive set of 2N_c + L mobile terminals, the only base stations that can help decode their transmissions are the corresponding base stations or the other L base stations with preceding indices that are connected to the set.

Using the Lemmas and definitions from Section <ref> we start our proof. We present cases on x, and δ.

Firstly we notice as each subnetwork has 2N_c + L mobile-terminals, base-stations pairs so there is a possibility of decoding more then 2N_c transmissions using subnetwork-only-decoding (SO-decoding) in L_k. We first show that only a maximum of 2N_c + 1 transmissions can be decoded using SO-decoding.By Lemma <ref>, to decode 2N_c + 2 transmissions using SO-decoding you need a subnetwork of 2*L + 1 mobile-terminal and base stations pairs which is larger than 2N_c + L. Thus L_k can decode at most 2N_c + 1 using SO-decoding

Thus, our first case is x = 1 and δ = 0. In L_k we have that the highest indexed base stations that can decode the 2N_c + 1 transmissions are the L + 1 + N_c + 1 = L + N_c + 2 highest indexed base stations. But to transmit and decode 2N_c +1 transmissions in L_k there will be at least 3 mobile terminals that achieve the maximum number of associations. By topology, only the lowest indexed such terminal is the one that blocks base stations in L_k-1. Specifically it blocks L  - 2N_c + 3 base stations, thus only 4N_c - 3 base stations are left to decode at least 2N_c + 1 - 1 base stations.

From these 4N_c - 3, L_k-1 needs at least L + 2 - (L - 2N_c + 3) base stations to decode N_c + 1 transmissions. But at least 2 of the mobile terminals transmitting to these base stations would meet its max constraint, and if you utilize exactly L + 2 - (L - 2N_c + 3) base stations then by Corollary <ref> only 1 of the highest L - 2N_c + 3 can be transmitting. Similar to above we only consider the lower indexed mobile terminal which reaches its maximum association constraint. This mobile terminal is indexed at most α_k-1 + 4N_c - 5, which results that including the base stations which are decoding transmissions and those which cant due association constraints at least L + 2 of the 4N_c - 3 base stations are used up. We are left with 4N_c - 3 - (L + 2) = 4N_c - L - 5 ≤ 2N_c - 6 base stations as N_c ≤L - 1/2. Now these 2N_c - 6 base stations must decode at least N_C - 1 transmissions, thus at least N_c - 2 of these transmissions must come from lowest indexed 2N_c - 7 mobile terminals, and the other transmission is transmitted from at most the α_k-1 + (2N_c + L - (L + 2)) - 1 = α_k-1 + (2N_c - 3). Which forces that in L_k-2 the highest indexed L + 1 - (2N_c - 2) = L - 2N_c + 3 can decode at most 1 transmission and the highest indexed L + 1 - (2N_c - 7) = L - 2N_c + 8 can decode at most 2 transmissions.

So in L_k-2 the remaining 2N_c + L - (L - 2N_c + 8) = 4N_c - 8 base stations must decode at least 2N_c - 2. Using a similar argument as above L + 2 of these available base stations are used to decode N_c + 1 transmissions. So in L_k-2, 4N_c - 8 - (L+2) = 4N_c - L - 10 ≤ 2N_c - 11 of the lowest indexed base stations must decode at least N_c - 3 transmissions. So compared to L_k-1 where the lowest indexed 2N_c - 6 base stations had to decode N_c - 1 transmissions, in L_k-2 the lowest indexed 2N_c - 11 have to decode at least N_c - 3 transmissions, so even though the number of needed transmissions decreased by 2, the number of available base stations decreased by 5. This propagation of interference would continue through all preceding subnetworks and the number of available base stations would keep decreasing faster than the number of transmissions to be decoded. So either we reach L_1 or we stop this propagation of interference at some L_i, where i < k. If the latter happens, then in L_i one could decode at most N_c + 1, which would bring the average number of decoded words between the k - i subnetworks to less than 2N_c/2N_c + L, so we just restart our argument from L_i - 1. If the former happens then the number of extra decoded transmissions did not scale with the network size, and thus the asymptotic puDOF is still 2N_c/2N_c + L.

If x = 1, and δ = 1, one observes that the interference from L_k to L_k-1 is actually worse than as described above. This is due to the fact that the extra active mobile terminal's transmission will be either decoded at one of the highest indexed L - 2N_c + 3 base stations of L_k-1 or a lower indexed base station. This either causes the same interference as described above or by Lemma 1 the effective max constraint of some of the higher indexed mobile terminals is reduced, which is worse than before. Thus a similar argument follows.

If x = 1, and δ > 1, then by Lemma 1, we have that the borrowed base stations in L_k-1 would be lower indexed than the highest indexed L - 2N_c + 3 base stations, thus the interference is worse than the first argument, which leads to the same conclusion that the asymptotic puDoF is still 2N_c/2N_c + L.

Now if x > 1, by Lemma 1, and the first argument we have that either the highest indexed L - 2N_c + 3 base stations are blocked in L_k-1 and some lower indexed base stations are borrowed, or all borrowed base stations have a lower index than the highest indexed L - 2N_c + 3 base stations, which from above arguments leads us to the same conclusion that the asymptotic puDoF is still 2N_c/2N_c + L.

So we have shown that We have shown that if any subnetwork has more than 2N_c active mobile terminals when N_c < L/2, either the number of extra active mobile terminals do not scale with size of the network, or the average over the whole network remains bounded by 2N_c active mobile terminals per subnetwork. This forces that the average number of decoded words per subnetwork is at most 2N_c, implying that the asymptotic puDoF during the uplink using zero forcing, τ_U^zf(L,N_c) ≤2N_c/2N_c + L. We have shown that τ_U^zf(L,N_c) ≥2N_c/2N_c + L, implying that τ_U^zf(L,N_c) = 2N_c/2N_c + L whenever N_c < L/2.



§ AVERAGE UPLINK-DOWNLINK DEGREES OF FREEDOM

In <cit.>, the puDoF value τ(L=1,N_c) was characterized. Here, we present zero-forcing schemes, with the goal of optimizing the average rate across both uplink and downlink for arbitrary values of L ≥ 2. We propose the following theorem 



The average uplink-downlink puDoF that can be achieved utilizing the interference avoidance schemes described in Section <ref> is characterized by

    τ^zf(L,N_c) ≥1/2(1 + γ_D(N_c, L))    L + 1≤ N_c, 
    2N_c/2N_c + L    1 ≤ N_c ≤L,

where γ_D(N_c, L) is the downlink component of the puDoF when N_c ≥ L+1, and is given by


    γ_D(L,N_c) =
                2(⌈L + 1/2⌉ + N_c - (L + 1))/L + 2(⌈L + 1/2⌉ + N_c - (L + 1)).



The coding scheme that achieves the inner bound for the second range of (<ref>) is essentially the union of the scheme described in Section <ref> and the scheme that achieves the third range of (<ref>). The network is split into disjoint subnetworks; each with consecutive 2N_c + L transmitter-receiver pairs. We consider two sets of base stations A_BS = {1,2,⋯, N_c} and B_BS = {N_c + 1,N_c + 2⋯, 2N_c}, and two sets of mobile terminals A_MT = {1,2,⋯, N_c} and B_MT = {N_c + L + 1,N_c + L + 2⋯, 2N_c + L}. Now for each i ∈ A_MT, C_i= A_BS. Similarly for each j ∈ B_MT, C_j= B_BS. Thus, for the downlink and uplink, we can get the optimal puDoF described in Sections <ref> and <ref> when N_c < L/2.
For the case where N_c ≥ L + 1, the coding scheme that achieves the inner bound in (<ref>) is as follows. First, we associate each mobile terminal with the L+1 base stations connected to it. This achieves the puDoF value of unity during the uplink in the same way as the scheme that achieves it in Section <ref>. Hence, we know so far that C_i⊇{i, i-1, i-2, ⋯, i - L}∩[K], ∀ i∈[K]. When sending messages from base stations to mobile terminals, the cooperative zero-forcing scheme works in a "downward" fashion as shown in  <cit.>. Due to the network topology, the uplink message passing scheme that achieves the unity puDoF works in an "upward" manner as shown in Section <ref>. So to maximize the downlink puDoF, we need to find a coding scheme that optimizes these opposing trends.

We define C_i^D as the set of extra associations that the downlink scheme requires for MT i. Thus, ∀ i ∈ [ K] we have that C_i =  C_i^D ∪{i, i-1, ⋯, i - L}. 

For the downlink, we divide the network into disjoint subnetworks; each consists of L + 2(⌈L + 1/2⌉ + N_c - (L + 1)) consecutive transmitter-receiver pairs. We define ϵ = ⌈L+1/2⌉, and κ = ϵ + N_c - (L+1). The cell association has a repeated pattern every 2κ + L BS-MT pairs, and hence, it suffices to describe it for the first 2κ+L BS-MT pairs. We consider two cases based on the parity of the connectivity parameter L. If L is odd, we partition the indices of mobile terminals in the subnetwork into three sets:

    S_1    =   {ϵ, ϵ + 1, ⋯, ϵ + κ -1},
    	
     S_2    =   {2ϵ + κ, 2ϵ + κ + 1 ⋯, 2ϵ + 2κ -1},
        
     S_3    =   {1, 2, ⋯, L + 2κ}∖ ( S_1 ∪ S_2).

The mobile terminals indexed in S_3 are kept inactive. The cell associations for downlink are given by the following description.

C_i^D=
{1, 2, ⋯, κ - 1},      ∀ i ∈ S_1,
{ϵ + κ, ϵ + κ + 1, ⋯, ϵ + 2κ - 1},     ∀  i ∈ S_2.


If L is even, we partition the indices of mobile terminals in the subnetwork into three sets:

    S'_1    =   {ϵ, ϵ + 1, ⋯, ϵ + κ -1},
    	
     S'_2    =   {2ϵ + κ - 1, 2ϵ + κ + 1 ⋯, 2ϵ + 2κ -2},
        
     S'_3    =   {1, 2, ⋯, L + 2κ}∖ ( S_1 ∪ S_2).


The mobile terminals indexed in S_3' are kept inactive. The cell associations are given by the following description.

C_i^D=
{1, 2, ⋯, κ - 1},      ∀ i ∈ S'_1,
{ϵ + κ, ϵ + κ + 1, ⋯, ϵ + 2κ - 1},     ∀  i ∈ S'_2.


So If L is odd we have a subnetwork of L + 2κ transmitter-receiver pairs and we decode 
    (ϵ + κ - 1 - ϵ + 1) + (2ϵ + 2κ -2 - (2ϵ + κ) + 1)= 2κ
 words during the downlink, and hence our average puDoF during the downlink is 
    2κ/L + 2κ = 2(L+1/2 + (N_c - (L+1)))/L + 2(L+1/2 + (N_c - (L+1))).
 A similar argument follows for the case when L is even. 
We have that (<ref>) is valid thus Theorem <ref> holds.

Figures <ref> and <ref> serve as examples for the average uplink-downlink inner bounds defined in this section .







































In the case of L = 1, the optimal puDoF is characterized in <cit.>. The findings there coincide with our findings, as L = 1 we find that for N_c ≥ L + 1, it directly implies that ϵ = 1 , andκ = (N_c - L) = N_c - 1 which results in γ_D(N_c, L) = 2(N_c - 1)/2(N_c -1) + 1.



§ AVERAGE UPLINK-DOWNLINK DOF WITH FULL UPLINK DOF

We show that the downlink puDoF as described in (<ref>) is optimal when we have unity DoF for uplink, i.e., each mobile terminal is associated with all the base stations connected to it. In other words, we are restricted in this section to cell associations that satisfy the following definition. 

 We say that an association scheme is called a Full coverage association if each mobile terminal is associated with all the base stations connected to it. More precisely, ∀i∈ K,   {i, i-1, ⋯ i-L}∈ C_i.


We then have the following theorem:


	Optimal downlink puDoF when we have a full coverage association and N_c > L is characterized as 
    γ_D(L,N_c) =
                2(⌈L + 1/2⌉ + (N_c - L + 1))/L + 2(⌈L + 1/2⌉ + (N_c - L + 1))
.


To help in proving Theorem <ref>, we define the following: ϵ = ⌈L+1/2⌉ and κ = ϵ + N_c - (L+1).
In order to prove Theorem <ref>, we first break up the network into subnetworks of 2κ + L consecutive base station (transmitter) and mobile terminal (receiver) pairs. Label each subnetwork L_k such that the lowest index of a base station and mobile terminal in L_k is k×(2κ + L). Our proof will be based on the concept that to break the puDoF described in Theorem <ref>, at least one subnetwork of 2κ + L consecutive MT-BS pairs must have more than 2κ + 1 active mobile terminals. 

First, we show that without borrowing base stations from preceding subnetworks, surpassing the puDoF value in Theorem <ref> is impossible. To do so, we first define a special class of downlink schemes and then formulate a lemma.



	We say that a downlink scheme relies on Subnetwork-only downlink decoding if transmissions from a subnetwork can only be decoded in the same subnetwork. 




	Utilizing subnetwork-only downlink when we already have a full coverage association scheme, a subnetwork can decode a maximum of 2κ words. Furthermore, for a subnetwork to decode 2κ words it must be that the last receiver in the subnetwork is active.


We prove Lemma <ref> by contradiction. Say 2κ + 1 words are decoded, then at least 2κ + 1 transmitters are active in the subnetwork. So there exists at least one active transmitter, say BS i, indexed between the indices of two sets of κ active transmitters.  Let ν_1 and ν_2 be the sets of indices of the active transmitters above and below i, with respect to order. Also, let x = maxν_1, and y = minν_2.

We first observe that the smallest possible value for x is κ and similarly the largest possible value of y is κ + ϵ, so we have i ∈ν_3 = {κ +1, κ + 2, ⋯, κ + ϵ -1}. For there to be 2κ + 1 words decoded words in the subnetwork, we also need 2κ + 1 active receivers, thus we can make corresponding disjoint index sets ν'_1, ν'_2, and ν'_3. Where receivers with indices in ν'_1 receive the first κ words, receivers with indices in ν'_2 receive the last κ words, and ν'_3 is the set of indices of receivers that can decode the extra word, say W_j. Now, x' = maxν'_1 and y' = minν'_2, so we observe that the smallest possible value for x' is κ + ϵ -1 and the largest possible value of y' is κ + L. Hence, we have j ∈ν'_3 = {κ + ϵ, ⋯κ + L -1}∩{i, i+1, ⋯ i+L}.

We then observe that at least ϵ + 1 active receivers indexed in the set ν' = ν'_1 ∪ν'_2 observe the extra transmitted signal. To aid in writing, we define μ'_j as the index set of receivers listening to transmitters which are transmitting W_j. Hence, for every receiver indexed μ'_j there must be a unique active transmitter indexed in ν = ν_1 ∪ν_3 such that the transmitter is associated with W_j to deliver the message at MT j and cancel interference at all other receivers indexed in ν'; call this set of indices μ_j. With just the full association scheme there are at most ϵ -1 such active transmitters, thus we must activate more transmitters indexed in ν_3, or add more active transmitters indexed in ν_1 ∪ν_2 to C_j. Either of those actions increases the size of μ'_j thus we must then increase the size of μ_j to cancel interference. So to cancel the interference of the extra transmitter we will always introduce more interference in the channel. So we cannot get more than 2κ words decoded through Subnetwork-only decoding. 

Additionally if the last receiver was not active we would have that the largest possible value of y' is κ + L - 1 which would imply that it would be observing transmissions from BS x, and to erase that interference an extra transmitter indexed in {κ + 1, ⋯ y'} must be activated during the downlink and associated with everything that BS x is associated with during the downlink. But this would mean at least one of the messages associated with BS x during the downlink would be associated with more than N_c base stations overall (uplink and downlink), which is not possible, thus the last active receiver must be active. This method of reasoning validates Lemma <ref>


If Theorem <ref> were not true then we would have that at least one subnetwork in the entire network must decode at least 2κ + x words. We present cases on x.

If x=1, let the first subnetwork that decodes 2κ + 1 words be L_k.  Lemma <ref> would then imply that at least one base station is active in L_k-1 such that this base station is either canceling the interference induced from the extra base stations active in L_k or is the extra base station that is carrying the extra word for L_k, but this implies that in L_k-1 at least the last mobile terminal cannot be active, which using Lemma <ref> implies that at most 2κ -1 words can be decoded in L_k-1. Thus the average over the two subnetworks is still 2κ words decoded per subnetwork. 

When x>1, one observes that the maximum number of base stations that can help from L_k-1 when trying to decode words in L_k is L - 1. These extra L - 1 base stations are being used to cancel interference induced from extra base stations in L_k. That would imply that the preceding subnetwork would only help with decoding a maximum of L + N_c - (L+1) words, so the smallest index of an active receiver in L_k that is being helped by transmitters in L_k-1 is at most L + N_c - (L+1), which implies that the next active receiver must have an index that is at least 2L + N_c - L+1, which leaves at most κ + ϵ - L receivers to decode at least 2κ + 1 - (L + N_c - (L+1)) = κ + ϵ - L + 1 words. Thus, in order to decode 2κ + x words where x > 1 in subnetwork L_k, we require that κ + ϵ - L receivers decode at least κ + ϵ - L + 1 words. Clearly this is impossible. Thus, only an average of 2κ words per subnetwork can be decoded, which results in 2κ/2κ + L words decoded per receiver, which is exactly what Theorem <ref> states.



	






























§ CONCLUDING REMARKS


In this work, we presented an effort to understand optimal cell association decisions in locally connected interference networks, focusing on optimizing for the average uplink-downlink puDoF problem. We consider a backhaul constraint that allows for associating each mobile terminal with N_c base stations (cells), and an interference network where each base station is connected to a corresponding mobile terminal as well as L mobile terminals with succeeding indices. We characterized the optimal association and puDoF for the uplink problem when zero-forcing schemes are considered. We also found that the characterization of the optimal association for the average uplink-downlink puDoF problem when N_c < L/2 follows from our uplink characterization and previous work for the downlink problem. We also presented the optimal zero-forcing downlink scheme if we fix the uplink scheme to the uplink-only-optimal scheme when N_c ≥ L+1. We conjecture that it is in fact optimal to have full DoF in the uplink when N_c ≥ L+1, and hence it would follow that the presented cell association and average puDoF are optimal in this case.



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