Flexible Backhaul Design and Degrees of Freedom for Linear Interference Networks
aa r X i v : . [ c s . I T ] J a n Flexible Backhaul Design and Degrees of Freedomfor Linear Interference Networks
Aly El Gamal and Venugopal V. Veeravalli
ECE Department and Coordinated Science LaboratoryUniversity of Illinois at Urbana-ChampaignEmail: { elgamal1,vvv } @illinois.edu Abstract —The considered problem is that of maximizing thedegrees of freedom (DoF) in cellular downlink, under a backhaulload constraint that limits the number of messages that canbe delivered from a centralized controller to the base stationtransmitters. A linear interference channel model is considered,where each transmitter is connected to the receiver having thesame index as well as one succeeding receiver. The backhaulload is defined as the sum of all the messages available at allthe transmitters normalized by the number of users. When thebackhaul load is constrained to an integer level B , the asymptoticper user DoF is shown to equal B − B , and it is shown that theoptimal assignment of messages to transmitters is asymmetricand satisfies a local cooperation constraint and that the optimalcoding scheme relies only on zero-forcing transmit beamforming.Finally, an extension of the presented coding scheme for the casewhere B = 1 is shown to apply for more general locally connectedand two-dimensional networks. I. I
NTRODUCTION
Managing wireless interference through infrastructural en-hancements is a major consideration for next generation cel-lular networks. One example of such an enhancement is incellular downlink through the assignment of one receiver’smessage to multiple base station transmitters and managinginterference through a Coordinated Multi-Point Transmission(CoMP) scheme. The cost of delivering messages to multipletransmitters over a backhaul link is highlighted in this work.In [2], the degrees of freedom (DoF) gain offered by CoMPtransmission in Wyner’s linear interference networks [1] wasstudied, under a cooperation constraint that limits the numberof transmitters at which each message can be available bya number M . The asymptotic limit of the per user DoF asthe number of users goes to infinity was shown to be M M +1 ,and was shown to be achieved by a simple coding schemethat relies only on zero-forcing transmit beamforming. It isto be noted that the maximum transmit set size constraintof M is not met tightly for all messages in the optimalmessage assignment scheme presented in [2]. In this work, wetherefore consider a cooperation constraint that is more generaland relevant to many scenarios of practical significance. Inparticular, we define the backhaul load constraint B as theratio between the sum of the transmit set sizes for all themessages and the number of users. In other words, we allowthe transmit set size constraints to vary across the messages,while maintaining a constraint on the average transmit set sizeof B . We establish in this paper that the asymptotic per userDoF in this new setting is B − B , which is larger than the per user DoF of B B +1 obtained with the more stringent permessage transmit set size constraint of B .Furthermore, we show that the scheme that achieves theoptimal DoF of B − B uses only zero-forcing beamforming atthe transmitters, and assigns messages non-uniformly acrossthe transmitters, with some messages being assigned to morethan B transmitters and others being assigned to fewer than B transmitters. We show that these insights can apply tomore general channel models than the simple linear modelconsidered in this work.We describe the system model in Section II. We thenprovide an illustrative example for the considered problem inSection III. The main result is proved in Section IV. We thendiscuss the result and its generalizations in Section V. Finally,we provide concluding remarks in Section VI.II. S YSTEM M ODEL AND N OTATION
We use the standard model for the K − user interferencechannel with single-antenna transmitters and receivers, Y i ( t ) = K X j =1 H i,j ( t ) X j ( t ) + Z i ( t ) , (1)where t is the time index, X j ( t ) is the transmitted signal oftransmitter j , Y i ( t ) is the received signal at receiver i , Z i ( t ) isthe 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 the time slot t . We remove the time index in the restof the paper for brevity unless it is needed. For any set A ⊆ [ K ] , we use the abbreviations X A , Y A , and Z A to denote thesets { X i , i ∈ A} , { Y i , i ∈ A} , and { Z i , i ∈ A} , respectively.Finally, we use [ K ] to denote the set { , , . . . , K } , and use φ to denote the empty set. A. Channel Model
Each transmitter is connected to its corresponding receiveras well as one following receiver, and the last transmitter isonly connected to its corresponding receiver. More precisely, H i,j = 0 iff i / ∈ { j, j + 1 } , ∀ i, j ∈ [ K ] , (2)and all non-zero channel coefficients are drawn independentlyfrom a continuous joint distribution. Finally, we assume thatglobal channel state information is available at all transmittersand receivers. The channel model is illustrated for K = 3 inFigure 1.ig. 1: Wyner’s linear asymmetric model for K = 3 . In thefigure, a solid line connects a transmitter-receiver pair if andonly if the channel coefficient between them is non-zero. B. Message Assignment
For each i ∈ [ K ] , let W i be the message intended forreceiver i , and T i ⊆ [ K ] be the transmit set of receiver i , i.e.,those transmitters with the knowledge of W i . The transmittersin T i cooperatively transmit the message W i to the receiver i .The average transmit set size is upper bounded by an integervalued backhaul load constraint B , P Ki =1 |T i | K ≤ B. (3) C. Message Assignment Strategy
A message assignment strategy is defined by a sequenceof supersets. The k th element in the sequence consists of thetransmit sets for a k − user channel. We use message assign-ment strategies to define a pattern for assigning messages totransmitters in large networks. D. Local Cooperation
We say that a message assignment strategy satisfies the localcooperation constraint, if and only if there exists a function r ( K ) such that r ( K ) = o ( K ) , and for every K ∈ Z + , thetransmit sets defined by the strategy for a K − user channelsatisfies the following, T i ⊆ { i − r ( K ) , i − r ( K ) + 1 , . . . , i + r ( K ) } , ∀ i ∈ [ K ] . (4) E. Degrees of Freedom
Let P be the average transmit power constraint at eachtransmitter, 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 decodingerror probabilities of all messages can be simultaneously madearbitrarily small for a large enough coding block length n , andthis holds for almost all channel realizations. The degrees offreedom 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 achievableDoF tuples. The total number of degrees of freedom ( η ) isthe maximum value of the sum of the achievable degrees offreedom, η = max D P i ∈ [ K ] d i .For a K -user channel, we define η ( K, B ) as the best achiev-able η over all choices of transmit sets satisfying the backhaulload constraint in (3). In order to simplify our analysis, wedefine the asymptotic per user DoF τ ( B ) to measure how η ( K, B ) scales with K while all other parameters are fixed, τ ( B ) = lim K →∞ η ( K, B ) K , (5)We call a message assignment strategy optimal for a se-quence of K − user channels, K ∈ { , , . . . } , if and only ifthere exists a sequence of coding schemes achieving τ ( B ) using the transmit sets defined by the message assignmentstrategy. III. E XAMPLE : B = 1 Before introducing the main result, we illustrate through asimple example that the potential flexibility in the backhauldesign according to the constraint in (3) can offer DoF gainsover a traditional design where all messages are assigned tothe same number of transmitters. We know from [2] that anyasymptotic per user DoF greater than cannot be achievedthrough assigning each message to one transmitter. We nowshow that τ ( B = 1) ≥ , by allowing few messages to beavailable at more than one transmitter at the cost of not trans-mitting other messages. Consider the following assignment ofthe first four messages, T = { , } , T = { } , T = φ ,and T = { } . Message W is transmitted through X to Y without interference. Since the channel state information isknown at the second transmitter, the transmit beam for W at X can be designed to cancel the interference caused by W at Y , and then W can be transmitted through X to Y without interference. Finally, W is transmitted through X to Y without interference. It follows that the sum DoF for thefirst four messages P i =1 d i ≥ . Since the fourth transmitteris inactive, the subnetwork consisting of the first four usersdoes not interfere with the rest of the network, and hence, wecan see that τ ( B = 1) ≥ through similar assignment ofmessages in each consecutive -user subnetwork.IV. M AIN R ESULT
We now characterize the asymptotic per user DoF τ ( B ) forany integer value of the backhaul load constraint. Theorem 1:
The asymptotic per user DoF τ ( B ) is given by, τ ( B ) = 4 B − B , ∀ B ∈ Z + . (6) Proof:
We provide the proof for the inner and outerbounds in Section IV-A and Section IV-B, respectively.
A. Coding Scheme
We treat the network as a set of subnetworks, each con-sisting of consecutive B transceivers. The last transmitter ofeach subnetwork is deactivated to eliminate inter-subnetwork interference. It then suffices to show that B − DoF canbe achieved in each subnetwork. Without loss of generality,consider the cluster of users with indices in the set [4 B ] . Wedefine the following subsets of [4 B ] , S = [2 B ] S = { B + 2 , B + 3 , . . . , B } We next show that each user in S ∪ S achieves one degreeof freedom, while message W B +1 is not transmitted. Let themessage assignments be as follows, T i = ( { i, i + 1 , . . . , B } , ∀ i ∈ S , { i − , i − , . . . , B + 1 } , ∀ i ∈ S , and note that P Bi =1 |T i | B = B , and hence, the constraintin (3) is satisfied. Now, due to the availability of channelstate information at the transmitters, the transmit beams foressage W i can be designed to cancel its effect at receiverswith indices in the set C i , where, C i = ( { i + 1 , i + 2 , . . . , B } , ∀ i ∈ S { i − , i − , . . . , B + 2 } , ∀ i ∈ S Note that both C B and C B +2 equal the empty set, asboth W B and W B +2 do not contribute to interfering signalsat receivers in the set Y S ∪ Y S . The above scheme for B = 2 is illustrated in Figure 2. We conclude that eachreceiver whose index is in the set S ∪ S suffers only fromGaussian noise, thereby enjoying one degree of freedom. Since |S ∪ S | = 4 B − , it follows that P Bi =1 d i ≥ B − .Using a similar argument for each following subnetwork, weestablish that τ ( B ) ≥ B − B , thereby proving the lower boundof Theorem 1.Fig. 2: Achieving / per user DoF with a backhaul constraint B = 2 . The figure shows only signals corresponding to thefirst subnetwork in a general K − user network. The signals inthe dashed boxes are deactivated. Note that the deactivationof X splits this part of the network from the rest.We note that the illustrated message assignment strategysatisfies the local cooperation constraint of (4). In other words,the network can be split into subnetworks, each of size B ,and the messages corresponding to users in a subnetworkcan only be assigned to transmitters with indices in the samesubnetwork. B. Upper Bound
We prove the converse of Theorem 1 in two steps. First,we provide an information theoretic argument in Lemma 2 toprove an upper bound on the DoF of any network that hasa subset of messages whose transmit set sizes are bounded.We then finalize the proof with a combinatorial argumentthat shows the existence of such a subset of messages inany assignment of messages satisfying the backhaul constraintof (3).In order to prove the information theoretic argument inLemma 2, we use Lemma from [2], which we restate below.For any set of receiver indices A ⊆ [ K ] , define U A as the setof indices of transmitters that exclusively carry the messages for the receivers in A , and its complement ¯ U A . More precisely, ¯ U A = ∪ i/ ∈A T i . Lemma 1 ([2]):
If there exists a set
A ⊆ [ K ] , a function f , and a function f whose definition does not depend onthe transmit power constraint P , and f ( Y A , X U A ) = X ¯ U A + f ( Z A ) , then the sum DoF η ≤ |A| .We also need [2, Corollary ] in the proof of Lemma 2; werestate it for the considered system model. Corollary 1 ([2]):
For any K − user linear interferencechannel, if the size of the transmit set |T i | ≤ M, i ∈ [ K ] , thenany element k ∈ T i such that k / ∈ { i − M, i − M + 1 , . . . , i + M − } can be removed from T i , without decreasing the sumrate.We now make the following definition to use in the proofof the following lemma. For any set S ⊆ [ K ] , let g S : S →{ , , . . . , |S|} be a function that returns the ascending orderof any element in the set S , e.g., g S (min { i : i ∈ S} ) = 1 and g S (max { i : i ∈ S} ) = |S| Lemma 2:
For any K − user linear interference channel withDoF η , if there exists a subset of messages S ⊆ [ K ] suchthat each message in S is available at a maximum of M transmitters, i.e., |T i | ≤ M, ∀ i ∈ S , then the DoF is boundedby, η ≤ K − |S| M + 1 + C K , (7)where lim K →∞ C K K = 0 . Proof:
We use Lemma 1 with a set A such that the size ofthe complement set | ¯ A| = |S| M +1 − o ( K ) . We define the set A such that ¯ A = { i : i ∈ S , g S ( i ) = (2 M +1)( j − M +1 , j ∈ Z + } .Now, we let s , s be the smallest two indices in ¯ A . We seethat g S ( s ) = M +1 , g S ( s ) = 3 M +2 . Note that X + Z H , = Y H , , and X + Z − H , H , Z H , = Y − H , H , Y H , . Similarly, it is clear how the first s − transmit signals X [ s − can be recovered from the received signals Y [ s − and linear combinations of the noise signals Z [ s − . In whatfollows, we show how to reconstruct a noisy version of thesignals { X s , X s +1 , . . . , X s − } , where the reconstructionnoise is a linear combination of the signals Z A . Then it willbe clear by symmetry how the remaining transmit signals canbe reconstructed.We now notice that it follows from Corollary 1 that message W s can be removed from any transmitter in T s whoseindex is greater than s + M − , without affecting the sumrate. Similarly, there is no loss in generality in assuming that ∀ s i ∈ S , s i = s , T s i does not have an element with indexless than s i − M . Since s i − s ≥ g S ( s i ) − g S ( s ) ≥ M + 1 ,it follows that X s + M ∈ X U A . The signal X s + M +1 + Z s M +1 H s M +1 ,s M +1 can be reconstructed from Y s + M +1 and X s + M . Then, it can be seen that the transmit sig-nals { X s + M +2 , X s + M +3 , . . . , X s − } can be reconstructedrom { Y s + M +1 , Y s + M +2 , . . . , Y s − } , and linear combina-tions of the noise signals { Z s + M +1 , Z s + M +2 , . . . , Z s − } .Similarly, since X s + M is known, the transmit signals { X s + M − , X s + M − , . . . , X s } can be reconstructed from { Y s + M , Y s + M − , . . . , Y s +1 } , and linear combinations ofthe noise signals { Z s + M , Z s + M − , . . . , Z s +1 } . By follow-ing a similar argument to reconstruct all transmit signals fromthe signals Y A , X U A , and linear combinations of the noisesignals Z A , we can show the existence of functions f and f of Lemma 1 to complete the proof.We now explain how Lemma 2 can be used to prove that τ ( B = 1) ≤ . For any message assignment satisfying (3)for a K − user channel, let R j be defined as follows for every j ∈ { , , . . . , K } , R j = | { i : i ∈ [ K ] , |T i | = j } | K . (8) R j is the fraction of users whose messages are available atexactly j transmitters. Now, if R + R ≥ , then Lemma 2 canbe used directly to show that η ≤ K + o ( K ) . Otherwise, morethan K users have their messages at two or more transmitters,and it follows from (3) that R ≥ P Kj =2 R j ≥ , and hence, η ≤ (1 − R ) K ≤ K .We generalize the above argument in the proof of the fol-lowing lemma to complete the proof that τ ( B ) ≤ B − B , ∀ B ∈ Z + . Lemma 3:
For any message assignment satisfying (3) for a K − user channel with an average transmit set size constraint B , there exists an integer M ∈ { , , . . . , K } , and a subset S ⊆ [ K ] whose size |S| ≥ M +14 B K , such that each messagein S is available at a maximum of M transmitters, i.e., |T i | ≤ M, ∀ i ∈ S . Proof:
Fix any message assignment satisfying (3) for a K − user channel with backhaul constraint B , and let R j , j ∈{ , , . . . , K } be defined as in (8). If P Kj =2 B R j ≤ B , thenmore than B − B K users have a transmit set whose size is atmost B − , and the lemma follows with M = 2 B − . Itthen suffices to assume that P Kj =2 B R j > B in the rest of theproof. We show in the following that there exists an integer M ∈ { , . . . , B − } such that P Mj =0 R j > M +14 B , therebycompleting the proof of the lemma.Define R ∗ j , j ∈ { , , . . . , B } such that R ∗ = R ∗ B = B ,and R ∗ j = B , ∀ j ∈ { , . . . , B − } . Now, note that P Bj =0 R ∗ j = 1 , and P Bj =0 jR ∗ j = B . It follows that if R j = R ∗ j , ∀ j ∈ { , . . . , B } , and R j = 0 , ∀ j ≥ B + 1 ,then the constraint in (3) is tightly met, i.e., P Ki =1 |T i | K = B .We will use this fact in the rest of the proof.We prove the statement by contradiction. Assume that P Kj =2 B R j > R ∗ B = B , and that ∀ M ∈ { , , . . . , B − } , P Mj =0 R j ≤ P Mj =0 R ∗ j = M +14 B . We know from (3) that P Kj =0 jR j ≤ P Bj =0 jR ∗ j = B . Also, since P Kj =0 R j = P Bj =0 R ∗ j = 1 and P Kj =2 B R j > R ∗ B , it follows thatthere exists an integer M ∈ { , , . . . , B − } such that R M > R ∗ M ; let m be the smallest such integer. Since P mj =0 R j ≤ P mj =0 R ∗ j , and ∀ j ∈ { , , . . . , m − } , R j ≤ R ∗ j , we can construct another message assignment by removingelements from some transmit sets whose size is m , such thatthe new assignment satisfies (3), and has transmit sets T ∗ i where ∀ j ∈ { , , . . . , m } , |{ i : i ∈ [ K ] , |T ∗ i | = j }| ≤ R ∗ j . Bysuccessive application of the above argument, we can constructa message assignment that satisfies (3), and has transmit sets T ∗ i where ∀ j ∈ { , , . . . , B − } , |{ i : i ∈ [ K ] , |T ∗ i | = j }| ≤ R ∗ j and |{ i : i ∈ [ K ] , |T ∗ i | ≥ B }| ≥ R ∗ B . Note that the newassignment has to violate (3) since P Bj =0 jR ∗ j = B , and wereach a contradiction.We now know from lemmas 2 and 3 that under the backhaulload constraint of (3), the DoF for any K − user channelis upper bounded by B − B K + o ( K ) . It follows that theasymptotic per user DoF τ ( B ) ≤ B − B , thereby proving theupper bound of Theorem 1.V. D ISCUSSION AND G ENERALIZATIONS
A. Maximum Transmit Set Size Constraint
In [2], we considered the problem where each transmitset size is bounded by a cooperation constraint M , i.e., |T i | ≤ M, ∀ i ∈ [ K ] . The DoF achieving coding scheme wasthen characterized for every value of M . We note that in theconsidered problem with an average transmit set size constraint B , the per user DoF τ ( B ) can be achieved using a combinationof the schemes that are characterized as optimal in [2] for thecases of M = 2 B − and M = 2 B . We note that even thoughthe maximum transmit set size constraint may not reflect aphysical constraint, the solutions in [2] provide a useful toolsetthat can be used to achieve the optimal per user DoF valueunder the more natural constraint on the total backhaul loadthat is considered in this work. B. Locally Connected Networks
Using a convex combination of the schemes that are optimalunder the maximum transmit set size constraint can alsoprovide good coding schemes for the more general locallyconnected channel model that is considered in [2], where eachreceiver can see interference from L neighbouring transmitters.More precisely, for the following channel model, H i,j is not identically , if and only if i ∈ (cid:20) j − (cid:22) L (cid:23) , j + (cid:24) L (cid:25)(cid:21) . (9)Let τ L ( B ) be the asymptotic per user DoF for a locallyconnected channel defined in (9) with connectivity parameter L . Then we can use a convex combination of the schemes thatare characterized as optimal in [2] to achieve the inner boundsstated in Table I for the case where B = 1 . L = 2 L = 3 L = 4 L = 5 L = 6 τ L ( B = 1) ≥
23 35 59 1121 12
TABLE I: Achievable per user DoF values for locally con-nected channels with a backhaul constraint P Ki =1 |T i | ≤ K .ow, we note that the inner bounds stated in Table I canbe achieved through the use of only zero-forcing transmitbeamforming. In other words, there is no need for the symbolextension idea required by the asymptotic interference align-ment scheme of [3]. In [2, Theorem ], it was shown thatfor L ≥ , by allowing each message to be available at onetransmitter, the asymptotic per user DoF is ; it was alsoshown in [2, Theorem ] that the per user DoF value cannotbe achieved through zero-forcing transmit forming for L ≥ .In contrast, in Table I it can be seen that for L ≤ , the per user DoF value can be achieved through zero-forcingtransmit beamforming and a flexible design of the backhaullinks, without incurring additional overall load on the backhaul( B = 1 ). C. Two-Dimensional Networks
The insights we have in this work on the backhaul designfor linear interference networks, may apply in denser networksby treating the denser network as a set of interfering linearnetworks. For example, consider the two-dimensional networkdepicted in Figure 3a where each transmitter is connectedto four cell edge receivers. The precise channel model fora K − user channel is as follows, H i,j is not identically , if and only if i ∈ n j, j + 1 , j + j √ K k , j + j √ K k + 1 o . (10)For this channel model, we can show that by assigning eachmessage to one transmitter, i.e., imposing the constraint |T i | ≤ , ∀ i ∈ [ K ] , the asymptotic per user DoF is at most , andthe use of only zero-forcing transmit beamforming can leadto at most per user DoF. However, under the backhaul loadconstraint P Ki =1 |T i | K ≤ , a per user DoF value of can beachieved using only zero-forcing transmit beamforming. Thiscan be done by deactivating every third row of transmitters, (a) (b) Fig. 3: Two dimensional interference network. In ( a ) , we plotthe channel model, with each transmitter being connected tofour surrounding cell edge receivers. In ( b ) , we show an exam-ple coding scheme where dashed red boxes and lines representinactive nodes and edges. The signals { X , . . . , X √ K } and { Y , . . . , Y √ K } form a linear subnetwork. Similarly, the sig-nals { X √ K +1 , . . . , X √ K } and { Y √ K +1 , . . . , Y √ K } form alinear subnetwork . and splitting the rest of the network into non-interfering linearsubnetworks (see Figure 3b). In each subnetwork, a backhaulload constraint of is imposed. For example, the followingconstraint is imposed on the first row of users, P ⌊ √ K ⌋ i =1 |T i | ⌊ √ K ⌋ ≤ .A convex combination of the schemes that are characterizedas optimal in [2] for the cases of maximum transmit set sizeconstraints M = 2 and M = 3 is then used to achieve per user DoF in each active subnetwork while satisfying abackhaul load constraint of . Since of the subnetworksare active, a per user DoF of is achieved while satisfying abackhaul load constraint of unity.VI. C ONCLUSION
We studied the potential gains offered by CoMP trans-mission in linear interference networks, through a backhaulload constraint that limits the average transmit set size acrossthe users. We characterized the asymptotic per user DoF,and showed that the optimal coding scheme relies only onzero-forcing transmit beamforming. The backhaul constraintis satisfied in the optimal scheme by assigning some messagesto more than B transmitters and others to fewer than B transmitters, where B is the average transmit set size. Weshowed that local cooperation is sufficient to achieve the DoFin large linear interference networks. We also noted that thecharacterized asymptotic per user DoF for linear interferencenetworks can be achieved by using a convex combination ofthe coding schemes that are identified as optimal in [2] undera cooperation constraint that limits the maximum size of atransmit set, as opposed to the average as we considered inthis work. We then illustrated that these results hold in moregeneral networks of practical relevance to achieve rate gainsand simplify existing coding schemes.R EFERENCES[1] A. Wyner, “Shannon-Theoretic Approach to a Gaussian Cellular Multiple-Access Channel,”
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