Degrees of Freedom of Interference Networks with Transmitter-Side Caches
Antonious M. Girgis, Ozgur Ercetin, Mohammed Nafie, Tamer ElBatt
11 Degrees of Freedom of Interference Networkswith Transmitter-Side Caches
Antonious M. Girgis, Ozgur Ercetin, Mohammed Nafie, and Tamer ElBatt
Abstract
This paper studies cache-aided interference networks with arbitrary number of transmitters andreceivers, whereby each transmitter has a cache memory of finite size. Each transmitter fills its cachememory from a content library of files in the placement phase. In the subsequent delivery phase,each receiver requests one of the library files, and the transmitters are responsible for delivering therequested files from their caches to the receivers. The objective is to design schemes for the placementand delivery phases to maximize the sum degrees of freedom (sum-DoF) which expresses the capacityof the interference network at the high signal-to-noise ratio regime. Our work mainly focuses on acommonly used uncoded placement strategy. We provide an information-theoretic bound on the sum-DoF for this placement strategy. We demonstrate by an example that the derived bound is tighter than thebounds existing in the literature for small cache sizes. We propose a novel delivery scheme with a higherachievable sum-DoF than those previously given in the literature. The results reveal that the reciprocal ofsum-DoF decreases linearly as the transmitter cache size increases. Therefore, increasing cache sizes attransmitters translates to increasing the sum-DoF and, hence, the capacity of the interference networks.
Index Terms
Coded caching, Interference networks, Degrees of freedom, Interference management.
I. I
NTRODUCTION
Wireless networks are experiencing an exponential growth of user traffic load due to the rapidproliferation of wireless devices. In particular, video content contributes to more than half ofthe overall data traffic, and it is expected that it will grow to about percent in the nextfew years [1]. Caching of popular content close to user terminals has the potential to tacklethis dramatic growth. In particular, on-demand video content is usually non-real-time, i.e., it isavailable prior to the transmission. The main idea of caching is to exploit the under-utilized Antonious M. Girgis is with Wireless Intelligent Networks Center (WINC), Nile University, Cairo. Ozgur Ercetin is withFaculty of Engineering and Natural Sciences, Sabanci University, Istanbul, Turkey. Mohammed Nafie is with Wireless IntelligentNetworks Center (WINC), Nile University, Cairo, Egypt and the Dept. of EECE, Faculty of Engineering, Cairo University, Giza,Egypt. Tamer ElBatt is with the Depart. of Computer Science and Engineering, American University, Cairo, Egypt. a r X i v : . [ c s . I T ] D ec network resources during off-peak hours by prefetching the most popular content in the edgeservers close to users without any knowledge of the future user demands. During the peak-hourswhen the network is congested, the caches at the edge servers can be exploited to partly servethe user requests without accessing the central server to improve the system performance as wellas the users’ experience (See [2]–[7] and references therein). It was shown in [2] that serving aportion of requests proactively and storing the data in the user caches yield a significant reductionin the outage probability. In [3], the authors proposed a proactive caching strategy under theobjective of minimizing the transmission costs. In [4], the authors developed distributed cachingnetworks to minimize the transmission latency with constraints on the capacity of the backhaullinks. The work in [5]–[7] aim to design caching systems under the objective of maximizing theenergy efficiency of the network.Recently, Maddah-Ali and Niesen introduced an information-theoretic approach, called codedcaching , for an error-free broadcast channel in which each user is equipped with an isolatedcache memory [8]. It was shown that the network traffic can be significantly reduced by jointlydesigning the content placement and delivery phases which in turn generates multicast codingopportunities in the delivery phase. Coded caching has received considerable attention for severalsettings [9]–[18]. The work in [9] characterized in closed form the memory-rate tradeoff for thecoded caching problem introduced in [8] under uncoded placement. Decentralized coded cachingwas introduced in [10], where each user randomly stores some bits from each file independentlyof each other. In [11]–[13], the authors studied the coded caching problem for an arbitrary filepopularity. The work in [14] and [15] studied decentralized coded caching under heterogeneouscache sizes at the users. Additionally, the concept of coded caching was extended for onlinecaching systems in [16], multi-server networks in [17], and hierarchical networks in [18].The role of caches at the transmitters of interference networks was first studied in [19].It was shown that the caches at transmitters can improve the sum degrees of freedom (sum-DoF) of the network by allowing cooperation between transmitters for interference mitigation.However, in [19], the authors considered only the case of three transmitters and three receivers.A lower bound on the transmission latency of cache-aided interference networks was derivedin [20], where the authors propose the normalized delivery time (NDT) as a performance metricwhich is proportional to the inverse of the sum-DoF. In [21], the authors derived the exactcharacterization of the NDT of Fog radio access networks (F-RAN) with caches equipped ateach transmitter. The extension for an arbitrary number of transmitters and receivers was studied files N bits F RX RX RX RX R K RX R K RX TX TX TX TX T K TX T K TX MF bits MF bits MF bits Fig. 1: Cache-aided interference network with K T transmitters and K R receivers.in [22]. The authors in [23] investigated the NDT of the binary-fading interference channelwith two receivers served by a small-cell base station with a finite-size cache and a macrobase station. The problem of interference networks with caches at both the transmitters and thereceivers was studied in [24]–[27]. In [24], the NDT of an F-RAN with two transmitters and anarbitrary number of receivers was characterized under a decentralized content placement scheme,where it was shown that the proposed coding scheme achieves a performance comparable to thederived lower bound. The authors in [25] focused on designing one-shot linear delivery schemesfor cache-aided interference networks in which the channel extension is not allowed. In [26], theapproximate characterization of the sum-DoF for interference channels with caches at both thetransmitters and receivers was investigated wherein the authors proposed a separation strategybetween the physical and the network layers. In [27], the authors proposed an order-optimaldelivery scheme for minimizing the NDT of cache-aided interference networks.In this paper, we consider a K T × K R interference network with a library of N files, whereeach transmitter is equipped with a cache memory of size M F bits. We study the fundamentallimits on the sum-DoF of interference networks as a function of the transmitter cache size M ,under an uncoded placement strategy that was originally introduced in [19] and [25]. Our maincontributions in this work are as follows: • We introduce a new communication paradigm in the delivery phase called cooperative X -networks in which each K T MN transmitters has a dedicated message for every receiver. We derive an information-theoretic bound on the sum-DoF for such network by using genie-aided, cut-setarguments. Hence, there is no delivery scheme for cache-aided interference networks achievinga sum-DoF higher than that of the derived upper bound using such uncoded caching scheme. • This derived bound gives important insights on the sum-DoF of the cache-aided interferencenetworks. We show that the delivery scheme in [27] is optimal in the case τ = K R − , τ = K T MN .Additionally, the derived bound is tighter than the upper bound in [20] (developed for arbitrarycaching schemes) for small cache sizes. Hence, by using an uncoded caching scheme, the boundin [20] is not achievable in general. • We propose a novel delivery scheme based on zero forcing (ZF) and interference alignment(IA) techniques. The main idea of the proposed delivery scheme is to first schedule the requestedbits into groups such that each group of bits (messages) would be sent in the same transmissionblocks. The scheduling strategy is designed in such a way that each transmitter has a dedicatedmessage to every receiver. In addition, the messages assigned to a transmitter is available as aside information at K T MN − consecutive transmitters, i.e., those are called cooperative (cognitive)transmitters. The benefits of cognitive messages on the sum-DoF have been studied in [28]–[30]for interference networks, and in [31] for multiple-input-multiple-output (MIMO) X -networkswith two transmitters and two receivers. Therefore, in each transmission block, we design two-layer precoding matrices at each transmitter. The first precoding layer is designed based on ZFto leverage the partial cooperation between transmitters in order to null out some interferencesignals at each receiver, while, the second precoding layer is designed based on IA to align theremaining interference signals at each receiver into smaller dimensions. • Although, the work in [25]–[27] have studied a similar setup, we show that our proposedcoding scheme achieves a higher sum-DoF than those of state-of-the-art schemes, where thedifferences between our results and those in [25]–[27] are discussed throughout the paper.Our analytical analysis indicates that the reciprocal of the sum-DoF decreases linearly as thetransmitter cache size increases. Moreover, we show that the achievable sum-DoF is within amultiplicative factor of from the upper bound which is derived without any restrictions on thecaching and delivery schemes, independent of system parameters.The rest of the paper is organized as follows. In Section II, we introduce the system model andthe problem formulation. In Section III, the main results of the paper are presented. A rigorouscomparison between our coding scheme and the state-of-the-art schemes is also presented inSection III. Section IV describes the uncoded placement. The converse results for cache-aided interference networks with this uncoded placement strategy is presented in Section IV. Weprove the achievable bound of the sum-DoF for a generic cache-aided interference network inSection V. Finally, the paper is concluded in Section VI. Some technical proofs are relegated toappendices. II. S YSTEM M ODEL
Notations:
The set of natural numbers and complex numbers are denoted by N + and C ,respectively. For a square matrix M , we use ( M ) − and det M to denote the inverse and thedeterminant, respectively. () T denotes transpose. We use calligraphic symbols for sets, e.g., S . |S| denotes the cardinality of set S . [ K ] denotes the set of integers { , . . . , K } , and [ a : b ] denotes the set of integer numbers between a and b : { a, . . . , b } for b ≥ a . For set [ i : j ] oftransmitters, the indices are taken to modulo K T which is the total number of transmitters suchthat [ i : j ] ⊆ [ K T ] . Similarly, for set [ i : j ] of receivers, the indices are taken to modulo K R which is the total number of receivers such that [ i : j ] ⊆ [ K R ] .We consider an interference network of K T transmitters connected to K R receivers over atime-varying Gaussian channel as depicted in Figure 1. There is a content library of N files, W (cid:44) { W , . . . , W N } , each of size F bits, where each file W f ∈ W is chosen independentlyand uniformly from (cid:2) F (cid:3) at random. Each transmitter TX i , i ∈ [ K T ] , has a local cache memory Z i of size M F bits, where M refers to the memory size in files, where M ≤ N .The system operates in two separate phases, a placement phase and a delivery phase . Inthe placement phase, the transmitters have access to the content library W , and hence, eachtransmitter fills its cache memory as an arbitrary function of the content library W under itscache size constraint. We emphasize that the caching decisions are taken without any priorknowledge of the future receiver demands and channel coefficients between the transmitters(TXs) and the receivers (RXs).In the delivery phase, receiver RX j requests a file W d j out of N files of the library. We consider d = [ d , . . . , d K R ] ∈ [ N ] K R as the vector of receiver demands. The transmitters are awareof all receiver demands . Thus, transmitter TX i , i ∈ [ K T ] , responds by sending a codeword X i (cid:44) ( X i ( t )) Tt =1 of block length T over the interference channel, where X i ( t ) ∈ C is thetransmitted signal of transmitter TX i at time t ∈ [ T ] . We impose an average transmit powerconstraint over the channel input T || X i || ≤ P . In this phase, each transmitter has only accessto its own cache contents, so that, codeword X i is determined by an encoding function of the receiver demands d , the cache contents Z i , and the channel coefficients between TXs and RXs.Afterwards, each receiver RX j implements a decoding function to estimate the requested file ˆ W d j from the received signal Y j (cid:44) ( Y j ( t )) Tt =1 which is given by Y j ( t ) = K T (cid:88) i =1 h ji ( t ) X i ( t ) + N j ( t ) , (1)where Y j ( t ) ∈ C is the received signal by receiver RX j at time t ∈ [ T ] , and N j ( t ) is the additivewhite Gaussian noise at receiver RX j at time t ∈ [ T ] . Let h ji ( t ) ∈ C represent the channel gainbetween transmitter TX i and receiver RX j at time t . We assume all channel coefficients aredrawn independently and identically distributed (i.i.d.) from a continuous distribution. For agiven coding scheme (caching, encoding, and decoding), the probability of error is obtained by Pe = max d ∈ [ N ] KR max j ∈ [ K R ] P (cid:16) ˆ W d j (cid:54) = W d j (cid:17) , (2)which is the worst-case probability of error over all possible demands d and over all receivers.The transmission rate for a given coding scheme for a fixed cache size of M F bits, and powerconstraint P is R ( M, P ) = K R F/T , where K R F is the number of bits required to be deliveredto receivers and T is the time required to send these bits. We say that the rate R ( M, P ) isachievable if and only if there exists a coding scheme that achieves R ( M, P ) such that Pe → as F → ∞ . The optimum rate R ∗ ( M, P ) is defined as the supremum of all achievable rates.Furthermore, we define the sum degrees of freedom of the network as a function of the cachesize M by DoF ( M ) = lim P →∞ R ∗ ( M, P ) F log ( P ) . (3)The sum-DoF is a performance metric that defines the pre-log capacity or the multiplexing gainof the network. In other words, the capacity can be expressed by R ∗ ( M, P ) =
DoF log ( P ) + o (log ( P )) at the high Signal-to-Noise-Ration (SNR) regime, where the o (log ( P )) term vanishesas P → ∞ . Our objective in this work is to characterize the trade-off between the cache size atthe transmitter and the sum degrees of freedom of the cache-aided interference network. Notethat the transmitter cache size should satisfy K T M ≥ N to guarantee that every bit of the librarycontent is stored, at least, at one of the transmitter caches in the network. Moreover, if the cachesize is larger than the library size, i.e., M > N , then each transmitter is able to cache all files andthe remaining cache memory would not be used, and hence, the problem will be meaningless.Therefore, we are interested in characterizing the
DoF of the network for cache sizes such that NK T ≤ M ≤ N . III. M
AIN R ESULTS
In this section, we present the main results of this paper. Our main contribution is proposing anovel delivery scheme for cache-aided interference networks that outperforms the coding schemesof prior work in the literature, for almost all values of the cache size. Moreover, we show thatthe proposed scheme is within a multiplicative factor of from the information-theoretic bound,independent of system parameters. Theorem 1.
For a general K T × K R cache-aided interference network with a library of N ≥ K R files, and cache size M ∈ [ N/K T : N ] files at each transmitter, the sum degrees of freedomsatisfies min (cid:26) K T K R K T + K R − K T M/N , K R (cid:27) ≤ DoF ( M ) ≤ min (cid:26) K T K R K T + K R − K T M/N , K R (cid:27) . (4)The proposed coding scheme that achieves the lower bound on the sum-DoF in Theorem 1is obtained by applying both zero-forcing and interference alignment techniques. In particu-lar, we schedule the bits needed to be transmitted to receivers as groups, where each groupcontains K R K T messages transmitted simultaneously in the same transmission block such thateach transmitter has a dedicated message intended to each receiver. Furthermore, the messagesassigned to a specific transmitter are available at K T M/N − adjacent transmitters which wecall cooperative or, equivalently cognitive transmitters. Thus, we design transmit beamformingmatrices comprising of two layers of precoders, i.e., two precoders are multiplied by each other.The first layer (ZF precoder) is designed to null out messages intended to a specific receiver at K T M/N − unintended, neighboring receivers. The second layer (IA precoder) is designed toalign the remaining interference signals at each receiver to occupy smaller dimensions of space.The detailed description of the achievable coding scheme is presented in Section V. Theorem 1also shows that the proposed coding scheme is within at most a multiplicative factor of fromthe information-theoretic bound for all system parameters (the number of transmitters K T , thenumber of receivers K R , the number of files N , and the cache size M ). The proof is providedin Appendix A.Before going into the details of our proposed scheme, we compare the achievable sum-DoF of our proposed coding scheme with those of the coding schemes in prior work [19], [25]–[27] .It is convenient to define a parameter τ = K T M/N to compare between the different schemesat corner points M = τ N/K T for τ ∈ [ K T ] . The reciprocal sum degrees of freedom / DoF ( M ) for M ∈ [ N/K T : N ] is obtained from the lower convex envelope of these corner points, since / DoF ( M ) is a convex function of cache size M (See [19, Lemma ]).In [19], the authors have exactly characterized the sum-DoF of cache-aided interferencenetworks for the special case of K T = K R = 3 at corner points τ ∈ [3] . When τ ∈ { , } ,the coding scheme in [19] and our proposed scheme achieve the same sum-DoF, where theoptimal sum-DoF can be achieved by only using interference alignment for τ = 1 , and thezero-forcing for τ = 3 . When τ = K R − , the authors in [19] have proposed a codingscheme that jointly uses both the zero-forcing and interference alignment techniques to achieve DoF = 18 / . Meanwhile our proposed scheme achieves DoF = 9 / by using a novel schedulingtechnique at the delivery phase. Although the scheme in [19] is optimal at the point τ = K R − ,this scheme cannot be generalized for the other corner points < τ < K R − for a general K T × K R interference network as we will elaborate more in Section IV.For comparison, we summarize the achievable sum-DoF of our proposed scheme and theschemes in [25]–[27] as follows DoF proposed =min (cid:26) K T K R K T + K R − τ , K R (cid:27) , (5) DoF [25] =min { τ, K R } , (6) DoF [26] = K T K R K T + K R − , (7) DoF [27] = K R τ ≥ K Rτ ( KTτ ) K R τ ( KTτ ) +1 τ = K R − (cid:26) max ≤ ` τ ≤ τ ` τ ( KT ` τ ) K R ` τ ( KT ` τ ) +( K R − ` τ ) ( KT ` τ − ) , τ (cid:27) τ < K R − . (8) In [25], the authors have introduced a one-shot linear scheme to achieve the sum-DoF in (6) byapplying the zero-forcing technique. It is easy to show that our proposed coding scheme achievesthe same sum-DoF when τ ≥ min { K T , K R } . Actually, the coding schemes that are basedon zero-forcing only can achieve the optimal DoF as τ ≥ min { K T , K R } (See Appendix A).However, our proposed coding scheme achieves a strictly higher sum-DoF when ≤ τ < It is worth mentioning that [25]–[27] investigate the problem of a general interference channel with caches at both thetransmitters and the receivers. Therefore, to compare their coding schemes to our proposed one, we consider the special caseof their results in which the caches are available only at the transmitters, i.e., the caches at receivers are equal to zero. M Proposed coding scheme Naderializadeh [25]Hachem [26]Fan Xu [27]
Fig. 2: Comparison of the reciprocal sum degrees of freedom versus the transmitter cache sizefor interference network with K T = 50 transmitters and K R = 50 receivers. min { K T , K R } which indicates that using zero-forcing alone is not sufficient to achieve a goodperformance for all cache size M .A delivery scheme using only interference alignment has been introduced in [26] to get thesum-DoF in (7), where the authors have neglected the gain that can be obtained from thecooperation between the transmitters when the cache size increases. In contrast, our codingscheme achieves a strictly higher DoF when < τ ≤ K T . Moreover, the multiplicative gapbetween the achievable sum-DoFs is given by DoF prposed
DoF [26] = 1 + τ − K T + K R − τ . Observe that this gapis approximately small when the number of transmitters or receivers is large. This is because,when K T → ∞ or K R → ∞ , the delivery scheme based on interference alignment achievesapproximately the optimal DoF of the K T × K R interference network which is min { K T , K R } .However, at a small or moderate number of transmitters or receivers, this gap increases as thetransmitter cache size increases, and hence, using both zero-forcing and interference alignmentbecome mandatory to obtain a better performance.Finally, [27] has developed a delivery scheme using both zero-forcing and interference align-ment to obtain the sum-DoF in (8). We show in Section IV that the scheme in [27] achieves theoptimal sum-DoF when τ = K R − . However, our proposed scheme outperforms their deliveryscheme when < τ < K R − . To see this, we multiply the numerator and denominator ofthe achievable sum-DoF in (5) with (cid:0) K T − τ − (cid:1) to get DoF proposed = τ ( KTτ ) K R τ ( KTτ ) +( K R − τ ) ( KT − τ − ) . The reasonbehind this is that our delivery scheme is designed to efficiently minimize the space dimensionsspanned by the interference signals at each receiver, compared to the scheme in [27]. Example 1.
Consider an interference network with K T = 50 transmitters, K R = 50 receivers,and a library of N = 50 files. Figure 2 shows the inverse of the sum-DoF of our proposed schemeand the other schemes in [25]–[27] as a function of the transmitter cache size. It is observedthat the proposed coding scheme outperforms the other schemes, specifically for moderate cachesizes. For example, when the transmitter cache size is M = 25 files, our proposed schemeachieves DoF proposed = 33 . , while ZF scheme in [25] achieves DoF [25] = 25 , IA scheme in [26]achieves
DoF [26] = 25 . , and the delivery scheme in [27] achieves DoF [27] = 25 . .IV. C ONTENT P LACEMENT AND T HE C OOPERATIVE X -N ETWORK
In this section, we first present the content placement strategy for cache-aided interferencenetworks that was introduced in [19] and [25]. Then, we show that for this content placementstrategy, there arises a new network topology in the delivery phase which we call the cooperative X -network . For the cooperative X -network, we derive an upper bound on the sum-DoF whichis considered an upper bound for any delivery scheme using such uncoded placement strategy. A. Content Placement
Assume each transmitter has a cache of size M = τ N/K T , where τ ∈ [ K T ] . In the placementphase, we split each file W f ∈ W into (cid:0) K T τ (cid:1) disjoint subfiles, each of size F/ (cid:0) K T τ (cid:1) bits. As aresult, file W f is represented by W f = { W f, S : S ⊆ [ K T ] , |S| = τ } . (9)For every file W f ∈ W , the subfile W f, S is stored at the cache of transmitter TX i , if i ∈ S .Thus, each transmitter stores N (cid:0) K T − τ − (cid:1) subfiles. Accordingly, the number of bits stored at eachtransmitter is equal to N (cid:0) K T − τ − (cid:1) F ( KTτ ) =
M F bits. Note that this placement strategy satisfies thecache size constraint for each transmitter. Moreover, we emphasize that the content placement isperformed without any prior knowledge of the receiver demands or channel gains in the deliveryphase, which is a practically relevant assumption, since there is a large time separation betweenthe placement and delivery phases. This placement strategy has been first proposed in [19]and [25] for cache-aided interference networks. For an illustration of the content placement,please refer to Example in Subsection IV-C. B. Cooperative X -network In the delivery phase, consider a demand vector d , where the receiver RX j requests file W d j , j ∈ [ K R ] . The transmitters should send the requested subfiles (cid:8) W d j , S : j ∈ [ K R ] , S ⊆ [ K T ] , |S| = τ (cid:9) to the receivers with a total of K R (cid:0) K T τ (cid:1) subfiles. In this case, each subset S of τ transmittershas a dedicated message W d j , S to be delivered to the receiver RX j , j ∈ [ K R ] . This transmissionproblem is called a cooperative X -network , where τ is the cooperation order between trans-mitters. Note that the cooperative X -network formulation involves several types of transmissionproblems discussed next: • ( X -networks ): When τ = 1 , the problem is reduced to the traditional X -network studiedin [32] in which each transmitter has a dedicated message for every receiver with no cooperationbetween the transmitters. The optimal sum-DoF of K T × K R X -networks is given by K T K R K R + K R − that is obtained by employing interference alignment. • ( MISO broadcast channel ): When τ = K T , all transmitters fully cooperate with each otherto deliver K R messages, one for each receiver. Thus, the problem becomes a MISO broadcastchannel with a single transmitter of K T antennas connected to K R receivers, each equipped with asingle antenna. The maximum sum-DoF of a MISO broadcast channel is given by min { K T , K R } which is obtained using zero-forcing [33]. • ( MISO X -networks ): For general < τ < K T , observe that the messages { W d j , S } K R j =1 are available at τ transmitters { TX i } , i ∈ S , i.e., these messages can be transmitted using τ antennas. Hence, the cooperative X -network can be seen as a MISO X -network with (cid:0) K T τ (cid:1) virtual transmitters each with τ antennas and K R single-antenna receivers. The MISO X -networkwas studied in [34], where the optimal sum-DoF for (cid:0) K T τ (cid:1) transmitters each with τ antennasand K R single-antenna receivers is determined as τ ( KTτ ) K R τ ( KTτ ) + K R − τ . However, there is a subtle, yet,important difference between the cooperative X -network and the actual MISO X -network. Thechannels between the multi-antenna transmitters and receivers in the actual MISO X -networkare i.i.d. drawn from continuous distributions; however, the channels between the (cid:0) K T τ (cid:1) virtualtransmitters and K R receivers in the cooperative X -network are dependent random variablesderived from only K R K T actual channels. Therefore, this raises an unsettled key question:Does the cooperative X -network with cooperation order < τ < K R have the same sum-DoF as the MSIO X -network? In [19], a delivery scheme for a × cooperative network wasproposed to achieve a sum-DoF of τ (cid:0) K T τ (cid:1) / (cid:0) τ (cid:0) K T τ (cid:1) + 1 (cid:1) for τ = K R − . Moreover, the authors in [27] proposed a delivery scheme for cooperative X -network achieving sum-DoF of τ (cid:0) K T τ (cid:1) / (cid:0) τ (cid:0) K T τ (cid:1) + 1 (cid:1) for τ = K R − for arbitrary number of transmitters and receivers. Theachievability proofs in [19] and [27] are mainly based on generating independent precodingfactors at the transmitters to deal with the dependence of the derived channel coefficients.Therefore, both cooperative X -networks and MISO X -networks have the same sum-DoF of τ (cid:0) K T τ (cid:1) / (cid:0) τ (cid:0) K T τ (cid:1) + 1 (cid:1) when τ = K R − . However, is it possible to achieve the same results forcooperative X -networks for all cooperation orders < τ < K R ? To answer this open question,we derive an upper bound on the sum-DoF of the cooperative X -networks presented in thefollowing theorem. Theorem 2.
The sum degrees of freedom of the cooperative X -network of K T transmitters, K R receivers, and K R (cid:0) K T τ (cid:1) messages satisfies DoF ( τ ) ≤ min τ ≤ σ ≤ min { K T ,K R } τ (cid:0) K T τ (cid:1) K R τ (cid:0) K T τ (cid:1) + ( K R − σ ) (cid:0) σ − τ − (cid:1) (10) Proof.
The proof is presented in Appendix B. (cid:4)
In Theorem 2, it is shown that the upper bound on the sum-DoF of the cooperative X -networkcoincides with the sum-DoF of the MISO X -network when τ = K R − . Hence, the achievableschemes in [19] and [27] are optimal when τ = K R − . However, the upper bound in Theorem 2shows that the cooperative X -network has a smaller sum-DoF than the MISO X -network when < τ < K R − . Therefore, we cannot proceed along the lines of the delivery scheme in [19]to achieve sum-DoF of τ ( KTτ ) K R τ ( KTτ ) + K R − τ for < τ < K R − which is fundamentally different fromthe case of τ = K R − .Theorem 2 not only gives an upper bound on the sum-DoF of cooperative X -networks, butalso gives an upper bound on the sum-DoF of cache-aided interference networks that uses theplacement strategy introduced in Subsection IV-A. In other words, there is no delivery scheme forcache-aided interference networks with transmitter caches of size M = τ N/K T files achievinga sum-DoF higher than the bound given in Theorem 2 under the content placement strategyintroduced in Subsection IV-A.In [20], an information-theoretic bound on the normalized delivery time (NDT) of the cache-aided interference networks is proposed. The NDT refers to the worst case delivery time forsatisfying the receivers requests. Thus, the relation between NDT and the sum-DoF is given by NDT = K R / DoF . Figure 3 depicts the inverse of the upper bound on the sum-DoF proposed MN DoF ∗ Achievable bound Theorem 1Upper bound Theorem 2Upper bound [20, Theorem 1]
Fig. 3: Comparison of the reciprocal of the upper bound on the sum-DoF proposed in [20] andthe reciprocal of the upper bound in Theorem 2 for the × cache-aided interference network.in [20, Theorem ] for the × cache-aided networks and the inverse of the upper boundon the sum-DoF of the cooperative X -networks in Theorem 2 for M = τ N/K T , τ ∈ [10] . Notethat, as demonstrated in Figure 3, the upper bound on the cooperative X -networks is tighter thanthe upper bound introduced in [20] for smaller cache sizes. This may be due to two possibleexplanations. First, it is possible that the content placement introduced in Section IV-A is notsufficient to achieve the upper bound in [20]. Second, the upper bound in [20] is not achievablefor small cache sizes. This observation motivates future work to find better placement strategiesor upper bounds for cache-aided interference networks.Before presenting the achievable scheme for the general case, we first illustrate the main ideaof the achievable scheme, using an example. C. Achievable scheme for × cache-Aided interference networks Consider a cache-aided interference network with K T = 3 transmitters each with a cache { Z i } i =1 of size M = 2 files, K R = 3 receivers, and a library of N = 3 files: W = { A, B, C } ,each consisting of F bits. In the placement phase , each file W f is split into (cid:0) (cid:1) = 3 subfiles of equal size F/ bits. Welabel each subfile with W f, S , where S ⊂ [3] and |S| = 2 such that the subfile W f, S is storedat transmitters TX i , i ∈ S . For example, file A is split as follows A = ( A , A , A ) , wheresubfile A is stored at transmitters TX and TX , subfile A at TX and TX , and subfile A at TX and TX . Hence, the cache contents at each transmitter are given by Z = ( A , A , B , B , C , C ) Z = ( A , A , B , B , C , C ) Z = ( A , A , B , B , C , C ) Hence, each transmitter stores ∗ (cid:0) − − (cid:1) = 6 subfiles with each of it having a size of F/ bitsresulting in a total of F bits that satisfies the memory size constraint. In the delivery phase , we assume that the receiver demand files W d = A , W d = B , and W d = C . Consider subfile W f, { ij } , i < j , ( i, j ) ∈ [3] stored at two transmitters TX i and TX j .We split subfile W f, { ij } into two disjoint smaller subfiles of equal sizes W if, { ij } , W jf, { ij } . Forsubfile W if, { ij } , the transmitter TX i is the primary (master) transmitter responsible for deliveringthis subfile, while transmitter TX j represents the cooperative transmitter that has this subfile asa side information. For example, subfile A is split into A = ( A , A ) , where subfile A isassigned to transmitter TX while it is also available at transmitter TX as a side information.Consequently, we schedule the small subfiles into sets with each set containing subfiles,such that the subfiles in the same set are delivered simultaneously in the same transmissionblock. In other words, it requires transmission blocks to complete the transmission, whereeach transmission block occurs over µ n -symbol extensions. The value of µ n will be specifiedlater. Figure 4 illustrates the scheduling strategy and the set of subfiles transmitted in eachtransmission block. In each transmission block, each transmitter has a distinct subfile intendedto each receiver, and also has the subfiles of an adjacent transmitter as a side information.For example, consider the first transmission block (Fig. 4a). Transmitter TX has subfiles A , B , and C intended to receivers RX , RX , and RX , respectively. In addition, the subfiles A , B , C are available as a side information at transmitter TX . Thus, we consider transmitterTX as a cooperative (cognitive) transmitter to the transmitter TX .We now present the delivery scheme at the first transmission block. The delivery scheme isbased on designing two-layer beamforming matrices at the transmitters, where the first layer isresponsible for applying zero-forcing and the second layer is responsible for applying asymptoticinterference alignment. For transmitter TX , the subfiles A , B , C are transmitted usingbeamforming matrices V a, , V b, , V c, , respectively, and subfiles A , B , C are transmittedusing beamforming matrices V a, , V b, , V c, , respectively. Each beamforing matrix consists TX TX RX RX RX RX RX RX TX TX A A A A A A B C B C B C B C B C B C (a) First transmission block TX TX RX RX RX RX RX RX TX TX A A A A A A B C B C B C B C B C B C (b) Second transmission block Fig. 4: Description of the delivery scheme for the × cache-aided interference network witha library of files and transmitter cache size M = 2 files.of two layers, e.g., V a, = V , ZF a, V IA a , where the first layer precoder V , ZF a, is different for eachtransmitter while the second precoder V IA a is the same for each transmitter.The first precoder is designed such that the messages intended to receiver RX i from alltransmitters are canceled at the adjacent receiver RX i +1 . In other words, the objective of thefirst layer is to zero-force the subfiles ( A , A , A ) at receiver RX , subfiles ( B , B , B ) at receiver RX , and ( C , C , C ) at receiver RX . For example, to cancel the interference ofsubfile A at receiver RX , we set precoders V , ZF a, and V , ZF a, as follows V , ZF a, ( u ) = h ( u ) , ∀ u ∈ [ µ n ] V , ZF a, ( u ) = − h ( u ) , ∀ u ∈ [ µ n ] , (11)where V , ZF a, ( u ) , V , ZF a, ( u ) are the u -th diagonal element of matrices V , ZF a, , V , ZF a, , respectively.Thus, subfile A is received with directions M V IA a , , M V IA a at receivers RX , RX , andRX , respectively, where M lmgk is a µ n × µ n diagonal matrix with the u -th element given by M lmgk ( u ) = det h gl ( u ) h gm ( u ) h kl ( u ) h km ( u ) . (12)The received directions of all subfiles after the first layer of precoding at all receivers is shown inTable I. Thus, each receiver has three desired signals and three interference signals. Therefore,we design the second layer of precoders V IA a , V IA b , V IA c such that the interference signals atreceiver RX are aligned at a space of dimension | V IA b | . Similarly, the interference signals at TABLE I: The received directions of the subfiles at each receiver after the first layer of precoding
Subfiles Received directionsRX RX RX A M V IA a V IA a A M V IA a V IA a A M V IA a V IA a B M V IA b M V IA b B M V IA b M V IA b B M V IA b M V IA b C V IA c M V IA c C V IA c M V IA c C V IA c M V IA c receiver RX are aligned at a space of dimension | V IA c | , and the interference signals at receiverRX are aligned at a space of dimension | V IA a | . We can satisfy these alignment conditions byapplying asymptotic interference alignment scheme as in [32]. The main challenge here is thatthe derived channel coefficients are polynomial functions of the original channel coefficients.Thus, we prove that the interference alignment conditions are still feasible in our scenario inAppendix D. For symmetry, we set | V IA a | = | V IA b | = | V IA c | = | V | . As a result, at each receiver,the desired signals occupy a space of dimension | V | and after applying the alignment strategy,the interference occupies a space of dimension | V | . Hence, if we consider µ n ≈ | V | symbolextensions such that each desired signal spans a space linearly independently of the other desiredsignals and the interference spaces, then each receiver achieves a DoF of | V | | V | . Thus, a total of / sum-DoF is achievable in the first transmission block. In a similar manner, we repeat thesame scheme in the second transmission block. Therefore, the sum-DoF of / is achievable,overall.In the rest of the paper, we focus on proving the achievability of the lower bound on thesum-DoF in Theorem 1 for corner points M = τ N/K T with τ ∈ [ K T ] .V. A CHIEVABLE S CHEME
In this section, we propose a new delivery scheme that achieves the lower bound in Theorem 1for corner points of transmitter cache size M = τ N/K T for τ ∈ [ K T ] . Without loss ofgenerality, consider a demand vector d in which the receiver RX j requests file W d j under thecontent placement strategy introduced in Subsection IV-A. Thus, the transmitters should send the requested subfiles (cid:8) W d j , S : j ∈ [ K R ] , S ⊆ [ K T ] , |S| = τ (cid:9) which means a total of K R (cid:0) K T τ (cid:1) subfiles are delivered to the receivers. A. When τ ≥ K R The transmission is performed over (cid:0) K T τ (cid:1) time slots. At each slot, the set S of τ transmitterscooperates to send subfiles (cid:8) W d j , S (cid:9) K R j =1 to the receivers, where the network becomes as a MISObroadcast channel with a single transmitter of τ antennas and K R single-antenna receivers.Thus, beamforming and zero-forcing are sufficient to achieve DoF = K R , where the sum-DoFis bounded by the number of receivers K R (See Appendix B).In the following, for a given set G , we use Π G to denote the set of |G| ! permutations of set G , and Π circ G to denote the set of ( |G| − circular permutations of set G . For a permutation π ∈ Π G , weuse π ( i ) = π ( i + |G| ) to denote the i th element in permutation π , and π [ i : j ] = [ π ( i ) , . . . , π ( j )] for j ≥ i . B. When τ < K R First, we split each subfile W d j , S for j ∈ [ K R ] , S ⊆ [ K T ] such that |S| = τ into τ ! ( K T − τ )! disjoint smaller subfiles as follows W d j , S (cid:44) (cid:110) W π (1) d j ,π, ˜ π : π ∈ Π S , ˜ π ∈ Π [ K T ] \S (cid:111) , (13)where the index π (1) of subfile W π (1) d j ,π, ˜ π refers to the primary (master) transmitter TX π (1) respon-sible for delivering this subfile, while transmitters π [2 : τ ] represent the cooperative transmittersthat have this subfile as a side information. Furthermore, subfile W π (1) d j ,π, ˜ π is not available at thecache of transmitters TX i , i ∈ ˜ π . This splitting strategy helps us to schedule the smaller subfilesrequired to be delivered to the receivers into ( K T − sets, where each set has K R K T smallersubfiles that would be delivered in the same transmission block. Lemma 1.
The set of smaller subfiles needed to be delivered to the receivers can be partitionedinto ( K T − disjoint sets each of size K R K T smaller subfiles as follows (cid:91) π ∈ Π circ [ KT ] (cid:110) W π ( i ) d j ,π [ i : i + τ − ,π [ i + τ : K T + i − : i ∈ [ K T ] , j ∈ [ K R ] (cid:111) . (14) A circular permutation of a set is an ordered selection of the set elements arranged along a fixed circle. For example, forthe set G = { , , } , Π circ G = { [1 , , , [1 , , } . Proof.
The proof is presented in Appendix C (cid:4)
According to Lemma 1, we have ( K T − sets, where each set is defined by a unique permu-tation π ∈ Π circ [ K T ] , and each set contains K R K T smaller subfiles that we call messages from now.For a given π ∈ Π circ [ K T ] , we sort the transmitters in the permutation order π . Thus, each transmitterTX π ( i ) , i ∈ [ K T ] , has a dedicated message intended to every receiver RX j , j ∈ [ K R ] . Furthermore,the messages assigned to transmitter TX π ( i ) , { W π ( i ) d j ,π [ i : i + τ − ,π [ i + τ : K T + i − } K R j =1 , are available atthe adjacent τ − cooperative transmitters { TX l : l ∈ π [ i + 1 : i + τ − } , while these mes-sages are not available at the remaining ( K T − τ ) transmitters TX l , l ∈ π [ i + τ : K T + i − .For any arbitrary n ∈ N + , we consider each transmission block occurs over µ n = K T n Γ +( K R − τ ) ( n + 1) Γ symbol extensions of the original channel, where Γ = K T ( K R − τ ) . We de-sign a coding scheme to deliver the messages in each block with a sum-DoF of K R K T n Γ K T n Γ ( K R − τ )( n +1) Γ . Note that as n → ∞ , we obtain DoF = K R K T K R + K T − τ . The main idea behind the proposed codingscheme in each transmission block is that each transmitter has K R messages, where each messageis required by a dedicated receiver, and these messages are available at τ − cooperativetransmitters as a side information. Thus, we develop two-layer beamforming matrices at eachtransmitter. In the first layer, each transmitter with the help of the cooperative transmitters nullsout the message intended to a specific receiver into the neighboring τ − receivers. Hence,for every receiver, there are K T desired messages, one from each transmitter, in addition to K T ( K R − τ ) interference messages. Then, we design an asymptotic interference alignmentscheme in the second layer to let the interference signals span a ( K R − τ ) dimensional space byaligning every K T interference signals into a single-dimensional space. Therefore, every receivercan decode the desired K T messages over K T + K R − τ dimensional spaces to get a DoF of K T K T + K R − τ per receiver.Without loss of generality, let us focus on the transmission block for π = [1 , . . . , K T ] fordelivering subfiles (cid:110) W id j , [ i : i + τ − , [ i + τ : K T + i − : i ∈ [ K T ] , j ∈ [ K R ] (cid:111) , where the delivery schemeof the remaining blocks is performed in the same manner. The input-output relation of theoriginal channel in (1) over µ n -symbol extensions is given by Y k = K T (cid:88) i =1 H ki X i + N k , (15)where Y k and N k represent µ n × column vectors of the received signal and the Gaussiannoise of the receiver RX k over µ n -symbol extension, respectively. X i is a µ n × column vector representing the transmitted vector of transmitter TX i . H ki is a µ n × µ n diagonal matrix ofchannel coefficients from transmitter TX i to receiver RX k over µ n symbol extension. H ki = h ki (1) 0 . . . h ki (2) . . . ... ... . . . ... . . . h ki ( µ n ) . (16)Each message W id j , [ i : i + τ − , [ i + τ : K T + i − , i ∈ [ K T ] , j ∈ [ K R ] , is encoded into n Γ independentstreams represented by a n Γ × column vector X j, [ i : i + τ − . Note that each vector X j, [ i : i + τ − can beconstructed at transmitters TX l , l ∈ [ i : i + τ − , i.e., it can be sent from τ transmitters. Hence,transmitter TX l uses beamforming matrix V lj, [ i : i + τ − for precoding the vector X j, [ i : i + τ − , where V lj, [ i : i + τ − is a µ n × n Γ matrix. Therefore, we can describe the transmitted vector of transmitterTX i as follows: X i = K R (cid:88) j =1 i (cid:88) l = i − τ +1 V ij, [ l : l + τ − X j, [ l : l + τ − , (17)where it can be verified that transmitter TX i is a cooperative transmitter for transmitters l ∈ [ i − τ + 1 : i ] , i.e., transmitter TX i has the messages of transmitters TX l , l ∈ [ i − τ + 1 : i ] , asa side information. Furthermore, the received signal at receiver RX k is given by Y k = K T (cid:88) i =1 H ki (cid:32) K R (cid:88) j =1 i (cid:88) l = i − τ +1 V ij, [ l : l + τ − X j, [ l : l + τ − (cid:33) + N k = K R (cid:88) j =1 K T (cid:88) i =1 i + τ − (cid:88) l = i H kl V lj, [ i : i + τ − X j, [ i : i + τ − + N k . (18)Note that the received signal spans a space of dimension µ n = K T n Γ + ( K R − τ ) ( n + 1) Γ .Our objective is to design the beamforming matrices (cid:110) V lj, [ i : i + τ − (cid:111) such that the interferencesignals occupy a subspace of dimension ( K R − τ ) ( n + 1) Γ , leaving the desired signals to spanan interference-free subspace of dimension K T n Γ out of µ n dimensions. We propose the beam-forming matrix V lj, [ i : i + τ − comprising of two layers (two precoder matrices) as follows V lj, [ i : i + τ − = V l, ZF j, [ i : i + τ − V IA j, [ i : i + τ − , (19)where V l, ZF j, [ i : i + τ − is a µ n × µ n diagonal matrix designed based on zero-forcing technique. Theobjective of these precoders is to null out the signals { X j, [ i : i + τ − } K T i =1 , intended to receiver RX j from all transmitters, at the neighboring τ − undesired receivers; however, these signals stillcause interference at the remaining ( K R − τ ) receivers. Observe that the precoders { V l, ZF j, [ i : i + τ − } are different at each transmitter TX l , l ∈ [ i : i + τ − , i.e., V l, ZF j, [ i : i + τ − (cid:54) = V ˜ l, ZF j, [ i : i + τ − for (cid:16) l, ˜ l (cid:17) ∈ [ i : i + τ − and l (cid:54) = ˜ l . The second precoder V IA j, [ i : i + τ − is a µ n × n Γ matrix designed toapply asymptotic interference alignment such that the signals intended to receiver RX j fromall transmitters are aligned at ( K R − τ ) undesired receivers into a single space of dimension ( n + 1) Γ , where this precoder is the same at all transmitters TX l , l ∈ [ i : i + τ − . In thefollowing, we give the design of these two-layers of precoders in detail.
1) Design of the Zero-forcing Precoder:
Given the beamforming matrix design in (19), wecan rewrite the received signal at receiver RX k as Y k = K R (cid:88) j =1 K T (cid:88) i =1 (cid:32) i + τ − (cid:88) l = i H kl V l, ZF j, [ i : i + τ − (cid:33) V IA j, [ i : i + τ − X j, [ i : i + τ − + N k . (20)Observe that the signal X j, [ i : i + τ − intended to receiver RX j is available at τ transmitters, andhence, these transmitters can cooperate with each other to null out this signal at τ − unintendedreceivers. Thus, we aim to design precoders { V l, ZF j, [ i : i + τ − } such that the signal X j, [ i : i + τ − doesnot cause interference at receivers RX k , k ∈ [ j + 1 : j + τ − . For given i ∈ [ K T ] , it is requiredfor precoders { V l, ZF j, [ i : i + τ − } , l ∈ [ i : i + τ − , to satisfy the conditions i + τ − (cid:88) l = i H kl V l, ZF j, [ i : i + τ − = (cid:80) i + τ − l = i h kl (1) V l, ZF j, [ i : i + τ − (1) . . . ... . . . ... . . . (cid:80) i + τ − l = i h kl ( µ n ) V l, ZF j, [ i : i + τ − ( µ n ) = µ n × µ n (21) for all k ∈ [ j + 1 : j + τ − , where V l, ZF j, [ i : i + τ − ( u ) is the u -th diagonal element of the precodermatrix V l, ZF j, [ i : i + τ − . Let H ( u ) denote the K R × K T channel matrix between K T transmitters and K R receivers at time slot u ∈ [ µ n ] . Moreover, H S T S R ( u ) represents |S R | × |S T | submatrix of thechannel matrix H ( u ) formed by taking rows indexed by S R and columns indexed by S T . Forarbitrary vector a = ( a , · · · , a τ ) , we consider the determinant of the following matrix usingcofactor expansion det h j +1 ,i ( u ) . . . h j +1 ,i + τ − ( u ) ... ... h j + τ − ,i ( u ) . . . h j + τ − ,i + τ − ( u ) a . . . a τ = τ (cid:88) l =1 a l c l ( u ) , (22) where c l ( u ) is the cofactor of element a l . By taking V l, ZF j, [ i : i + τ − ( u ) = c l ( u ) , we can verify that i + τ − (cid:88) l = i h kl ( u ) V l, ZF j, [ i : i + τ − ( u ) = det h j +1 ,i ( u ) . . . h j +1 ,i + τ − ( u ) ... ... h j + τ − ,i ( u ) . . . h j + τ − ,i + τ − ( u ) h ki ( u ) . . . h k,i + τ − ( u ) = if k ∈ [ j + 1 : j + τ − M [ i : i + τ − { k }∪ [ j +1: j + τ − ( u ) otherwise (23)where M [ i : i + τ − { k }∪ [ j +1: j + τ − ( u ) is the determinant of the submatrix H [ i : i + τ − { k }∪ [ j +1: j + τ − ( u ) . Therefore,the conditions in (21) are satisfied. Thus, the received signal at receiver RX k can be rewrittenas Y k = K T (cid:88) i =1 M [ i : i + τ − k : k + τ − V IA k, [ i : i + τ − + K T (cid:88) i =1 K R + k − τ (cid:88) j = k +1 M [ i : i + τ − { k }∪ [ j +1: j + τ − V IA j, [ i : i + τ − + N k , (24)where M { k }∪ [ j +1: j + τ − is a µ n × µ n diagonal matrix with the u -th diagonal element M [ i : i + τ − { k }∪ [ j +1: j + τ − ( u ) for u ∈ [ µ n ] . As a result, the signals intended to receiver RX j from all transmitters are can-celed at receivers k ∈ [ j + 1 : j + τ − , while these signals interfere at receivers RX k , k ∈ [ j + τ : K R + j − . Remark 1. ( Full rank of the first-layer precoder ): Note that the entries of the channel matrix H ( u ) are drawn i.i.d. from a continuous distribution. Hence, H ( u ) has full rank with probabilityone for all u ∈ [ µ n ] . Furthermore, any square submatrix from H ( u ) also has full rank withprobability one. Thus, cofactors { c l ( u ) } µ n u =1 have non-zero values, and they are independentof each other, since they are formed from i.i.d. channel coefficients. Therefore, the first-layerprecoder V l, ZF j, [ i : i + τ − is a full rank matrix almost surely.
2) Design of the Interference Alignment Precoder:
The received signal at receiver RX k given in (24) consists of two terms: the first term represents K T desired data streams, andthe second term is the interference signals from the data streams intended to receivers RX j , j ∈ [ k + 1 : K R + k − τ ] . Our objective is to align the interference signals at receiver RX k into asubspace of dimension ( K R − τ ) ( n + 1) Γ in order to let the remaining µ n − ( K R − τ ) ( n + 1) Γ = K T n Γ dimensions to the desired signals. Consider the signals desired by receiver RX j that causesinterference at ( K R − τ ) receivers RX k , k ∈ [ j + τ : K R + j − . We would like to align thesesignals into a single subspace of dimension ( n + 1) Γ at receivers RX k , k ∈ [ j + τ : K R + j − . Towards this objective, precoders V IA j, [ i : i + τ − , i ∈ [ K T ] , are chosen to satisfy the followingalignment conditions. M [ i : i + τ − { k }∪ [ j +1: j + τ − V IA j, [ i : i + τ − ≺ V IA j , ∀ i ∈ [ K T ] , ∀ k ∈ [ j + τ : K R + j − , (25)where P ≺ Q means that the column space of matrix P is a subspace of the column space ofthe matrix Q . First, we choose V IA j, [ i : i + τ − = V IA j . (26)Therefore, the problem is reduced to finding precoders V IA j , V IA j for all j ∈ [ K R ] satisfying M [ i : i + τ − { k }∪ [ j +1: j + τ − V IA j ≺ V IA j , ∀ i ∈ [ K T ] , ∀ k ∈ [ j + τ : K R + j − . (27)In (27), there are Γ = K T ( K R − τ ) conditions that V IA j and V IA j are required to satisfy. First,we generate random vectors a j = [ a j (1) , . . . , a j ( µ n )] T , j ∈ [ K R ] , i ∈ [ K T ] , where its elementsare i.i.d. drawn from a continuous distribution. Then, we choose V IA j = (cid:40) K T (cid:89) i =1 K R + j − (cid:89) k = j + τ (cid:16) M [ i : i + τ − { k }∪ [ j +1: j + τ − (cid:17) α [ k,i ] j a j : 0 ≤ α [ k,i ] j ≤ n (cid:41) , V IA j = (cid:40) K T (cid:89) i =1 K R + j − (cid:89) k = j + τ (cid:16) M [ i : i + τ − { k }∪ [ j +1: j + τ − (cid:17) α [ k,i ] j a j : 0 ≤ α [ k,i ] j ≤ n − (cid:41) . (28)Thus, we can verify that the conditions in (27) are satisfied and the precoder matrices V IA j and V IA j , j ∈ [ K R ] , have full column rank of ( n + 1) Γ and n Γ , respectively. The proof is presentedin Appendix D. Now consider receiver RX k . The desired streams of receiver RX k have arrivedwith directions of K T n Γ column vectors D k = (cid:104) M [1: τ ][ k : k + τ − V IA k M [2: τ +1][ k : k + τ − V IA k · · · M [ K T : K T + τ − k : k + τ − V IA k (cid:105) , (29)while the interference signals have arrived after alignment with directions of ( K R − τ ) ( n + 1) Γ column vectors I k = (cid:2) V IA j (cid:3) , j ∈ [ k + 1 : K R + k − τ ]= (cid:2) V IA k +1 · · · V IA K R + k − τ (cid:3) (30)To ensure that the receiver RX k can decode the desired streams, we should maintain that thedirections of all desired streams are linearly independent of each other and independent of alldirections of the interference streams. This can be ensured if the following matrix R k = [ D k I k ] (31) has full rank of µ n almost surely for all channel realizations. We left the proof of full rank toAppendix D. As a result, each receiver RX k can decode K T n Γ desired streams over µ n -symbolextensions, and hence, a total of K R K T n Γ K T n Γ +( K R − τ )( n +1) Γ DoF is achievable in each transmission block.VI. C
ONCLUSION
In this paper, we have studied the degrees of freedom of interference networks with caches atthe transmitters. Our main result is the characterization of the sum degrees of freedom of cache-aided interference channels within a multiplicative factor of from the information-theoreticbound, independent of all system parameters. To achieve this result, we have proposed a noveldelivery scheme for arbitrary number of transmitters and receivers that outperforms state-of-the-art schemes in literature, for almost all transmitter cache sizes. The achievability proof dependsmainly on scheduling the messages needed to be delivered to receivers into groups, where thegroups are delivered in subsequent transmission blocks. Then, we construct two-layer precodingmatrices based on ZF and IA techniques to deliver each group of messages in each transmissionblock. Moreover, we have derived an upper bound on the sum-DoF of delivery schemes that usesthe uncoded caching scheme. We have shown that the derived upper bound is tighter than theupper bound that exists in the literature under small cache sizes. Even with these improvements,there still exists a multiplicative gap of at most a factor of between the achievable scheme andupper bounds. It is an interesting point to find more sophisticated bounds to narrow this gap.A PPENDIX AP ROOF OF M ULTIPLICATIVE F ACTOR In this Appendix, we present the converse proof of Theorem 1. We first derive a simpleupper bound on the sum-DoF for general cache-aided networks without any restrictions on thecaching and delivery schemes. Then, we show that the multiplicative gap between the achievablelower bound in Theorem 1 and the derived upper bound is less than independent of all systemparameters. Assume that there exists full cooperation among K T transmitters to construct a singletransmitter of K T antennas that has access to cache memories of all K T transmitters. Moreover,assume that K R receivers are allowed to fully cooperate with each other to construct a singlereceiver with K R antennas. Note that this cooperation can not reduce the sum-DoF, and hence,this assumption does not violate the upper bound. The constructed multi-antenna transmitter has the K R requested files of the single, multi-antenna receiver. Thus, the system becomes apoint-to-point MIMO channel [35]. Hence the total sum-DoF is bounded by DoF ∗ ≤ min { K T , K R } . (32)Let τ = K T M/N . When τ ≥ min { K T , K R } , it is easily to see that both the upper and lowerbounds are the same, i.e., the achievable scheme is optimal in this case.When ≤ τ ≤ min { K T , K R } , the proposed coding scheme achieves DoF = K T K R K T + K R − τ .Hence, the multiplicative gap between the upper bound and the achievable DoF is given by DoF ∗ DoF = min { K T , K R } / K T K R K T + K R − τ ≤ . (33)Therefore, the achievable sum-DoF is within a factor of from the derived upper bound.A PPENDIX BC ONVERSE OF C OOPERATIVE X - NETWORK
Here, we present an upper bound on the sum-DoF of the cooperative X -network defined inSubsection IV-B. The upper bound is mainly based on genie-aided and cut-set arguments. Insuch cooperative X -networks, we have K R (cid:0) K T τ (cid:1) messages defined by W (cid:44) { W j, S : j ∈ [ K R ] , S ⊆ [ K T ] , |S| = τ } , such that each subset S ⊆ [ K T ] of τ transmitters has a dedicated message to every receiverRX j , j ∈ [ K R ] . Let R j, S ( P ) = | W j, S | T denote the rate of the codeword encoding the message W j, S , where the DoF associated with message W j, S is given by d j, S = lim P →∞ R ∗ j, S ( P )log( P ) . Thus,the sum-DoF of the cooperative X -network is given by DoF (cid:80) = (cid:80) S⊆ [ K T ] |S| = τ (cid:80) j ∈ [ K R ] d j, S . Let Y K be a |K| T × concatenated vector of the received signals of receivers RX j , j ∈ K , and X K bea |K| T × concatenated vector of the transmitted signals of transmitters in set K . Moreover, weconsider H is a K R T × K T T channel matrix between transmitters and receivers over T slots,and H K t K r is a |K r | T × |K t | T submatrix of H . Lemma 2.
The degrees of freedom of cooperative X -networks satisfies (cid:88) S⊆ [ K T ] |S| = τ (cid:88) j ∈S r d j, S + (cid:88) S⊆S t |S| = τ (cid:88) j ∈S r d j, S ≤ σ (34)for a subset S t of transmitters such that |S t | = σ and a subset S r of receivers such that |S r | = σ ,where τ ≤ σ ≤ min { K T , K R } . Moreover S t = [ K T ] \ S t , and S r = [ K R ] \ S r . Proof.
We first define the following three disjoint subsets of messages: W S t = { W j, S : j ∈ [ K R ] , S ⊆ S t , |S| = τ } , W S r = { W j, S : j ∈ S r , S ⊆ [ K T ] , S (cid:54)⊆ S t , |S| = τ } , W = (cid:8) W j, S : j ∈ S r and S (cid:54)⊆ S t , |S| = τ (cid:9) . (35)Now assume that a genie provides receivers in set S r with messages W , and receivers in set S r with messages W (cid:83) W S r . Since the messages W are given to all receivers, we are onlyconcerned of the degrees of freedom of the remaining messages W S t (cid:83) W S r . Assume that thereis full cooperation between receivers RX j , j ∈ S r . The main idea of the proof is to show thatthe set S r of σ receivers can decode the messages W S r (cid:83) W S t from its received signals usingmessages W as a side information. We represent the received signals of S r and S r receivers as Y S r = H S t S r X S t + H S t S r X S t + Z S r , Y S r = H S t S r X S t + H S t S r X S t + Z S r . (36)Consider receivers in set S r that can decode messages W S r . Using genie-aided messages W anddecoded messages W S r , receivers j ∈ S r can compute the transmitted signals X i , i ∈ S t andsubtract it from the received signal. Similarly, receivers in set S r have messages W (cid:83) W S r as aside information, and hence, they can compute the transmitted signals X i , i ∈ S t and subtractit from the received signal. Therefore, we can rewrite the signals of S r and S r receivers as ˜Y S r = H S t S r X S t + Z S r , ˜Y S r = H S t S r X S t + Z S r , (37)where receivers j ∈ S r are able to decode their intended messages { W j, S : j ∈ S r , S ⊆ S t } from the received signal vector ˜Y S r , and receivers j ∈ S r are able to decode their intendedmessages { W j, S : j ∈ S r , S ⊆ S t } from the received signal vector ˜Y S r . Notice that the σT × σT submatrix channel H S t S r is invertable almost surely. Thus, by reducing noise at receivers S r andmultiplying the constructed signal ˜Y S r at receivers S r by H S t S r (cid:0) H S t S r (cid:1) − , we have ˜Y (cid:48)S r = H S t S r X S t + ˜Z S r , (38)which is a degraded version of ˜Y S r , where ˜Z S r represents the reduced noise at receivers S r .Thus, receivers S r can decode all messages W S t . Thus, by using Fano’s inequality, we have H (cid:0) W S t | Y S r , W (cid:1) ≤ H (cid:0) W S t | Y S r , W , W S r (cid:1) ≤ |W S t | T (cid:15). (39) Notice that the applied assumptions (genie-aided information, cooperation between subset ofreceivers, reducing noise) cannot hurt the coding scheme. Thus, we have H ( W S t , W S r )= (cid:88) S⊆S t |S| = τ (cid:88) j ∈S r R ∗ j, S T + (cid:88) S⊆ [ K t ] |S| = τ (cid:88) j ∈S r R ∗ j, S T ( a ) = H (cid:0) W S t , W S r |W (cid:1) ( b ) = I (cid:0) W S t , W S r ; Y S r |W (cid:1) + H (cid:0) W S t , W S r | Y S r , W (cid:1) (40) ( c ) ≤ I (cid:0) X [ K T ] ; Y S r (cid:1) + H (cid:0) W S t , W S r | Y S r , W (cid:1) ( d ) ≤ T σ log ( P ) + H (cid:0) W S r | Y S r , W (cid:1) + H (cid:0) W S t | Y S r , W , W S r (cid:1) ( e ) ≤ T σ log ( P ) + |S r | T (cid:15) + |S t | T (cid:15) where ( a ) follows from the fact that the messages are independent. Step ( b ) follows from thechain rule. Step ( c ) follows from data processing inequality, where the signal X [ K T ] is a functionof messages W S t (cid:83) W S r . Step ( d ) follows from the bound of the degrees of freedom of multipleaccess channel (MAC) with K T single-antenna transmitters and a receiver with |S r | antennas.Finally, step ( e ) follows from Fano’s inequality. By diving on T log ( P ) and taking P → ∞ and (cid:15) → , we get (34). (cid:4) Repeating the inequality (34) for every subset S r ⊆ [ K R ] and every subset S t ⊆ [ K T ] , or forsimplicity consider a symmetric case where { d j, S = d } , we have DoF (cid:80) ≤ τ (cid:0) K T τ (cid:1) K R τ (cid:0) K T τ (cid:1) + ( K R − σ ) (cid:0) σ − τ − (cid:1) (41)To get the best tighter bound we optimize the bound in (41) on σ such that τ ≤ σ ≤ min { K T , K R } . This completes the proof.A PPENDIX CP ROOF OF L EMMA d , we have K R (cid:0) K T τ (cid:1) subfiles that need to be delivered to thereceives. In (13), we split each subfile into τ ! ( K T − τ )! smaller subfiles. Thus, a total of K R K T ! smaller subfiles needed to be delivered to all receivers. In Lemma 1, K R K T ! smaller subfiles arepartitioned into groups such that each group of smaller subfiles is defined by a unique permutation π ∈ Π circ [ K T ] of total ( K T − groups. Furthermore inside each group, we have K R K T smallersubfiles. Accordingly, the total number of smaller subfiles in all groups is given by ( K T − K R K T = K R K T ! (42) which is equal to the total number of smaller subfiles required to be delived, i.e., this partitioningstrategy contains all smaller subfiles needed to be delivered. However, we should ensure that thisgroups contains disjoint smaller subfiles, and each smaller subfile is an actual smaller subfileneeded to be delivered to maintain the correctness of the partitioning strategy.From splitting strategy in (13), we can see that each smaller subfile W π (1) d j ,π [1: τ ] ,π [ τ +1: K T ] , for π ∈ Π [ K T ] , j ∈ [ K R ] , is required to be delivered to the receivers. Moreover, for a given circularpermutation π ∈ Π circ [ K T ] , if we circularly shift π with ≤ i < K T times, it generates anotherpermutation ˜ π , where ˜ π ∈ Π [ K T ] but, ˜ π / ∈ Π circ [ K T ] . Therefore, for a given circular permutation π ∈ Π circ [ K T ] , smaller subfile W π ( i ) d j ,π [ i : i + τ − ,π [ i + τ : K T + i − , for i ∈ [ K T ] , j ∈ [ K R ] , is an actualsmaller subfiles required to be delivered to the receivers. Now, we prove that the groups aredisjoint. To prove this, consider an arbitrary subfile W π ( i ) d j ,π [ i : i + τ − ,π [ i + τ : K T + i − . We should ensurethat this subfile does not appear at another group, where we prove this by contradiction. Assumethat there exists a smaller subfile satisfying that W π ( i ) d j ,π [ i : i + τ − ,π [ i + τ : K T + i − ≡ W ˜ π ( ˜ i ) d j , ˜ π [ ˜ i :˜ i + τ − ] , ˜ π [ ˜ i + τ : K T +˜ i − ] (43)for ˜ π, π ∈ Π circ [ K T ] , ˜ π (cid:54) = π , ≤ i, ˜ i ≤ K T . The condition in (43) is satisfied if and only if π [ i : K T + i −
1] = ˜ π (cid:2) ˜ i : K T + ˜ i − (cid:3) . In other words, we can construct a circular permutationfrom another by circular shift process; however, this condition is not achievable. Therefore, eachsmaller subfile appears only at a single group, i.e., the groups are disjoint. Hence, the proof iscompleted. A PPENDIX DF ULL R ANK OF M ATRICES V IA j AND R k In this Appendix, we prove that the matrices V IA k , V IA k have full column rank of ( n + 1) Γ , n Γ , respectively. Then, we prove that the matrix R k has full rank of µ n . Before that, we presentan important Lemma that is essential in the proofs. Lemma 3.
Let H be a K R × K T channel matrix whose entries are i.i.d. generated from acontinuous distribution. For given j ∈ [ K R ] , π ∈ Π circ [ K T ] , and ≤ τ ≤ min { K T , K R } , any subsetof monomials of M π [ i : i + τ − { k }∪ [ j +1: j + τ − for i ∈ [ K T ] , k ∈ [ j + τ : K R + j ] , are linearly independent. Proof.
Note that monomials of { M π [ i : i + τ − { k }∪ [ j +1: j + τ − } K T i =1 are linearly independent of monomials of { M π [ i : i + τ − { l }∪ [ j +1: j + τ − } K T i =1 for l ∈ [ j + τ : K R + j ] , l (cid:54) = k , since the channel coefficients { h kπ ( i ) } K T i =1 onlycontribute in { M π [ i : i + τ − { k }∪ [ j +1: j + τ − } K T i =1 and the channels are i.i.d. variables. Thus, it is only required to prove the linearly independence of monomials of { M π [ i : i + τ − { k }∪ [ j +1: j + τ − } K T i =1 . Observe that eachminor M π [ i : i + τ − { k }∪ [ j +1: j + τ − is a polynomial of degree τ . Therefore, monomials of { M π [ i : i + τ − { k }∪ [ j +1: j + τ − } K T i =1 with different degrees are linearly independent due to the different exponents of the channelcoefficients. This implies that we only need to prove the linearly independence of monomials of M π [ i : i + τ − { k }∪ [ j +1: j + τ − with the same degree. Consider a linear combination of a set of N monomialsof degree L . N (cid:88) n =1 c n K T (cid:89) i =1 (cid:16) M π [ i : i + τ − { k }∪ [ j +1: j + τ − (cid:17) α i ( n ) = 0 (44) where (cid:80) K T i =1 α i ( n ) = L , and c = [ c , . . . , c N ] ∈ C N . If these monomials are not linearlyindependent, then, there exists c (cid:54) = satisfying (44). Note that the linear combination in (44) canbe factored into multiplied terms, where each term is a linear combination of { M π [ i : i + τ − { k }∪ [ j +1: j + τ − } K T i =1 . As a result monomials of { M π [ i : i + τ − { k }∪ [ j +1: j + τ − } K T i =1 of the same degree are linearly independent ifand only if the minors { M π [ i : i + τ − { k }∪ [ j +1: j + τ − } K T i =1 are linearly independent. Using cofactor expansion,we represent M π [ i : i + τ − { k }∪ [ j +1: j + τ − = (cid:80) i + τ − l = i h kπ ( l ) C π [ i : i + τ − \ π ( l )[ j +1: j + τ − , where C π [ i : i + τ − \ π ( l )[ j +1: j + τ − is the cofactor of h kπ ( l ) . Thus, the linear combination of these minors is given by K T (cid:88) i =1 c i M π [ i : i + τ − { k }∪ [ j +1: j + τ − = K T (cid:88) l =1 h kπ ( l ) (cid:32) l (cid:88) i = l − τ +1 c i C π [ i : i + τ − \ π ( l )[ j +1: j + τ − (cid:33) = 0 (45) Since { h kπ ( l ) } K T l =1 are i.i.d. variables, (45) is satisfied if and only if (cid:80) li = l − τ +1 c i C π [ i : i + τ − \ π ( l )[ j +1: j + τ − = 0 for every l ∈ [ K T ] . Note that if τ = 2 , then { C π [ i : i + τ − \ π ( l )[ j +1: j + τ − } li = l − τ +1 are i.i.d. channel coefficients.Otherwise, we can use cofactor expansion to represent { C π [ i : i + τ − \ π ( l )[ j +1: j + τ − } li = l − τ +1 . Then, the sameargument can be iteratively applied until we reach to a linear combination of some channelsthat are i.i.d. random variables. Therefore, we reach to the result that { c i } K T i =1 must be zero tosatisfy (45), i.e., { M π [ i : i + τ − { k }∪ [ j +1: j + τ − } K T i =1 are linearly independent. The details are omitted here dueto the space limitations. (cid:4) Consider first the µ n × ( n + 1) Γ matrix V IA j given in (28). At any row u ∈ [ µ n ] , the en-tries of different columns are different monomials in (cid:110) M π [ i : i + τ − { k }∪ [ j +1: j + τ − ( u ) (cid:111) for i ∈ [ K T ] , k ∈ [ j + τ : K R + j − . In other words, (cid:104) α [ k,i ] j ( l ) (cid:105) i ∈ [ K T ] k ∈ [ j + τ : K R + j − (cid:54) = (cid:104) α [ k,i ] j ( l (cid:48) ) (cid:105) i ∈ [ K T ] k ∈ [ j + τ : K R + j − for ( l, l (cid:48) ) ∈ (cid:104) ( n + 1) Γ (cid:105) with l (cid:54) = l (cid:48) . Furthermore, the exponents of monomials at the same column with differentrows are equal. Thus, using [29, Lemma ], the matrix V IA j has full rank almost surely if andonly if (cid:110) M π [ i : i + τ − { k }∪ [ j +1: j + τ − (cid:111) for i ∈ [ K T ] , k ∈ [ j + τ : K R + j − , are linearly independent. UsingLemma 3, matrix V IA j has full rank of ( n + 1) Γ . Similarly, we can prove that matrix V IA j hasfull rank of n Γ , where it is designed in the same manner as matrix V IA j . Now, consider matrix R k given in (31). Using [29, Lemma ], matrix R k is full rank if and only if there does notexist an annihilating polynomial satisfying K T (cid:88) i =1 n Γ (cid:88) l =1 c kil M π [ i : i + τ − k : k + τ − V IA k ( l ) + K R + k − τ (cid:88) j = k +1 ( n +1) Γ (cid:88) l =1 c jl V IA j ( l ) = 0 , (46) where { c kil } and { c jl } are non-zero factors. V IA k ( l ) is the entry in the l -th column and an arbitraryrow of matrix V IA k , and V IA j ( l ) is the entry in the l -th column and an arbitrary row of matrix V IA j . 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