Fundamental Storage-Latency Tradeoff in Cache-Aided MIMO Interference Networks

Transcription

1 Fundamental Storage-Latency radeoff in Cache-Aided MIMO Interference etworks Youlong Cao, Meixia ao, Fan Xu and Kangqi Liu Abstract arxiv:609086v [csi] 4 May 07 Caching is an effective technique to improve user perceived experience for content delivery in wireless networks Wireless caching differs from traditional web caching in that it can exploit the broadcast nature of wireless medium and hence opportunistically change the network topologies his paper studies a cache-aided MIMO interference network with transmitters each equipped with M antennas and receivers each with antennas With caching at both the transmitter and receiver sides, the network is changed to hybrid forms of MIMO broadcast channel, MIMO X channel, and MIMO multicast channels We analyze the degrees of freedom (DoF) of these new channel models using practical interference management schemes Based on the collective use of these DoF results, we then obtain an achievable normalized delivery time (D) of the network, an information-theoretic metric that evaluates the worst-case delivery time at given cache sizes he obtained D is for arbitrary M, and any feasible cache sizes It is shown to be optimal in certain cases and within a multiplicative gap of from the optimum in other cases he extension to the network with arbitrary number of transmitters and receivers is also discussed A Motivation Index erms Coded caching, degrees of freedom, interference management, multicast, linear transmission scheme I IRODUCIO Over the last decade, the ever-growing mobile cellular traffic has undergone a fundamental shift from voices and messages to rich content distribution, such as video streaming In particular, video traffic amounts for more than 50% of the total mobile data traffic in 05 and is foreseen to contribute 75% in 00 [] An important feature of video contents is that they are cachable and the same content can be requested by many users Wireless caching is to prefetch the popular contents at the wireless edge, such as local base stations or mobile users, during the off-peak time in order to reduce the peak data traffic and improve user perceived experience Caching at the wireless edge can be regarded as an effective way to trade the scarce communication bandwidth with the more sustainable storage size through traffic time shifting It has attracted significant attention from both academia and industry recently, see for example [] [5] and references therein raditional caching has been long proposed in computer networks for reducing the downloading delay [6] since a requested file can be obtained in the local cache without resorting to a remote server Wireless caching differs from traditional caching in that it can exploit the broadcast nature of wireless medium and hence opportunistically change the network topologies A fundamental question about wireless caching is what and how much gain it can achieve his has driven the study of fundamental limits of caching in various wireless systems, including broadcast channel [7], interference networks [8] [], partially connected networks [], device-to-device networks [4] [6], and fog radio access networks [7] his work aims to investigate the fundamental limits of caching in wireless MIMO interference networks where each node is equipped with both a local cache and multiple antennas he system operates in two phases In the cache placement phase, which usually takes places in a large time scale (eg a day or an hour), each node prefetches some file bits from a library into its local cache In the content delivery phase, which happens in a small time scale (eg second), each transmitter sends the messages according his work is presented in part at the 06 IEEE Global elecommunications Conference [] he authors are with the Department of Electronic Engineering, Shanghai Jiao ong University, Shanghai, China s: {caoyoulong, mxtao, xxiaof}@sjtueducn, kliucn@ieeeorg

2 to the receiver requests, cache states, and the MIMO channel conditions Our goal is to characterize the storage-latency tradeoff through the careful design of cache placement and content delivery B Related Works he fundamental limits of caching at the receiver side were first studied in [8] for a shared link with one server and multiple cache-aided receivers he study in [8] shows that caching can exploit multicast opportunities even when user demands are different, and hence greatly reduces the traffic load over the shared link his is enabled by proper file splitting during the cache placement phase and coded transmission during the content delivery phase, known as coded caching he benefits of caching at the transmitter side were studied in [8] for a interference channel It is shown that caching can induce transmitter cooperation and hence allows interference coordination for throughput enhancement he limits of caching when equipped at both the transmitter and receiver sides were investigated in [9] [] very recently, which all considered a general interference network but with different restrictions and performance metrics he works [9], [0] characterized the tradeoff between storage size and content delivery time, in terms of an information-theoretic metric, normalized delivery time (D) An achievable D is obtained in [9] for an interference network with arbitrary number of transmitters and arbitrary number of receivers at any feasible cache sizes heir achievable D is optimal at certain cache size regions and is within a bounded multiplicative gap to a theoretical lower bound at other cache size regions he study in [9] reveals that, with a novel cooperative transmitter and receiver caching strategy, the interference network can be turned opportunistically into more favorable channels, including X channel, broadcast channel, multicast channel, and a hybrid form of these channels In [], an order-optimal approximation on the system performance for arbitrary number of transmitters and receivers was presented But their analysis is limited to the case where the accumulated cache size at the transmitter side is large enough to cache all the files and only hybrid X-multicast channel is considered he work [], on the other hand, adopted the standard sum degrees of freedom (DoF) to characterize the performance and their analysis is restricted to one-shot linear transmission schemes he aforementioned studies on fundamental limits of caching at both transmitter and receiver sides are limited to the single-antenna interference network ote that a crucial step in analysing the performance of cache-aided interference networks is to derive the DoF, a capacity approximation at high signal-to-noise ratio (SR) regime, of the new network topologies formed by caching, for example the X-multicast channel [9], [] DoF characterizations for a wide variety of MIMO channels have been considered recently, in particular for MIMO interference channel [9] [] and MIMO X channel [] [5] However, the DoF results of these MIMO channels with multicast traffic and/or transmitter cooperation remain unsolved in general C Our Contribution In this paper, we study a cache-aided MIMO interference network with three transmitters and three receivers, as shown in Fig Each transmitter is equipped with M antennas and a local cache of normalized size µ, and each receiver with antennas and a local cache of normalized size µ R he performance is characterized by D, the same information-theoretic metric applied in [9], [7], [6] his work is a non-trivial extension of [9] due to the deployment of multiple antennas Preliminary results in the special case with symmetric antenna configuration M = are presented in the conference paper [] his journal paper considers the more general case with arbitrary M and arbitrary he main contributions and findings of this paper are summarized as follows: An achievable D: We adopt the same cooperative x/rx caching scheme proposed in [9], [0] for file placement, but design different and more practical transmission schemes for content delivery An achievable D is obtained by solving a linear programming problem of file splitting or, equivalently, memory sharing coefficients he achievable D is for any number of transmit antennas M, any number

3 LF bits bits x Library Rx bits bits x Rx bits bits x Rx bits the Cache Placement Phase the Content Delivery Phase Fig : he cache-aided MIMO interference network of receive antennas, and any feasible cache size tuple (µ R,µ ) Its closed-form expression is piecewiselinearly decreasing with the normalized cache sizes, which reflects the caching gain Each additive item in the expression is inversely proportional to the number of antennas, which reflects the spatial multiplexing gain induced by MIMO It also found, interestingly, that the traditional equal file splitting strategy [9] [] is not always optimal at integer points DoF of new MIMO channel models: A crucial step in analyzing the achievable D is to derive the DoF of the new network topologies formed by different file placement patterns during the content delivery phase In this work, several new channel models are formed, including partially cooperative MIMO X channel, MIMO X-multicast channel, and partially or fully cooperative MIMO X-multicast channel We derive the achievable DoF per user of these channels by using linear precoding based interference management schemes with finite symbol extensions such as interference alignment, interference neutralization and zero forcing We would like to remark that a related but different effort is the study of DoF region of MIMO interference network with general message demands in [] Our channel models differ from [] in that () each transmitter has multiple messages to send and can cooperate with each other; () the antenna configurations at the transmitter and receiver sides are asymmetric Another related effort is the study of DoF region of X channel with multicast traffic in [7], which, however only considers single antenna A lower bound of the minimum D: We also obtain a theoretical lower bound of the minimum D of the considered cache-aided MIMO interference network by using a cut-set like argument his lower bound has no restriction on the linearity of MIMO transmission schemes and allows arbitrary intra-file coding but not inter-file coding at the cache placement phase With this lower bound, we show that our achievable upper bound is optimal for certain antenna configurations and cache size regions Analysis also shows that the maximum multiplicative gap between the upper and lower bounds is otations: x, x, X and X denotes scalar, vector, matrix and set, respectively Θ(x) denotes that lim x Θ(x) x = ( ) denotes the transpose of a matrix tr(x) and null(x) stand for the trace and the null space of the matrix X [n] denotes the set {,,,n} where n is an integer II SYSEM MODEL AD PERFORMACE MERICS We consider a cache-aided MIMO interference network as illustrated in Fig, where each transmitter is equipped with M antennas and each receiver is equipped with antennas Each node he equal file splitting strategy is to split each file into ( µ R )( µ ) equal-sized subfiles, each cached inµr receivers andµ transmitters when µ R,µ { 0,,,} Due to that µ R = m and µ = n with m,n being integers, these points are called integer points [9] or corner points [8]

4 4 has a local cache of finite size Consider a library consisting of L files, denoted by {W,W,,W L } hroughout this study, we focus on the case where the number of files L is larger than or equal to the number of receivers, ie, L Each file has the same length of F bits Each transmitter can cache Q F bits and each receiver can cache Q R F bits, where Q,Q R L he normalized cache sizes at the transmitter and receiver sides are defined, respectively, as µ = Q L, and µ R = Q R L () his work focuses on the feasible cache size region [9] [0]: { 0 µr,µ, () µ R +µ he communication involves two phases, the cache placement phase, which takes place in a large time scale, and the content delivery phase, which happens in a small time scale During the cache placement phase, each transmitter i has a caching function φ i : [ F ] L [ FQ ], () mapping the L files in the library to its local cache content U i φ i (W,W,,W L ), for i [] Each receiver j also has a caching function ψ j : [ F ] L [ FQ R ], (4) mapping the L files to its local cache content V j ψ j (W,W,,W L ), for j [] As in [9], [6], it is assumed that the caching functions {φ i,ψ j } allow arbitrary intra-file coding, but do not allow inter-file coding In the content delivery phase, each receiver j requests a file W dj from the library, where d j [L] We denote d [d,d,d ] as the demand vector Each transmitter further consists of an encoding function Λ i : [ FQ ] [L] C M C M, (5) where is the block length of the code and depends on the receiver demand d and the network channel state information (CSI) H = {H i,j C M : i [],j []} Each H i,j is the channel matrix from each transmitter i to each receiver j, whose entries are drawn independently and identically distributed (iid) from a continuous distribution, and remain invariant within each codeword transmission ransmitter i uses Λ i to map its local cache content U i, receiver demands d and the network CSI H to the signal vectors [ x i (t) ] Λ t= i(u i,d,h), which is subject to a power constraint tr [ x i (t)x H i (t) ] P In each time slot t [], the received signal at each receiver j, denoted as y j (t) C, can be expressed as y j (t) = H ij x i (t)+n j (t), j [], (6) i= where n j (t) denotes the additive white Gaussian noise (AWG) vector at receiver j, with each element being independent and having zero mean and unit variance In this paper, we assume that the network CSI is available at all transmitters and receivers he decoding function Γ j at receiver j can be defined as: Γ j : [ FQ R ] C C M [L] [ F ] (7) Each receiver j uses Γ j to estimate Ŵj Γ j (V j, [ y j (t) ] t=,h,d) of its desired file W d j, with its cached content V j and the channel realization H he worst-case error probability is max d [L] max j [] P(Ŵj W dj ) (8)

5 5 x Rx x Rx / / x Rx Fig : File splitting and cache placement at µ R =, µ = with M = he given caching and coding functions {φ i,λ i,ψ j,γ j } are said to be feasible if, for almost all channel realizations H, the worst-case error probability approaches 0 when F In this work, we adopt the following performance metric to characterize the fundamental storage-latency tradeoff Definition [6]: For any given feasible caching and coding scheme at given normalized cache sizes µ and µ R, the normalized delivery time (D) is defined as he minimum D is defined as max τ(µ R,µ ) lim lim sup d P F F/logP (9) τ (µ R,µ ) = inf{τ(µ R,µ ) : τ(µ R,µ ) is achievable} (0) ote that F/logP is the delivery time of transmitting one file of F bits over a point-to-point Gaussian channel with single antenna in the high signal-to-noise ratio (SR) regime An D of τ thus indicates that the worst-case time required to serve any possible demand vector d is τ times of this reference time period Remark [9]: Let R denote the worst-case traffic load per user with respect to the file size F, and d logp +o(logp) denote the per-user capacity of the network in the high SR regime at a given caching and coding scheme By Definition, D can be approximately expressed as τ = RF/(d logp) F/logP = R d () hus, D characterizes the asymptotic delivery time of the actual traffic load per user, R, at a transmission rate specified by the DoF per user, d, when both transmit power P and file size F go to infinity Example Consider a MIMO interference network with the normalized cache sizes µ R = and µ = under symmetric antenna setting M = he cache placement strategy is shown in Fig, where each file is split into two subfiles, one with F bits and cached in all receivers, the other with F bits and cached in all transmitters During the delivery phase, consider the worst case where the three receivers request distinct files, denoted as A, B, C, respectively hen each receiver only needs the subfile with the length of F bits that it does not cache, which is available at all the three transmitters he traffic load per user is R = he network topology can be viewed as a virtual MIMO broadcast channel where the virtual transmitter has M antennas and each receiver has M antennas he DoF per user of this channel is d = M By Remark, the achievable D is τ = M he performance metric D is first proposed in [6] for wireless networks with transmitter caches only It is then scaled by the number of receivers and renamed as fractional delivery time (FD) by taking receiver caches into account in [], [9], [0] as well as the prior version of this paper During the paper revision, we have removed the scaling and changed back to D for consistency with [6] as suggested by reviewers

6 6 III MAI RESULS In this section, we present our main findings on the minimum D in the cache-aided MIMO interference network heorem (Upper Bound) Consider the cache-aided MIMO interference network where each transmitter is equipped with M antennas and a cache of normalized size µ, and each receiver is equipped with antennas and a cache of normalized size µ R An achievable D based on linear transmission schemes with finite symbol extensions is given by τ u, the optimal solution of the following linear programming problem: P : τ u min {β mn} st (m,n) A (m,n) A (m,n) A β mn m/ d mn β mn =, β mn µ o mn µ, (a) (b) (c) 0 β mn, (m,n) A, (d) where A = {(m,n) : m+n,m,n {0,,,}}; µ = [µ R,µ ] denotes any feasible point in the cache size region; µ o mn = [m, n ] denotes the integer point with (µ R = m,µ = n ) in the cache size region; β mn is the (memory sharing) parameter to be optimized; and d mn is given below: { } d 0 = min, M d 0 = M, k ξ,, k + + ξ ( ] 0, M (, ] 5 M ] 5, M ( 5, 5 ( 5, ) M d = d = d = min{,m},, d 0 = min{m,}, d = min{,m}, 6, 7 6M, d = M, 7, (0,] M (, ], (0,] 9 M M, M 7,], d = ( ], M 7 (, ), (,], M M, (, ) M M, M ( 9 where k = min{m,}, k + = max{m,} and ξ = k k + k Remark : he linear programming (LP) problem P in heorem can be solved efficiently he explicit and closed-form, but somewhat tedious expression of τ u is given in Appendix A It is seen from Appendix A that the achievable D decreases piecewise linearly with the normalized cache sizes and each additive item of D is, in general, inversely proportional to the number of antennas he latter property explicitly shows the multiplexing gain induced by MIMO Moreover, the antenna configuration (ie, the ratio /M) determines the partition of the cache size region Remark : In the special case with symmetric antenna configuration, ie, M =, the achievable D reduces to the results in [] Furthermore, when M = =, the obtained D is numerically at most times of the one in the single antenna case [0] he slight increase in the achievable D is due to that we only use linear precoding based interference management schemes with finite symbol extensions heorem (Lower Bound) Consider the cache-aided MIMO interference network where each transmitter is equipped with M antennas and a cache of normalized sizeµ, and each receiver is equipped with antennas and a cache of normalized size µ R he minimum D is lower bounded by τ τ l max { ( µ R),max s [] s M ( sµ R) } ()

7 7 Remark 4: By comparing the closed-form upper bound in Appendix A and the lower bound in heorem, it can be seen that the achievable D is optimal under the following conditions: ) ( 0, M ] and (µr,µ ) {(µ R,µ ) : µ R +µ,µ R,µ }; ) (0,] and (µ M R,µ ) {(µ R,µ ) : µ R +µ,µ R,µ }; ) (0,] and (µ M R,µ ) {(µ R,µ ) : µ R +µ,µ R +µ 5,µ R,µ }; 4) (0, ) and (µ M R,µ ) {(µ R,µ ) : µ R +µ, µ R,µ } Corollary (Multiplicative Gap) he multiplicative gap between the upper and lower bounds of the minimum D for the considered network is at most he proof of heorem will be given in Sections IV and V he proofs of heorem and Corollary will be given in Section VI IV CACHIG AD DELIVERY SCHEME he achievable upper bound of minimum D in heorem is based on the same cache placement strategy in [9] but with different delivery scheme due to the deployment of multiple antennas In this section, we first review the file splitting and caching strategy proposed in [9] for self-completeness hen we present the delivery scheme in detail Since each transmitter and receiver can decide whether to cache each bit of each file, there are 6 = 64 possible cache states for each bit ot every cache state is, however, legitimate, due to that every bit of the file which is not cached simultaneously in all receivers must be cached in at least one transmitter his results in a total of 57 legitimate cache states for each bit and the feasible domain of µ R and µ, given in () We now split each file into 57 subfiles, each corresponding to a unique cache state and having a possibly different length to be optimized Define transmitter subset I [] and receiver subset J [] hen, let W κrj t I denote the subfile split from W κ that is cached in receiver subset J and transmitter subset I For example, W κrøt is the subfile cached in none of the three receivers but in all three transmitters and W κr t is the subfile cached in receiver, and transmitters, Similarly, we denote W κti as the collection of all subfiles cached in I, ie W κti = J W κrj t I As in [9], the sizes of the subfiles with the same cardinality of transmitter and receiver subsets are assumed to be equal Let m = J and n = I denote the cardinalities of J and I respectively, and define a mn as the file splitting ratio to be optimized hen each subfile W κrj t I contains Fa mn bits he splitting ratios must satisfy the following constraints: a 0 +a +a +a +9a +9a +a +9a +9a +a +a 0 +a 0 +a 0 =, (4) a 0 +a +a +a +6a +6a +a +a +a +a µ R, (5) a +a +a +a +6a +a +a +6a +a +a 0 +a 0 +a 0 µ (6) Constraint (4) comes from the total file size limit, where the multiplier of each splitting ratio a mn indicates the total number of subfiles that have the same length a mn Constraints (5) and (6) come from the cache size limit in receiver and transmitter, respectively, where the multiplier of each splitting ratio a mn indicates the total number of subfiles that are stored in a same receiver or transmitter, and have the same length a mn In the delivery phase, we consider the worst-case scenario where all user demands are distinct When user demands are not all distinct, the same delivery strategy can be applied either directly or by treating the demands as being different ow, without loss of generality, we assume that receiver,, request file W, W, and W respectively Similar to [9], we first divide all the subfiles to be transmitted into groups according to the number of transmitters and receivers where they are cached, then deliver each group separately However, the ote that 57 is the maximum number of legitimate subfiles by exhausting all the possible combinations he actual number of subfiles can be much less after optimization since not all possible combinations are needed

8 8 x x x Fig : he delivery of subfiles {W κrøt p : κ,p []} Here, we use A, B and C to denote the W, W and W, respectively Each file is divided into three subfiles of equal size, eg, A = (A,A,A ), with the subscript indicating at which transmitter this subfiles is to be cached specific delivery strategy for each group is significantly different from [9] due to the deployment of multiple antennas amely, as the crucial step in analyzing the achievable D, the DoF analysis of the new channel models formed by the different subfile groups in this work is for multiple antennas, while the DoF analysis in [9] is for single antenna A Delivery of Subfiles Cached in Zero Receiver and One ransmitter Consider the delivery of subfiles {W κrøt p : κ,p []}, each of which is cached at one transmitter but none of the receivers and has fractional length a 0 he network topology in this case can be seen as a MIMO X channel Previous study on the DoF of MIMO X channel can be found in [4], but the results require infinite symbol extensions which limits its practical use [8] In this work, we treat the MIMO X channel as a MIMO interference channel instead, whose optimal DoF is obtained in [], [9] and only requires linear transmission scheme with finite symbol extensions he conversion from MIMO X channel to MIMO interference channel is shown in Fig, where three phases are needed to deliver the subfiles, and in each phase each transmitter sends an independent message to a different receiver he DoF per user of the MIMO interference channel, denoted as d 0, is [], [9] 4, d 0 = min { k /ξ, k + +/ξ }, (7) where k = min{m,}, k + = max{m,} and ξ = k k + k Given that the total amount of bits per user to deliver in each phase is a 0 F bits, by (), the D over the three-phase transmission can be computed as τ = a 0 d 0 B Delivery of Subfiles Cached in Zero Receiver and wo ransmitters Consider the delivery of subfiles {W κrøt pq : κ,p,q [], p < q}, each of which is cached in two transmitters and none of receivers and has fractional length a 0 he network topology in this case can be viewed as a partially cooperative MIMO X channel, where every set of two transmitters forms a transmit cooperation group and has an independent message to send to each receiver Lemma For the partially cooperative MIMO X channel, the achievable DoF per user, denoted as d 0, is given in (8), ( ] 0, M M d 0 =, (, ] 5 M 5, ( 5, ] 5 (8) M M, ( 5, ) M 4 In case d 0 is not an integer, the achievable scheme needs t-symbol extension such that td 0 is an integer

9 9 x x x Fig 4: he delivery of subfiles {W κrøt pq : κ,p,q [], p < q} Each file is divided into three subfiles of equal size, eg, A = (A,A,A ), with the subscript indicating at which transmitters this subfiles is to be cached Proof: he achievable scheme takes three phases, as shown in Fig 4 In each phase, each of the three transmit cooperation groups ({, }, {, } and {, }) sends one independent message intended to a different receiver he two interference signals seen by each receiver are cancelled by interference neutralization with linear transmit and receive processing he detailed proof is given in Appendix B Based on Lemma, the D of these subfiles is τ = a 0 d 0 C Delivery of Subfiles Cached in Zero Receiver and hree ransmitters Consider the delivery of subfiles {W κrøt : κ []}, each of which has fractional length a 0 Since each subfile is cached in all the three transmitters, the transmitters can fully cooperate he delivery in this case can be regarded as an MIMO broadcast channel where the virtual transmitter has M antennas, and each receiver has antennas he optimal DoF per user of this channel is d 0 = min{m,} [0] herefore, the D of these subfiles is τ = a 0 d 0 he delivery scheme in Example shown in Section II belongs to this case where a 0 = D Delivery of Subfiles Cached in One Receiver and One ransmitter Consider the delivery of subfiles {W κrk t p : κ,k,p [], k κ}, each of which has fractional length a Since each subfile desired by one receiver is already cached in one of the other receivers, we can use coded multicasting in the delivery phase For example, transmitter can generate message W κ jk t W jrk t W krj t desired by receiverj and k, where denotes the bit-wise XOR ow each XORed message is desired by two receivers he network topology of sending coded subfiles {W r jk t p : j,k,p [], j < k} becomes a MIMO X-multicast channel, where every set of two receivers forms a receive multicast group and each transmitter has an independent message to send to each receive multicast group Lemma For the MIMO X-multicast channel, the achievable DoF per user, denoted as d, is given in (9) 6, (0,] 7 M 6M d =, ( ], 9 7 M 7, ( 9,] (9) M 7 M, (, ) M Proof: When the antenna configuration satisfies 9 M, we use linear interference alignment 7 technique so that all the interference signals at each receiver can be aligned in the same direction When the antenna configuration satisfies > 9 M, the delivery includes three phases and in each phase, each 7 transmitter sends one independent message to a different receive multicast group he receiver combining matrices are designed to cancel the interference signals he detailed proof is given in Appendix C Based on Lemma, the D of these subfiles is τ = 6a d

10 0 E Delivery of Subfiles Cached in One Receiver and wo ransmitters Similar to Subsection D, coded multicasting gain can be exploited in the delivery of subfiles {W κrk t pq : κ,k,p,q [], k κ, p < q}, each of which has fractional length a he difference is that each subfile is available at two transmitters and hence transmitter cooperation gain can be exploited For example, transmitter and transmitter can generate a coded message W κ jk t W jrk t W krj t he delivery of coded subfiles {W r jk t pq : j,k,p,q [], j < k, p < q} can be viewed as a partially cooperative MIMO X-multicast channel, where every set of two receivers forms a receive multicast group, every set of two transmitters forms a transmit cooperation group, and each transmit cooperation group has an independent message for each receive multicast group Lemma For the partially cooperative MIMO X-multicast channel, the achievable DoF per user, denoted as d, is given in (0), (0,] M M, d = ( ], M, (,] (0) M M, (, ) M Proof: When the antenna configuration satisfies M, we use the linear interference naturalization by designing the precoding matrices of each transmit cooperation group When > M, the achievable scheme takes three phases and in each phase, each transmit cooperation group sends one independent message to a different receive multicast group Each receiver applies zero-forcing processing for interference cancellation he detailed proof is given in Appendix D Based on Lemma, the D of these subfiles is τ = 6a d F Delivery of Subfiles Cached in One Receiver and hree ransmitters Similar to Subsections D and E, the coded multicasting scheme can also be exploited in the delivery of subfiles {W κrk t : κ,k [], k κ} he difference is that each subfile is available at all the transmitters For example, all the transmitter can generate message W κ jk t W jrk t W krj t he delivery of coded subfiles {W r jk t : j,k [], j < k} can be regarded as a fully cooperative MIMO X-multicast channel, where every set of two receivers forms a receive multicast group, all the transmitters forms a transmit cooperative group, and the transmit cooperation group has an independent message to send to each receive multicast group Lemma 4 For the fully cooperative MIMO X-multicast channel, the achievable DoF per user, denoted as d, is given in () d = min{,m} () Proof : he achievable scheme of this channel is to design the receiver combining matrices he detailed proof is given in Appendix E Based on Lemma 4, the D of these subfiles is a d G Delivery of Subfiles Cached at wo Receivers and One or More ransmitters Consider the delivery of subfiles {W κrk,l : κ,k,l [], k,l κ, k < l} Since each subfile desired by one receiver is already cached at the other two receivers, we can similarly use coded multicasting For example, transmitter can generate messages W t W r t W r t W r t, W t W r t W r t W r t, () W t W r t W r t W r t

11 Each of the above coded message is desired by all the three receivers, yielding a MIMO multicast channel We first give the delivery scheme of coded messages {Wt p : p []}, each with fractional length a When M, by antenna deactivation [], we let each transmitter use antennas and transmit data streams5 Each user can decode data streams using antennas, and the DoF per user is When > M, we let each transmitter use M antennas and transmit M data streams By the antenna deactivation, each user can decode M data streams using M antennas, and M DoF per user can be achieved For the other two coded messages {Wt pq : p,q [], p < q} and {Wt }, we can use the similar scheme as the one used in coded messages {Wt p : p []}, because each coded message is simultaneously at more than one transmitter ( So the DoF per user d = d = d = min{,m}, and a the D can be computed as τ = d + a d + a d ) Summing up the Ds obtained in all the above subsections yields the total D as: τ = a 0 d 0 + a 0 d 0 + a 0 d 0 + 6a d + 6a d + a d + a d + a d + a d () V OPIMIZAIO OF FILE SPLIIG RAIOS AD COECIO WIH MEMORY SHARIG In this section, we study the optimization of the file spitting ratios {a mn } to minimize the total D in () subject to the constraints (4) (5) (6) his can be formulated as the following LP problem: P : min τ(µ R,µ ) (4) {a J I } st (4)(5)(6) (5) Clearly, by defining a new optimization variable β mn as: ( )( ) β mn = a mn, (m,n) A, (6) m n where A = {(m,n) : m+n,m,n {0,,,}}, P can be equivalently expressed as P in heorem Here, constraint (4) is equivalent to constraint (b), and constraints (5) and (6) are equivalent to constraint (c) By solving P, heorem is then proved he significance of rewriting P as P is that P can be interpreted as memory sharing optimization his is detailed as below First, consider an integer pointµ o mn = [ m, n ] with(µ R = m,µ = n ) in the cache size region Assume that equal file splitting strategy is adopted hat is, each file is split into ( m)( n) amn equal-sized subfiles, each cached simultaneously at m receivers and n transmitters In that case, we have a mn = / ( ) m)( n and all the rest a m n = 0 By the delivery scheme introduced in Section IV, the D at µo mn can be computed as τmn o = m/ d mn hen, consider any feasible point µ = [µ R,µ ] in the cache size region he given µ can always be expressed as a convex combination of all the feasible integer points, ie, µ = β mn µ o mn (7) (m,n) A We now adopt the memory sharing strategy for cache placement amely, we split the transmitter and receiver cache sizes as in (7) with memory sharing parameter β mn For each β mn fraction of the memory, we take β mn fraction of each file, split and cache it according to the equal file splitting strategy at the integer point µ o mn hen, a total achievable D can be obtained as τ = β mn τmn o (8) (m,n) A 5 hroughout this paper, if the number of antennas after deactivation or the number of data streams sent from each transmitter (or received by each receiver), denoted as d, is not an integer, we can use t-symbol extensions such that td is an integer

12 We can minimize the total D by finding the optimal memory sharing parameters {β mn } his is expressed mathematically in P in heorem Both P and P are standard LP problems By using linear equation substitution and other manipulations, we obtain the closed-form but somewhat tedious expression of the optimal solution µ u for any µ R,µ,M, in Appendix A he antenna configuration is divided into 0 cases, and for each case the feasible cache size region is partitioned into several regions as shown in Fig 5 In each region, the achievable τ u is a linear decreasing function of µ R and µ and hence can be achieved by memory sharing of the integer points within that region Remark 5: In the single antennas case [9], [0], the equal file splitting strategy at integer points is shown to be optimal (in the sense of achieving the optimal solution of the linear programming problem P ) though not unique But in the MIMO case, the equal file splitting is not always optimal For example, consider integer point µ o 0 = [0, ] with = 5,M =, ie, Case 7 in Fig 5 If equal file splitting strategy is adopted, the D is On the other hand, from the optimal solution of P in heorem, the optimal memory sharing coefficients are β 0 = β 0 = his means that a half of cache size shall be used to adopt the caching scheme at integer point µ o 0 = [0, ] and the other to adopt caching scheme at integer point µ o 0 = [0,] he corresponding D shall be + = 5 < VI COVERSE AD MULIPLICAIVE GAP In this section, we present the proof of the D lower bound in heorem and the proof of the maximum multiplicative gap in Corollary A Converse We first introduce the following Lemma to help bound the entropy of received signals Lemma 5 For the cache-aided MIMO interference network, the differential entropy of the received signals at any l antennas, which can be equipped at different receivers is upper bounded by ( ) h(y [:l] ) l log πe(cp +), (9) where the parameter c is a function of the channel coefficient Proof : See Appendix F ow, we begin the proof he method of the proof follows the similar cut-set argument in [8] Consider any s [] users requesting L = s L different files during Z = s L requests Given the s receivers s caches and received signals during Z requests, these L files can be decoded successfully in the high-sr regime hus, we have: Fǫ F = H(W,,W L y [:s],,yz [:s],v [:s]) (0a) = H(W,,W L) I(W,,W L;y [:s],,y Z [:s],v [:s] ) (0b) = LF h(y [:s],,yz [:s] V [:s]) H(V [:s] )+h(y [:s],,yz [:s] V [:s],w,,w L) +H(V [:s] W,,W L) LF h(y[:s],,yz [:s] ) sµ RLF +H(V [:s] W,,W L) LF h(y[:s],,yz [:s] ) sµ RLF +s(l L)µ R F (0c) (0d) where (0a) follows from the Fano s inequality [, heorem 0]; (0d) comes from the fact that conditioning reduces entropy; (0e) comes from the fact that each user caches µ R F bits of each file on average and the s receivers know the L files of the total L files Using Lemma 5, we further bound (0e) as (0e) Fǫ F LF szθ(logp) sµ R LF ()

13 Alternatively, using the data-processing inequality in [, heorem 8]: we can bound (0e) as h(x [:],,xz [:] ) h(y [:s],,yz [:s]), () Fǫ F LF h(x [:],,x Z [:]) sµ R LF LF ZMΘ(logP) sµ R LF (a) (b) Rearranging () and (b) and taking P and F, the minimum D is lower bounded by { } τ = lim lim P F F/logP max max s [] ( sµ s R), M ( sµ R) { } = max ( µ s R),max s [] M ( sµ R), (4) which completes the proof of heorem B Multiplicative Gap o assist the analysis, we first relax the lower bound τ l in (4) as follows: τ l (µ R,µ ) ˆτ l (µ R,µ ) = { ( µ R), M ( µ R), (0,] M (5) (, ) he relaxed lower bound can be rewritten as the convex combination at all integer points: ( ˆτ l (µ R,µ ) =γ 0ˆτ l 0, ( )+γ 0ˆτ l 0, ( ) ( )+γ 0ˆτ l 0, +γ ˆτ l, ) ( +γ ˆτ l, ) ( ) ( +γ ˆτ l, +γ ˆτ l, ) ( +γ ˆτ l, ( ) )+γ ˆτ l, ( +γ 0ˆτ l (,0 )+γ ˆτ l, ( )+γ ˆτ l, ( ) )+γ ˆτ l, }{{} where the combination coefficients satisfy herefore, we have (m,n) A =0 γ mn = and M (m,n) A γ mn µ o mn = [µ R,µ ] ρ τ u τ min {β 0τ0 o +β 0τ0 o + +β τ o } u {β mn} = (6a) τ l ˆτ l γ 0ˆτ l (0,/)+γ 0ˆτ l (0,/)+ +γ ˆτ l (/,) γ 0 τ0 o +γ 0 τ0 o + +γ τ o (6b) γ 0ˆτ l (0,/)+γ 0ˆτ l (0,/)+ +γ ˆτ l (/,) { τ0 o max ˆτ l (0,/), τ0 o ˆτ l (0,/),, τ o } (6c) ˆτ l (/,) { where (6c) is due to the inequality x +x + +x n x y +y + +y n max y, x y,, xn y n } τmn Define ρ mn o he upper bounds of{ρ ˆτ l (m/,n/) mn} can be obtained using simple mathematical deduction and are summarized in able I hus, for any antenna configuration, we haveρ max{ρ 0,,ρ }, which completes the proof of Corollary

14 ABLE I: he multiplicative gap at any antenna configurations 4 M ρ ρ 0ρ ρ ρ ρ ρ ρ ρ ( 0, ] (, ] (,] (, 5 ] ( 5,] (,] (, ) ρ ρ 7 6 ρ 0 5 ρ 0 ρ 0 VII EXESIO O ARBIRARY UMBER OF RASMIERS AD RECEIVERS In this section, we discuss the extension of the D analysis to the more general R networks with transmitters and R receivers We first extend the theoretical lower bound on the minimum D in the following theorem he proof is very similar to that of heorem and hence ignored heorem (Lower bound for R networks) Consider the R cache-aided MIMO interference network where each transmitter is equipped with M antennas and a cache of normalized size µ, and each receiver is equipped with antennas and a cache of normalized size µ R he minimum D is lower bounded by { τ max ( µ R), max s [ R ] } s M ( sµ R) (7) ext we discuss the extension of the achievable scheme By using the same file splitting and caching strategy as in [9], an achievable D for the R networks can be similarly expressed in the form of an LP problem as in heorem : P : τ u min {β mn} st (m,n) A (m,n) A (m,n) A β mn m/ R d mn β mn =, β mn µ o mn µ, (8a) (8b) (8c) 0 β mn, (m,n) A (8d) Here, A = {(m,n) : m + R n R,m {0,,, R },n {0,,, }}, [ and d mn ] is the DoF per user of the channel formed by the cache state at each integer point µ mn = m n R, in the cache size region At a general integer point µ mn, the newly formed channel is referred to as a ( R ) ( m ) n cooperative MIMO X-multicast channel, where every set ofmout of the total R receivers forms a receive multicast group, every set of n out of the total transmitters forms a transmit cooperation group, and each transmit cooperation group has an independent message for each receive multicast group he DoF per user of this channel can be obtained ] in some special cases as follows d 0 at integer point µ 0 [0, = : In this case, the ( R ) ( m ) n cooperative MIMO X-multicast channel degenerates to an R MIMO X channel where each transmitter (receiver) is equipped with M () antennas By using [5, heorem ] and antenna deactivation, an achievable DoF per user of this channel is given by 6 : { } M min qm d 0 = R, +q R, if q M { } = q min,, if M = q, (9) where q is any positive integer q q + R q 6 In the general case, we remove the limitation of linear transmission with finite symbol extensions on the achievable schemes

15 5 d 0 at integer point µ 0 = [0,]: he channel in this case is a MIMO broadcast channel where the virtual transmitter is equipped with M antennas and each receiver is equipped with antennas he optimal DoF per user of this channel is { } M d 0 = min, (40) R he DoF results at other integer points remain unknown in general, to our best knowledge evertheless, it is still possible to obtain an achieve D based on these special cases ote that the convex region formed by integer pointsµ 0, µ 0, µ R 0 and µ R is the whole cache size region hus, by substituting these four integer points as well as the corresponding DoF values in the LP problem P, an achievable D for the general R networks can be obtained as d 0 ( µ R ), (µ R,µ ) R τ [ d 0 ( d 0 d 0 )] ( µ R ) ( d 0 d 0 ) where { R = {(µ R,µ ) : µ R +µ,µ R,µ } R = {(µ R,µ ) : µ R +µ <,µ R 0,µ R + µ } µ, (µ R,µ ) R, (4) (4) By comparing (4) with (7), it is found that the achievable D is optimal when () (0, M R ], ( ] and () M R, R and (µ R,µ ) R It is also found that the multiplicative gap ρ is less than when R and = M In a more general setting of (M,,, R ), the gap does not converge to a constant Further investigation is needed but beyond the scope of this paper VIII COCLUSIO In this paper, we study the storage-latency tradeoff in the cache-aided MIMO interference network With different file splitting patterns, the MIMO interference channel can be turned to MIMO broadcast channel, MIMO multicast channel, MIMO X channel, or hybrid forms of these channels We propose linear transmission schemes and obtain the DoF results of these channels We obtain the achievable upper bound of minimum D by solving a linear programming problem he achievable D decreases piecewise linearly with the normalized cache sizes and each additive item is inversely proportional to the number of antennas his finding reveals that the MIMO gain and cache gain are cumulative in the considered wireless network We also give a lower bound of minimum D It is shown that the achievable D is optimal in certain cases and is within a multiplicative gap of to the optimal in other cases Although this work mainly focuses on the network with three transmitters and three receivers, the results have been extended, in certain ways, to the more general network with arbitrary number of transmitters and receivers he main challenge for further investigation would be the DoF analysis of the new class of cooperative MIMO X-multicast channels APPEDIX A: HE CLOSED FORM EXPRESSIO OF τ u I HEOREM Case : M ( ] 0, τ u (µ R,µ ) = ( µ R) (4) Case : M (, ] 4 9 { τ u (µ R,µ ) = ( µ R) (µ R,µ ) R ( +µ R +µ )+ M ( µ (44) R µ ) (µ R,µ ) R

16 6 Case : M ( 4 9, ] ( µ R) (µ R,µ ) R (5 5µ R µ ) (µ R,µ ) R τ u (µ R,µ ) = ( µ min{ /ξ, M +/ξ } R µ )+ ( +7µ R +9µ ) (µ R,µ ) R ( µ min{ /ξ, M +/ξ } R µ )+ 7 ( +µ R +µ ) (µ R,µ ) R 4 Case 4: M (, ] 0 7 ( µ R) (µ R,µ ) R ( µ R µ )+ M ( +µ R +µ ) (µ R,µ ) R 7 τ u (µ R,µ ) = ( µ R µ )+ 9 M ( +µ R +µ ) (µ R,µ ) R min{ /ξ, M ( µ +/ξ } R µ )+ 7 µ R + M ( +µ ) (µ R,µ ) R 4 ( µ min{ /ξ, M +/ξ } R µ )+ 7 ( +µ R +µ ) (µ R,µ ) R 5 Case 5: M ( 0 7,] ( µ R) (µ R,µ ) R (4 4µ R µ ) (µ R,µ ) R 7 τ u (µ R,µ ) = µ R + 9 M ( µ R µ )+ ( +µ R +µ ) (µ R,µ ) R ( µ min{ /ξ, M +/ξ } R µ )+ 7 µ R + M ( +µ ) (µ R,µ ) R 4 ( µ min{ /ξ, M +/ξ } R µ )+ 7 ( +µ R +µ ) (µ R,µ ) R 5 Case 6: M ( ], 4 ( µ R) (µ R,µ ) R ( +4µ R +µ )+ M (5 6µ R µ ) (µ R,µ ) R ( µ max{ 6 7 τ u (µ R,µ ) = M, } R µ )+ ( +µ R +µ )+ M ( +µ ) (µ R,µ ) R max{ 6 }µ 7 M, R + ( µ min{ M /ξ, +/ξ } R µ )+ M ( +µ ) (µ R,µ ) R 4 max{ 6 7 M, } ( +µ R +µ )+ ( µ min{ M /ξ, +/ξ } R µ ) (µ R,µ ) R 5 max{ 6 7 M, } ( µ R µ )+ ( +µ R +µ ) (µ R,µ ) R 6 Case 7: M ( 4,] Case 8: M ( ], 5 ( µ R) (µ R,µ ) R 6 ( µ R µ ) (µ R,µ ) R τ u (µ R,µ ) = (+4µ R µ )+ M ( µ R) (µ R,µ ) R µ R + 4M (7 8µ R µ ) (µ R,µ ) R 4 ( +µ R +µ )+ 9 M ( µ R µ ) (µ R,µ ) R 5 ( µ R µ ) (µ R,µ ) R 6 ( µ R) (µ R,µ ) R ( +µ R)+ M ( µ R) (µ R,µ ) R 6 ( µ R 9µ )+ 6M ( +µ ) (µ R,µ ) R τ u (µ R,µ ) = ( µ )+ M ( 4µ +µ ) (µ R,µ ) R 4 (5 8µ R 5µ )+ M ( +µ R +µ ) (µ R,µ ) R 5 (7 µ R µ ) (µ R,µ ) R 6 ( µ R µ ) (µ R,µ ) R 7 ( µ R µ ) (µ R,µ ) R 8 (45) (46) (47) (48) (49) (50)

17 7 / / 0 / / Case Case Case Case 4 Case 5 Case 6 Case 7 Case 8 Case 9 Case 0 Fig 5: Cache regions of the different number of antennas Case 9: M ( 5,] ( µ R) (µ R,µ ) R ( +µ R)+ M ( µ R) (µ R,µ ) R 6 ( µ R 9µ )+ 6M ( +µ ) (µ R,µ ) R min{ τ u (µ R,µ ) = 5,M} ( µ R)+ (+µ R µ )+ 6M ( +µ R +µ ) (µ R,µ ) R 4 min{ 5,M} ( µ )+ M ( µ R +µ ) (µ R,µ ) R 5 ( µ R µ )+ M ( +µ ) (µ R,µ ) R 6 ( µ R µ ) (µ R,µ ) R 7 ( µ R µ ) (µ R,µ ) R 8 Case 0: M (, ] M ( µ R) (µ R,µ ) R 9M (5 6µ R) (µ R,µ ) R τ u (µ R,µ ) = M ( µ R) (µ R,µ ) R M ( µ R µ ) (µ R,µ ) R 4 M ( µ R µ ) (µ R,µ ) R 5 he cache size regions {R i } of each case are illustrated in Fig 5 (5) (5) APPEDIX B: PROOF OF LEMMA hroughout this Appendix and Appendices C, D and E, we adopt the DoF plane introduced in [] to present the DoF results he DoF per user of the partially cooperative MIMO X channel shown in (8) of Lemma is illustrated in Fig 6(a) o prove its achievability, it suffices to prove the achievability of points {Q,Q } in the DoF plane, by [, Lemma ] he achievable scheme of the partially cooperative MIMO X channel needs three phases as shown in Fig 4 In each phase, the transmission scheme is similar and we take the phase I for an example Let the d vectors s rt, s rt, s rt denote the actual transmitted signal vectors of messages A, B, C, intended for receivers,, and, respectively Here, d is the desired DoF per user Due to the symmetry of the three receivers, we take receiver as an example Its received signal (ignoring noise for brevity) can be written as y =H (V rt s rt +V rt s rt )+H (V rt s rt +V rt s rt ) +H (V rt s rt +V rt s rt ), where V rk t pqi is the M d precoding matrix of signal s rk t pq at transmitter i {p,q} ext, we give the detailed design method of transmitter precoding matrices and, if necessarily, receiver combining matrices (5)

18 8 () the achievability of Q : his is to show that the DoF per user d = M is achievable at antenna configuration = M For receiver, s rt and s rt are the interference signals We can design the M M precoding matrices V rt, V rt, V rt and V rt to satisfy: H V rt = H V rt (54) H V rt = H V rt In this way, the interferences froms rt ands rt will be cancelled at receiver, which is known as interference neutralization Similarly, the interferences at receiver and can be neutralized by the following design: H V rt = H V rt, H V rt = H V rt, H V rt = H V rt, H V rt = H V rt ote that each precoding matrix needs to meet two conditions from (54) and (55) he existence is justified as follows We take V rt and V rt as examples hey can be designed as: [ ] [ ] Vrt H H null (56) H H V rt Given that each H ji is a M M matrix, using the null space theorem, we can obtain such precoding matrices V r t and V rt that satisfy (55) with probability one In this way, all the interferences are cancelled and each receiver can decode M data streams () the achievability of Q : We intend to achieve DoF per user d = M at antenna configuration = 5M In this case, we need to jointly design the transmit precoding matrices and the receive combining matrices for interference neutralization In specific, we first design the M 5 M combining matrices, denoted as P j, for each receiver j as follows: p p p M p M + p M + p M null[ H H null[ H H ], ], p M + p M + p M p p p M null[ H H null[ H H ], ], p p p M p M + p M + p M null[ H H null[ H H where p m j denotes the m-th row of P j hen, we design the M M transmit precoding matrices to meet the same conditions in (54) and (55) with each channel matrix H ji replaced by the effective channel matrix P j H ji he existence of such precoding matrices is justified using the similar null space theorem as in (56) By doing so, all the interferences are neutralized and M data streams can be decoded by each receiver APPEDIX C: PROOF OF LEMMA he DoF per user of the MIMO X-multicast channel, shown in (9) of Lemma, is illustrated in Fig 6(b) () the achievability of Q : his is to show that the DoF per user d = 6M 7 is achievable at antenna configuration = M Let s rjk t p denote the M M 7 transmitted signal vector for W r jk t p, intended to receive multicast group {j,k} from transmitter p Let V rjk t p denote the M M 7 precoding matrix of s r jk t p at transmitter p At receiver, the received signal can be expressed as (ignoring the noise for brevity) y = H (V rt s rt +V rt s rt +V rt s rt ) +H (V rt s rt +V rt s rt +V rt s rt ) (58) +H (V rt s rt +V rt s rt +V rt s rt ) Receiver desires signalss rt,s rt,s rt,s rt,s rt, ands rt, and it wants to align the interference signalss rt,s rt, and s rt along a same direction so as to cancel them all at once: H V rt = H V rt = H V rt V (59) ], ], (55) (57)

19 9 0 (a) 0 (b) 0 (c) 0 (d) Fig 6: he DoF planes: (a) the partially cooperative MIMO X channel, (b) the MIMO X-multicast channel, (c) the partially cooperative MIMO X-multicast channel, (d) the fully cooperative MIMO X-multicast channel x x x Fig 7: he alternating transmission scheme in the MIMO X-multicast channel At receiver and, the similar equations can be obtained: H V rt = H V rt = H V rt V, (60) H V rt = H V rt = H V rt V (6) We need to further design V, V and V to ensure the decodability of desired signals at each receiver We give an achievable method as below: V = V = V = diag{ 7, 7,, 7 }, (6) }{{} M M 7 where 7 denotes the 7 vector with all elements being one In this way, all desired signals are linearly independent of each other and each receiver can decode its desired signals successfully So the 6M 7 DoF per user can be obtained () the achievability of Q : We intend to achieve DoF per user d = M at antenna configuration = M In this case, we use the alternating transmission scheme as shown in Fig 7 We take phase I as an example For receiver, the post-processed received signal (ignoring the noise for brevity) after the M M combining matrix P can be expressed as ŷ = P (H s rt +H s rt +H s rt ) (6)