Fundamental Storage-Latency Tradeoff in Cache ... - Semantic Scholar

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Fundamental Storage-Latency Tradeoff in Cache-Aided MIMO Interference Networks Youlong Cao, Meixia Tao, Fan Xu and Kangqi Liu

arXiv:1609.01826v1 [cs.IT] 7 Sep 2016

Abstract 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. This paper studies the fundamental limits of caching in a MIMO interference network with 3 cache-aided transmitters each equipped with M antennas and 3 cache-aided receivers each with N antennas. The performance is characterized by fractional delivery time (FDT), an information theoretic metric evaluating the worst-case relative delivery time with respect to a point-to-point baseline system without cache. By using a newly proposed cooperative Tx/Rx caching strategy for file placement, the MIMO interference network during content delivery is turned to hybrid forms of MIMO X channel and MIMO multicast channel with partial or full transmit cooperation. We derive the degrees of freedom (DoF) of these channels through linear precoding based interference management schemes. Based on these results, we obtain an achievable FDT for arbitrary M , N and any feasible cache sizes. The achievable FDT is optimal at certain antenna configurations and cache size regions, and is within a multiplicative gap of 3 from the optimum at other cases. Index Terms Cache-aided MIMO wireless network, degrees of freedom, interference management, multicast, linear transmission scheme.

I. I NTRODUCTION A. Motivation 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 2015 and is foreseen to contribute 75% in 2020 [2]. 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 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 experiences. 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 [3]–[5] and references therein. Traditional caching has been long proposed in the computer network for reducing the downloading delay [6] due to that the 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. This has driven the study of fundamental limits of caching in various wireless systems, including broadcast channel [7], [8], interference networks [9]–[13], partially connected networks [14], device-to-device networks [15]–[17] and fog radio access networks [18]. This work will be presented in part at the 2016 IEEE Global Telecommunications Conference [1]. The authors are with the Department of Electronic Engineering, Shanghai Jiao Tong University, Shanghai, China. Emails: {caoyoulong, mxtao, xxiaof}@sjtu.edu.cn, [email protected].

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This 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. The system operates in two phases. In the cache placement phase, which usually takes places in a large time scale (e.g. 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 (e.g. second), each transmitter sends the messages according to the requests of receivers, cache status, and the MIMO channel conditions. Our goal is to characterize the storage-latency tradeoff through the careful design of the cache placement and content delivery phases. B. Related Works The fundamental limits of caching at the receiver side are first studied in [7] for a shared link with one server and multiple cache-aided receivers. The study in [7] shows that caching can exploit multicast opportunities even when user demands are different, and hence greatly reduces the traffic load over the shared link. This is enabled by proper file splitting during the cache placement phase and coded transmission during the content delivery phase, known as coded caching. With coded caching, both local caching gain and global caching gain are achieved. The benefits of caching at the transmitter side are studied in [9] for a 3 × 3 interference channel. It is shown that caching can induce transmitter cooperation and hence allows interference coordination for throughput enhancement. The limits of caching when equipped at both the transmitter and receiver sides are investigated in [10]–[13] very recently, which all considered a general interference network but with different restrictions and performance metrics. The works [10], [11] characterize the tradeoff between storage size and content delivery time, in terms of an information-theoretic metric, fractional delivery time (FDT). An achievable FDT is obtained in [10] for an interference network with arbitrary number of transmitters and up to three receivers at any feasible cache sizes. Their achievable FDT is optimal at certain cache size regions and has a maximum multiplicative gap of 2 to a theoretical lower bound at other cache size regions. The studies in [10] reveal 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 [12], an order-optimal approximation on the system performance for arbitrary number of transmitters and receivers is 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. The work [13], on the other hand, adopts the standard sum degrees to freedom (DoF) to characterize the performance and their analysis is restricted to one-shot linear transmission schemes. The aforementioned studies of fundamental limits of caching at both transmitter and receiver sides are limited to the single-antenna interference network. Note that a crucial step in analysing the fundamental limits of caching in interference networks is to derive the DoF, a capacity approximation at high signal-to-noise ratio (SNR) region, of the new network topologies formed by caching, for example the X-multicast channel [10], [12]. DoF characterizations for a wide variety of MIMO wireless channels have recently been obtained, in particular MIMO interference channel [19]–[22] and MIMO X channel [23]–[25]. In general, the DoF results of these MIMO channels with multicast traffic and/or transmitter cooperation for arbitrary antenna setting remain unsolved. C. Our Contribution In this paper, we study the fundamental storage-latency tradeoff in a cache-aided MIMO interference network with three transmitters and three receivers, as shown in Fig. 1. Each transmitter is equipped with M antennas and a local cache of normalized size µT , and each receiver with N antennas and a local cache of normalized size µR . The tradeoff is characterized by FDT, the same information theoretic metric applied in [10]. This work is a non-trivial extension of [10] due to the deployment of multiple antennas. The main contributions and findings are summarized as follows: • An achievable FDT: We adopt the same cooperative Tx/Rx caching scheme proposed in [10], [11] for file placement, but design linear precoding based transmission schemes for content delivery due to

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the deployment of multiple antennas. An achievable FDT is obtained by solving a linear programming problem of file splitting or, equivalently, memory sharing coefficients. The achievable FDT is for any number of transmit antennas M, any number of receive antennas N, and any feasible cache size regions (µR , µT ). Its closed-form expression preserves the property of being piecewise linear decreasing with the normalized cache sizes as in [10], [11], which reflects the caching gain. Additionally, each additive item in the expression is inversely proportional to the number of antennas, which reflects the spatial multiplexing gain induced by MIMO. An interesting finding from our achievable results is that the traditional widelyadopted equal file splitting strategy [10]–[13] is not always a good choice at integer points.1 • DoF of new MIMO channel models: A crucial step in analyzing the achievable FDT 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 3 × 3 partially cooperative MIMO X channel, 3 × 3 MIMO X-multicast channel, and 3 × 3 partially or fully cooperative MIMO X-multicast channel. We derive the achievable DoF per user of these channels for any antenna configurations M and N by using linear precoding based interference management schemes 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 [22]. Our channel models differ from [22] in that (1) each transmitter has multiple messages to send and can cooperate with each other; (2) 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 in [26], which, however only considers single antenna. • A lower bound of the minimum FDT: We also obtain a theoretical lower bound of the minimum FDT of the considered 3 × 3 cache-aided MIMO interference network by using a cut-set like argument. This 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, although all the transmission schemes are linear. Analysis also shows that the maximum multiplicative gap between the upper and lower bounds is 3. Notations: x, x, X and X denotes scalar, vector, matrix and set, respectively. Θ(x) denotes that lim Θ(x) = 1. (·)T denotes the transpose of a matrix . tr(X) and null(X) stand for the trace and the x x→∞

null space of the matrix X. X⊥ denotes the orthogonal complement of the row space of X. [n] denotes the set {1, 2, · · · , n} where n is an integer. II. S YSTEM M ODEL AND AND P ERFORMANCE M ETRICS We consider a cache-aided MIMO interference network with three transmitters and three receivers, as illustrated in Fig. 1, where each transmitter is equipped with M antennas and each receiver is equipped with N antennas. Each node has a local cache of finite size. Consider a library consisting of L files, denoted by {W1 , W2 , . . . , WL }. Throughout this study, we focus on the case where the number of files L is larger than or equal to the number of users in the system, i.e., L ≥ 3. Each file has a size of F bits. Each transmitter can cache QT F bits and each receiver can cache QR F bits, where QT , QR ≤ L. We define the normalized transmitter cache size and normalized receiver cache size as QR QT , and µR = . (1) µT = L L This work focuses on the feasible cache size region [10] [11]:  0 ≤ µR , µT ≤ 1, (2) µR + 3µT ≥ 1.

The equal file splitting strategy means that each file is split into 3µ3R 3µ3T equal-sized subfiles, each cached in 3µR receivers and 3µT  1 2 transmitters when µR , µT ∈ 0, 3 , 3 , 1 . Due to that 3µR = m and 3µT = n with m, n being integers, these points are called integer points [10] or corner points [7]. 1

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The communication involves two phases, the cache placement phase, which takes place in a large time scale (e.g., on a daily or hourly basis) and the content delivery phase, which happens in a small time scale (e.g., second). During the cache placement phase, each transmitter i has a caching function φi : [2F ]L → [2⌊F QT ⌋ ],

(3)

mapping the L files in the library to its local cache content Ui , φi (W1 , W2 , . . . , WL ). Each receiver j also has a caching function ψj : [2F ]L → [2⌊F QR ⌋ ], (4) mapping the L files in the library to its local cache content Vj , ψj (W1 , W2 , . . . , WL ). The caching functions {φi , ψj } are assumed to be known globally by all nodes. Similar to [10], [27], we do not allow inter-file coding but allow arbitrary intra-file coding for caching function {φi , ψj } in this paper. In the content delivery phase, each receiver j requests a file Wdj from the library, where dj ∈ [L]. We denote d , [d1 , d2 , d3 ]T as the demand vector. Each transmitter further consists of an encoding function Λi : [2⌊F QT ⌋ ] × [L]3 × C3N ×3M → CM ×T .

(5)

where T is the block length of the code and depends on the receiver demand d and the channel state information (CSI) H. Here, H collects the channel matrices Hi,j ∈ CN ×M from each transmitter i to each receiver j, which are assumed to be fixed during each codeword transmission but can vary independently from one codeword to another. Transmitter  i uses T Λi to map its local cache content Ui , receiver demands d and the CSI H to the signal vectors xi (t) t=1 , Λi (Ui , d, H), which is subject to a power constraint N ×1 tr xi (t)xH , i (t) ≤ P . In each time slot, the received signal at each receiver j, denoted as yj (t) ∈ C can be expressed as 3 X yj (t) = Hji xi (t) + nj (t), ∀j = 1, 2, 3. (6) i=1

where nj (t) denotes the additive white Gaussian noise (AWGN) vector at receiver j, with each element being independent and having zero mean and unit variance. In this paper, we assume that the perfect channel state information is available at both transmitters and receivers. The decoding function Γj at receiver j can be defined as: Γj : [2⌊F QR ⌋ ] × CN ×T × C3N ×3M × [L]3 → [2⌊F ⌋ ].

(7)

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  ˆ j , Γj (Vj , yj (t) T , H, d) of its desired file Wd , with its cached Each receiver j uses Γj to estimate W j t=1 content Vj and the channel realization H. The worst-case error probability is ˆ j 6= Wd ) max max P(W j

(8)

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The given caching and coding functions {φi , Λi , ψj , Γj } are said to be feasible if the worst-case error probability approaches 0 when F → ∞. In this work, we adopt the following performance metric originally proposed in [27], then refined in [10] [11] to characterize the fundamental storage-latency tradeoff. Definition [27] [10]: For any given feasible caching and coding scheme at given normalized cache sizes µT and µR , the fractional delivery time (FDT) is defined as max EH (T d,H ) τ (µR , µT ) , lim lim sup P →∞ F →∞

d

3F · 1/ log P

.

(9)

The minimum FDT is defined as τ ∗ (µR , µT ) = inf{τ (µR , µT ) | τ (µR , µT ) is achievable.}

(10)

The FDT τ is an asymptotic approximation of the worst-case relative delivery time with respect to that of delivering the total 3F requested bits in an interference-free single-antenna point-to-point baseline system with transmission rate log P when both transmit power P and file size F go to infinity [10]. It R is not difficult to see that the FDT can also be expressed as τ (µR , µT ) = 3·d , where R is the worst-case 2 traffic load per user with respect to file size over the air [7] and d is the standard DoF per user of the communication channel formed by the given caching strategy. Example 1. Consider a 3 × 3 MIMO interference network with the normalized cache sizes µR = 13 and µT = 32 under symmetric antenna setting M = N. The cache placement strategy is shown in Fig. 2, where each file is split into two subfiles, one with 31 F bits and cached in all receivers, the other with 32 F bits and cached in all transmitters. During the delivery phase, consider the worst case where the three receivers request different files, denoted as A, B, C, respectively. Then each receiver only needs the subfile with the length of 23 F bits that it does not cache, which is available at all the three transmitters. The network topology can be viewed as a virtual MIMO broadcast channel where the virtual transmitter has 3M antennas and each receiver has M antennas. The DoF per user of this channel is M. By definition, 1− 1 2 the achievable FDT is τ = 3×M3 = 9M . 2 The traffic load per user is not necessarily equal to the total traffic load divided by the number of users. For example, in a multicast channel with 2 receivers, the traffic load per user is the same as the total traffic load.

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III. M AIN RESULTS In this section, we present our main findings on the minimum FDT in the 3 × 3 cache-aided MIMO interference network. Theorem 1 (Upper Bound): Consider the 3 × 3 cache-aided MIMO interference network where each transmitter is equipped with M antennas and a cache of normalized size µT , and each receiver is equipped with N antennas and a cache of normalized size µR . An achievable FDT based on linear transmission schemes is given by τu , the optimal solution of the following problem: X 1 − m/3 P1 : τu , min βmn (11a) {βmn } 3dmn (m,n)∈A X s.t. βmn = 1, (11b) (m,n)∈A

X

βmn µomn ≤ µ,

(11c)

(m,n)∈A

0 ≤ βmn ≤ 1,

∀(m, n) ∈ A.

(11d)

where A = {(m, n) | m + 3n ≥ 3, m, n ∈ {0, 1, 2, 3}}; µ = [µR , µT ]T denotes any feasible point in the cache size region; µomn = [ m3 , n3 ]T denotes the integer point with (µR = m3 , µT = n3 ) in the cache size region; and βmn is the (memory sharing) parameter to be optimized; dmn is given below: ) ( k+ k− , , d03 = min{M, N}, d21 = d22 = d23 = min{N, 3M}, d01 = min 2 − 1ξ 2 + 1ξ  6N    N N N N, ∈ 0, 23 , M ∈ (0, 1] ∈ (0, 1] N,       M 7 M (12)        6M , N ∈ 1, 9 M,  2M , N ∈ 2 , 5 3 N ∈ 1, 3 M 3 3 , d 7 M 7 , d M 2 . d02 = 2N 11 = 12 = d13 = N 5 5 N 9 N 3 2N 2N    , ∈ , , ∈ , 3 , ∈ , 3    M 3 2
M 7 M 2     3  3  5 N N N ∈ 52 , ∞ ∈ (3, ∞) ∈ (3, ∞) M, M 2M, M 2M, M −

where k − = min{M, N}, k + = max{M, N} and ξ = ⌈ k+k−k− ⌉. Remark 1: Problem P1 in Theorem 1 is a linear programming problem. Its optimal solution τu can be obtained efficiently. The explicit and closed-form, but somewhat tedious expression of τu is given in Appendix A. It is seen from Appendix A that the achievable FDT decreases piecewise linearly with the normalized cache sizes and each additive item of FDT is, in general, inversely proportional to the number of antennas. The latter property explicitly shows the multiplexing gain induced by MIMO. Moreover, the antenna configuration (i.e., the ratio N/M) determines the partition of the cache size region. Remark 2: In the special case with symmetric antenna configuration, i.e., M = N, the achievable FDT reduces to the results in [1]. Furthermore, when M = N = 1, the obtained FDT is numerically at most 1.2 times of the one in the single antenna case [11]. The slight increase in the FDT is due to that we only use linear precoding based interference management schemes with finite symbol extension. Theorem 2 (Lower Bound): Consider the 3 × 3 cache-aided MIMO interference network where each transmitter is equipped with M antennas and a cache of normalized size µT , and each receiver is equipped with N antennas and a cache of normalized size µR . The minimum FDT is lower bounded by   1 s ∗ τ ≥ τl , max (1 − µR ), max (1 − sµR ) . (13) s∈[3] 9M 3N Remark 3: By comparing the closed-form upper bound in Appendix A and the lower bound in Theorem 2, it can be seen that the achievable FDT is optimal under the following conditions:  N ∈ 0, 31 and (µR , µT ) ∈ {(µR , µT ) : µR + 3µT ≥ 1, µR ≤ 1, µT ≤ 1}; 1) M N 2) M ∈ (0, 1] and (µR , µT ) ∈ {(µR , µT ) : µR + µT ≥ 1, µR ≤ 1, µT ≤ 1};

7 N 3) M ∈ (0, ∞) and (µR , µT ) ∈ {(µR , µT ) : µR + µT ≥ 1, 32 ≤ µR ≤ 1, µT ≤ 1}. Corollary 3 (Multiplicative Gap): The multiplicative gap between the upper bound and the lower bound of the minimum FDT for the considered channel is at most 3. The proof of Theorem 1 will be given in Sections IV and V. The proofs of Theorem 2 and Corollary 3 will be given in Section VI.

IV. C ACHING AND DELIVERY S CHEME The achievable upper bound of minimum FDT in Theorem 1 is based on the same cache placement strategy in [10] 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 [10] for self-completeness. Then we present the proposed delivery scheme in detail. Since each transmitter and receiver can decide whether to cache each bit of each file, there are 26 = 64 possible cache states for each bit. Not 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. This results in a total of 57 legitimate cache states for each bit and the feasible domain of µR and µT , given in (2). 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 ⊆ [3] and receiver subset J ⊆ [3]. Then, let WκrJ tI denote the subfile split from Wκ that is cached in receiver subset J and transmitter subset I. For example, Wκrø t123 is the subfile cached in none of the three receivers but in all three transmitters and Wκr12 t12 is the subfile cached in receiver 1, 2 and S transmitters 1, 2. Similarly, we denote WκtI as the collection of all subfiles cached in I, i.e. WκtI = WκrJ tI . As in [10], the sizes of J

the subfiles with the same cardinality of transmitter and receiver subsets are assumed to be equal. Let |J | and |I| denote the cardinalities of J and I respectively, and define a|J ||I| as the file splitting ratio to be optimized. Then each subfile WκrJ tI contains F a|J ||I| bits. The splitting ratios must satisfy the following constraints: a30 + 3a31 + 3a32 + a33 + 9a21 + 9a22 + 3a23 + 9a11 + 9a12 + 3a13 + 3a01 + 3a02 + a03 = 1, a30 + 3a31 + 3a32 + a33 + 6a21 + 6a22 + 2a23 + 3a11 + 3a12 + a13 ≤ µR , a31 + 2a32 + a33 + 3a21 + 6a22 + 3a23 + 3a11 + 6a12 + 3a13 + a01 + 2a02 + a03 ≤ µT .

(14) (15) (16)

Constraint (14) comes from the constraint of total file size. Constraints (15) and (16) come from the cache size limit in receiver and transmitter, respectively. In the delivery phase, without loss of generality, we assume that receiver 1, 2, 3 request file W1 , W2 , and W3 respectively. The detailed delivery schemes are given below. A. Delivery of Subfiles Cached in Zero Receiver and One Transmitter Consider the delivery of subfiles {Wκrø tp | κ, p ∈ [3]}, each of which is cached at one transmitter but none of the receivers and has fractional length a01 . The network topology in this case can be seen as a 3 × 3 MIMO X channel. Previous study on the DoF of MIMO X channel can be found in [24], but the results are limited to the symmetric antenna configuration and require non-linear operation. In this work, we treat the MIMO X channel as a MIMO interference channel instead, whose optimal DoF is obtained in [21], [28] and only requires linear transmission scheme. We use three time slots to deliver the subfiles as shown in Fig. 3, where in each time slot, each transmitter sends an independent message to a different receiver. Thus, the channel in each time slot can be regarded as a 3 × 3 MIMO interference channel where each transmitter has M antennas and each receiver has N antennas. The DoF per user in this channel, defined as d01 , is,   k− k+ d01 = min , (17) , 2 − 1/ξ 2 + 1/ξ −

where k − = min{M, N}, k + = max{M, N} and ξ = ⌈ k+k−k− ⌉. Given that the total amount of bits to 01 deliver to each user is 3a01 F , by definition, the FDT of these subfiles can be computed as τ = 13 · 3a . d01

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B. Delivery of Subfiles Cached in Zero Receiver and Two Transmitters Consider the delivery of subfiles {Wκrø tpq | κ, p, q ∈ [3], p < q}, each of which is cached in two transmitters and none of receivers and has fractional length a02 . The network topology in this case can be viewed as a 3 × 3 partially cooperative MIMO X channel, where every set of two transmitters forms a cooperation group and has an independent message to send to each receiver. Lemma 1: For the 3 × 3 partially cooperative MIMO X channel, the achievable DoF per user, denoted as d02 , is given in (18).   N ∈ 0, 23 N,   M   2M , N ∈ 2 , 5  3 M 3 3 (18) d02 = 2N N 5 5  , ∈ ,2  5 M 3 
 N ∈ 52 , ∞ M, M

Proof: The achievable scheme takes three time slots, as shown in Fig. 4. In each time slot, each of the three transmitter cooperation groups ({1, 2}, {1, 3} and {2, 3}) sends one independent message intended to a different receiver. The two interference signals seen by each receiver are cancelled by interference neutralization with linear transmit and receiver processing. The detailed proof is given in Appendix B. 02 . Based on Lemma 1, the FDT of these subfiles is τ = 13 · 3a d02

C. Delivery of Subfiles Cached in Zero Receiver and Three Transmitters Consider the delivery of subfiles {Wκrø t123 | κ ∈ [3]}, each of which has fractional length a03 . Since each subfile is cached in all the three transmitters, the transmitters can fully cooperate. The delivery in this case can be regarded as an MIMO broadcast channel where the virtual transmitter has 3M antennas, and each receiver has N antennas. The optimal DoF per user of this channel is d03 = min{M, N} [29]. Therefore, the FDT of these subfiles is τ = 13 · ad03 . The delivery scheme in Example 1 shown in Section 03 2 II belongs to this case where a03 = 3 .

9

D. Delivery of Subfiles Cached in One Receiver and One Transmitter Consider the delivery of subfiles {Wκrk tp | κ, k, p ∈ [3], k 6= κ}, each of which has fractional length a11 . 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 1 can generate message Wκ⊕jk t1 , Wjrk t1 ⊕Wkrj t1 desired by receiver j and k, where ⊕ denotes the bit-wise XOR. Now each XORed message is desired by two receivers. The network topology of sending coded subfiles {Wr⊕jk tp | j, k, p ∈ [3], j < k} becomes a 3 × 3 MIMO X-multicast channel, where every set of two receivers forms a multicast group and each transmitter has an independent message to send to each multicast group. Lemma 2: For the 3 × 3 MIMO X-multicast channel, the achievable DoF per user, denoted as d11 , is given in (19).  6N N , M ∈ (0, 1]   7   6M , N ∈ 1, 9  7 M 7 d11 = 2N (19) N 9  , ∈ , 3  3 M 7   N ∈ (3, ∞) 2M, M Proof: When the antenna configuration satisfies N ≤ 97 M, we use linear interference alignment technique so that all the interference signals at each receiver can be aligned in the same direction. When the antenna configuration satisfies N > 97 M, the achievable scheme takes three time slots and in each time slot, each transmitter sends one independent message to a different multicast group. The receiver combining matrices are designed to cancel the interference signals. The detailed proof is given in Appendix C. 11 Based on Lemma 2, the FDT of these subfiles is τ = 13 · 6a . d11

E. Delivery of Subfiles Cached in One Receiver and Two Transmitters Similar to Subsection D, coded multicasting gain can be exploited in the delivery of subfiles {Wκrk tpq | κ, k, p, q ∈ [3], k 6= κ, p < q}, each of which has fractional length a12 . The difference is that each subfile is available at two transmitters and hence transmitter cooperation gain can be exploited. For example, transmitter 1 and transmitter 2 can generate a coded message Wκ⊕jk t12 , Wjrk t12 ⊕ Wkrj t12 . The delivery of coded subfiles {Wr⊕jk tpq | j, k, p, q ∈ [3], j < k, p < q} can be viewed as a 3 × 3 partially cooperative MIMO X-multicast channel, where every set of two receivers forms a multicast group, every set of two transmitters forms a cooperation group, and there is an independent message to be communicated from each cooperation group to each multicast group. Lemma 3: For the 3 × 3 partially cooperative MIMO X-multicast channel, the achievable DoF per user, denoted as d12 , is given in (20).  N N, ∈ (0, 1]   M   M, 3 N ∈ 1, M 2 d12 = 2N (20) N 3  , ∈ , 3  3 M 2   N 2M, M ∈ (3, ∞)

Proof: When the antenna configuration satisfies N ≤ 23 M, we use the linear interference naturalization by designing the precoding matrices of each cooperation group. When N > 32 M, the achievable scheme takes three time slots and in each time slot, each cooperation group sends one independent message to a different multicast group. Each receiver applies zero-forcing processing to cancel the interference signals. The detailed proof is given in Appendix D. 12 . Based on Lemma 3, the FDT of these subfiles is τ = 13 · 6a d12

10

F. Delivery of Subfiles Cached in One Receiver and Three Transmitters Similar to Subsections D and E, the coded multicasting scheme can also be exploited in the delivery of subfiles {Wκrk t123 | κ, k ∈ [3], k 6= κ}. The difference is that each subfile is available at all the transmitters. For example, all the transmitter can generate message Wκ⊕jk t123 , Wjrk t123 ⊕ Wkrj t123 . The delivery of coded subfiles {Wr⊕jk t123 | j, k ∈ [3], j < k} can be regarded as a 3 × 3 fully cooperative MIMO X-multicast channel, where every set of two receivers forms a multicast group, all the transmitters forms a cooperative group, and there is an independent message to be communicated from the cooperation group to each multicast group. Lemma 4: For the 3 × 3 fully cooperative MIMO X-multicast channel, the achievable DoF per user, denoted as d13 , is given in (21). (  N min{M, N}, ∈ 0, 23 M
d13 = (21) N min{ 2N , 2M}, M ∈ 23 , ∞ 3 Proof : See Appendix E. Based on Lemma 4, the FDT of these subfiles is τ =

1 3

·

2a13 . d13

G. Delivery of Subfiles Cached at Two Receivers and One or More Transmitters Consider the delivery of subfiles {Wκrk,l | κ, k, l ∈ [3], k, l 6= 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 1 can generate messages Wt⊕1 , W1r23 t1 ⊕ W2r13 t1 ⊕ W3r12 t1 , Wt⊕12 , W1r23 t12 ⊕ W2r13 t12 ⊕ W3r12 t12 , Wt⊕123

(22)

, W1r23 t123 ⊕ W2r13 t123 ⊕ W3r12 t123 .

Each of the above coded message is desired by all the three receivers, yielding an MIMO multicast channel. We first give the delivery scheme of coded messages {Wt⊕p | p ∈ [3]}, each with fractional length a21 . When N ≤ 3M, by antenna deactivation [30], we let each transmitter use N3 antennas and transmit N3 data streams3 . Each user can decode N data streams using N antennas, and the DoF per user is N. When N > 3M, we let each transmitter use M antennas and transmit M data streams. By the antenna deactivation, each user can decode 3M data streams using 3M antennas, and 3M DoF per user can be achieved. For the other two coded messages {Wt⊕pq | p, q ∈ [3], p < q} and {Wt⊕123 }, we can use the similar scheme as the one used in coded messages {Wt⊕p | p ∈ [3]}, because each coded message is simultaneously at more than one transmitter. So the DoF  per user d21 = d22 = d23 = min{N, 3M}, and  3a22 a23 3a21 1 the FDT can be computed as τ = 3 · d21 + d22 + d23 . Summing up the FDTs obtained in all the above subsections yields the total FDT as: a01 a02 a03 2a11 2a12 2a13 a21 a22 a23 τ= + + + + + + + + . (23) d01 d02 3d03 d11 d12 3d13 d21 d22 3d23 V. O PTIMIZATION OF F ILE S PLITTING R ATIOS AND C ONNECTION WITH M EMORY S HARING In this section, we study the optimization of the file spitting ratios {a|J ||I|} to minimize the total FDT in (23) subject to the constraints (14) (15) (16). This can be formulated as the following linear programming problem: P2 :

min τ (µR , µT )

{a|J ||I| }

s.t. (14)(15)(16)

(24) (25)

3 Throughout 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 extension such that td is an integer [21].

11

Clearly, by defining a new optimization variable βmn as:    3 3 amn , ∀(m, n) ∈ A, βmn = m n

(26)

where A = {(m, n) | m+3n ≥ 3, m, n ∈ {0, 1, 2, 3}}, P2 can be equivalently expressed as P1 in Theorem 1. Here, constraint (14) is equivalent to constraint (11b), and constraints (15) and (16) are equivalent to constraint (11c). By solving P1 , Theorem 1 is then proved. The significance of rewriting P2 as P1 is that P1 can be interpreted as memory sharing optimization. This is detailed as below. T First, consider an integer point µomn = m3 , n3 with (µR = m3 , µT = n3 ) in the

cache size region. Assume that equal file splitting strategy is adopted. That is, each file is split into m3 n3 amn equal-sized subfiles,

each cached simultaneously at m receivers and n transmitters. In that case, we have amn = 1/ m3 n3 and all the rest a|J ||I| = 0 . By the delivery scheme introduced in Section IV, the FDT at µomn with such o equal file splitting and caching strategy can be computed as τmn = 1−m/3 . 3dmn T Then, consider any feasible point µ = [µR , µT ] in the cache size region. The given µ can always be expressed as a convex combination of all the feasible integer points, i.e., X µ= βmn µomn . (27) (m,n)∈A

We now adopt the memory sharing strategy for cache placement. Namely, we split the transmitter and receiver cache sizes as in (27) with memory sharing parameter βmn . For each βmn fraction of the memory, we take βmn fraction of each file, split and cache it according to the equal file strategy at the integer point µomn . Then, a total achievable FDT can be obtained as X o τ= βmn τmn . (28) (m,n)∈A

We can minimize the total FDT by finding the optimal memory sharing parameters {βmn }. This is expressed mathematically in P1 in Theorem 1. Both P1 and P2 are standard linear programming problems. By using linear equation substitution and other manipulations, we obtain the closed-form but somewhat tedious expression of the optimal solution µu for any µR , µT , M, N in Appendix A. The antenna configuration is divided into 10 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 τu is a linear decreasing function of µR and µT and hence can be achieved by memory sharing of the integer points within that region. Remark 3: In the single antennas case [10], [11], 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 P1 ) though not unique. But in the MIMO case, the equal file splitting is not always a good choice. For example, consider integer point µo02 = [0, 32 ]T with N = 5, M = 3, i.e., Case 6 in Fig. 5. If equal file 1 splitting strategy is adopted, the FDT is 3·2 = 61 . On the other hand, from the optimal solution of P1 in Theorem 1, the optimal memory sharing coefficients are β01 = β03 = 12 . This means that a half of cache size shall be used to adopt the caching scheme at integer point µo01 = [0, 13 ]T and the other to adopt 1 1 5 caching scheme at integer point µo03 = [0, 1]T . The corresponding FDT shall be 3·2 · 21 + 3·3 · 21 = 36 < 16 . VI. C ONVERSE AND M ULTIPLICATIVE G AP In this section, we present the proof of the FDT lower bound in Theorem 2 and the proof of the maximum multiplicative gap in Corollary 3.

12

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 from any l antennas, which can be equipped at different receivers is upper bounded as   h(y[1:l] ) ≤ lT log 2πe(cP + 1) , (29)

where the parameter c is a function of the channel coefficient. Proof : See Appendix F. Now, we begin to prove the converse. The method of the proof is similar to [7], but we consider both the receiver side and the transmitter side. To fully utilize receiver cache, we let s ∈ [3] users request ˜ = s⌊ L ⌋ files during Z = ⌊ L ⌋ requests. Given the s receivers’ cache and received signals during Z L s s ˜ files can be decoded successfully in the high-SNR regime. By using this argument, we requests, these L have: 1 Z F ǫF = H(W1 , . . . , WL˜ | y[1:s] , . . . , y[1:s] , V[1:s] )

(30a)

1 Z = H(W1 , . . . , WL˜ ) − I(W1 , . . . , WL˜ ; y[1:s] , . . . , y[1:s] , V[1:s]) (30b) 1 Z 1 Z ˜ − h(y[1:s] | V[1:s]) − H(V[1:s]) + h(y[1:s] , . . . , y[1:s] | V[1:s] , W1 , . . . , WL˜ ) = LF , . . . , y[1:s]

+ H(V[1:s] | W1 , . . . , WL˜ ) 1 Z ˜ − h(y[1:s] ) − sµR LF + H(V[1:s] | W1 , . . . , WL˜ ) ≥ LF , . . . , y[1:s] ˜ − h(y1 , . . . , yZ ) − sµR LF + s(L − L)µ ˜ RF ≥ LF

(30d)

˜ − sZT NΘ(log P ) − sµR LF ˜ ≥ LF

(30f)

[1:s]

[1:s]

(30c) (30e)

where (30a) follows from the Fano’s inequality [31]; (30d) comes from the fact that conditioning reduces 1 Z entropy and we can recover y[1:s] , . . . , y[1:s] from W1 , . . . , WL˜ ; (30e) comes from the fact that each user ˜ files of the total L files; (30f) is due cache µR F bits of a file on average and the s receivers know the L to Lemma 5. Rearranging (30f), we have F T ≥ (1 − sµR ). (31) N log P Then we consider from the transmitter side. Using the data-processing inequality [31], (30e) can be rewritten as ˜ − h(x1 , . . . , xZ ) − sµR LF ˜ F ǫF ≥ LF ˜ − 3ZT MΘ(log P ) − sµR LF. ˜ ≥ LF Rearranging (32b), we have T ≥

sF (1 − sµR ). 3M log P

Combining (31) and (33) and taking P → ∞ and F → ∞, the FDT is lower bounded by   1 s 1 τ ≥ max max (1 − sµR ), (1 − sµR ) 3 s∈[3] N 3M   s 1 1 (1 − µR ), max (1 − sµR ) , = max s∈[3] 3M 3 N which completes the proof of Theorem 2.

(32a) (32b)

(33)

(34)

13

TABLE I T HE MULTIPLICATIVE GAP AT ANY ANTENNA CONFIGURATIONS . N ❍ ❍ M ρ ❍❍ ❍

ρ30 ρ31 ρ32 ρ33 ρ21 ρ22 ρ23 ρ12 ρ13 ρ11 ρ03 ρ02 ρ01

0,

1 3

1 2 , 3 3





2 ,1 3



1,

5 2



5 ,3 2



(3, ∞)

1 1 3 2 3 2

1 7 6

1

3 5 2

1 1

3 3

B. Multiplicative Gap To assist the analysis, we first relax the lower bound τl in (34) as follows: ( N 1 (1 − µR ), M ∈ (0, 3] τl (µR , µT ) ≥ τˆl (µR , µT ) = 3N . 1 N (1 − µR ), M ∈ (3, ∞) 9M

(35)

The relaxed lower bound can be rewritten as the convex combination at all integer points:           2 1 1 1 2 1 + γ02 τˆl 0, + γ03 τˆl 0, 1 + γ11 τˆl , + γ12 τˆl , τˆl (µR , µT ) =γ01 τˆl 0, 3 3 3 3 3 3         2 2 2 2 1 1 + γ22 τˆl , + γ23 τˆl , 1 + γ13 τˆl , 1 + γ21 τˆl , 3 3 3 3 3 3         2 1 + γ32 τˆl 1, + γ33 τˆl 1, 1 + γ30 τˆl 1, 0 + γ31 τˆl 1, 3 3 | {z } =0

where the combination coefficients satisfy

P

(m,n)∈A

Therefore, we have

γmn = 1 and

P

γmn µomn = [µR , µT ]T .

(m,n)∈A

o o o min {β01 τ01 + β02 τ02 + · · · + β23 τ23 } τu τu {βmn } ρ, ≤ = τl τˆl γ01 τˆl (0, 1/3) + γ02 τˆl (0, 2/3) + · · · + γ23 τˆl (2/3, 1) o o o γ01 τ01 + γ02 τ02 + · · · + γ23 τ23 ≤ γ01 τˆl (0, 1/3) + γ02 τˆl (0, 2/3) + · · · + γ23 τˆl (2/3, 1) o n τo o o τ02 τ23 01 , ,··· , ≤ max τˆl (0, 1/3) τˆl (0, 2/3) τˆl (2/3, 1) o n x1 +x2 +···+xn x1 x2 xn where (36c) is due to the inequality y1 +y2 +···+yn ≤ max y1 , y2 , · · · , yn .

(36a) (36b) (36c)

o

τmn . The upper bounds of {ρmn } can be obtained using simple mathematical deducDefine ρmn , τˆl (m/3,n/3) tion and are summarized in Table I. Thus, for any antenna configuration, we have ρ ≤ max{ρ01 , . . . , ρ23 } ≤ 3, which complete the proof of Corollary 3.

VII. CONCLUSION In this paper, we study the storage-latency tradeoff in the 3×3 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 some linear transmission schemes and obtain the DoF results of these channels. We obtain the achievable upper bound of minimum FDT by solving a linear programming problem. The achievable FDT decreases

14

piecewise linearly with the normalized cache sizes and each additive item is inversely proportional to the number of antennas. This finding means that the MIMO gain and cache gain are cumulative in the considered wireless network. We also give a lower bound of minimum FDT. It is shown that the achievable FDT is optimal at certain cases and is at most 3 times of the lower bound at other cases. To characterize the FDT in the network with more than 3 transmitters and receives is challenging. The difficulty comes from analysing the DoF of those hybrid MIMO channels formed by the delivery of different subfiles.

Case 1:

Case 2:

N M

N M

A PPENDIX A:  ∈ 0, 31 ∈

1 4 , 3 9



τu (µR , µT ) = Case 3:

N M

∈

4 2 , 9 3

(

THE CLOSED FORM EXPRESSION OF τu IN

1 3N 1 3N

τu (µR , µT ) =

T HEOREM 1

1 (1 − µR ) 3N

(1 − µR ) (−1 + µR + 3µT ) +

1 M

(37)

(µR , µT ) ∈ R1 (2 − 2µR − 3µT ) (µR , µT ) ∈ R2



 1 (1 − µR )  3N   1   9N (5 − 5µR − 3µT ) 1 1 τu (µR , µT ) = (−3 + 7µR + 9µT ) (2 − 3µR − 3µT ) + 9N N M  3 min{ 2−1/ξ , 2+1/ξ }    1 7  (1 − µR − 2µT ) + 9N (−1 + µR + 3µT ) min{ N , M }

(µR , µT ) ∈ R1 (µR , µT ) ∈ R2 (µR , µT ) ∈ R3

(38)

(39)

(µR , µT ) ∈ R4

2−1/ξ 2+1/ξ

Case 4:

N M

∈

2 20 , 3 27



 1  (1 − µR )  3N   1 1  (3 − 3µR − 3µT ) + 2M (−2 + 2µR + 3µT )    3N 3 7 τu (µR , µT ) = 9N (2 − 2µR − 3µT ) + 2M (−1 + µR + 2µT ) 1 7 1   (2 − 3µR − 3µT ) + 9N µR + 2M (−1 + 3µT ) N M  3 min{ 2−1/ξ , 2+1/ξ }    1   (1 − µ − 2µ ) + 7 (−1 + µ + 3µ ) N M min{ 2−1/ξ , 2+1/ξ }

Case 5:

N M

∈

20 ,1 27

R

T

9N

R

T



 1  (1 − µR )  3N   1  (4 − 4µR − µT )    9N 1 1 7 τu (µR , µT ) = 9N µR + 2M (3 − 6µR − 3µT ) + 3N (−2 + 3µR + 3µT ) 1 1 7   µR + 2M (−1 + 3µT ) (2 − 3µR − 3µT ) + 9N N M  3 min{ 2−1/ξ , 2+1/ξ }    1   (1 − µ − 2µ ) + 7 (−1 + µ + 3µ ) N M min{ 2−1/ξ , 2+1/ξ }

R

T

9N

R

T

(µR , µT ) ∈ R1 (µR , µT ) ∈ R2 (µR , µT ) ∈ R3 (µR , µT ) ∈ R4

(40)

(µR , µT ) ∈ R5

(µR , µT ) ∈ R1 (µR , µT ) ∈ R2 (µR , µT ) ∈ R3 (µR , µT ) ∈ R4 (µR , µT ) ∈ R5

(41)

15

Case 6:

N M

τu (µR , µT ) =

 ∈ 1, 35  1  (1 − µR )  3N   1 1  µ + 6M (2 − 3µR )   6N R   2   3 max{ 6 M, 2 N } (1 − µR − µT ) + 7

N M

1 (−1 + 3µT ) 6M 1 (−1 + 3µT ) 6M

(−1 + 2µR + µT ) +

2 µ + 2 min{ M1 , N } (1 − 2µR − µT ) +  3 max{ 67 M, 32 N } R  2−1/ξ 2+1/ξ   2  (−1 + µ + 3µT ) + min{ M1 , N } (1 − µR − 2µT )  2 6 R  3 max{ 7 M, 3 N }  2−1/ξ 2+1/ξ   1 2  (−1 + µR + 2µT ) (1 − µ − µ ) + R T 3N 3 max{ 6 M, 2 N } 7

Case 7:

3

1 6N

∈

5 ,2 3

3



(µR , µT ) ∈ R1 (µR , µT ) ∈ R2 (µR , µT ) ∈ R3 (µR , µT ) ∈ R4 (µR , µT ) ∈ R5 (µR , µT ) ∈ R6 (42)

 1  (1 − µR ) (µR , µT ) ∈ R1  3N   1  (5 − 6µR ) (µR , µT ) ∈ R2   9N   1 µ + 1 (1 − 3µ ) (µR , µT ) ∈ R3 R R 3M τu (µR , µT ) = N1 1  µ + 12M (7 − 18µR − 3µT ) (µR , µT ) ∈ R4  N R   3 1  (−1 + µR + 3µT ) + 2M (1 − µR − 2µT ) (µR , µT ) ∈ R5   N   1 (2 − 2µ − µ ) (µR , µT ) ∈ R6 R T 3N  N ∈ 2, 25 Case 8: M  1  (1 − µR ) (µR , µT ) ∈ R1  3N   1  (5 − 6µR ) (µR , µT ) ∈ R2  9N   1 1   (µR , µT ) ∈ R3  N µR + 3M (1 − 3µR ) 1 1 τu (µR , µT ) = 2N (5 − 8µR − 5µT ) + 3M (−2 + 3µR + 3µT ) (µR , µT ) ∈ R4   1  (7 − 12µR − 3µT ) (µR , µT ) ∈ R5  6N   1   (2 − 2µR − 3µT ) (µR , µT ) ∈ R6    N1 (2 − 2µR − µT ) (µR , µT ) ∈ R7 3N  N ∈ 52 , 3 Case 9: M  1  (1 − µR ) (µR , µT ) ∈ R1  3N   1  (5 − 6µR ) (µR , µT ) ∈ R2   9N   1 µ + 1 (1 − 3µ ) (µR , µT ) ∈ R3 R R 3M τu (µR , µT ) = N1 1  (2 − 2µR − 3µT ) + 3M (−1 + 3µT ) (µR , µT ) ∈ R4  N   1  (2 − 2µR − 3µT ) (µR , µT ) ∈ R5   N   1 (2 − 2µ − µ ) (µR , µT ) ∈ R6 R T 3N

Case 10:

N M

(43)

(44)

(45)

∈ (3, ∞]

 1 (1 − µR )  9M   1    27M (5 − 6µR ) 1 τu (µR , µT ) = 3M (1 − 2µR )   1  (2 − 2µR − 3µT )    3M 1 (2 − 2µR − µT ) 9M

(µR , µT ) ∈ R1 (µR , µT ) ∈ R2 (µR , µT ) ∈ R3 (µR , µT ) ∈ R4 (µR , µT ) ∈ R5

The cache size regions {Ri } of each case are illustrated in Fig. 5.

(46)

16

ߤܶ 1

࣬͵

࣬ʹ

1/3

0

1/3

1 ߤܴ

2/3

Case 1

࣬Ͷ

࣬ʹ

࣬͵

࣬ͷ

࣬ͳ ࣬͸

Case 6

Fig. 5.

࣬ͳ

࣬ͳ

2/3

࣬Ͷ

࣬Ͷ

࣬ʹ ࣬ͷ

࣬ʹ

࣬ʹ

࣬ͳ ࣬͸

࣬Ͷ

࣬͵

࣬ͷ

Case 7

࣬ʹ ࣬͸

࣬Ͷ

࣬ͷ

Case 4

࣬ͳ ࣬͹

࣬͵

࣬͵

Case 3

Case 2

࣬͵

࣬Ͷ

࣬ͳ

࣬͵

࣬Ͷ

Case 8

࣬ʹ ࣬ͷ

࣬ͳ ࣬ʹ ࣬ͷ

Case 5

࣬ͳ ࣬͸

Case 9

࣬͵

࣬ʹ ࣬Ͷ

࣬ͳ

࣬͸

Case 10

Cache regions of the different number of antennas.

A PPENDIX B: PROOF OF L EMMA 1 Throughout this Appendix and Appendices C and D, we adopt the DoF plane introduced in [32] to present the DoF results. The DoF per user of the 3 × 3 partially cooperative MIMO X channel shown in (18) of Lemma 1 is illustrated in Fig. 6(a). To prove its achievability, it suffices to prove the achievability of points {Q1 , Q2 } in the DoF plane, by Lemma 2 of [32]. The achievable scheme of the 3 × 3 partially cooperative MIMO X channel needs three time slots as shown in Fig. 4. In each time slot, the transmission scheme is similar and we take the time slot I for an example. Let the d × 1 vectors sr1 t12 , sr2 t13 , sr3 t23 denote the actual transmitted signal vectors of messages A12 , B13 , C23 , intended for receivers 1, 2, and 3, respectively. Here, d is the desired DoF per users. Due to the symmetry of the three receivers, we take receiver 1 for example. Its received signal (ignoring noise for brevity) can be written as y1 =H11 (Vr1 t12 1 sr1 t12 + Vr2 t13 1 sr2 t13 ) + H12 (Vr1 t12 2 sr1 t12 + Vr3 t23 2 sr3 t23 ) + H13 (Vr2 t13 3 sr2 t13 + Vr3 t23 3 sr3 t23 ),

(47)

where Vrk tpq i is the M × d precoding matrix of signal srk tpq at transmitter i ∈ {p, q}. Next, we give the detailed design method of transmitter precoding matrices and, if necessarily, receiver combining matrices. (1) the achievability of Q1 : For receiver 1, sr2 t13 and sr3 t23 are the interference signals. We can design the M × 2M precoding matrices Vr2 t13 1 , Vr2 t13 3 , Vr3 t23 2 and Vr3 t23 3 to satisfy: 3 H11 Vr2 t13 1 = −H13 Vr2 t13 3 H12 Vr3 t23 2 = −H13 Vr3 t23 3

(48)

In this way, the interferences from sr2 t13 and sr3 t23 will be cancelled at receiver 1, which is known as interference neutralization. Similarly, the interferences at receiver 2 and 3 can be neutralized by the following design: H21 Vr1 t12 1 = −H22 Vr1 t12 2 , H31 Vr2 t13 1 = −H33 Vr2 t13 3 ,

H22 Vr3 t23 2 = −H23 Vr3 t23 3 , H31 Vr1 t12 1 = −H32 Vr1 t12 2 .

(49)

data streams. In this way, all the interferences are cancelled and each receiver can decode 2M 3 (2) the achievability of Q2 : Under this antenna configuration, we need to jointly design the transmit precoding matrices and the receive combining matrices for interference neutralization. In specific, we first

17

݀ ‫ܯ‬

ܳʹ

1 ʹ ͵

݀ ‫ܯ‬

ܳͳ ͸ ͹

0

ܳʹ

2

ʹ ͵

(a)

ͷ ͵

ͷ ʹ

ܰ ‫ܯ‬

݀ ‫ܯ‬

ܳͳ

0

1

ͻ ͹

3 (b)

ܰ ‫ܯ‬

ܳʹ

2

1

ܳͳ

0

1

͵ ʹ

3 (c)

ܰ ‫ܯ‬

Fig. 6. The DoF planes: (a) the 3 × 3 partially cooperative MIMO X channel, (b) the 3 × 3 MIMO X-multicast channel, (c) the 3 × 3 partially cooperative MIMO X-multicast channel.

design the M × 25 M combining matrices, denoted as Pj , for each receiver j to compress the received signal space. Taking P1 for example, we design P1 as follows: T  M  1 T +1 p1 p12    p21    T  p M2 +2    .   ⊆ null H12 H13 T 1 (50) ,  .  ⊆ null H11 H13   ..   .   . M p12 pM 1

where pm 1 denotes the m-th row of P1 . Then, we design the M × M transmit precoding matrices to meet the same conditions in (48) and (49) with each channel matrix Hji replaced by the effective channel matrix Pj Hji after compression. By doing so, all the interferences are neutralized and M data streams can be decoded for each receiver. A PPENDIX C: PROOF OF L EMMA 2 The DoF per user of the 3 × 3 MIMO X-multicast channel, shown in (19) of Lemma 2, is illustrated in Fig. 6(b). (1) the achievability of Q1 : Let the M × M7 vector srjk tp be the transmitted signal vector for Wr⊕jk tp , intended to receiver multicast group {j, k} from transmitter p. Let Vrjk tp denote the M × M7 precoding matrix of srjk tp at transmitter p. At receiver 1, the received signal can be expressed as (ignoring the noise for brevity) y1 = H11 (Vr12 t1 sr12 t1 + Vr23 t1 sr23 t1 + Vr13 t1 sr13 t1 ) +H12 (Vr12 t2 sr12 t2 + Vr23 t2 sr23 t2 + Vr13 t2 sr13 t2 ) +H13 (Vr12 t3 sr12 t3 + Vr23 t3 sr23 t3 + Vr13 t3 sr13 t3 ).

(51)

18

•‫ ͳ ݐ ͵ͳݎ• ͳݐ ͵ʹݎ• ͳݐ ʹͳݎ‬

•‫ͳݐ ʹͳݎ‬

•‫ͳݐ ͵ʹݎ‬

•‫ͳݐ ͵ͳݎ‬

Tx 2

•‫ ʹ ݐ ͵ͳݎ• ʹݐ ͵ʹݎ• ʹݐ ʹͳݎ‬

•‫ʹݐ ͵ʹݎ‬

•‫ʹݐ ͵ͳݎ‬

•‫ʹ ݐ ʹͳݎ‬

Tx 3

•‫ ͵ ݐ ͵ͳݎ• ͵ݐ ͵ʹݎ• ͵ݐ ʹͳݎ‬

•‫͵ݐ ͵ͳݎ‬

•‫͵ݐ ʹͳݎ‬

•‫͵ݐ ͵ʹݎ‬

7LPHVORW,

7LPHVORW,,

7LPHVORW,,,

Tx 1

&DFKHGFRQWHQW

Fig. 7.

The alternating transmission scheme in the 3 × 3 MIMO X-multicast channel.

Receiver 1 desires signals sr12 t1 , sr13 t1 , sr12 t2 , sr13 t2 , sr12 t3 , and sr13 t3 , and it wants to align the interference signals sr23 t1 , sr23 t2 , and sr23 t3 along a same direction so as to cancel them all at once: H11 Vr23 t1 = H12 Vr23 t2 = H13 Vr23 t3 , V1 .

(52)

At receiver 2 and 3, the similar equations can be obtained: H21 Vr13 t1 = H22 Vr13 t2 = H23 Vr13 t3 , V2 ,

(53)

H31 Vr12 t1 = H32 Vr12 t2 = H33 Vr12 t3 , V3 .

(54)

We need to further design V1 , V2 and V3 to ensure the decodability of desired signals at each receiver. We give an achievable method as below: V1 = V2 = V3 = diag {17×1 , 17×1 , · · · , 17×1 }, {z } |

(55)

y ˆ1 = P1 (H11 sr12 t1 + H12 sr23 t2 + H13 sr13 t3 ).

(56)

M× M 7

where 17×1 denotes the 7 × 1 vector with all elements being one. In this way, all desired signals are linearly independent of each other. So the 6M DoF per user can be obtained. 7 (2) the achievability of Q2 : In this case, we use the alternating transmission scheme as shown in Fig. 7. We take time slot I as an example. For receiver 1, the post-processed received signals (ignoring the noise for brevity) after the 2M × 3M combining matrix P1 can be expressed as

We can design the zero-forcing receive matrix P1 to cancel the interference signal sr23 t2 as   PT1 ⊆ null HT12 .

(57)

The other receive matrices can be designed in the similar way. Thus, each receiver can obtain its desired signals without interference and hence achieves 2M DoF per user.

A PPENDIX D: PROOF OF L EMMA 3 The DoF per user of the 3 × 3 partially cooperative MIMO X-multicast channel, shown in (20) of Lemma 3, is illustrated in Fig. 6(c). (1) the achievability of Q1 : Let the M × 1 vector srjk tpq be the transmitted signal vector for Wr⊕jk tpq , intended to receiver multicast group {j, k} from transmitter cooperation group {p, q}. Let the M × M

19

matrix Vrjk tpq i be the precoding matrix of srjk tpq at transmitter i ∈ {p, q}. The received signal at receiver 1 is y1 = H11 (Vr12 t12 1 sr12 t12 + Vr23 t12 1 sr23 t12 + Vr13 t12 1 sr13 t12 +Vr12 t13 1 sr12 t13 + Vr23 t13 1 sr23 t13 + Vr13 t13 1 sr13 t13 ) +H12 (Vr12 t12 2 sr12 t12 + Vr23 t12 2 sr23 t12 + Vr13 t12 2 sr13 t12 +Vr12 t23 2 sr12 t23 + Vr23 t23 2 sr23 t23 + Vr13 t13 2 sr13 t23 ) +H13 (Vr12 t23 3 sr12 t23 + Vr23 t23 3 sr23 t23 + Vr13 t23 3 sr13 t23 +Vr12 t13 3 sr12 t13 + Vr23 t13 3 sr23 t13 + Vr13 t13 3 sr13 t13 ) where sr12 t12 , sr13 t12 , sr12 t23 , sr13 t23 , sr12 t13 and sr13 t13 are desired by receiver 1; sr23 t12 , sr23 t23 and sr23 t13 are the interference signals. The precoding matrices can be designed to satisfy : H11 Vr23 t12 1 = −H12 Vr23 t12 2 = [E, 0, 0, 0, 0, 0]T H11 Vr23 t13 1 = −H13 Vr23 t13 3 = [0, E, 0, 0, 0, 0]T

(58)

H12 Vr23 t23 2 = −H13 Vr23 t23 3 = [0, 0, E, 0, 0, 0]T where E and 0 denote the identity matrix and null matrix, respectively. The equations for receiver 2 and 3 can be obtained similarly. In this way, all the interferences are neutralized and the desired signals can be successfully decoded at each receiver by using a zero-forcing matrix. Thus, M DoF per user can be obtained. (2) the achievability of Q2 : In this case, we use the alternating schemes as shown in Fig. 8. In each time slot, each signal is sent simultaneously from two transmitters. If we let different transmitters send different signals, we can use the same transmission scheme as the one in the 3 × 3 MIMO X-multicast channel at point Q2 in the proof of Lemma 2. So the DoF per user is equal to 2M. A PPENDIX E: PROOF OF L EMMA 4 According to antenna configuration, we divide the achievable scheme into two cases: N (1) M ∈ (0, 32 ]: We define k − = min{M, N} and let each node only use k − antennas by antenna deactivation. The delivery in this case can be viewed as a two-channel use of MIMO broadcast channel where the virtual transmitter is equipped with 3k − antennas and each receiver is equipped with k − antennas. The virtual transmitter first delivers Wr⊕12 t123 , Wr⊕23 t123 and Wr⊕13 t123 to receiver 1, 2 and 3 respectively at the first time slot, and then delivers Wr⊕13 t123 , Wr⊕12 t123 and Wr⊕23 t123 to receiver 1, 2 and 3 respectively at the second time slot. Thus, the DoF per user of this channel is k − . N (2) M ∈ ( 23 , ∞): We let each transmitter and each receiver use η and 3η antennas, respectively, where η = min{ N3 , M}. All the transmitters cooperatively send η data streams for every subfiles, Wr⊕12 t123 , Wr⊕23 t123 and Wr⊕13 t123 . There are 3η data streams sent from the transmitter side. Each receiver can decode the desired 2η data streams from the total 3η data streams, by using 3η antennas. In this way, each receiver can obtain their desired message without interference. 2η DoF per user is thus achieved.

20

Tx 1

Tx 2

Tx 3

Fig. 8.

•‫ ʹͳݐ ͵ͳݎ• ʹͳ ݐ ͵ʹݎ• ʹͳݐ ʹͳݎ‬ •‫ ͵ͳݐ ͵ͳݎ• ͵ͳ ݐ ͵ʹݎ• ͵ͳݐ ʹͳݎ‬

•‫ ʹͳݐ ʹͳݎ‬ •‫ ͵ͳݐ ͵ʹݎ‬

•‫ ʹͳݐ ͵ʹݎ‬ •‫͵ͳݐ ͵ͳݎ‬

•‫ ͵ͳݐ ͵ʹݎ‬ •‫ ͵ʹݐ ͵ͳݎ‬

•‫ ͵ͳݐ ͵ͳݎ‬ •‫ ͵ʹݐ ʹͳݎ‬

•‫ ʹͳݐ ͵ͳݎ• ʹͳ ݐ ͵ʹݎ• ʹͳݐ ʹͳݎ‬ •‫ ͵ʹݐ ͵ͳݎ• ͵ʹ ݐ ͵ʹݎ• ͵ʹݐ ʹͳݎ‬

•‫ ʹͳݐ ʹͳݎ‬ •‫ ͵ʹݐ ͵ͳݎ‬

&DFKHGFRQWHQW

7LPHVORW,

•‫ ͵ʹݐ ͵ͳݎ• ͵ʹ ݐ ͵ʹݎ• ͵ʹݐ ʹͳݎ‬ •‫ ͵ͳݐ ͵ͳݎ• ͵ͳ ݐ ͵ʹݎ• ͵ͳݐ ʹͳݎ‬

•‫ ʹͳݐ ͵ͳݎ‬ •‫ ͵ͳݐ ʹͳݎ‬

•‫ ʹͳݐ ͵ʹݎ‬ •‫ ͵ʹݐ ʹͳݎ‬

•‫ ʹͳݐ ͵ͳݎ‬ •‫ ͵ʹݐ ͵ʹݎ‬

7LPHVORW,,

7LPHVORW,,,

•‫ ͵ͳݐ ʹͳݎ‬ •‫ ͵ʹݐ ͵ʹݎ‬

The alternating transmission scheme in the 3 × 3 partially cooperative MIMO X-multicast channel.

A PPENDIX F: PROOF OF L EMMA 5 This is an extension of Lemma 1 in [27] to MIMO case. The differential of the received signals from any l antennas can be upper bounded by X    X T 3M l X T h hιm xm (t) + nι (t) (59a) h y[1:l] ≤ ≤ = ≤

ι=1 t=1 T l X X

ι=1 t=1 l X T X

ι=1 t=1 T l X X ι=1 t=1

≤

T l X X ι=1 t=1

m=1

  X 3M hιm xm (t) + nι (t) log 2πeVar 

m=1 X 3M

 log 2πe Var

m=1



hιm xm (t) + 1

(59b)



(59c)

  X 3M q       X 2 (59d) hιm hιn Var xm (t) Var xn (t) + 1 log 2πe hιm Var xm (t) + m=1

m6=n

      X 3M   X Var xm (t) + Var xn (t) 2 +1 log 2πe hιm Var xm (t) + hιm hιn 2 m=1 m6=n

   l X T X 3(M + 1) ≤ log 2πe c˜P +1 2 ι=1 t=1    ≤ lT log 2πe cP + 1

(59e) (59f) (59g)

where (59c) comes from the fact that the noise is uncorrelated with transited signals and is i.i.d; (59d) comes from Cauchy-Schwartz Inequality; c˜ is defined as c˜ = max h2ιm ; c is defined as 3˜c(M2 +1) . m

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