Fundamental Tradeoff between Storage and Latency in Cache ... - arXiv

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Fundamental Tradeoff between Storage and Latency in Cache-Aided Wireless Interference

arXiv:1605.00203v1 [cs.IT] 1 May 2016

Networks Fan Xu, Meixia Tao, Senior Member, IEEE, Kangqi Liu, Student Member, IEEE

Abstract This paper studies the fundamental tradeoff between storage and latency in a general wireless interference network with caches equipped at all transmitters and receivers. The tradeoff is characterized by an information-theoretic metric, fractional delivery time (FDT), which is defined as the delivery time of the actual load normalized by the total requested bits at a transmission rate specified by degrees of freedom (DoF) of the given channel. We obtain achievable upper bounds of the minimum FDT for the considered network with NT (≥ 2) transmitters and NR = 2 or NR = 3 receivers. We also obtain a theoretical lower bound of the minimum FDT for any NT transmitters and NR receivers. Both upper and lower bounds are convex and piece-wise linear decreasing functions of transmitter and receiver cache sizes. The achievable bound is partially optimal and is at most

4 3

and

3 2

times of the

lower bound in the considered NT × 2 and NT × 3 network, respectively. To show the achievability, we propose a novel cooperative transmitter/receiver coded caching strategy. It offers the freedom to adjust file splitting ratios for FDT minimization. Based on this caching strategy, we then design the delivery phase carefully to turn the considered network opportunistically into more favorable channels, including X channel, broadcast channel, multicast channel, or a hybrid form of these channels. Receiver local caching gain, coded multicasting gain, and transmitter cooperation gain (interference alignment and interference neutralization) are thus leveraged in different cache size regions and different channels. Index Terms Wireless cache network, coded caching, content delivery, multicast, and interference management.

This paper will be presented in part at IEEE ISIT 2016. Fan Xu, Meixia Tao, and Kangqi Liu are with the Department of Electronic Engineering, Shanghai Jiao Tong University, Shanghai, China (Emails: [email protected], [email protected], [email protected])

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I. I NTRODUCTION Over the last decades, mobile data traffic has been shifting from connection-centric services, such as voice, e-mails, and web browsing, to emerging content-centric services, such as video streaming, push media, application download/updates, and mobile TV [1]–[3]. These contents are typically produced well ahead of transmission and can be requested by multiple users at different times. This allows us to cache the contents at the edge of networks, e.g, base stations and user devices, during periods of low network load. The local availability of contents at the network edge has significant potential of reducing user access latency and alleviating wireless traffic. Recently, there have been increasing interests from both academia and industry in characterizing the impact of caching on wireless networks [4]–[7]. Caching in a shared link with one server and multiple cache-enabled users is first studied by Maddah-Ali and Niesen in [8]. It is shown that caching at user ends brings not only local caching gain but also global caching gain. The latter is achieved by a carefully designed cache placement and coded delivery strategy, which can create multicast chances for content delivery even if users demand different files. The idea of coded caching in [8] is then extended to the decentralized coded caching in a large network in [9], which achieves a rate close to the optimal centralized scheme. Taking file popularity into consideration, the authors in [10]–[12] showed that the peak traffic load is close to optimal within a constant gap by partitioning users into two groups and then using the decentralized scheme in [9] for content delivery. In [13], the authors considered the wireless broadcast channel with imperfect channel state information at the transmitter (CSIT) and showed that the gain of coded multicasting scheme can offset the loss due to the imperfect CSIT. In specific, if the user cache size is large enough, the content delivery latency approaches to the case with perfect CSIT and no cache. Besides the shared link or broadcast channels mentioned above, coded multicasting is also investigated in other network topologies, such as hierarchical cache networks [14], device-to-device cache networks [15], and multi-server networks [16]. Caching at transmitters is studied in [17]–[19] to exploit the opportunities for transmitter cooperation and interference management. In specific, the authors in [17] exploited the MIMO cooperation gain via joint beamforming by caching the same erasure-coded packets at all edge nodes in a backhaul-limited multi-cell network. The authors in [18] studied the degrees of

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freedom (DoF) and clustered cooperative beamforming in cellular networks with edge caching via a hypergraph coloring problem. The authors in [19] studied the transmitter cache strategy in the cache-aided interference channel under an information-theoretical framework. It is shown that splitting contents into different parts and caching each part in different transmitters can turn the interference channel into broadcast channel, X channel, or hybrid channel and hence increase the system throughput via interference management. The authors in [20] presented a lower bound of delivery latency in a general interference network with transmitter cache and showed that the scheme in [19] is optimal in certain region of cache size. The above literature reveals that caching at the receiver side can bring local caching gain and coded multicasting gain, and that caching at the transmitter side can induce transmitter cooperation for interference management and load balancing. It is thus of theoretical importance and practical interest to investigate the impact of caching at both transmitter and receiver sides. In this paper, we aim to study the fundamental limits of caching in a general wireless interference network with caches equipped at all transmitters and receivers as shown in Fig. 1. The performance metric to characterize the gains of caching varies in the existing works. For the broadcast channel with receiver cache, the authors in [8] characterized the gain by memory-rate tradeoff, where the rate is defined as the load of the shared link normalized by the file size in the delivery phase. For the interference channel with transmitter cache, the authors in [19] characterized the gain by the standard DoF from the information-theoretic studies in the delivery phase. In [20], the authors proposed the storage-latency tradeoff, where the latency is defined as the relative delivery time with respect to an ideal baseline system with unlimited cache and no interference in the high signal-to-noise ratio (SNR) region. In our considered wireless interference network with both transmitter and receiver caches, the standard DoF is unable to capture the potential reduction in the traffic load due to receiver caching, and the rate is unable to capture the potential DoF enhancement due to cache-induced transmitter cooperation. Interestingly, the latency-oriented performance metric introduced in [20] can reflect not only the load reduction due to receiver cache but also the DoF enhancement due to transmitter cache, since it evaluates the delivery time of the actual load at a transmission rate specified by the given DoF. As such, we adopt the storage-latency tradeoff to characterize the fundamental limits of caching in this work. In specific, we measure the performance by the fractional delivery time (FDT), denoted as τ (µR , µT ), which is a function of the normalized receiver cache size µR and the normalized

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transmitter cache size µT . Our preliminary results on the latency-storage tradeoff study in the special case with NT = 3 transmitters and NR = 3 receivers are presented in [21]. Note that an independent work on the similar problem with both transmitter and receiver caches is studied in [22]. We shall discuss the differences with [22] at appropriate places throughout the paper presentation. The main contributions and results of this work are as follows: • A novel file splitting and caching strategy: We propose a novel file splitting and caching strategy for any transmitter and receiver numbers and at any feasible normalized cache sizes. This strategy is more general than the existing file splitting and caching strategy in [8], [19], [22]. It offers the freedom to adjust the file splitting ratios for caching gain optimization. • Achievable storage-latency tradeoff for NT ×2 and NT ×3 networks: Based on the proposed file splitting and caching strategy, we obtain an upper bound of the minimum FDT for the interference networks with NT transmitters and NR = 2 or 3 receivers. The main idea is to design the delivery phase carefully so that the network topology can be opportunistically changed to more favorable ones, including broadcast channel, multicast channel, X channel, or a hybrid form of these channels. Interference neutralization, interference alignment, and real interference alignment are used in different channels to increase the system DoF. The obtained FDT is a convex and piece-wise linear decreasing function of the transmitter and receiver cache sizes. Our analysis shows that the transmitter cooperation gain, local caching gain, and coded multicasting gain can be leveraged in different cache size regions. Our analysis also shows that the optimal file spitting ratios are not unique. • Lower bound of storage-latency tradeoff : We also obtain a lower bound of the minimum FDT for the general NT × NR interference network by using cut-set bound and genie-message approach. With this lower bound, we show that the achievable FDT upper bound in the NT × 2 and NT × 3 networks is optimal at certain regions. The ratio of the upper bound over the lower bound is at most

4 3

and 32 , respectively, for NR = 2 and NR = 3, and it approaches to one when

NT → ∞. The remainder of this paper is organized as follows. Section II introduces our system model and performance metric. Section III presents the main results of this paper. Section IV describes the cache placement strategy. Section V and VI illustrate the content delivery strategy for the NT × 2 and NT × 3 interference network, respectively. Section VII presents some discussions of

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Tx 1

Rx 1 Cache 9 1

Cache 8 1

Tx 2 Cache 8 2

Г Г Г

Rx 2 Г Г Г

Tx NT

Cache 9 2

Rx NR Cache 915

Cache 817

Fig. 1: Cache-aided wireless interference network with NT transmitters and NR receivers.

our results. Section VIII proves the lower bound of FDT, and Section IX concludes this paper. Notations: (·)T denotes the transpose. [K] denotes set {1, 2, · · · , K}. ⌊x⌋ denotes the largest T + integer no greater than x. (xj )K j=1 denotes vector (x1 , x2 , · · · , xK ) . (x) denotes the maximum

of x and 0, (x)+ = max{0, x}. A1∼S denotes set {A1 , A2 , · · · , AS }. CN (m, σ 2 ) denotes the complex Gaussian distribution with mean of m and variance of σ 2 . II. S YSTEM D ESCRIPTION A ND P ERFORMANCE M ETRIC A. System Description Consider a general cache-aided wireless interference network with NT (≥ 2) transmitters and NR (≥ 2) receivers as illustrated in Fig. 1, where each node is equipped with a cache memory of finite size. Each node is assumed to have single antenna. The communication link between each transmitter and each receiver experiences channel fading and is corrupted with additive white Gaussian noise. The communication at each time slot t over this network is modeled by Yj (t) =

NT X

hjp (t)Xp (t) + Zj (t), j = 1, 2, · · · , NR ,

p=1

where Yj (t) ∈ C denotes the received signal at receiver j, Xp (t) ∈ C denotes the transmitted signal at transmitter p, hjp (t) ∈ C denotes the channel coefficient from transmitter p to receiver j, and Zj (t) denotes the noise at receiver j distributed as CN (0, 1). The channel is assumed to be quasi-static fading so that it remains constant during a transmission frame that contains many time slots but varies independently from one frame to another.

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Consider a database consisting of L files (L >> NR ), denoted by {W1 , W2 , · · · , WL }. Each file is chosen independently and uniformly from [2F ] = {1, 2, · · · , 2F } randomly, where F is the file size in bits. Each transmitter has a local cache able to store MT F bits and each receiver has a local cache able to store MR F bits. The normalized cache sizes at each transmitter and receiver are defined, respectively, as MT MR , µR , . L L The network operates in two phases, cache placement phase and content delivery phase. µT ,

During the cache placement phase, each transmitter p designs a caching function φp : [2F ]L → [2⌊F MT ⌋ ], mapping the L files in the database to its local cached content Up , φp (W1 , W2 , · · · , WL ). Each receiver j also designs a caching function ψj : [2F ]L → [2⌊F MR ⌋ ], mapping the L files to its local cached content Vj , ψj (W1 , W2 , · · · , WL ). The caching functions {φp , ψj } are assumed to be known globally at all nodes. In the delivery phase, each receiver j NR R requests a file Wdj from the database. We denote d , (dj )N as the demand vector. j=1 ∈ [L]

Each transmitter p has an encoding function Λp : [2⌊F MT ⌋ ] × [L]NR × CNT ×NR → CT . Transmitter p uses Λp to map its cached content Up , receiver demands d and channel realization H to the signal (Xp [t])Tt=1 , Λp (Up , d, H), where T is the block length of the code. Note that T may depend on the receiver demand d and channel realization H, and thus can also be denoted as T d,H (With a slight abuse of notation, we will use T again to denote the average worstcase delivery time in Definition 1). Each codeword (Xp [t])Tt=1 has an average transmit power constraint P . Each receiver j has a decoding function Γj : [2⌊F MR ⌋ ] × CT × CNT ×NR × [L]NR → [2F ]. We denote (Yj [t])Tt=1 as the signal received at receiver j. Upon receiving (Yj [t])Tt=1 , each receiver ˆ j , Γj (Vj , (Yj [t])T , H, d) of its desired file Wd using its cached content j uses Γj to decode W t=1

j

Vj and the channel realization H as side information. The worst-case error probability is ˆ j 6= Wd ). Pǫ = max max P(W j d∈[L]NR j∈[NR ]

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The given caching and coding scheme {φp , ψj , Λp , Γj } is said to be feasible if the worst-case error probability Pǫ → 0 when F → ∞. Note that the cache placement phase and the content delivery phase take place on different timescales. In general, cache placement is in much large timescale (e.g. on a daily or hourly basis) while content delivery is in a much shorter timescale. As such, the caching functions designed in the cache placement phase are unaware of the future content requests, but the coding functions during the content delivery phase are dependent on the caching functions. B. Performance Metric In this work, we adopt the following latency-oriented performance metrics originally proposed in [20]. Definition 1: The delivery time (DT) for a given feasible caching and coding scheme is defined as T , lim lim max EH (T d,H ). P →∞ F →∞

(1)

d

Definition 2: The fractional delivery time (FDT) for a given feasible caching and coding scheme is defined as max EH (T d,H ) τ (µR , µT ) , lim lim sup P →∞ F →∞

d

NR F · 1/ log P

.

(2)

Moreover, the minimum FDT at given normalized cache sizes µT and µR is defined as τ ∗ (µR , µT ) = inf{τ (µR , µT ) : τ (µR , µT ) is achievable}.

(3)

The above performance metrics are defined in the asymptomatic sense when P → ∞ and F → ∞. It is clear that the FDT and DT are related by τ =

T log P NR F

. The FDT τ (µR , µT ) can

be regarded as the relative time with respect to delivering the total NR F requested bits in an interference-free baseline system with transmission rate log P in the high SNR region. Remark 1: Our definition of FDT τ is slightly different from the normalized delivery time (NDT) δ in [20] in that our FDT is further normalized by NR , the number of receivers (content requesters). That is, τ = δ/NR . With such normalization, the FDT is defined for the total NR F bits requested in the network rather than the F bits requested by a single receiver as in [20]. As a result, the range of FDT is 0 ≤ τ ≤ 1, which is truly normalized.

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Remark 2: Compared to the “load” R defined for the shared link in [8], the FDT can be expressed as τ =

R , NR ·DoF

where DoF is the sum DoF of the considered channel. Comparing to

the standard DoF adopted for interference channel with transmitter cache only in [19], we have τ (µR = 0, µT ) = 1/DoF. As a result, the FDT evaluates the delivery time of the actual load at a transmission rate specified by DoF of the given channel, and hence is particularly suitable to characterize the performance of the wireless network with both transmitter and receiver caches. Remark 3 (Feasible domain of FDT): The FDT introduced above is able to measure the fundamental tradeoff between the cache storage and content delivery latency. However, not all normalized cache sizes are feasible. Given fixed L and MT , all the transmitters together can store at most NT MT F bits of files, which leaves LF − NT MT F bits of files to be stored in all receivers. Thus we must have MR F ≥ LF − NT MT F . This gives the feasible region for the normalized cache sizes as:   0 ≤ µ ,µ ≤ 1 R T .  µR + NT µT ≥ 1

(4)

Throughout this paper, we study the FDT only in the feasible domain (4). III. M AIN R ESULTS In this section, we present our main results on the fundamental storage-latency tradeoff for the cache-aided wireless interference network. The first two theorems below present achievable upper bounds of the minimum FDT for NR = 2 and 3 receivers, respectively. The last theorem presents a lower bound of the minimum FDT for any NT and any NR . Theorem 1 (Achievable FDT for the NT × 2 network): For the cache-aided interference network with NT ≥ 2 transmitters and NR = 2 receivers, the minimum FDT is upper bounded by τ ∗ (µR , µT ) ≤

  

1 − 21 µR , 2 NT +2 T +2 − N2N µR 2NT T

(µR , µT ) ∈ R1NT 2 − 12 µT , (µR , µT ) ∈ R2NT 2

,

where {RiNT 2 }2i=1 are given below and sketched in Fig. 2.   R1 = {(µ , µ ) : 2µ + N µ ≥ 2, 0 ≤ µ ≤ 1, µ ≤ 1} R T R T T R T NT 2 .  R2 = {(µR , µT ) : 2µR + NT µT < 2, µR ≥ 0, µR + NT µT ≥ 1} NT 2

(5)

9 1


T




  2/NT

 

1/NT 0 0

1/2

1


R

Fig. 2: Cache size regions in the NT × 2 network. Theorem 2 (Achievable FDT for the NT × 3 network): For the cache-aided interference network with NT ≥ 2 transmitters and NR = 3 receivers, the minimum FDT is upper bounded by 1) When NT = 2,

τ ∗ (µR , µT ) ≤

                  

1 3 1 2

− 31 µR ,

(µR , µT ) ∈ R123

− 21 µR − 91 µT ,

(µR , µT ) ∈ R223

11 18 5 6 7 6

− 65 µR − 91 µT , (µR , µT ) ∈ R323 ,

− 67 µR − 31 µT ,

(µR , µT ) ∈ R423

− 67 µR − µT ,

(µR , µT ) ∈ R523

(6)

where {Ri23 }5i=1 are given below and sketched in Fig. 3(a).   R123 = {(µR , µT ) : 3µR + 2µT ≥ 3, µR ≤ 1, µT ≤ 1}      2    R23 = {(µR , µT ) : 3µR + 2µT < 3, 3µR ≥ 1, 3µR + 4µT > 3}

. R323 = {(µR , µT ) : 3µR + 2µT ≥ 2, µT ≤ 1, 3µR < 1}     R423 = {(µR , µT ) : 3µR + 2µT < 2, µR ≥ 0, 2µT > 1}      5 R23 = {(µR , µT ) : 2µT ≤ 1, 3µR + 4µT ≤ 3, µR + 2µT ≥ 1}

2) When NT = 3,

τ ∗ (µR , µT ) ≤

                  

1 3

− 31 µR ,

(µR , µT ) ∈ R133

4 9

− 94 µR − 91 µT ,

(µR , µT ) ∈ R233

1 − 95 µR − 61 µT , 2 13 − 98 µR − 21 µT , 18

(µR , µT ) ∈ R333 ,

8 9

(µR , µT ) ∈ R533

− 98 µR − µT ,

(µR , µT ) ∈ R433

(7)

10

where {Ri }5i=1 are given below and sketched in Fig. 3(b).   R133 = {(µR , µT ) : µR + µT ≥ 1, µR ≤ 1, µT ≤ 1}      2    R33 = {(µR , µT ) : µR + µT < 1, 2µR + µT ≥ 1, µR + 2µT > 1} . R333 = {(µR , µT ) : 3µR + 3µT ≥ 2, 2µR + µT < 1, µR ≥ 0}     R433 = {(µR , µT ) : 3µR + 3µT < 2, µR ≥ 0, 3µT > 1}      5 R33 = {(µR , µT ) : 3µT ≤ 1, µR + 2µT ≤ 1, µR + 3µT ≥ 1} 3) When NT ≥ 4,  1  − 13 µR ,  3     1 1 1 1 1    3 + NT (NT −1) − ( 3 + NT (NT −1) )µR − 3(NT −1) µT , NT −3 1 T −5 T −5 τ ∗ (µR , µT ) ≤ + 3N2N − ( 13 + 3N2N )µR − 3(N µT , 3 T (NT −1) T (NT −1) T −1)    NT −2 1 T −2  + N1T + 3NTN(N − ( 31 + 3N5 T )µR − ( 31 + 3(N )µT ,   3 T −1) T −1)    1 + 3N5 T − ( 13 + 3N5 T )µR − µT , 3

(µR , µT ) ∈ R1NT 3 (µR , µT ) ∈ R2NT 3 (µR , µT ) ∈ R3NT 3 , (µR , µT ) ∈ R4NT 3 (µR , µT ) ∈ R5NT 3 (8)

where {Ri }5i=1 are given below and sketched in Fig. 3(c).   R1NT 3 = {(µR , µT ) : 3µR + NT µT ≥ 3, 0 ≤ µR ≤ 1, µT ≤ 1}      2    RNT 3 = {(µR , µT ) : µR ≥ 0, 3µR + NT µT < 3, 2µR + NT µT ≥ 2}

R3NT 3 = {(µR , µT ) : 2µR + NT µT < 2, 3µR + NT µT ≥ 2, 3µR + 2NT µT > 3} .     R4NT 3 = {(µR , µT ) : NT µT > 1, µR ≥ 0, 3µR + NT µT < 2}      5 RNT 3 = {(µR , µT ) : NT µT ≤ 1, 3µR + 2NT µT ≤ 3, µR + NT µT ≥ 1}

Theorem 3 (Lower bound of FDT for the NT × NR networks): For the cache-aided wireless interference network with NT ≥ 2 transmitters and NR ≥ 2 receivers, the minimum FDT is lower bounded by ∗

τ (µR , µT ) ≥ max



 1 (1 − µR ), τl , NR

(9)

where 1 τl = max l=1,2,··· ,min{NT ,NR } NR l

   2NR − l + 1 2 NR − (NT − l)(NR − l)µT − · l + (NR − l) µR 2    2NR − l + 2 + . (10) (l − 1) + (NR − l) (1 − NT µT ) 2

Remark 4 (Optimality): It can be seen from the above theorems that both the upper and lower bounds of the minimum FDT are convex and piece-wise linear decreasing functions of

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1

1










2/3


T


T


 1/2










1/3


0

0 0

1/3

2/3


R

1

(a)

0

1/3


R

2/3

1

(b)

1




 


T

3/NT

2/NT



 

 

1/NT



 

 

0 0

1/3


R

2/3

1

(c)

Fig. 3: Cache size regions in the NT × 3 network. (a) NT = 2, (b) NT = 3, (c) NT ≥ 4. normalized cache sizes µR and µT . By comparing Theorems 1, 2 with Theorem 3, it is seen that when (µR , µT ) lies in region R1 (R1NT 2 , R123 , R133 and R1Nt 3 in Theorem 1, 2, respectively) or on the bottom boundary line µR + NT µT = 1 (see in Figs. 2, 3), our achievable FDT is optimal. In other regions, numerical results in Fig. 4 show that the maximum ratio between the achievable upper bound and the theoretical lower bound of minimum FDT, denoted as g =

τachievable , τlower bound

is

within 4/3 and 3/2, respectively, in the NT × 2 network and NT × 3 network, and g decreases as NT increases. Below we discuss some special cases and the connections with existing works. 1) Transmitter cache only (µR = 0):

4/3 1.3

3/2

Maximum Ratio

Maximum Ratio

12

1.2 1.1 1

1.4 1.3 1.2 1.1 1

2

10

20

30

40

50

2

10

20

30

NT

NT

(a)

(b)

40

50

Fig. 4: Maximum ratio g between the achievable upper bound and the theoretical lower bound of minimum FDT in the NT × NR interference networks. (a) NR = 2, (b) NR = 3. In the special case when µR = 0 (transmitter cache only), the achievable FDT for the 3 × 3 network in Theorem 2 reduces to   13/18 − µ /2, 1/3 ≤ µ ≤ 2/3 T T ∗ τ (0, µT ) ≤ ,  1/2 − µT /6, 2/3 < µT ≤ 1

which is the same as the upper bound of 1/DoF in [19]. Also, when µR = 0, given that (1 − NT µT )+ = 0 from the feasible domain, τl in Theorem 3 reduces to 1 (NR − (NT − l)(NR − l)µT ) . l=1,2,··· ,min{NT ,NR } NR l

τl =

max

Thus, Theorem 3 reduces to the same lower bound δ ∗ (µ) in [20] as a special case. 2) Full transmitter cache (µT = 1): When µT = 1, each transmitter can cache all the files and hence the network can be viewed as a virtual broadcast channel as in [8] except that the server (transmitter) has NT distributed antennas. From Theorem 1 and Theorem 3, we obtain that the optimal FDT of the NT × 2 network is τ ∗ =

1−µR . 2

Comparing to the result in [8], i.e., τ =

can see that our FDT is better when 0 ≤ µR
0 and a03 + a12 > 0, while in R2NT 3 , our achievable scheme exploits transmitter cooperation gain and local caching gain only because only file splitting ratios a02 , a30 , a03 > 0 in our solution. In R3 (R323 , R333 , R3NT 3 , respectively) and R4 (R423 , R433 , R4NT 3 , respectively), coded muticasting gain, transmitter cooperation gain and local caching gain are all exploited since our solutions all satisfy a11 > 0 and a02 + a03 > 0. In R5 (R523 , R533 , R5NT 3 , respectively), local caching gain and multicasting gain are exploited and no transmitter cooperation gain can be exploited since our solution satisfies amn = 0 (∀n ≥ 2). C. On The Differences With [22] Although the similar caching problem is considered in [22], their performance metric, caching scheme, and conclusion are significantly different from ours. First, we adopt the FDT as the performance metric while [22] used the standard DoF. As we noted in Remark 2 in Section

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II, FDT is particularly suitable for the considered network because it reflects not only the load reduction due to receiver cache but also the DoF enhancement due to transmitter cache. In specific, we can express the FDT as τ =

R , NR ×DoF

where R is the traffic load normalized by

each file size. To illustrate this in detail, we consider the integer points (µR = 13 , µT = 23 ) and (µR = 23 , µT = 31 ) in the 3 × 3 interference network. In [22], they have DoF = 3 at both points. However, the actual delivery time at these two points is different. At point (µR = 31 , µT = 32 ), they split each file into 9 equal-sized subfiles, each cached in one receiver and two transmitters, i.e. a12 =

1 9

and other ratios are 0. Thus, each receiver caches 3 out of 9 subfiles of its desired file and

only needs the other 6 subfiles in the delivery phase. The corresponding FDT is τ =

3×6×a12 3×3

= 29 .

On the other hand, at point (µR = 32 , µT = 13 ), they also split each file into 9 equal-sized subfiles, but each cached in two receivers and one transmitter, i.e. a21 =

1 9

and other ratios are 0. Thus,

each receiver caches 6 out of 9 subfiles of its desired file and only needs the other 3 subfiles in the delivery phase, yielding the corresponding FDT τ =

3×3×a21 3×3

= 19 . Clearly, the DoF alone

is unable to fully capture the gains of joint transmitter and receiver caching as FDT does. Due to such inefficiency, the achievable DoF at non-integer points cannot simply be obtained as a linear combination of the DoFs at integer points through memory and file partitions as in [22]. Second, the file splitting ratios in [22] are pre-determined at each given cache size point (µR , µT ) as noted in Remark 5 of Section IV. However, our file splitting ratios are obtained by solving a linear programming problem at each given cache size point and thus are probably optimal under the given caching strategy. Another difference between our work and [22] is that the transmission scheme in [22] is restricted to one-shot linear processing, while we allow interference alignment which may require infinite symbol extension. As such, our FDT analysis is more complete than the DoF analysis in [22]. VIII. L OWER B OUND O F T HE M INIMUM FDT F OR NT × NR N ETWORKS . In this section, we present the proof of the lower bound of the minimum FDT for any NT and NR in Theorem 3. The main idea of the proof is based on cut-set bound and geniemessage approach. Without loss of generality, we assume that receiver 1, 2, · · · , NR request files W1 , W2 · · · WNR , respectively. By the converse assumption, each receiver j can decode its intended message Wj with the

29

received signal Yj and its local cache content Vj as side information. Therefore, we have the following bound, where εF is a function of F and εF → 0 when F → ∞. F εF = H(Wj \VjWj |Yj , Vj )

(39a)

= H(Wj \VjWj |VjWj ) − I(Wj \VjWj ; Yj , VjW1∼jWL \VjWj |VjWj ) = (1 − µR )F − H(Yj , VjW1∼jWL \VjWj |VjWj ) − H(Yj , VjW1 ∼jWL \VjWj |Wj )
≥ (1 − µR )F − H(Yj , VjW1∼jWL \VjWj |VjWj ) − H(VjW1∼jWL \VjWj |Wj )

(39b)


= (1 − µR )F − H(VjW1∼jWL \VjWj |VjWj ) + H(Yj |Vj ) − H(VjW1∼jWL \VjWj |Wj ) = (1 − µR )F − H(Yj |Vj )

(39c) (39d)


(39e) (39f)

≥ (1 − µR )F − H(Yj )

(39g)

≥ (1 − µR )F − T log(2πe(c · P + 1)),

(39h)

where VjWs denotes the content of file Ws cached in receiver j, Ws \VjWs denotes the content of file Ws not cached in receiver j and c is only related to channel coefficients. Here, (39a) comes from the Fano’s inequality; (39c) comes from the fact that given each receiver has MR F cache storage, it can cache µR F bits of each file Wj on average, thus, its non-cached part Wj \VjWj has size (1 − µR )F on average; (39e) comes from the chain rule; (39g) comes from conditioning reducing entropy and (39h) comes from Lemma 1 in [20]. Therefore, when F → ∞ and P → ∞, we have T log P ≥ (1 − µR )F from (39h), which implies τ≥

1 − µR . NR

(40)

Thus, the first term on the right side in (9) is proved. To prove τ ∗ ≥ τl , we use the geniemessage approach similar to [20] except that in our method, we take the receiver cache content into account. Under converse assumption, we can make the following statement. Given received signals Y from any l receivers, local caches U from any NT − l transmitters, and all the local caches V1∼NR at receivers, the desired files for all receivers can be successfully decoded in the high-SNR regime by the genie. To explain it, for example, we consider the case when the genie has received signals Y1∼l from receivers 1, 2, · · · , l, transmitter caches Ul+1∼NT from transmitters l + 1, l + 2, · · · , NT and receiver caches V1∼NR at all receivers. Given transmitter caches Ul+1∼NT , the genie can construct transmitted signals Xl+1∼NT at transmitters l + 1, l + 2, · · · , NT . Given transmitted signals Xl+1∼NT and received signals Y1∼l , it can further obtain

30 All desired files decoded

All transmitted signals obtained

(Nt-l) transmitter caches

All receiver caches

l received signals

Tx 1

Rx 1

Cache

Cache Г

Cache Г

Cache

Tx 2

Rx 2

Г

Г

Tx l+1

Rx l

Г

Г

Tx Nt

Rx Nr

Cache

Cache Г

Cache Г

Cache

Fig. 7: Given received signals Y1∼l from receivers 1, 2, · · · , l, transmitter caches Ul+1∼NT from transmitters l + 1, l + 2, · · · , NT and receiver caches V1∼NR at all receivers, the genie can successfully decode desired files for all receivers.

the remaining unknown transmitted signals X1∼l almost surely as stated in Lemma 3 in [20], neglecting noise in the high SNR regime. After solving all the transmitted signals X1∼NT , it can obtain all the received signals Y1∼NR and can further decode all the desired files successfully together with local caches V1∼NR . This statement is shown in Fig. 7. To obtain the lower bound of FDT, without loss of generality, we assume that receiver 1, 2, · · · , NR want file W1 , W2 , · · · , WNR , respectively. Using Lemma 2 in [20] and from the analysis above, we can have H(W1∼NR |Y1∼l , Ul+1∼NT , V1∼NR , WNR +1∼L ) = F εF + T εP log P,

(41)

where εP is a function of power P and εP → 0 when P → ∞. Then, the entropy of desired

31

files W1 , W2 , · · · , WNR can be expressed as NR F = H(W1∼NR |WNR +1∼L ) = H(W1∼NR |Y1∼l , Ul+1∼NT , V1∼NR , WNR +1∼L ) + I(W1∼NR ; Y1∼l , Ul+1∼NT , V1∼NR |WNR +1∼L ) = I(W1,∼NR ; Y1∼l , Ul+1∼NT , V1∼NR |WNR +1∼L ) + F εF + T εP log P.

(42)

The first term in (42) is upper bounded by: I(W1∼NR ; Y1∼l , Ul+1∼NT , V1∼NR |WNR +1∼L ) = H(Y1∼l , Ul+1∼NT , V1∼NR |WNR +1∼L ) − H(Y1∼l , Ul+1∼NT , V1∼NR |W1∼L )

(43a)

= H(Y1∼l , Ul+1∼NT , V1∼NR |WNR +1∼L ) − T εP log P

(43b)

≤ H(Y1∼l |WNR +1∼L ) + H(Ul+1∼NT , V1∼NR |Y1∼l , WNR +1∼L )

(43c)

≤ lT log(2πe(c · P + 1)) + H(V1∼NR |Y1∼l , WNR +1∼L ) + H(Ul+1∼NT |V1∼NR , Y1∼l , WNR +1∼L ) (43d) ≤ lT log(2πe(c · P + 1)) + H(V1∼NR |Y1∼l , WNR +1∼L ) + H(Ul+1∼NT |W1∼l , WNR +1∼L ) + F εF (43e) = lT log(2πe(c · P + 1)) + H(V1∼NR |Y1∼l , WNR +1∼L ) + (NT − l)(NR − l)µT F + F εF . (43f) Here, the second term in (43b) comes from the noise of received signal Y ; (43d) comes from the chain rule and Lemma 1 in [20]; (43e) comes from the Fano’s inequality; (43f) comes from the fact that given each transmitter has MT F cache storage, it can cache µT F bits of each file on average.

32

Further, H(V1∼NR |Y1∼l , WNR +1∼L ) in (43f) is upper bounded by H(V1∼NR |Y1∼l , WNR +1∼L ) = H(V1|Y1∼l , WNR +1∼L ) + H(V2∼NR |Y1∼l , WNR +1∼L , V1 ) ≤ NR µR F + H(V2 |Y1∼l , WNR +1∼L , V1 , W1 ) +

l X

(44a)

H(Vi |Y1∼l , WNR +1∼L , V1∼i−1 , W1∼i−1 )

i=3

+ H(Vl+1∼NR |Y1∼l , WNR +1∼L , V1∼l , W1∼l ) + F εF ≤ NR µR F + (NR − 1) µR − (1 − NT µT )

+




F+

l X

(44b) H(Vi |Y1∼l , WNR +1∼L , V1∼i−1 , W1∼i−1 )

i=3

+ H(Vl+1∼NR |Y1∼l , WNR +1∼L , V1∼l , W1∼l ) + F εF ≤ NR µR F + (NR − 1) µR − (1 − NT µT ) +

NR X

+




F+

l X i=3

(44c)
(NR − i + 1) µR − (1 − NT µT )+ F

H(Vi |Y1∼l , WNR +1∼L , V1∼l , W1∼l ) + F εF

(44d)

i=l+1

≤

l X i=2



(NR − i + 1) µR − (1 − NT µT )+ F + (NR − l)(NR − l) µR − (1 − NT µT )+ F

+ NR µR F + F εF (44e)     2NR − l 2NR − l + 1 l + (NR − l)2 µR F − (l − 1) + (NR − l)2 (1 − NT µT )+ F = 2 2 + F εF .

(44f)

Here, (44b) comes from the Fano’s inequality and the fact that given receiver 1 has MR F cache storage, it can cache µR F bits of each file on average; (44c) comes from the fact that all receivers must cache at least the same (1 − NT µT )F bits of each file on average to guarantee the feasibility of the scheme if NT MT F ≤ LF ; (44d) comes from the fact that each Vi (i ≤ l) has (µR −(1−NT µT )+ )F unknown bits of each file in Wi∼NR on average since we already know files W1∼i−1 and WNR +1∼L ; (44e) comes from the fact that each Vi (i > l) has (µR − (1 − NT µT )+ )F unknown bits of each file in Wl+1∼NR on average since we already know files W1∼l and WNR +1∼L . Note that the inequalities (44) are the main novelty of the proof of the converse.

33

Combining (42)(43f)(44f), we have 

 2NR − l + 1 2 l + (NR − l) µR F 2

NR F ≤lT log(2πe(c · P + 1)) + (NT − l)(NR − l)µT F +   2NR − l 2 − (l − 1) + (NR − l) (1 − NT µT )+ F + F εF + T ǫP log P. 2

(45)

Dividing NR F on both sides of (45) and letting F → ∞ and P → ∞, we have T log P τ = lim lim P →∞ F →∞ NR F    1 2NR − l + 1 2 ≥ NR − (NT − l)(NR − l)µT − · l + (NR − l) µR NR l 2    2NR − l + 2 + . (l − 1) + (NR − l) (1 − NT µt ) 2

(46)

Optimizing the bound in (46) over all possible choices of l = 1, 2, · · · , min{NT , NR } and combining (40) complete the proof of Theorem 3. IX. C ONCLUSION In this paper, we have studied the storage-latency tradeoff for the cache-aided interference network with NT transmitters and NR receivers. We characterized the tradeoff by the fractional delivery time. We have obtained the achievable upper bounds of the minimum FDT for the NT × 2 and NT × 3 interference networks. We have also obtained the theoretical lower bound for the general NT × NR interference network. Both upper and lower bounds are convex and piece-wise linear decreasing functions of normalized cache sizes µR and µT . The achievable bound is optimal at certain cache size regions and its maximum ratio to the lower bound is at most

4 3

and

3 2

for NR = 2 and NR = 3 receivers, respectively. The two bounds approach to each

other when NT → ∞. The proposed caching strategy is general for any transmitter and receiver numbers and at any feasible cache size regions. The optimal file splitting ratios are obtained by solving a linear programming problem for FDT minimization under the given caching strategy. Analysis shows that the optimal ratios are not unique, unlike existing works where the ratios are all pre-determined. This offers freedoms to choose appropriate caching strategy in practical systems with different constraints. As a byproduct, the achievable DoFs of three new types of channels are obtained, namely, the X channel with partial transmitter cooperation, the hybrid X-multicast channel, and the hybrid X-multicast channel with partial transmitter cooperation.

34

The extension of the proposed FDT analysis to the cache-aided interference network with NR > 3 receivers is challenging due to that the channels formed by the delivery of different subfiles are much more complicated. A PPENDIX A: P ROOF O F L EMMA 1 Consider the NT × 3 X channel with partial transmitter cooperation shown in Fig. 6(b). Note that in the case with NT = 3, the achievable DoF is already studied in [25]. Here, we extend the proof to the general case for arbitrary NT using the similar method.
In this channel, each receiver wants N2T independent messages, each of which is available at

two different transmitters. Denote Apq , Bpq , Cpq as the message available at transmitters p, q (We let p < q to avoid the overlap of messages, i.e., A12 and A21 are the same message.) and needed by receiver 1, 2, 3, respectively. Since each message is available at two transmitters, transmitter cooperation can be applied to neutralize interference at receivers. In specific, each message needed by receiver i is split into two equal-sized submessages. The first part is neutralized at receiver j, j ∈ {1, 2, 3}\{i}, and the second part is neutralized at receiver k, k ∈ {1, 2, 3}\{i, j}. Note that we use the superscript of each submessage to denote the receiver index at which the submessage will be neutralized, e.g. submessages A2pq and A3pq are neutralized at receiver 2 and 3, respectively. In what follows, we use the method similar in [25] to design the precoding factors so that each submessage is neutralized at a specified receiver. For example, for the submessage A2pq , we design the precoding factor h2q at transmitter p and −h2p at transmitter q. With this precoding operation, a single virtual source instead of two transmitters is transmitting the submessage A2pq . We refer to this virtual source node as the virtual transmitter for the submessage A2pq . It is seen that the equivalent channel coefficients from the virtual transmitter for A2pq to receiver 1, 2, 3 can be expressed as (h1p h2q − h1q h2p ), 0, and (h3p h2q − h3q h2p ), respectively, and A2pq is exactly neutralized at receiver 2. Similarly, we can obtain the equivalent channel coefficients from the virtual transmitter for each submessage to receiver 1, 2, 3, respectively, as shown in TABLE I. Here, all the complex scaling factors α are independent and γ is related to the power constraint,

which will be explained later. From TABLE I, it is worth mentioning that there are totally 3NT (NT − 1) submessages transmitted from the virtual transmitters. For each receiver, there are NT (NT − 1) desired

35

TABLE I: Equivalent channel coefficients for submessages Equivalent Channel Coefficient

Submessage (∀p, ∀q > p)

at Receiver 1

A2pq

γα2Apq (h1p h2q

A3pq

γα3Apq (h1p h3q

1 Bpq 3 Bpq

− h1q h3p )

0 γα3Bpq (h1p h3q

1 Cpq 2 Cpq

− h1q h2p )

− h1q h3p )

− h1q h2p )

− h2q h3p )

γα1Bpq (h2p h1q

− h2q h1p )

γα3Bpq (h2p h3q

− h2q h3p )

0

at Receiver 3 γα2Apq (h3p h2q

0 γα3Apq (h2p h3q

γα1Cpq (h2p h1q

0 γα2Cpq (h1p h2q

at Receiver 2

− h2q h1p )

− h3q h2p )

0 γα1Bpq (h3p h1q

− h3q h1p )

0 γα1Cpq (h3p h1q

− h3q h1p )

γα2Cpq (h3p h2q

− h3q h2p )

submessages and 2NT (NT −1) undesired submessages as interference. From the aforementioned precoding factor design method, NT (NT − 1) undesired submessages are neutralized at each receiver. There are additional NT (NT − 1) undesired submessages remaining at each receiver. In what follows, we aim to align these NT (NT − 1) additoinal undesired submessages using real interference alignment [26]. To use real interference alignment, we encode each submessage into ΩNT (NT −1) independent data streams where Ω is an integer and to be determined later. Each data stream is modulated over integer constellation ZQ , Z ∩ [−Q, Q] and scaled with a unique monomial from set VΩ . The monomial set is defined as follows: Definition 3: For a complex set X with s complex elements, i.e., X = {x1 , x2 , · · · , xs }, its monomial set is VΩ (X ) , {xω1 1 xω2 2 · · · xωs s : ∀ω1 , ω2 · · · ωs ∈ Z ∩ [1, Ω]}, where ω1 , ω2 · · · ωs are all integers between 1 and Ω, and the cardinality of VΩ (X ) is |VΩ (X )| = Ωs . 3 2 In specific, each data stream in Bpq and Cpq (∀p, ∀q > p) is scaled with a unique monomial

in set VΩ (X1 ) with  3 2 X1 = αBpq (h1p h3q − h1q h3p ), αCpq (h1p h2q − h1q h2p ) : ∀p, ∀q > p .

1 Each data stream in A3pq and Cpq (∀p, ∀q > p) is scaled with a unique monomial in set VΩ (X2 )

with  3 1 X2 = αApq (h2p h3q − h2q h3p ), αCpq (h2p h1q − h2q h1p ) : ∀p, ∀q > p .

36 1 Each data stream in A2pq and Bpq (∀p, ∀q > p) is scaled with a unique monomial in set VΩ (X3 )

with  2 1 X3 = αApq (h3p h2q − h3q h2p ), αBpq (h3p h1q − h3q h1p ) : ∀p, ∀q > p .

2 3 and Cpq are both encoded to the points in the Thus, for all p and all q > p, submessages Bpq P 1 are both encoded to the points signal constellation v∈VΩ (X1 ) (vZQ ), submessages A3pq and Cpq P 1 in the signal constellation v∈VΩ (X2 ) (vZQ ), and submessages A2pq and Bpq are both encoded to P the points in the signal constellation v∈VΩ (X3 ) (vZQ ).

Now we consider the received signal constellation at receiver 1, which can be expressed as y =C1 + C2 + C3 ,

where C1 =

NX T −1

NT X

NX T −1

NT X

NX T −1

NT X

p=1 q=p+1

C2 =

p=1 q=p+1

C3 =

p=1 q=p+1

+

NX T −1



2 γαApq (h1p h2q − h1q h2p )



3 γαApq (h1p h3q − h1q h3p )



3 γαBpq (h1p h3q − h1q h3p )

NT X

p=1 q=p+1



(vZQ ) ,

X

(vZQ ) ,

X

(vZQ )

v∈VΩ (X3 )

v∈VΩ (X2 )

v∈VΩ (X1 )

2 γαCpq (h1p h2q − h1q h2p )



X

X



v∈VΩ (X1 )





(vZQ ) .

The received constellations C1 and C2 are the desired data streams encoded from desired submessages A2pq and A3pq , respectively, and the received constellation C3 are the interference encoded 3 2 from Bpq and Cpq (∀p, ∀q > p). Using similar arguments in [25], we have X
C3 ⊂ γ vZNT (NT −1)Q ,

(47)

v∈VΩ+1 (X1 )

3 2 which implies the interference Bpq and Cpq (∀p < q) are aligned into a constellation of cardinality

much smaller than the multiplication of the transmitted constellation cardinalities. The received signal constellation at receiver 2 and 3 can be obtained similarly and thus omitted here. Next, we need to assure the monomials forming the received constellation are linearly independent as functions of h, α and γ, so that Theorem 3 in [25] can be applied to prove the decodebility of this signal constellation.

37 2 1 It can be seen that the monomials forming C1 are functions of αApq and αBpq (∀p, ∀q > p), 3 1 the monomials forming C2 are functions of αApq and αCpq , and the monomials forming C3 3 2 are functions of αBpq and αCpq . Since none of them are overlapped, we can guarantee the

independence across the monomials forming constellations C1 , C2 and C3 . Then, it remains to prove the linear independence of the monomials within each constellation Ci . Without loss of generality, we take C1 as an example. Note that monomials forming C1 are 2 {γαApq (h1p h2q − h1q h2p )VΩ (X3 ) : ∀p, ∀q > p}. In what follows, we use contradiction to prove

the linear independence of the monomials across different p, q. We assume that a monomial 2 αA12 (h11 h22 −h12 h21 )·v1 in the constellation of A212 can be linearly expressed by other monomials

in the constellation of A2pq with different p, q, where v1 = γ

NY T −1

NT Y

2 αApq (h3p h2q − h3q h2p )

p=1 q=p+1

2
ωpq

1 (h3p h1q − h3q h1p ) αBpq

1
ωpq

.

2 Then, to linearly express αA12 (h11 h22 − h12 h21 ) · v1 , the monomials of the constellations of

A2pq with different p, q should have the same power of each scaling factor α in the monomial 2 αA12 (h11 h22 −h12 h21 )·v1 due to the fact that each α is independent, i.e., these monomials should

have the following power form of the scaling factor α: 2 αA12

NY T −1

NT Y

p=1 q=p+1

2 αApq

2
ωpq

1 αBpq

1
ωpq

.

(48)

Equation (48) implies that for any constellation of A2pq with (p, q) 6= (1, 2), it only remains monomial 2 2 αApq (h1p h2q − h1q h2p )αA12 (h31 h22 − h32 h21 ) · v1 2 αApq (h3p h2q − h3q h2p )

which can contribute to the linear expression. Thus, the linear expression can be written as 2 αA12 (h11 h22 − h12 h21 ) · v1

=

NX T −1

NT X

p=1 q=p+1((p,q)6=(1,2))

2 2 αApq (h1p h2q − h1q h2p )αA12 (h31 h22 − h32 h21 ) · v1 , cpq 2 αApq (h3p h2q − h3q h2p )

where cpq is the scalar. Dividing v1 and all α in (49), it is equivalent to say that

(49)

h11 h22 −h12 h21 h31 h22 −h32 h21

can be linearly expressed as NX T −1 h11 h22 − h12 h21 = h31 h22 − h32 h21 p=1

NT X

q=p+1((p,q)6=(1,2))

cpq

h1p h2q − h1q h2p . h3p h2q − h3q h2p

(50)

38

Note that both sides of (50) are functions of all channel coefficients {hpq : p 6= q}. Doubling h11 and h12 while fixing other channel coefficients in (50), given the assumed linear dependence, we can have N

N

T T 2(h11 h22 − h12 h21 ) X 2h11 h2q − h1q h21 X 2h12 h2q − h1q h22 = + c1q c2q h31 h22 − h32 h21 h31 h2q − h3q h21 h32 h2q − h3q h22 q=3 q=3

+

NX T −1

NT X

cpq

p=3 q=p+1

h1p h2q − h1q h2p . h3p h2q − h3q h2p

(51)

Subtracting (50) from (51), we have N

N

T T X h11 h22 − h12 h21 X h11 h2q h12 h2q = + . c1q c2q h31 h22 − h32 h21 h h − h h h h − h h 31 2q 3q 21 32 2q 3q 22 q=3 q=3

(52)

By changing h23 only and fixing other channel coefficients, it is easy to see that c13

h12 h23 h11 h23 + c23 h31 h23 − h33 h21 h32 h23 − h33 h22

remains constant as a function of h23 , which implies c13 = c23 = 0. Thus, similarly, for all q > 2, we can have c1q = c2q = 0 which is contradicted with the assumption of linear dependence. Therefore, we proved the linear independence of the monomials across constellations of A2pq with different p, q. Similar arguments can be applied to constellations of C2 and C3 . Given linear independence of monomials in each monomial set VΩ (X1 ), VΩ (X3 ) and VΩ (X3 ), the linear independence of the monomials of the received constellations, as functions of h, α and γ, is thus proved. By setting I , NT (NT − 1)ΩNT (NT −1) + (Ω + 1)NT (NT −1) , 1−ε 1 P I+2ε , NT (NT − 1) I ς , − 1 + ε, 2

Q,

I−2+4ε

γ , c1 P 2(I+2ε) , where c1 , ε are constants and c1 , ε > 0, and using Theorem 3 in [25] and results in [27], we can have the minimum distance ∆ ≥ c2 γ (NT (NT − 1)Q)−ς = c2 c1 P ε/2

(53)

39

for almost all channel coefficients h and scaling factors α. Here c2 is a constant independent with P and is positive for almost all h. From (53), the decodability of the signal constellation is guaranteed for almost all channel coefficients h as ∆ → ∞ when P → ∞. The average transmit power is upper bounded by c3 c21 P. (54) (NT (NT − 1))2 Then, the DoF for one receiver under constellation modulation can be calculated similarly in c3 γ 2 Q2 =

[26] as follows: 1 − exp(−∆2 /8) log |desired signal cardinality| P →∞ log P
log(2Q + 1) = lim NT (NT − 1)ΩNT (NT −1) P →∞ log P
NT (NT − 1)ΩNT (NT −1) (1 − ε) = . NT (NT − 1)ΩNT (NT −1) + (Ω + 1)NT (NT −1) + 2ε lim

Letting Ω → ∞ and ǫ → 0, we obtain that the sum DoF is 3 ·

NT (NT −1) . NT (NT −1)+1

(55) Thus, Lemma 1

is proved. A PPENDIX B: P ROOF O F L EMMA 2 Consider the NT × 3 hybrid X-multicast channel shown in Fig. 6(d). We denote Ap , Bp , Cp as the message available at transmitter p and needed by receivers {2, 3}, receivers {1, 3} and receivers {1, 2}, respectively. By using a (2NT + 1)-symbol extension, the effective channel between transmitter p and receiver j over (2NT + 1) channel uses, denoted by Hjp, becomes a (2NT + 1) × (2NT + 1) diagonal matrix with diagonal entries chosen i.i.d from some continuous distribution. We further denote vAp , vBp , vCp as the (2NT + 1) × 1 precoding vector for the transmitted symbol xAp , xBp , xCp , which are encoded from Ap , Bp , Cp , respectively. Then, the received signal at receiver 1 can be expressed as y1 =

NT X p=1


H1p vAp xAp + vBp xBp + vCp xCp + n1 ,

where n1 denotes the (2NT + 1) × 1 additive white Gaussian noise (AWGN) vector at receiver 1. Note that xBp and xCp are the desired signals and xAp is the interference at receiver 1. To T this end, we need to design the precoding vectors {vAp }N p=1 as

H1p vAp = v, ∀p ∈ [NT ] ⇒ vAp = H−1 1p v,

(56)

40

where v is a (2NT + 1) × 1 vector to be determined later, so that all the interference can be aligned in a same subspace at receiver 1. Similarly, for receiver 2 and 3, we need to design NT T {vBp }N p=1 and {vCp }p=1 such that

H2p vBp = v, ∀p ∈ [NT ] ⇒ vBp = H−1 2p v,

(57)

H3p vCp = v, ∀p ∈ [NT ] ⇒ vCp = H−1 3p v.

(58)

Let v be the (2NT + 1) × 1 vector with each element non-zero, for example v = (1, 1, · · · , 1)T , we can construct all the precoding vectors according to (56)-(58). Next, we need to ensure the space spanned by the desired signals is linearly independent of the space spanned by the interference at each receiver. Without loss of generality, we take receiver 1 as an example. At receiver 1, the span space of the 2NT desired data streams can be expressed as span (HD ) , span



H11 H−1 31 v

H11 H−1 21 v

···

H1NT H−1 3NT v

H1NT H−1 2NT v

and the span space of the interference can be expressed as




,

span (HI ) , span ([v]) . Since each channel matrix is chosen independently of each other, [HD HI ] is a full-rank matrix with probability 1, which guarantees the decodability of the desired data streams. Similarly, receiver 2, 3 can also decode its desired signals. As a result, 3NT data streams are transmitted over (2NT + 1) channel uses, which indicates that the sum DoF is

3NT . 2NT +1

Thus, Lemma 2 is

proved. A PPENDIX C: P ROOF O F L EMMA 3 Consider the NT × 3 partially cooperative hybrid X-multicast channel where each transmitter subset Ψ (|Ψ| = i ≥ 2) has three independent messages, each of which is needed by two receivers. The special case for |Ψ| = 2 is shown in Fig. 6(e), and we will prove the general case that |Ψ| = i ≥ 2 here. We denote AΨ , BΨ , CΨ as the message available at transmitters in Ψ needed by receivers {2, 3}, receivers {1, 3} and receivers {1, 2}, respectively. By using a

2 NiT -symbol extension, the effective channel between transmitter p and receiver j over 2 NiT

channel uses, denoted by Hjp , becomes a 2 NiT × 2 NiT diagonal matrix with diagonal entries
p p p NT ×1 chosen i.i.d from some continuous distribution. We further denote vA , v , v as the 2 B C i Ψ Ψ Ψ

41

precoding vector for the transmitted symbol xAΨ , xBΨ , xCΨ , which are encoded from message AΨ , BΨ , CΨ , respectively, at transmitter p ∈ Ψ. Then, the signal received at receiver 1 can be expressed as y1 =

X X Ψ

p p x + vB x + vCp Ψ xCΨ H1p vA Ψ AΨ Ψ BΨ

p∈Ψ

!


+ n1 .

Note that xBΨ and xCΨ are the desired data streams and xAΨ are the interference at receiver 1. To this end, we need to design the precoding vectors as X p H1p vA = 0, Ψ

(59)

p∈Ψ

so that all the interference are neutralized. Similarly, for receiver 2 and 3, we need to design the precoding vectors such that X

p H2p vB = 0, Ψ

(60)

X

H3p vCp Ψ = 0.

(61)

p∈Ψ

p∈Ψ

p p Besides the three constraints above, we need to further design vA , vB , vCp Ψ so that receivers Ψ Ψ
can successfully decode its desired message. Define ei = (0, · · · , 0, 1, 0, · · · , 0)T as the 2 NiT ×1
vector with its i-th element being 1 and other elements being 0. Due to the fact that there are NiT

p p different choices of Ψ, we should design the precoding vector vA , vB , vCp Ψn for transmitter Ψn Ψn
p at set Ψn (n ∈ [ NiT ]) as p vA = wAp Ψn en , Ψn

(62)

p p vB = wB e N , Ψn Ψn n+( T )

(63)

i

vCp Ψn = wCp1Ψn en + wCp2Ψn en+(NT ) ,

(64)

i

where factor w is designed to satisfy interference neutralization (59)-(61), respectively. Then, we consider received signal at receivers. Without loss of generality, we take receiver 1

42

as an example. The signal received at receiver 1 can be expressed as NT ! (X i )   X p p + n1 + vCp Ψn xCΨn x + vB x H1p vA y1 = Ψn BΨn Ψ n AΨ n n=1

p∈Ψn

NT i

=

=

(X)

!

n=1

p∈Ψn

  p p H1p vBΨn xBΨn + vCΨn xCΨn

NT (X i )

X

X

!     p1 p2 p + n1 . H1p wBΨn en+(NT ) xBΨn + wCΨn en + wCΨn en+(NT ) xCΨn

n=1

p∈Ψn

+ n1

(65)

i

i

p From (60) and (61), we can see wB , wCp1Ψn and wCp2Ψn is independent with H1p , thus none of Ψn

the received factors of desired signals xBΨn and xCΨn degenerates to zero with probability one. Therefore, receiver 1 can first decode xCΨn using zero-forcing along vector en , and then decode 
 xBΨn along en+(NT ) (∀n ∈ NiT ) with xCΨn as side information. Similarly, receiver 2, 3 can i
also successfully decode its desired signals. As a result, 3 NiT data streams are transmitted over
2 NiT channel uses, which indicates that the sum DoF is 23 . Thus, Lemma 3 is proved. R EFERENCES [1] F. Boccardi, R. W. Heath, A. Lozano, T. L. Marzetta, and P. Popovski, “Five disruptive technology directions for 5G,” IEEE Communications Magazine, vol. 52, no. 2, pp. 74–80, February 2014. [2] H. Liu, Z. Chen, X. Tian, X. Wang, and M. Tao, “On content-centric wireless delivery networks,” Wireless Communications, IEEE, vol. 21, no. 6, pp. 118–125, 2014. [3] Cisco, cast

“Cisco update

visual 2015-2020,”

networking White

index: Paper,

Global

mobile

Feb

2016.

data

traffic

[Online].

foreAvailable:

http://www.cisco.com/c/en/us/solutions/collateral/service-provider/visual-networking-index-vni/mobile-white-paper-c11-520862.html [4] N. Golrezaei, A. F. Molisch, A. G. Dimakis, and G. Caire, “Femtocaching and device-to-device collaboration: A new architecture for wireless video distribution,” IEEE Communications Magazine, vol. 51, no. 4, pp. 142–149, April 2013. [5] E. Bastug, M. Bennis, and M. Debbah, “Living on the edge: The role of proactive caching in 5G wireless networks,” IEEE Communications Magazine, vol. 52, no. 8, pp. 82–89, Aug 2014. [6] X. Wang, M. Chen, T. Taleb, A. Ksentini, and V. C. M. Leung, “Cache in the air: exploiting content caching and delivery techniques for 5G systems,” IEEE Communications Magazine, vol. 52, no. 2, pp. 131–139, February 2014. [7] G. S. Paschos, E. Bastug, I. Land, G. Caire, and M. Debbah, “Wireless caching: Technical misconceptions and business barriers,” 2016. [Online]. Available: http://arxiv.org/abs/1602.00173 [8] M. Maddah-Ali and U. Niesen, “Fundamental limits of caching,” in IEEE International Symposium on Information Theory (ISIT), July 2013, pp. 1077–1081. [9] ——, “Decentralized coded caching attains order-optimal memory-rate tradeoff,” IEEE/ACM Trans. on Networking, vol. 23, no. 4, pp. 1029–1040, Aug 2015.

43

[10] U. Niesen and M. A. Maddah-Ali, “Coded caching with nonuniform demands,” in IEEE Conference on Computer Communications Workshops (INFOCOM WKSHPS), April 2014, pp. 221–226. [11] M. Ji, A. M. Tulino, J. Llorca, and G. Caire, “Order-optimal rate of caching and coded multicasting with random demands,” 2015. [Online]. Available: http://arxiv.org/abs/1502.03124 [12] J. Zhang, X. Lin, and X. Wang, “Coded caching under arbitrary popularity distributions,” in Information Theory and Applications Workshop (ITA), Feb 2015, pp. 98–107. [13] J. Zhang and P. Elia, “Fundamental limits of cache-aided wireless BC: interplay of coded-caching and CSIT feedback,” 2015. [Online]. Available: http://arxiv.org/abs/1511.03961 [14] N. Karamchandani, U. Niesen, M. A. Maddah-Ali, and S. Diggavi, “Hierarchical coded caching,” IEEE Trans. on Information Theory, vol. PP, no. 99, pp. 1–1, 2016. [15] M. Ji, G. Caire, and A. F. Molisch, “Fundamental limits of caching in wireless D2D networks,” IEEE Trans. on Information Theory, vol. 62, no. 2, pp. 849–869, Feb 2016. [16] S. P. Shariatpanahi, A. S. Motahari, and B. H. Khalaj, “Multi-server coded caching,” 2015. [Online]. Available: http://arxiv.org/abs/1503.00265 [17] A. Liu and V. K. N. Lau, “Exploiting base station caching in mimo cellular networks: Opportunistic cooperation for video streaming,” IEEE Trans. on Signal Processing, vol. 63, no. 1, pp. 57–69, Jan 2015. [18] B. Azari, O. Simeone, U. Spagnolini, and A. M. Tulino, “Hypergraph-based analysis of clustered co-operative beamforming with application to edge caching,” IEEE Wireless Communications Letters, vol. 5, no. 1, pp. 84–87, Feb 2016. [19] M. Maddah-Ali and U. Niesen, “Cache-aided interference channels,” in IEEE International Symposium on Information Theory (ISIT), June 2015, pp. 809–813. [20] A. Sengupta, R. Tandon, and O. Simeone, “Cache aided wireless networks: Tradeoffs between storage and latency,” 2015. [Online]. Available: http://arxiv.org/abs/1512.07856 [21] F. Xu, K. Liu, and M. Tao, “Cooperative Tx/Rx caching in interference channels: A storage-latency tradeoff study,” IEEE International Symposium on Information Theory (ISIT), 2016, to appear. [22] N. Naderializadeh, M. A. Maddah-Ali, and A. S. Avestimehr, “Fundamental limits of cache-aided interference management,” 2016. [Online]. Available: http://arxiv.org/abs/1602.04207 [23] V. Cadambe and S. Jafar, “Interference alignment and the degrees of freedom of wireless X networks,” IEEE Trans. on Information Theory, vol. 55, no. 9, pp. 3893–3908, Sept 2009. [24] H. Weingarten, Y. Steinberg, and S. Shamai, “The capacity region of the Gaussian multiple-input multiple-output broadcast channel,” IEEE Trans. on Information Theory, vol. 52, no. 9, pp. 3936–3964, Sept 2006. [25] M.

A.

Maddah-Ali

and

U.

Niesen,

“Cache-aided

interference

channels,”

2015.

[Online].

Available:

http://arxiv.org/abs/1510.06121 [26] A. S. Motahari, S. Oveis-Gharan, M. A. Maddah-Ali, and A. K. Khandani, “Real interference alignment: Exploiting the potential of single antenna systems,” IEEE Trans. on Information Theory, vol. 60, no. 8, pp. 4799–4810, Aug 2014. [27] J. R. Munkres, Analysis on manifolds.

Adison-Wesley, 1990.