Multi-antenna Interference Management for Coded Caching

Multi-antenna Interference Management for Coded Caching

Antti Tölli, Seyed Pooya Shariatpanahi, Jarkko Kaleva and Babak Khalaj
Centre for Wireless Communications, University of Oulu, University of Oulu, P.O. Box 4500, 90014, Finland
School of Computer Science, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran
Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran.
firstname.lastname@oulu.fi, pooya@ipm.ir,khalaj@sharif.edu
Abstract

A single cell downlink scenario is considered where a multiple-antenna base station delivers contents to multiple cache-enabled user terminals. Using the ideas from multi-server coded caching (CC) scheme developed for wired networks, a joint design of CC and general multicast beamforming is proposed to benefit from spatial multiplexing gain, improved interference management and the global CC gain, simultaneously. Utilizing the multiantenna multicasting opportunities provided by the CC technique, the proposed method is shown to perform well over the entire SNR region, including the low SNR regime, unlike the existing schemes based on zero forcing (ZF). Instead of nulling the interference at users not requiring a specific coded message, general multicast beamforming strategies are employed, optimally balancing the detrimental impact of both noise and inter-stream interference from coded messages transmitted in parallel. The proposed scheme is shown to provide the same degrees-of-freedom at high SNR as the state-of-art methods and, in general, to perform significantly better than several base-line schemes including, the joint ZF and CC, max-min fair multicasting with CC, and basic unicasting with multiuser beamforming.

I Introduction

The pioneering work of [1] considers an information theoretic framework for the caching problem, through which a novel coded caching (CC) scheme is proposed. In the coded caching scheme the idea is that, instead of simply replicating high-popularity contents near-or-at end-users, one should spread different contents at different caches. This way, at the content delivery phase, common coded messages could be broadcast to different users with different demands, that would benefit all of the users. This global caching gain relies on the observation that almost in all communication scenarios, broadcasting is much simpler than unicasting. Follow-up works extend the coded caching scheme proposed in [1] to other setups such as online coded caching [2], hierarchical coded caching [3], and multi-server scenarios [4].

All the above works consider the gains of coded caching in wired networks, far from mobile delivery requirements. Thus, the specific characteristics of wireless networks must be investigated to be able to implement the original idea of [1] in mobile delivery scenarios. To this end, the authors in [5] consider the effect of delayed channel state information at the transmitter (CSIT). Moreover, the work [6] considers a cache-enabled wireless interference channel while [7] treats the same setup with mixed-CSIT. In addition, the authors in [8] investigate coded caching schemes in wireless device-to-device network. Furthermore, [9] assumes interference management in a cellular setup with coded caching, and [10] considers using the rate-splitting idea along with coded caching.

While all these papers consider a wireless networks, the analysis is for the high signal-to-noise-ratio (SNR) regime, and in terms of degrees-of-freedom (DoF). As high SNR analysis is not always a good indicator for practical implementations performance, there is still a gap which should be filled in with finite SNR analysis of the CC idea. To fill this gap, the papers [11] and [12] propose different CC schemes in a wireless multiple-input single-output broadcast channel (MISO-BC), and provide a finite SNR analysis, in different system operating regimes. While the main idea in [11] is to use rate-splitting along with CC, the authors in [12] propose a joint design of CC and zero-forcing (ZF) to benefit from the spatial multiplexing gain and the global gain of CC, at the same time. While the ideas in [12] originally came from adapting the multi-server CC scheme of [4] (which is almost optimal in terms of DoF as shown in [6]) to a Gaussian MISO-BC, the interesting observations in [12] reveal that careful code and beamformer design modifications should be further considered having significant effects on the finite SNR performance.

In this paper, extending the joint interference nulling and CC concept originally proposed in [12], a joint design of CC and generic multicast beamforming is introduced to benefit from spatial multiplexing gain, improved management of inter-stream interference from coded messages transmitted in parallel, and the global caching gain, simultaneously. The general signal-to-interference-plus-noise ratio (SINR) expressions are handled directly to optimally balance the detrimental impact of both noise and inter-stream interference at low SNR. As the resulting optimization problems are not necessarily convex, successive convex approximation (SCA) methods are used to devise efficient iterative algorithms similarly to existing multicast beamformer design solutions [13].

Ii System Model

Downlink transmission from a single -antenna BS serving cache enabled single-antenna users is considered. The BS is assumed to have access to a library of files , each of size bits. Every UE is assumed to be equipped with a cache memory of capacity bits. Furthermore, each user has a message stored in its cache, where denotes a function of the library files with entropy not larger than bits. This operation is referred to as the cache content placement, and it is performed once and at no cost, e.g. during network off-peak hours.

Upon a set of requests at the content delivery phase, the BS multicast coded signals, such that at the end of transmission all users can reliably decode their requested files. Notice that user decoder, in order to produce the decoded file , makes use of its own cache content as well as of its own received signal from the wireless channel.

The received signal at user terminal at time instant can be written as

(1)

where the channel vector between the BS and UE is denoted by , denotes the multicast beamformer dedicated to users in subset of set of users, and is the corresponding multicast message chosen from a unit power complex Gaussian codebook at time instant . In the following, the time index is ignored for simplicity. The receiver noise is assumed to be circularly symmetric zero mean . Finally, the CSIT of all users is assumed to be perfectly known at the BS.

Iii Multiantenna Coded Caching for Finite SNR

In this work, we focus on the worst-case (over the users) delivery rate at which the system can serve any users requesting any file of the library. Multicasting opportunities due to the coded caching [1, 4, 12] are utilized to devise efficient multiantenna multicast beamforming method that perform well over the entire SNR region.

In the following, for the sake of easy exposure, we introduce the basic concept for two simple scenarios and discuss the generalization of the proposed scheme afterwards.

Iii-a Scenario 1: , , and

Consider a content delivery scenario illustrated in Fig. 1, where a transmitter with antennas should deliver requests arising at users from a library of size files each of bits. Suppose that in the cache content placement phase each user can cache files, without knowing the actual requests beforehand. In the content delivery phase we suppose each user requests one file from the library. Following the same cache content placement strategy as in [1] the cache contents of users are as follows

where each file is divided into three non-overlapping equal-sized subfiles.

Fig. 1: Scenario 1: , , and

At the content delivery phase suppose that the 1st, the 2nd, and the 3rd user request files , , and , respectively. In the simple broadcast scenario in [1], the following coded messages are sent by the transmitter one after another

(2)

where represents summation in the corresponding finite field. In this coding scheme of [1], each coded message is heard by all the three users, but is only beneficial to two users. For example, is useful for the first and second user only. This multicasting gain is called as the Global Caching Gain. It can be easily checked that after the transmission is concluded all the users can decode their requested files. Moreover, for every possible combination of the users requests the scheme works, with the same cache content placament, but with another set of coded delivery messages.

Now, in the given Scenario 1 we can combine the spatial multiplexing gains provided by transmit antennas, and the global caching gain following the scheme proposed in [12] (see also [4, 6]). The basic idea in [12] is to force the unwanted messages at each user to zero by sending

(3)

where stands for the modulated version of , chosen from a unit power complex Gaussian codebook [12]. Although, this scheme is order-optimal in terms of Degrees-of-Freedom (DoF) [6] it is suboptimal at low SNR regime [12].

Instead of nulling interference at unwanted users, general multicast beamforming vectors are defined as

(4)

Then, the received signals at users will be

where the desired terms are underlined. Let us focus on user who is interested in decoding both , and while appears as Gaussian interference. Thus, from receiver 1 perspective, is a Gaussian Multiple Access Channel (MAC). Suppose now user 1 can decode both of its required messages with the equal rate111Symmetric rate is imposed to minimize the time needed to receive both messages , and .

(5)

Thus, the total useful rate is . Since the user 1 needs to receive the missing bits ( and ), the time needed to decode file is

(6)

As all the users decode their files in parallel, the total time needed to complete the decoding process is constrained by the worst user as

(7)

Then, the Symmetric Rate (Goodput) per user will be

(8)

which, when optimized with respect to the beamforming vectors, can be found as

(9)

The symmetric rate maximization problem for users is given as

(10)

which can be equally presented in an epigraph from as

(11)

Problem (11) is non-convex due to the SINR constraints. Similarly to [13], successive convex approximation (SCA) approach can be used to devise an iterative algorithm that is able to converge to a local solution. To begin with, the SINR constraint for can be reformulated as

(12)
(13)

Now, the R.H.S of (13) is convex quadratic-over-linear function and it can be linearly approximated (lower bounded) as

(14)

where denote the fixed values (points of approximation) for the corresponding variables from the previous iteration. Using (III-A), the approximated problem is written as

(15)

This is a convex problem that can be readily solved using existing convex solvers. However, the logarithmic functions require further approximations to be able to apply the convention of convex programming algorithms. Problem (15) can be equally formulated as computationally efficient second order cone problem (SOCP). To this end, we note that the sum rate constraint can be bounded as

Now, the equivalent SOCP reformulation follows as

(16)

Finally, a solution for the original problem (III-A) can be found by solving (16) in an iterative manner using SCA, i.e, by updating the points of approximations in (III-A) after each iteration. As each difference-of-convex constraint in (13) is lower bounded by (III-A), the monotonic convergence of the objective of (16) is guaranteed. Note that the final symmetric rates are achieved by time sharing between the rate allocations corresponding to different points (decoding orders) in the sum rate region of the MAC channel.

As a lower complexity alternative, a zero forcing solution, denoted as CC with ZF, is also proposed. By assigning , , , the interference terms are cancelled and (III-A) becomes:

(17)

where . This is readily a convex power optimization problem with three real valued variables, and hence it can be solved in an optimal manner.

In the following, we introduce three baseline schemes that are used as reference cases for the proposed multiantenna caching scheme.

Iii-A1 1st Baseline Scheme: CC with ZF (equal power) [12]

If the multicast transmit powers are made equal, , the resulting scheme is the same as originally published in [12].

Iii-A2 2nd Baseline Scheme: MaxMinSNR Multicasting

The message is multicast to the users 1 and 2, without any interference (orthogonally), by sending the signal . A single transmit beamformer is found to minimize the time needed for multicasting the common message:222This multicast maxmin problem is NP-hard in general, but near-optimal solutions can be obtained by a semidefinite relaxation (SDR) approach, see [12] and the references therein.

(18)

Similarly, the messages and should be delivered to the users with corresponding times and . Finally the resulting symmetric rate (Goodput) per user will be

(19)

Note that in this scheme only the coded caching gain is exploited, while the multiple transmit antennas are used just for the beamforming gain.

Iii-A3 3rd Baseline Scheme: MaxMinRate Unicast

In this scheme, only the local caching gain is exploited and the CC gain is ignored altogether. The BS simply sends two parallel independent streams to the users at each time instant.

Now, let us consider users 1 and 2 in time slot 1. The transmitted signal to deliver and to users 1 and 2, respectively, is given as . Thus the delivery time of bits is

(20)

where

(21)

The minimum delivery time in (19) can be equivalently formulated as a maxmin SINR problem and solved optimally. By repeating the same procedure for the subsets and , the symmetric rate expression is equivalent to (19).

Iii-B Scenario 2: , , and

In this scenario, we assume that the BS transmitter has antennas, and there are users each with cache size , requesting files from a library of files. Following the same cache content placement strategy as in [1] the cache contents of users are as follows

where here each file is divided into four non-overlapping equal-sized subfiles.

At the content delivery phase suppose that the users request files , respectively. Here, we have and the subsets and will be of size , and , respectively (for details see [4]). Following the approach of Scenario 1, the transmit signal vector is

(22)

where

It can be easily verified that if each multicast message is delivered to all the members of then all the users can decode their requested files.

The received signals at each user are

where the desired terms are underlined. Thus, as in Scenario 1, each user faces a MAC channel with three desired signals, three Gaussian interference terms, and one noise term. Suppose that user can decode each of its desired signals with the rate . Then this user receives useful information with the rate , and the time required to fetch the entire file is . Following the same steps as in (7)–(8), the symmetric rate per user can be found as

(23)

where we have used the notation . As the 3-dimensional MAC rate region for each user is formed by 7 rate constraints, the following optimization problem is solved to find the symmetric rate per stream:

In order to solve the above non-convex problem, the SCA method is again used and the SINR constraints are approximated similarly to (13)–(III-A).

The Second Baseline Scheme similarly as in the Scenario 1. In summary, each is being delivered to the users in the subset , without interference. Thus, in total six time slots are needed to transmit the corresponding multicast messages.

The Third Unicasting Baseline Scheme is also the same as in Scenario 1. Let´s first consider the transmitted symbols to the first the subset of users with the corresponding minimum transmission time

(24)

where the rates are given as in (21). Similarly, the transmitted signals to three remaining user subsets , and are , and , respectively. Finally, the Symmetric Rate will be .

Iii-C General , , and

The guidelines for constructing multicast messages for multi-server/antenna coded caching with any , , and are given in [4]. For example, the case , , and would require altogether multicast messages and each user should be able to decode 4 multicast messages. Thus, the total number of rate constraints would be while the number of SINR constraints would be . In general, if , , are increased with the same ratio, the number of rate constraints grows exponentially, i.e., per user. As an efficient way to limit the complexity of the problem (with a certain performance loss), we may want to limit the size of the subsets benefiting from a common transmitted signal to three, for example. Consequently, the number of multicast messages to be decoded at each user would be two, as in (III-A). This can be achieved by splitting each subfile into minifiles and using slotted transmission to serve different combinations of users in each slot. More detailed development of the reduced complexity scheme will be included in the journal version of this work.

Iv Numerical Examples

The numerical examples are simulated for Scenarios 1 and 2. The channels are considered to be i.i.d. complex Gaussian. The average performance is attained over independent channel realizations. The SNR is defined as , where is the power budget and is the fixed noise floor.

Fig. 2: Coded caching with multiantenna transmission, and

Fig. 2 shows the performance of the interference coordination with coded caching in Scenario 1, i.e., with users and transmit antennas. It can be seen that the proposed CC-BF-SCA scheme achieves dB gain at low SNR as compared to the ZF with equal power loading [12]. At high SNR, the ZF with optimal power loading in (III-A) achieves comparable performance while other schemes have significant performance gap. At low SNR regime, the simple MaxMin SNR multicasting with CC has similar performance as the proposed CC-BF-SCA scheme. This is due to the fact that, at low SNR, an efficient strategy for beamforming is to concentrate all available power to a single (multicast) stream at a time and to serve different users/streams in TDMA fashion. Note that (18) is usually solved via reformulation as a semidefinite programming (SDP) problem. Due to simultaneous global coded caching gain and inter-stream interference handling, both CC-BF-SCA and CC-ZF schemes achieve an additional DoF, which was already shown (for high SNR) in [4, 12]. The unicasting scheme does not perform well in this scenario as it does not utilize the global caching gain (only the local cache) and the spatial DoF is limited to two.

In Fig. 3, the number of transmit antennas is increased to . This provides more than dB additional gain for the CC-BF at low SNR, when compared to the antenna scenario, while the DoF is the same for all the compared schemes. The optimal ZF multicast beamformer solution is no longer trivial, as the additional antenna makes the interference free signal space two-dimensional for the ZF schemes. A heuristic solution is used where orthogonal projection is first employed to get interference free signal space and then the strongest eigenvector of the stacked user channel matrix, projected to null space, is used to get a sufficiently good direction within the interference free signal space. It can be seen that the ZF scheme does achieve the same DoF as CC-BF method, but there is a constant performance gap at high SNR. Interestingly, the CC-BF scheme with antennas has better performance than MaxMin SINR unicast with antennas. Both schemes have the same DoF, but the global caching gain is more beneficial than the additional spatial DoF of the unicast method.

Fig. 3: Coded caching with multiantenna transmission, and

The performance of different schemes in Scenario 2 is illustrated in Fig. 4. The CC-BF-SCA achieves dB gain to CC-ZF scheme at low SNR, which is considerable more than in the less complex Scenario 1. Again, at high SNR, the CC-ZF with optimal power loading provides comparable performance with lower computational complexity. Similarly to Scenario 1, the performance of the MaxMin SNR multicasting overlaps with CC-BF-SCA at low SNR but there is a large gap at high SNR due to the DoF difference.

Fig. 4: Coded caching with multiantenna transmission, and

V Conclusions

Multiantenna multicasting opportunities provided by caching at user terminal were utilized to devise an efficient multiantenna transmission with coded caching. General multicast beamforming strategies were employed together with CC, optimally balancing the detrimental impact of both noise and inter-stream interference from coded messages transmitted in parallel. The proposed scheme was shown to perform significantly better than several base-line schemes over the entire SNR region.

References

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