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Nonsmooth Optimization Algorithms for Multicast Beamforming inContent-Centric Fog Radio Access Networks
Nguyen, H. T., Tuan, H. D., Duong, T. Q., Poor, H. V., & Hwang, W-J. (2020). Nonsmooth OptimizationAlgorithms for Multicast Beamforming in Content-Centric Fog Radio Access Networks. IEEE Transactions onSignal Processing. https://doi.org/10.1109/TSP.2020.2964250
Published in:IEEE Transactions on Signal Processing
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Download date:21. May. 2021
Nonsmooth Optimization Algorithms for MulticastBeamforming in Content-Centric Fog Radio Access
NetworksHuy T. Nguyen, Hoang Duong Tuan, Trung Q. Duong, H. Vincent Poor, and Won-Joo Hwang
Abstract—This paper considers a content-centric fog radioaccess network (F-RAN). Its multi-antenna remote radio heads(RRHs) are capable of caching and executing signal processingfor content delivery to its users. The fronthaul traffic is thus savedsince its baseband processing unit (BBU) needs to transfer onlythe cache-missed content items to the RRHs via limited-capacityfronthaul links. The problem of beamforming design maximizingthe energy efficiency in content delivery subject to the quality-of-content-service constraints in terms of content throughputand fronthaul limited-capacity is addressed. Unlike the user’sthroughput in user-centric networks, the content throughput incontent-centric networks is no longer a differentiable functionof the beamforming vectors. The problem is inherently high-dimensional due to the involvement of many beamforming vectorseven in simple cases of three RRHs serving three users. Path-following algorithms, which invoke a simple convex quadraticoptimization problem to generate a better feasible point, are pro-posed for computation of this nonsmooth and high-dimensionaloptimization problem. We also employ the generalized zero-forcing beamforming, which forces the multi-content interferenceto zero or nearly to zero to reduce the problem dimensionality forcomputational efficiency. The numerical results are provided todemonstrate their computational effectiveness. They also revealthat when the fronthaul traffic becomes more flexible, the hard-transfer fronthauling is more energy efficient than the soft-transfer fronthauling.
Index Terms—Fog radio access network (F-RAN), multi-inputsingle output (MISO), hard-transfer fronthauling, soft-transferfronthauling, unstructured beamforming, structured beamform-ing, content throughput
I. INTRODUCTION
Wireless communications are continuously evolving toachieve higher-rate and lower-latency [1]. With the exponentialgrowth of wireless devices for the Internet-of-things, the wire-less service is shifted from the traditional user-centric trending,such as emails, phone calls, and web browsing, to the emerging
This work was supported in part by the U.K. Royal Academy of Engineer-ing Research Fellowship under Grant RF1415\14\22.
Huy T. Nguyen is now with School of Computer Science andEngineering, Nanyang Technological University, Singapore (email:[email protected]).
Won-Joo Hwang is with the School of Biomedical Convergence Engi-neering, Pusan National University, Gyeongsangnam-do 50612, Korea (email:[email protected]).
Hoang Duong Tuan is with the School of Electrical and Data Engineering,University of Technology Sydney, Broadway, NSW 2007, Australia (email:[email protected]).
Trung Q. Duong is with the School of Electronics, Electrical Engineeringand Computer Science, Queen’s University Belfast, Belfast BT7 1NN, U.K.(email: [email protected]).
H. Vincent Poor is with the Department of Electrical Engineering, PrincetonUniversity, Princeton, NJ 08544, USA (email: [email protected])
content-centric trending, like video streaming, file sharing, andmobile TV [2], [3]. Cloud radio access network (C-RAN) of acentralized baseband signal processing unit (BBU) linked withmultiple low-complexity remote radio heads (RRHs) throughfronthaul links [4] has emerged as a promising architecture forsupporting the quality-of-content-service. The dense deploy-ment of RRHs brings the network closer to its users and thushelps to improve spectral and energy efficiency [5], [6], whichcan be further optimized by appropriate resource allocation(see e.g. [7], [8] for weighted sum-rate optimization and[9]–[11] for transmission cost minimization). Achieving theattractive spectral and energy efficiency, C-RAN still hardlymeets the latency requirements for new emerging applications.In fact, its throughput is rapidly downgraded during the high-peak traffic periods because of the limitation of fronthaulcapacity.
Fog radio access network (F-RAN) is a C-RAN, where theRRHs are upgraded with local storages and signal processingunits to cache and process content items for their delivery tothe users [12]. The most frequently requested content itemsare often pre-fetched at the RRHs’ cache during the off-peak traffic periods [13]–[15]. In real time content service,the cache-missed content items are transferred from the BBUto the assigned RRHs through the fronthaul links. The newfeatures in F-RAN promise the gain in both spectral efficiencyand transmission latency [16], [17]. More importantly, thefronthaul consumption is remarkably saved in low-latencyapplications since the BBU now transfers only the cache-missed content items to the RRHs. Unlike RRHs in C-RAN,which simply forward the received signals from the BBU[5], the RRHs in F-RAN transmit both the self-processedsignals for the cached content items and the received signalsof the cache-missed content items to the desired users. Thedesign of optimal resource allocation in F-RAN, which mustconcurrently incorporate fronthaul, precoding, and cachinginto account is much more complex than that in C-RAN[18]. Unlike the user’s throughput in user-centric networks,the content throughput is no longer a smooth (differentiable)function of the beamforming vectors. This means that thepopular techniques of smooth function optimization, such aswater-filling, duality between downlink and uplink channels orconvex relaxation for user throughput function optimization,are no longer efficient for content throughput function opti-mization. In [19] and [20], optimization in the beamformingvectors is recast as optimization in the matrices of theirouter products, which is then relaxed by dropping the rank-
2
constraint on these outer products. The authors then use d.c.(difference of two convex) iterations [21] for computation,which is still of high computational complexity as in eachiteration they involve logarithm determinant function opti-mization with no known solver of polynomial complexity. Itis not surprising at all that numerical examples in [19] arerestricted on the simplest and lowest-dimensional cases ofpower allocation for three single antenna RRHs in servingthree users. It should be mentioned that d.c. iterations canbe used for direct optimization in the beamforming vectors[22], [23] but their computational complexity is still high. Onthe other hand, the successive convex function approximationused in [10, eq. (28)] does not lead to the convergence of itsiterations to a suboptimal solution, each of which still involveslogarithm function optimization with no solver of polynomialcomplexity. The issue of high computational complexity is alsopresented in [24]–[26].
On the background that energy efficiency is a pivot forpromoting wireless networks [27]–[31], the design problemof multicast content beamforming in a content-centric F-RANunder both hard-transfer fronthauling (HTF) and soft-transferfronthauling (SFT) for maximizing the energy efficiency incontent delivering to the users end is considered the firsttime in this paper. The optimization objective is the minimumratio between the content throughput and its consumed power.Besides the common constraints to cap RRHs’ transmis-sion power and fronthaul capacity limitation, the quality-of-content-service constraints in terms of content throughputsare also incorporated into account to secure the full deliv-ery of all requested content items. As expected, they leadto nonconvex and nonsmooth optimization problems, whichalready are high-dimensional for simple scenarios of threeRRHs serving three users with many beamforming vectorsinvolved, especially under STF [16]. The paper contributionis two-fold:
• To develop path-following algorithms, which generate abetter feasible point at each iteration by solving a sim-ple convex quadratic optimization of low computationalcomplexity, for their computation;1
• To address the issue of problem high dimension, thegeneralized zero-forcing beamforming, which forces themulti-content interference to zero or nearly to zero isfurther employed. The design of beamforming vectorsrelegates to their power allocation only to reduce theproblem dimensionality. Path-following algorithms fortheir optimal power allocation is then accordingly devel-oped.
The numerical results are provided to demonstrate the viabilityof the proposed computational solutions, which also reveal thatSTF helps to achieve a better quality-of-content-service thanHTF in the low fronthaul capacity regime but the former iseasily outperformed by the latter once the fronthaul links aresupplied with enough capacity.
1One should distinguish the path-following algorithms from the sequentialconvex algorithms, which aim to solve the penalized problems by a sequenceof convex problems to find a good feasible point.
Fig. 1. F-RAN
The rest of this paper is organized as follows. Section IIsketches the F-RAN modeling. Content multicast beamform-ing under HTF and STF are respectively considered in SectionIII and Section IV. Numerical results are presented in SectionV and conclusions are drawn in Section VI.
Notation. Notation is standard but only optimization vari-ables are boldfaced to emphasize their appearance in nonlinearfunctions. Also |A| for the set A is its cardinality.
II. F-RAN MATHEMATICAL MODELING
Fig. 1 illustrates an F-RAN linking a BBU with NR nt-antenna array RRHs (indexed by i ∈ NR , {1, . . . , NR})through the capacity limited fronthaul links to serve contentsto Nu single-antenna users (UEs) (indexed by k ∈ Nu ,{1, . . . , Nu}). Each RRH is equipped with a cache of capacityNc. Denote by C the capacity of the fronthaul link betweenthe BBU and RRH i.
The UEs request contents from a library of F files of thetotal capacity Fc . The files f ∈ F , {1, . . . , F} are labelledin order of their popularity, which is distributed according toa Zipf’s distribution [32] P (f) = f−γ/
∑j∈F j
−γ with thepopularity exponent γ > 0. The larger γ is the more concentra-tion is on most popular contents. Under uncoded strategy, eachfile f is split into L subfiles (f, `), ` ∈ L , {1, . . . , L}. With-out loss of generality, the fractional caching capacity at eachRRH is fixed at µ = Nc/Fc, so each RRH can store a fractionµ of each file in the pre-fetching phase [33]. For binary-valuedcif,` such that cif,` = 1 if and only if (f, `) is cached by RRHi, it is true that
∑Ff=1
∑L`=1 c
if,`/L ≤ µF, i ∈ NR.
The set of requested files is defined as
Freq , {fν ∈ F : ν = 1, . . . , Nreq},
3
so 1 ≤ Nreq ≤ Nu, where Nreq is the number of requestedfiles. For each ν, define Kν as the set of UEs, who requestfile fν :
Kν , {1ν , . . . , |Kν |ν}. (1)
In what follows, for convenience of presentation, we usethe notation ciν,` to refer to the above cifν ,` and Nreq ={1, . . . , Nreq}.
The subfile (fν , `) requested by UE kν is transferred on thefronthaul links to NF RRHs with the largest channel gainsfrom them to this UE among the RRHs that do not have (fν , `)on their cache. Accordingly, define
diν,` =
{1 if subfile (fν , `) is transferred to RRH i,
0 otherwise.(2)
For i = 1, . . . , NR, the set of subfiles that are either in RRHi’s cache or are received by RRH i from the BBU is definedas
Ni , {(ν, `) : ν ∈ Nreq, and ciν,` = 1 or diν,` = 1}. (3)
III. MULTICAST BEAMFORMING UNDER HTF
Under HTF, the BBU transfers hard information of thecache-missed content items to the RRHs through the fronthaullinks. Thus the transmit signal at RRH i is
xi =∑
(ν,`):(ν,`)∈Ni
viν,`sfν ,`, (4)
where viν,` ∈ Cnt is the baseband beamformer for sfν ,` ∈ Cof unit power, which encodes (fν , `). The signal received atUE kν is
ykν =
NR∑i=1
hHkν ,i∑
(ν′,`)∈Ni
viν′,`sfν′ ,` + zkν
=
L∑`=1
NR∑i=1
hHkν ,i∑
ν:(ν,`)∈Ni
viν,`sfν ,`
+∑
ν′∈Nreq\{ν}
L∑`=1
∑i:(ν′,`)∈Ni
hHkν ,iviν′,`sfν′ ,`
+zkν , kν ∈ Kν , (5)
where hkν ,i ∈ Cnt is the channel from RRH i to UE kν , andzkν ∈ C is the background noise with covariance σ.
For v , {viν′,` : (ν′, `) ∈ Ni, i ∈ NR}, upon successiveinterference cancellations (SIC) decoding in the ascendingdecoding order, the throughput of sfν ,` is
rν,`(v) , mink=1,...,|Kν |
rkν ,`(v), (6)
where
rkν ,`(v) , ln
(1 +
|Lkν ,`(v)|2
φkν ,`(v) + σ
), (7)
withLkν ,`(v) ,
∑i:(ν,`)∈Ni
hHkν ,iviν,` (8)
and
φkν ,`(v) ,L∑
`′=`+1
∣∣∣∣∣∣∑
i:(ν,`′)∈Ni
hHkν ,iviν,`′
∣∣∣∣∣∣2
+∑
ν′∈Nreq\{ν}
L∑`=1
∣∣∣∣∣∣∑
i:(ν′,`)∈Ni
hHkν ,iviν′,`
∣∣∣∣∣∣2
. (9)
Let Pnon be the total non transmission power consumptionsat the RRHs. The total power consumption for delivering filefν is
πν(v) ,NR∑i=1
∑`:(ν,`)∈Ni
||viν,`||2 + Pnon. (10)
Accordingly, the energy efficiency in delivering the particularfile fν is calculated by log2(e)ϕν(v) for
ϕν(v) ,L∑`=1
rν,`(v)/πν(v),
while the energy efficiency in delivering allthe requested files is log2(e)ϕ(v) for ϕ(v) ,∑Nreqν=1
∑L`=1 rν,`(v)/
∑Nreqν=1 πν(v).
We consider the following problem of max-min energyefficiency optimization:
maxv
minν=1,...,Nreq
ϕν(v) s.t. (11a)
Nreq∑ν=1
L∑`=1
diν,`rν,`(v) ≤ log2(e)C, i ∈ NR, (11b)
rmin/ log2(e) ≤ rν,`(v), (ν, `) ∈ Nreq × L, (11c)Nreq∑ν=1
∑`:(ν,`)∈Ni
||viν,`||2 ≤ P, i ∈ NR. (11d)
Recalling (2) for the definition of diν,`, the fronthaul constraintis expressed by (11b), which means that the hard transfer ofsubfiles to RRH i is constrained by the fronthaul capacityC. The quality-of-content-service constraint (11c) sets theminimum required rate rmin of the file requests [34] toguarantee the delivery of all requested files within an exacttime. For a given power budget P > 0, (11d) constrains thetransmit power at each RRH.
Note that [10] considered the problem of power consump-tion minimization when all RRHs are equipped with a singleantennas and have no cache capacity while the files are notsplit, which is different from the above problem (11). Theabove problem is also totally different from the followingproblem of sum rate maximization in [19]:
maxv,Rfk ,k=1,...,Nu
mink=1,...,Nu
L∑`=1
Rk,` s.t. (11d) (12a)
Nu∑k=1
L∑`=1
dik,`Rk,` ≤ C, i ∈ NR, (12b)
Rk,`/ log2(e) ≤ rk,`(v), (k, `) ∈ Nu × L, (12c)Rk,` ≤ S/L, (k, `) ∈ Nu × L, (12d)
4
which may leave many subfiles undelivered with the almostzero delivery rates for maximizing the sum rate. As rν,`(v)is a very complex function of v, [19] used the outer productsWi
f,` = vif,`(vif,`)
H to express it as a d.c. functions in Wif,`,
whose rank-one constraint is then dropped to facilitate d.c.iterations [21] for computation. The optimal solution of thisrelaxed problem cannot result in a feasible point for (12) unlessit is of rank-one, which rarely happens. Also, the d.c. iterationsstill involve optimization of logarithm determinant functionswith no known solver of polynomial complexity.
The content throughput functions rν,`(v) defined from (6)are not only non-concave but non smooth (non-differentiable).As such, (11) is a nonconvex problem of maximizing the non-concave and non-smooth objective function in (11a) subject tonon-convex and non-smooth constraints (11b) and (11c) of thelimited fronthaul capacity and the quality-of-content-service.The next subsections present its computation.
A. Unstructured beamforming
Let v(κ) , [viν′,`](ν′,`)∈Ni,i∈NR be the feasible point for(11) that is found from the (κ − 1)th iteration. To providetractable inner approximations for the nonconvex constraints(11b) and (11c) we need to develop both lower bounding con-vex approximation and upper bounding concave approximationfor the content throughput function rν,`(v).
We apply the inequality (75) in the appendix for x =Lkν ,`(v), ||y||2 = φkν ,`(v) and x = Lkν ,`(v
(κ)), ||y||2 =φkν ,`(v
(κ)) to obtain the following concave lower boundingapproximation for rkν ,`(v):
rkν ,`(v) ≥rlw,(κ)kν ,`
(v)
,α(κ)kν ,`−
β(κ)kν ,`
2<{L∗kν ,`(v(κ))Lkν ,`(v)} − |Lkν ,`(v(κ))|2
− χ(κ)kν ,`
φkν ,`(v), (13)
over the trust region
2<{L∗kν ,`(v(κ))Lkν ,`(v)} − |Lkν ,`(v(κ))|2 > 0, (14)
for
α(κ)kν ,`
, ln(1 +|Lkν ,`(v(κ))|2
φkν ,`(v(κ)) + σ
)
+|Lkν ,`(v(κ))|2
|Lkν ,`(v(κ))|2 + φkν ,`(v(κ)) + σ
×(
2− σ
φkν ,`(v(κ)) + σ
),
0 ≤ β(κ)kν ,`
,|Lkν ,`(v(κ))|4
|Lkν ,`(v(κ))|2 + φkν ,`(v(κ)) + σ
,
0 ≤ χ(κ)kν ,`
,|Lkν ,`(v(κ))|2(
|Lkν ,`(v(κ))|2 + φkν ,`(v(κ)) + σ
)× 1(
φkν ,`(v(κ)) + σ
) . (15)
Therefore, a lower bounding approximation for rν,`(v) is
rlw,(κ)ν,` (v) , min
k=1,...,|Kν |rlw,(κ)kν ,`
(v), (16)
which is a concave function as the minimum of a family ofconcave functions rlw,(κ)
kν ,`(v) [35]. The non-convex constraint
(11b) is thus innerly approximated by the following convexconstraint
rlw,(κ)ν,` (v) ≥ rmin/ log2(e), (ν, `) ∈ Nreq × L. (17)
To avoid unnecessary excessively complex approximation,note that by (16), the fronthaul constraint (11b) is guaranteedby
Nreq∑ν=1
L∑`=1
diν,`rν,`(v) ≤ log2(e)C, i ∈ NR, (18)
with
rν,`(v) ,1
|Kν |
|Kν |∑k=1
rkν ,`(v). (19)
We apply the inequality (78) in the appendix for x =Lkν ,`(v), ||y||2 = φkν ,`(v) and x = Lkν ,`(v
(κ)), ||y||2 =φkν ,`(v
(κ)) to obtain the following convex upper boundingapproximation for rkν ,`(v):
rkν ,`(v) ≤ rup,(κ)kν ,`
(v)
, η(κ)kν ,`
+ µ(κ)kν ,`
|Lkν ,`(v)|2
2λkν ,`(v)− φkν ,`(v(κ)) + σ(20)
over the trust region
2λkν ,`(v)− φkν ,`(v(κ)) > 0, k = 1, . . . , |Kν |, (21)
for
η(κ)kν ,`
, ln(1 +|Lkν ,`(v(κ))|2
φkν ,`(v(κ)) + σ
)
− |Lkν ,`(v(κ))|2
[Lkν ,`(v(κ))]2 + φkν ,`(v
(κ)) + σ,
µ(κ)kν ,`
,1− |Lkν ,`(v(κ))|2
|Lkν ,`(v(κ))|2 + φkν ,`(v(κ)) + σ
,
λkν ,`(v) ,L∑
`′=`+1
<
∑i:(ν,`′)∈Ni
hHkν ,ivi,(κ)ν,`′
∗
×
∑i:(ν,`′)∈Ni
hHkν ,iviν,`′
+
∑ν′∈Nu\{ν}
L∑`=1
<
∑i:(ν′,`)∈Ni
hHkν ,ivi,(κ)ν′,`
∗
×
∑i:(ν′,`)∈Ni
hHkν ,iviν′,`
. (22)
Therefore, a convex upper bounding approximation for rν,`(v)is
rup,(κ)ν,` (v) ,
1
|Kν |
|Kν |∑k=1
rup,(κ)kν ,`
(v), (23)
and the non-convex constraint (11b) is innerly approximatedby the following convex constraint
Nreq∑ν=1
L∑`=1
diν,`rup,(κ)ν,` (v) ≤ log2(e)C, i ∈ NR. (24)
5
Set
θ(κ) = minν=1,...,Nreq
∑L`=1 rν,`(v
(κ))
πν(v(κ)). (25)
Particularly, θ(0) = minν=1,...,Nreq
∑L`=1 rν,`(v
(0))/πν(v(0)).At the κ-th iteration we solve the following convex optimiza-tion problem to generate the next feasible point v(κ+1) for(11):
maxv
Φ(κ)(v) , minν=1,...,Nreq
[
L∑`=1
rlw,(κ)ν,` (v)− θ(κ)πν(v)]
s.t. (11d), (14), (17), (21), (24). (26)
The computational complexity of (26) is
O(n2m2.5 +m3.5), (27)
where n = Nt∑NRi=1 |Ni|, which the number of its scalar
variables, and m = NR(2+∑NRi=1 |Ni|)+NreqL+
∑Nreqν=1 |Kν |,
which is the number of its constraints.Note that any feasible point for (26) is feasible for (11). By
(25), Φ(κ)(v(κ)) = 0 so Φ(κ)(v(κ+1)) > 0, which means
θ(κ+1) = minν=1,...,Nreq
∑L`=1 rν,`(v
(κ+1))
πν(v(κ+1))
≥ minν=1,...,Nreq
∑L`=1 r
lw,(κ)ν,` (v(κ+1))
πν(v(κ+1))> θ(κ).
As such {v(κ)} is a sequence of improved feasible points for(11), which converges at least to a local optimal solution of(11) satisfying the Karush-Kuh-Tucker (KKT) condition [21].
To locate an initial feasible point for (12), initialized by anyfeasible point for the convex constraint (11d) we iterate
maxv
min
{min
(ν,`)∈Nreq×L[rlw,(κ)ν,` (v)
rmin/ log2(e)− 1],
mini∈NR
[1− 1
log2(e)C
Nreq∑ν=1
L∑`=1
diν,`rup,(κ)ν,` (v)]
s.t. (11d), (14), (21), (28)
until the value of the objective in (28) is more or equal to0. If no such value is found, the problem (12) is declaredinfeasible. However, in our simulations we never experiencedwith the infeasibility issue. The pseudo code for the proposedcomputational procedure is given by Algorithm 1.
Algorithm 1 Multicast Unstructured Beamforming Algorithmfor HTF
1: Set κ = 0. Iterate (28) for a feasible point v(0) for (11).2: Repeat until convergence.3: Solve the convex optimization problem (26) to generate
the next feasible point v(κ+1) for (11). Set κ := κ+ 1.4: Output v(κ) as the optimal solution of (11).
B. Structured beamforming
To reduce the computational complexity of unstructuredbeamformers, which prevents them to work for large numbersof RRHs or UEs, this subsection follows [36]2 to focus onstructured beamformers. For this, set
vν,` , Col(viν,`)i:(ν,`)∈Ni ∈ Cnt|Qν,`|, (29)
which consists of beamformers for sfν ,` arranged in a singlevector, where Qν,` , {i ∈ NR : (ν, `) ∈ Ni}, which is theset of RRHs that either have (fν , `) in their cache or receive(fν , `) from the BBU, and |Qν,`| is its cardinality.
The interference of sfν ,` to UE kν′ (ν′ 6= ν) is∑i:(ν,`)∈Ni h
Hkν′ ,i
viν,` = ψH(ν,`),kν′vν,`, where ψ(ν,`),kν′
,
Col(hkν′ ,i)i:(ν,`)∈Ni ∈ Cnt|Qm,`|. Rewrite (5) as
ykν =
L∑`=1
∑i:(ν,`)∈Ni
hHkν ,iviν,`sfν ,`
+∑
ν′∈Nreq\{ν}
L∑`=1
ψH(ν′,`),kνvν′,`sf ′
ν ,` + zk. (30)
Define the interference matrix
ψν,` = Row(ψ(ν,`),k
){kν′ :ν′∈Nreq\{ν}}
∈ Cnt|Qν,`|×(Nreq−1),
(31)which consists of ψ(ν,`),k arranged in block row, where theindex set {kν′ : ν′ ∈ Nreq \ {ν}} is understood as {kν′ :k = 1, . . . , |Kν′ |, ν′ ∈ Nreq \ {ν}}.
The interference of (fν , `) to all other UEs kν′ is expressedby ψHν,`vν,`. There are two possibilities:• The null space of the matrix ψHν,` is nonzero. Then vν,`
is sought in the class of zero-forcing
vν,` = Nν,`uν,`, (32)
where Nν,` ∈ Cnt|Qν,`|×(nt|Qν,`|−ην,`) is the base of thenull space of ψHν,`, uν,` ∈ Cnt|Qν,`|−ην,` is the vectorvariable, and ην,` is the rank of ψν,`.
• The null space of ψHν,` is zero. Make SVD ψHν,` =
Q(l)ν,`Σν,`Q
(r)ν,` with unitary Q
(l)ν,` ∈ C(Nreq−1)×(Nreq−1)
and Q(r)ν,` ∈ C(nt|Qν,`|)×(nt|Qν,`|) and rectangular diago-
nal Σν,` ∈ C(Nreq−1)×(nt|Qν,`|) with the decreasing orderof singular values on its diagonal. Let Σν,` retain onlythe ην,` largest singular values of ψHν,`, rounding othersingular values to zero, so ψHν,` , Q
(l)ν,`Σν,`Q
(r)ν,` is the best
rank ην,` approximation of ψHν,`. By the Eckart and Youngtheorem, ‖ψHν,` − ψHν,`‖F = ξην,`+1, where ξην,`+1 is the(ην,`+1)-th largest singular eigenvalue of ψHν,`. Then vν,`is sought in the class of the approximated zero-forcing
vν,` = Nν,`uν,`, (33)
where Nν,` ∈ C(nt|Qν,`|)×(nt|Qν,`|−ην,`) is the base ofthe null space of ψHν,` and ufν ∈ Cnt|Qν,`|−ην,` is the
2It should be emphasized that [36] considers the problem of maximizingthe cost efficiency for ultra dense networks with hard-transfer backhauling.There is nor imposed backhaul capacity neither file split in [36], leading toquite different optimization issues
6
vector variable. For simplicity, we choose ην,` such thatthe dimension of uν,` is L.
For Nν,` defined from (32) or (33), write
Nν,` = Col(N iν,`)i:(ν,`)∈Ni , (34)
where N iν,` ∈ Cnt×(nt|Qν,`|−ην,`) is such that the partition
(34) is consistent with the partition (29), under which
viν,` = N iν,`uν,` for (ν, `) ∈ Ni. (35)
Based on (7)-(9), the throughput of sfν ,` is
rν,`(u) = mink=1,...,|Kν |
rkν (u), (36)
with
rkν ,`(u) = ln(
1 + |Lkν ,`(u)|2/ (ϕkν ,`(u) + σ)), (37)
where
Lkν ,`(u) ,NR∑i=1
hHkν ,i∑
i:(ν,`)∈Ni
N iν,`uν,`, (38)
and
ϕkν ,`(u) ,L∑
`′=`+1
∣∣∣∣∣∣∑
i:(ν,`′)∈Ni
hHkν ,iNiν,`′uν,`′
∣∣∣∣∣∣2
+∑
ν′∈Nreq\{ν}
L∑`=1
∣∣∣∣∣∣∑
i:(ν′,`)∈Ni
hHkν ,iNiν′,`uν′,`
∣∣∣∣∣∣2
.
(39)
Hence, the optimization problem (11) is reduced to the fol-lowing problem
maxu
minν=1,...,Nreq
∑L`=1 rν,`(u)
τν(u)s.t. (40a)
Nreq∑ν=1
L∑`=1
diν,` ¯rν,`(u) ≤ log2(e)C, i ∈ NR, (40b)
rmin/ log2(e) ≤ rν,`(u), (ν, `) ∈ Nreq × L, (40c)Nreq∑ν=1
∑`:(ν,`)∈Ni
||N iν,`uν,`||2 ≤ P, i ∈ NR, (40d)
with τν(u) ,∑NRi=1
∑`:(ν,`)∈Ni ||N
iν,`uν,`||2 + Pnon, and
¯rν,`(u) , 1|Kν |
∑|Kν |k=1 rkν (u).
Let u(κ) be the feasible point for (40) that is found fromthe (κ− 1)th iteration. Similarly to (13):
rkν ,`(u) ≥rlw,(κ)kν ,`
(u)
,α(κ)kν ,`−
β(κ)kν ,`
2<{L∗kν ,`(u(κ))Lkν ,`(u)} − |Lkν ,`(u(κ))|2
− χ(κ)kν ,`
ϕkν ,`(u), (41)
over the trust region
2<{L∗kν ,`(u(κ))Lkν ,`(u)} − |Lkν ,`(u(κ))|2 > 0, (42)
where
α(κ)kν ,`
, ln(1 +|Lkν ,`(u(κ))|2
ϕkν ,`(u(κ)) + σ
)
+|Lkν ,`(u(κ))|2
|Lkν ,`(u(κ))|2 + ϕkν ,`(u(κ)) + σ
×(
2− σ
ϕkν ,`(u(κ)) + σ
),
0 ≤ β(κ)kν ,`
,|Lkν ,`(u(κ))|4
|Lkν ,`(u(κ))|2 + ϕkν ,`(u(κ)) + σ
,
0 ≤ χ(κ)kν ,`
,|Lkν ,`(u(κ))|2(
|Lkν ,`(u(κ))|2 + ϕkν ,`(u(κ)) + σ
)× 1(
ϕkν ,`(u(κ)) + σ
) . (43)
Therefore, a concave lower bounding approximation forrν,`(u) is
r(κ)ν,` (u) , min
k=1,...,|Kν |rlw,(κ)kν ,`
(u),
and the non-convex constraint (40c) is innerly approximatedby the convex constraint
rlw,(κ)ν,` (u) ≥ rmin/ log2(e), (ν, `) ∈ Nreq × L. (44)
Similarly to (20), a convex upper bounding approximation for¯rν,`(u) is given by
¯rup,(κ)ν,` (u) ,
1
|Kν |
|Kν |∑k=1
rup,(κ)kν ,`
(u) (45)
over the trust region
2λkν ,`(u)− ϕkν ,`(u(κ)) > 0, k = 1, . . . , |Kν |, (46)
where
rup,(κ)kν ,`
(u) , η(κ)kν ,`
+µ(κ)kν ,`
|Lkν ,`(u)|2
2λkν ,`(u)− ϕkν ,`(u(κ)) + σ(47)
and
η(κ)kν ,`
, ln(1 +|Lkν ,`(u(κ))|2
ϕkν ,`(u(κ)) + σ
)
− |Lkν ,`(u(κ))|2
|Lkν ,`(u(κ))|2 + ϕkν ,`(u(κ)) + σ
,
µ(κ)kν ,`
,1− |Lkν ,`(u(κ))|2
|Lkν ,`(u(κ))|2 + ϕkν ,`(u(κ)) + σ
,
λkν ,`(u) ,L∑
`′=`+1
<
∑
(ν,`′)∈Ni
hHkν ,iNiν,`′u
(κ)ν,`
∗
×
∑i:(ν,`′)∈Ni
hHkν ,iNiν,`′uν,`′
+
∑ν′∈Freq\{ν}
L∑`=1
<
∑i:(ν′,`)∈Ni
hHkν ,iNiν′,`u
(κ)ν′,`
∗
×
∑i:(ν′,`)∈Ni
hHkν ,iNiν′,`uν′,`
. (48)
7
The nonconvex constraint (40b) is innerly approximated bythe convex constraint
Nreq∑k=1
L∑`=1
diν,` ¯rup,(κ)ν,` (u) ≤ log2(e)C, i ∈ NR. (49)
Following the same steps as in the previous subsection, atthe κth iteration we solve the following convex optimizationproblem to generate u(κ+1),
maxu
mink=1,...,Nreq
[
L∑`=1
rlw,(κ)ν,` (u)− θ(κ)τν(u)]
s.t. (40d), (42), (44), (46), (49) (50)
with θ(κ) = mink=1,...,Nreq
∑L`=1 rν,`(u
(κ))/τν(u(κ)).The computational complexity of (50) withn =
∑Nri=1
∑(ν,`)∈Ni(nt|Qν,`| − ην,`) and
m = NR(2 +∑Nri=1 |Ni|) +NreqL+
∑Nreqν=1 |Kν |.
To find an initial feasible point for (50), initialized by anyfeasible point for the convex constraint (40d) we iterate
maxu
min
{min
(ν,`)∈Nreq×L[rlw,(κ)ν,` (u)
rmin/ log2(e)− 1],
mini∈R
[1− 1
log2(e)C
Nreq∑k=1
L∑`=1
diν,` ¯rup,(κ)ν,` (u)]
s.t. (40d), (42), (46), (51)
until the value of the objective in (51) is more or equal to 0.The pseudo code for the proposal computational procedure isgiven by Algorithm 2.
Algorithm 2 Multicast Structured Beamforming Algorithm forHTF
1: Set κ = 0. Iterate (51) for a feasible point u(0) for (40).2: Repeat until convergence.3: Solve the convex optimization problem (50) to generate
the next feasible point u(κ+1) for (40). Set κ := κ+ 1.4: Output u(κ) as the optimal solution of (40).
IV. MULTICAST BEAMFORMING UNDER STF
Let Ξi , Ni \ {(ν, `) : ciν,` = 1}. Instead of transferringhard information of {sfν ,` : (ν, `) ∈ Ξi} under HTF, theBBU transfers the following noisy version of the beamformedsubfiles to RRH i under STF:
ξi =∑
(ν,`)∈Ξi
viν,`sfν ,` + υi, (52)
where υi ∈ CN (0,∆∆∆i) is the independent quantization noise.For vi , (viν,`)(ν,`)∈Ni and Λi(v
i) ,∑
(ν,`)∈Ξi[viν,`]
2 �0, the fronthaul constraint is the following reliably recoveredconstraint3
µi(vi,∆∆∆i) , ln |Int + Λi(v
i)∆∆∆−1i | ≤ log2(e)C, i ∈ NR,
(53)
3For simplicity we assume that the quantizer is lossless [37]
which also means that a better quantization is also more costly.Compared to (4), the transmit signal at RRH i is
xi =∑
(ν,`)∈Ni
viν,`sfν ,` + υi, (54)
which contains the quantization noise υi.For each ν ∈ Nreq let ιν be the set of RRHs that do not
have the entire file ν in their cache, i.e.
ιν , {i ∈ NR : ciν,` 6≡ 1}.
The power consumption by transmitting the signal carrying fνfrom RRH i is
NR∑i=1
∑`:(ν,`)∈Ni
||viν,`||2 +∑i∈ιν
〈∆∆∆i〉.
Here and after, the notation 〈.〉 stands for the matrix trace.For ∆∆∆ , {∆∆∆i : i ∈ NR}, the total power consumption in
delivering fν is log2(e)πν(v,∆∆∆) with
πν(v,∆∆∆) ,NR∑i=1
∑`:(ν,`)∈Ni
||viν,`||2 +∑i∈ιν
〈∆∆∆i〉+ Pnon. (55)
Instead of (5), the receive equation at user kν is now
ykν =
NR∑i=1
hHkν ,i
∑(ν′,`)∈Ni
viν′,`sfν′ ,` + υi
+ zkν
=
L∑`=1
∑i:(ν,`)∈Ni
hHkν ,iviν,`
sfν ,`+
∑ν′∈Nreq\{ν}
NR∑i=1
∑`:(ν′,`)∈Ni
hHkν ,iviν′,`sfν′ ,`
+
NR∑i=1
hHkν ,iυi + zkν . (56)
The throughput of sfν ,` upon SC decoding is
rν,`(v,∆∆∆) = mink=1,...,|Kν |
rkν ,`(v,∆∆∆),
with
rkν ,`(v,∆∆∆) = ln
(1 +
|Lkν ,`(v)|2
ψkν ,`(v,∆∆∆) + σ
), (57)
where ψkν ,`(v,∆∆∆) , φkν ,`(v) +∑Nri=1(hkν ,i)
H∆∆∆ihkν ,i withLkν ,`(v) and φkν ,`(v) defined from (8) and (9), which isa convex function. It is obvious that a better quantizationin terms of smaller error covariance ∆∆∆i leads to a higherthroughput but the soft-transfer of the quantized signals ismore costly and more importantly, the capacity of STF isconstrained by (53).
8
Similarly to (11), the problem of energy efficiency optimiza-tion is formulated by
maxv,∆∆∆=[∆∆∆i]i∈NR
minν=1,...,Nreq
∑L`=1 rν,`(v,∆∆∆)
πν(v,∆∆∆)s.t. (53), (58a)
∆∆∆i � 0, i ∈ NR, (58b)rmin/ log2(e) ≤ rν,`(v,∆∆∆), (ν, `) ∈ Nreq × L, (58c)
Nreq∑ν=1
∑`:(ν,`)∈Ni
||viν,`||2 + 〈∆∆∆i〉 ≤ P, i ∈ NR.(58d)
The semi-definite constraint (58b) on the quantization noisecovariance matrix ∆∆∆i means that ∆∆∆i is positive-definite.
Remark. By exploiting a trade-off between quantization andreliable recovery, STF makes the fronthaul constraint (53) andQoS constraint (58c) more flexible than their HTF’s counter-parts (11b) and (11c). As confirmed through simulations, suchSTF’s flexibility helps STF to outperform HTF under weakerfronthaul capacities C.
A. Unstructured beamforming
The constraints (53) and (58c) in (58) are seen non-convex.For inner approximation of (53) we need an upper boundingconvex approximation for the non-convex function µi(vi,∆∆∆i)in (53), while for inner approximation of (58c) we need alower bounding concave approximation for the non-concavefunction rν,`(v,∆∆∆) in (58c).
Let (v(κ),∆(κ)) be the feasible point for (58) that is foundfrom the (κ−1)th iteration. Since the function ln |∆∆∆i+Λi(v
i)|is concave in ∆∆∆i + Λi(v
i) � 0 while the function ln |∆∆∆−1i | is
concave in ∆∆∆i, we can obtain
µi(vi,∆∆∆i)
= ln |∆∆∆i + Λi(vi)|+ ln |∆∆∆−1
i |≤ ln |∆(κ)
i + Λi(vi,(κ))|+ ln |(∆(κ)
i )−1|+ 〈∆(κ)i ,∆∆∆−1
i 〉− nt + 〈(∆(κ)
i + Λi(vi,(κ)))−1,∆∆∆i + Λi(v
i)〉 − nt= µi(v
i,(κ),∆(κ)i )− 2nt + 〈∆(κ)
i ,∆∆∆−1i 〉
+ 〈(∆(κ)i + Λi(v
i,(κ)))−1,∆∆∆i + Λi(vi)〉
= µi(vi,(κ),∆
(κ)i )− 2nt + 〈∆(κ)
i ,∆∆∆−1i 〉
+∑
(ν,`)∈Ni\Ξi
(viν,`)H(∆
(κ)i + Λi(v
i,(κ)))−1viν,`
+ 〈(∆(κ)i + Λi(v
i,(κ)))−1,∆∆∆i〉, µ
up,(κ)i (vi,∆∆∆i), (59)
where the function µup,(κ)i (vi,∆∆∆i) is already convex. The non-
convex constraint (53) in (58) is then innerly approximated bythe following convex constraint:
µup,(κ)i (vi,∆∆∆i) ≤ log2(e).C, i ∈ NR. (60)
Furthermore, applying the inequality (75) for x = Lkν ,`(v),||y||2 = ψkν ,`(v,∆∆∆) and x = Lkν ,`(v
(κ)), ||y||2 =
ψkν ,`(v(κ),∆(κ)) yields the following lower bounding concave
approximation for rkν ,`(v,∆∆∆):
rkν ,`(v,∆∆∆) ≥rlw,(κ)kν ,`
(v,∆∆∆)
,α(κ)kν ,`
−β
(κ)kν ,`
2<{L∗kν ,`(v(κ))Lkν ,`(v)} − |Lkν ,`(v(κ))|2
− χ(κ)kν ,`
ψkν ,`(v,∆∆∆), (61)
over the trust region (14), where
α(κ)kν ,`
, ln(1 +|Lkν ,`(v(κ))|2
φkν ,`(v(κ)) + σ
)
+|Lkν ,`(v(κ))|2
|Lkν ,`(v(κ))|2 + ψkν ,`(v(κ),∆(κ)) + σ
×(
2− σ
ψkν ,`(v(κ),∆(κ)) + σ
),
0 ≤ β(κ)kν ,`
,|Lkν ,`(v(κ))|4
|Lkν ,`(v(κ))|2 + ψkν ,`(v(κ),∆(κ)) + σ
,
0 ≤ χ(κ)kν ,`
,|Lkν ,`(v(κ))|2(
|Lkν ,`(v(κ))|2 + ψkν ,`(v(κ),∆(κ)) + σ
)× 1(
ψkν ,`(v(κ),∆(κ)) + σ
) . (62)
Hence, the following function provides a lower boundingconcave approximation for rν,`(v):
rlw,(κ)ν,` (v) , min
k=1,...,|Kν |rlw,(κ)kν ,`
(v,∆∆∆),
leading to the following inner approximation for the non-convex constraint (58c) in (58)
rlw,(κ)ν,` (v,∆∆∆) ≥ rmin/ log2(e), (ν, `) ∈ Nreq × L. (63)
For θ(κ) , minν=1,...,Nreq
∑L`=1 rν,`(v
(κ),∆(κ))/πν(v(κ)),we solve the following convex optimization problem tt the κ-thiteration to generate the next feasible point (v(κ+1),∆(κ+1))for (58)
maxv,∆∆∆
minν=1,...,Nreq
[
L∑`=1
rlw,(κ)ν,` (v,∆∆∆)− θ(κ)πν(v,∆∆∆)]
s.t. (14), (58b), (58d), (60), (63). (64)
The computational complexity of (64) is (27) with n =Nt(∑NRi=1 |Ni| + NR(Nt + 1)/2) and m = NR(
∑NRi=1 |Ni| +
Nt +NreqL+ 1).To find an initial feasible point for (58), initialized by any
feasible point for the convex constraints (58b), (58d) we iterate
maxv,∆∆∆
min
{min
(ν,`)∈Nreq×L[rlw,(κ)ν,` (v,∆∆∆)
rmin/ log2(e)− 1],
mini∈NR
[1− 1
log2(e)Cµup,(κ)i (vi,∆∆∆i)]
}s.t. (14), (58b), (58d), (65)
until the value of the objective in (65) is more or equal to 0.Analogously to Algorithm 1, it can be shown that the proposedAlgorithm 3 converges at least to a locally optimal solution of(58).
9
Algorithm 3 Multicast Unstructured Beamforming Algorithmfor STF
1: Set κ = 0. Iterate (65) for a feasible point (v(0),∆(0)) for(58).
2: Repeat until convergence.3: Solve the convex optimization problem (64) to generate
the next feasible point (v(κ+1),∆(κ+1)) for (58). Set κ :=κ+ 1.
4: Output (v(κ),∆(κ)) as the optimal solution of (58).
B. Structured beamforming
In this case viν,` is defined from (35), while the quantizationnoise covariances are diagonal: ∆∆∆i = Xi = diag[xi,j ]
ntj=1,
xi,j > 0. Then, for X = (Xi)i∈NR , like (36) we obtainrν,`(u,X) = mink=1,...,|Kν | rkν (u,X) with
rkν ,`(u,X) , ln(
1 + |Lkν ,`(u)|2/ (ζkν ,`(u,X) + σ)),
where4 ζkν ,`(u,Xi) , ϕkν ,`(u) +∑Nri=1
∑ntj=1 xi,j |hkν ,i(j)|2, and Lkν ,`(u) and ϕkν ,`(u)
are defined from (38) and (39). Hence, the problem (58) isreduced to
maxu,X
minν=1,...,Nreq
∑L`=1 rν,`(u,X)
τν(u,X)s.t. (66a)
xi,j > 0, i = 1, . . . , NR; j = 1, . . . , nt, (66b)
µi(ui,Xi) , ln |Int + Λ(ui)X−1
i | ≤ log2(e)C, i ∈ NR,(66c)rmin/ log2(e) ≤ rν,`(u,X), (ν, `) ∈ Nreq × L, (66d)
Nreq∑ν=1
∑`:(ν,`)∈Ni
||N iν,`uν,`||2 +
nt∑j=1
xi,j ≤ P, i ∈ NR, (66e)
where ui , (uiν,`)(ν,`)∈Ni , Λ(ui) ,∑
(ν,`)∈Ξi[N i
ν,`uiν,`]
2,and
τν(u,X) ,NR∑i=1
∑`:(ν,`)∈Ni
||N iν,`u
iν,`||2 +
∑i∈ιν
nt∑j=1
xi,j +Pnon.
Let (u(κ), X(κ)) be the feasible point that is found from the(κ− 1)th iteration. Like (60), the non-convex constraint (66c)in (66) is innerly approximated by the convex constraint
µup,(κ)i (ui,Xi) ≤ log2(e)C, i ∈ NR, (67)
with
µup,(κ)i (ui,Xi) , µi(u
i,(κ), X(κ)i )− 2nt
+∑
(ν,`)∈Ni\Ξi
(N iν,`u
iν,`)
H(X(κ)i + Λi(u
i,(κ)))−1N iν,`u
iν,`
+ 〈(X(κ)i + Λi(u
i,(κ)))−1,Xi〉+ 〈X(κ)i ,X−1
i 〉, (68)
while similarly to (63), the non-convex constraint (66d) in (66)is innerly approximated by the convex constraint
rmin/ log2(e) ≤ rlw,(κ)ν,` (u,X), (ν, `) ∈ Nreq × L, (69)
4hkν ,i(j) is the jth entry of hkν ,i
over the trust region (42), with rlw,(κ)ν,` (u,X) ,
mink=1,...,|Kν | rlw,(κ)kν ,`
(u,X), and
rlw,(κ)kν ,`
(u,X) ,
α(κ)kν ,`−
β(κ)kν ,`
2<{L∗kν ,`(u(κ))Lkν ,`(u)} − |Lkν ,`(u(κ))|2
−χ(κ)kν ,`
ϕkν ,`(u,X), (70)
and
α(κ)kν ,`
, ln(1 +|Lkν ,`(u(κ))|2
ζkν ,`(u(κ), X(κ)) + σ
)
+|Lkν ,`(u(κ))|2
|Lkν ,`(u(κ))|2 + ζkν ,`(u(κ), X(κ)) + σ
×(
2− σ
ζkν ,`(u(κ), X(κ)) + σ
),
0 ≤ β(κ)kν ,`
,|Lkν ,`(u(κ))|4
|Lkν ,`(u(κ))|2 + ζkν ,`(u(κ), X(κ)) + σ
,
0 ≤ χ(κ)kν ,`
,|Lkν ,`(u(κ))|2(
|Lkν ,`(u(κ))|2 + ζkν ,`(u(κ), X(κ)) + σ
)× 1(
ζkν ,`(u(κ), X(κ)) + σ
) . (71)
For θ(κ) = minν=1,...,Nreq
∑L`=1 rν,`(u
(κ), X(κ))/τν(u(κ), X(κ)),we solve the following convex optimization problem at the κthiteration to generate the next feasible point (u(κ+1), X(κ+1))for (66).
maxu,X
minν=1,...,Nreq
[
L∑`=1
rlw,(κ)ν,` (u,X)− θ(κ)τν(u,X)]
s.t. (42), (66b), (66e), (67), (69). (72)
The computational complexity of (72) is (27) with n =∑Nri=1
∑(ν,`)∈Ni(nt|Qν,`| − ην,`) + NRNt and m =
NR(∑NRi=1 |Ni|+Nt +NreqL+ 1).
To find an initial feasible point for (66) we iterate thefollowing convex optimization problem until the value of theobjective in (73) is more or equal to 0:
max min
{min
(ν,`)∈Nreq×L[rlw,(κ)ν,` (u,X)
rmin/ log2(e)− 1],
mini∈NR
[1− 1
log2(e)Cµup,(κ)i (ui,Xi)]
}s.t. (66b), (66e), (42). (73)
Analogously to Algorithm 2, it can be shown that the proposedAlgorithm 4 converges at least to a locally optimal solution of(66).
V. SIMULATION RESULTS
The effectiveness of the proposed algorithms along withtheir convergence is demonstrated through the numerical re-sults in this section. Let us consider a circular cell with radius500 m. Each RRH is equipped with nt = 4 antennas. Thechannel vector hk,i between the RRH i to UE k at the distance
10
Algorithm 4 Multicast Structured Beamforming Algorithm forSTF
1: Set κ = 0. Iterate (73) for a feasible point (u(0), X(0))for (66).
2: Repeat until convergence.3: Solve the convex optimization problem (72) to generate
the next feasible point (u(κ+1), X(κ+1)) for (66). Set κ :=κ+ 1.
4: Output (u(κ), X(κ)) as the optimal solution of (66).
dk,i in km is modeled as hk,i =√
10−ρk,i/10hk,i, whereρk,i = 148.1+37.6 log10(dk,i) (dB) expresses the pathloss andhk,i with independent and identical distributed complex entriesexpresses small-scale fading [38]. The non-transmit power isPnon = NRntPa where Pa is the antenna circuit power. Theerror tolerance for convergence is set as ε = 1e-3. The otherparameters are set as in Table I. There exist three caching
TABLE IPARAMETER SETTINGS
Parameter ValueRadius of cell 500 mCarrier frequency/ Bandwidth(Bw) 2 GHz / 10 MHzRRH maximum transmit power 46 dBmShadowing standard deviation 8 dBNoise power density −174 dBm/HzThe circuit power per antenna 5.6*1e-3 W
strategies:• Caching most popular files (CMP): The Zipf’s distri-
bution is concentrated at most popular files for largerpopularity exponent γ. Each RRH stores Fi most popularfiles, so |Fi| = bµF c and cif,` = 1 if and only if f ≤ |Fi|.
• Caching fractions of distinct files (CFD): Each RRHstores up to bµLc fragments of each file that are randomlychosen, so cif,` = 1 if and only if ` ∈ Lif , where Lif isa set of bµLc random numbers picked randomly fromL , {1, . . . , L}, which are independent across the file fand the RRH index i. This strategy enforces collaborationbetween RRHs when their cache capacity C is small.
• Random caching (RanC): Each RRH stores a set Fi ofdistinct files, which are arbitrarily selected among F files,so |Fi| = bµF c and cif,l = 1 if and only if f ∈ Fi.
Each UE is allowed for requesting only one file at each time.
A. F-RAN with three RRHs and three UEs
There are 3 RRHs and 3 UEs, which are uniformly dis-tributed. The file library consists of total F = 10 distinct files,each of which is split into L = 6 subfiles of similar capacityaccording to the fragment strategy. Thus there are total 60 sub-file items and the optimization problem (11) or (58) involves atleast 60 complex beamforming vectors of dimension nt = 4.The UEs requests are generated by the Zipf’s distribution withγ = 1. Accordingly, CFD with µ = 1/3 is adopted to studythe impact of fronthaul capacity. Note that this example ismuch more complicated than the numerical examples of [19],which involve only 30 real variables of power allocation for
20 30 40 50 60 707
8
9
10
11
12
C=50 Mbps
C=100 Mbps
Hard-transfer Fronthauling Soft-transfer Fronthauling
Ener
gy E
ffici
ency
(Mbi
ts/Jo
ule)
Minimum required rate, rmin (Mbps)
Fig. 2. The energy efficiency versus rmin by unstructured beamforming.
30 subfile items in scenarios of three single antenna RRHsand three UEs.
Fig. 2 plots the achievable energy efficiency versus rmin
by unstructured beamforming with NF = 3, which meansthat all RRHs collaborate in delivering the cache-missed itemsto the UEs. This figure shows that the achievable energyefficiency decreases under both HTF and STF as the quality-of-content-service in terms of rmin is set higher. A higher rmin
demands much more transmit power. The energy efficiency isbetter under HTF when the fronthaul capacity C is strong(C = 100 Mbps), which allows the BBH to transfer allcache-missed items to all RRHs at high rates. However, for Cmoderate (C = 50 Mbps), the energy efficiency under HTF isdramatically degraded, which becomes worse than that underSTF as rmin = 4 Mbps is set. In addition, rmin > 4 Mbps isnot reached to HTF.
Fig. 3 plots the achievable energy efficiency versus C byunstructured beamforming with rmin kept fixed at 4 Mbps,which particularly shows the energy efficiency is on uptrend byincreasing C. rmin is not reached to HTF under C < 40 Mbpsand the numbers NF = 2 and NF = 3 of RRHs serving eachcache-missed items. However, for C ≥ 60 Mbps, the energyefficiency is better as NF is larger. The RRHs equally sharemost items when NF is small, so the fronthaul consumptionis significantly reduced. A larger NF under HTF needs morefronthaul capacity for receiving the cache-missed items. Again,STF is a better choice in achieving high rmin for small C butit cannot compete with HTF once C is sufficiently high.
Fig. 4 plots the achievable energy efficiency by unstructuredand structured beamforming for their comparison versus rmin
with C and NF fixed at 100 Mbps and 1, respectively. Ofcourse, unstructured beamforming performs better than struc-tured beamforming, where the latter cannot grant rmin = 10Mbps. Fig. 5 shows the convergence behaviors of Algorithms1-4 in simulating Fig. 4, where the energy efficiency isimproved after each iteration. Alg. 2 and 4 for structured
11
30 40 50 60 70 80 90 100 110 1209
10
11
12
NF = 2
NF = 3
Hard-transfer Fronthauling Soft-transfer Fronthauling Full Caching
En
ergy
Effi
cien
cy (M
bits/
Joul
e)
Fronthaul Capacity C (Mbps)
NF = 1
Fig. 3. The energy efficiency versus C by unstructured beamforming.
40 50 60 70 80 90 100
3
6
9
12
Unstructured Beamforming
Hard-transfer Fronthauling Soft-transfer Fronthauling
Ener
gy E
ffici
ency
(Mbi
ts/Jo
ule)
Minimum required rate, rmin (Mbps)
Structured Beamforming
Fig. 4. The comparison between unstructured and structured beamformingschemes, C = 100 Mbps and NF = 1.
beamforming converge more rapidly than Alg. 1 and Alg. 3 forunstructured beamforming as the former involve less numbersof decision variables than the latter.
B. F-RAN with more RRHs and UEs
We exploit the lower computational complexity of structuredbeamforming to realize the energy efficiency for cases of 5RRHs (NR = 5) and 5 UEs (Nu = 5), which are randomlydistributed in the cell. Also, each file is fragmented intoL = 4 subfiles. Fig. 6 plots the achievable energy efficiencyversus the fractional caching capacity µ to show the impactof fractional caching under rmin = 4 Mbps and NF = 3. Theenergy efficiency under HTF is better than that under STF.With the small µ = 0 and µ = 1/4, and the small C = 50
Mbps, the targeted rmin is not reached to HTF. The numberof subfile items, which are fetched via fronthaul links areexceedingly increased when µ gets lower. The targeted rmin isthen not guaranteed once the required fronthaul consumptionexceeds C. In contrast, the influence of rmin on the fronthaulconsumption is minimal under STF, making it reachable.
The effect of the popularity exponent γ on energy efficiencyunder different caching strategies is revealed by Fig. 7. Forplotting the achievable energy efficiency versus γ in this figurewe fix C = 50 Mbps, rmin = 4 Mbps, and NF = 3.The total number of files in the library is set as F = 8,and µ = 1/4. Under CFD, a fraction of all files is equallycached by all RRHs. Therefore, the cache-hit ratio for eachUE request is always equal to µ for all values of γ. As suchCFD is not sensitive to γ. On the other hand, the achievableenergy efficiency under RanC is unpredictable since eachRRH selects the arbitrary files to store in their caches. UErequests are concentrated at the popular files with higher γ,which under CMP are probably transmitted by all RRHs toenjoy the full collaboration gain. For small γ, the cache hitratio is severely truncated since the UEs tend to request theless popular files, putting more burdens on fronthauling andcausing the downgraded energy efficiency under CMP.
VI. CONCLUSIONS
We have considered the design problem of multicast beam-forming for maximizing the energy efficiency in providing thequality-of-content-service by an F-RAN with limited-capacityfronthaul links, which is modeled by high dimensional andnonconvex nonsmooth optimization problems and thus posesmany computational challenges. We have developed path-following algorithms for their computation to figure out thepros and cons of fronthauling types, caching strategies, andbeamforming classes. Namely, hard-transfer fronthauling, un-der which the baseband processing unit (BBU) transfers thehard information of the cache-missed content items to theRRHs is efficient with sufficient strong fronthauling capacity.Otherwise, soft-transfer fronthauling, under which the BBUtransfers the quantized signals carrying the cache-missed con-tent items is preferred. Caching most popular files at the RRHsworks very well when the content requests are concentratedon the most popular contents in their Zipf’s distributionwith large popularity exponent. Unstructured beamformingperforms better than structured beamforming but the latter ispractical for F-RANs with more RRHs and users thanks toits lower computational complexity. An extension to physicalsecurity layer for content service by F-RAN is currently understudy.
12
0 10 20 30 40
3
6
9
12
Ener
gy E
ffici
ency
(Mbi
ts/Jo
ule)
Number of iterations
rmin= [4,5,6,7] Mbps
(a) Convergence of Alg. 1
0 10 20 302
4
6
8
10
Ener
gy E
ffici
ency
(Mbi
ts/Jo
ule)
Number of iterations
rmin= [4,5,6,7] Mbps
(b) Convergence of Alg. 2
0 10 20 30 40 50 600
2
4
6
8
10
12
Ener
gy E
ffici
ency
(Mbi
ts/Jo
ule)
Number of iterations
rmin= [4,5,6,7] Mbps
(c) Convergence of Alg. 3
0 10 20 30 402
4
6
8
10
Ener
gy E
ffici
ency
(Mbi
ts/Jo
ule)
Number of iterations
rmin= [4,5,6,7] Mbps
(d) Convergence of Alg. 4
Fig. 5. The convergence behaviors of the proposed schemes.
APPENDIX: CONCAVE UPPER BOUNDING AND CONVEXLOWER BOUNDING APPROXIMATIONS
Given σ > 0, the following inequalities hold true for allx ∈ C, y ∈ Cn and x ∈ C, y ∈ Cn:
ln(1 + |x|2/(||y||2 + σ)) ≥
al(x, y)− bl(x, y)
|x|2− cl(x, y)||y||2 ≥ (74)
al(x, y)− bl(x, y)
2<{x∗x} − |x|2− cl(x, y)||y||2, (75)
over the trust region
2<{x∗x} − |x|2 > 0, (76)
and
ln(1 + |x|2/(||y||2 + σ)) ≤
au(x, y) + bu(x, y)|x|2
||y||2 + σ≤ (77)
au(x, y) + bu(x, y)|x|2
2<{yHy} − ||y||2 + σ, (78)
over trust region
2<{yHy} − ||y||2 > 0, (79)
13
0 1/4 2/4 3/4 14.5
5.0
5.5
6.0
6.5
C = 100 Mbps
Hard-transfer Fronthauling Soft-transfer Fronthauling Full Caching
Ener
gy E
ffici
ency
(Mbi
ts/Jo
ule)
Fractional caching capacity
C = 50 Mbps
Fig. 6. The achievable energy efficiency versus fractional caching capacityµ.
0.4 0.8 1.2 1.6 2.0 2.45
6
7
8
9
10
CMP scheme CFD scheme RanC scheme
Ener
gy E
ffici
ency
(Mbi
ts/Jo
ule)
Popularity exponent
Fig. 7. The achievable energy efficiency versus popularity exponent γ, C =50 Mbps, rmin = 4 Mbps, and NF = 3.
where
al(x, y) = ln(1 + |x|2/(||y||2 + σ))
+|x|2
|x|2 + ||y||2 + σ(2− σ
||y||2 + σ),
0 < bl(x, y) =|x|4
|x|2 + ||y||2 + σ,
0 < cl(x, y) =|x|2
(|x|2 + ||y||2 + σ2)(||y||2 + σ2),
and
au(x, y) = ln(1 + |x|2/(||y||2 + σ))− |x|2
|x|2 + ||y||2 + σ,
bu(x, y) =1− |x|2
|x|2 + ||y||2 + σ.
The function defined by the right hand side (RHS) of (75) isconcave over the trust region (76) while the functions definedby the RJS of (78) is convex over the trust region (79). Theleft hand side (LHS) and the RHS of (75) and (78) match at(x, y).
The inequality (74) follows by substituting |x|2 = z,||y||2 = t and |x|2 = z, ||y||2 = t in the following inequality[39]
ln(1 + z/t) ≥ ln(1 + z/t) +z
t+ z(2− z
z− t
t),
∀ (z, t) ∈ R2+, (x, t) ∈ R2
+, (80)
and then the inequality (75) holds true because |x|2 ≥2<{x∗x} − |x|2 for all x ∈ C and x ∈ C that results in1/|x|2 ≤ 1/(2<{x∗x} − |x|2) over the trust region (76).
The inequality (77) follows by substituting t =|x|2/(||y||2 +σ)) and t = |x|2/(||y||2 +σ)) in the well-knowninequality
ln(1 + t) ≤ ln(1 + t) +1
1 + t(t− t) ∀t ≥ 0, t ≥ 0
and then the inequality (78) holds true because ||y||2 ≥2<{yHy} − ||y||2 for all y ∈ Cn and y ∈ Cn that results in1/||y||2 ≤ 1/(2<{yHy} − ||y||2) over the trust region (79).
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Huy T. Nguyen received the B.S. degree (Hons.)in Computer Science and Engineering from HCMCUniversity of Technology, Vietnam, in 2013. Hereceived the M.S. degree and Ph.D. degree fromthe Department of Information and CommunicationSystem, Inje University, South Korea in 2016, and2019, respectively. He is currently a Research Fel-low at Nanyang Technological University (NTU),Singapore. He was a visiting student with Queen’sUniversity of Belfast, U.K. in 2017 and internshipPh.D student in National Institute of Information and
Communications Technology (NICT), Japan in 2018. His research interestsinclude performance analysis and optimization techniques in signal processingfor wireless communications.
Hoang Duong Tuan received the Diploma (Hons.)and Ph.D. degrees in applied mathematics fromOdessa State University, Ukraine, in 1987 and 1991,respectively. He spent nine academic years in Japanas an Assistant Professor in the Department ofElectronic-Mechanical Engineering, Nagoya Univer-sity, from 1994 to 1999, and then as an AssociateProfessor in the Department of Electrical and Com-puter Engineering, Toyota Technological Institute,Nagoya, from 1999 to 2003. He was a Professor withthe School of Electrical Engineering and Telecom-
munications, University of New South Wales, from 2003 to 2011. He iscurrently a Professor with the School of Electrical and Data Engineeringand a Core Member of the Global Big Data Technologies Centre, Universityof Technology Sydney. He has been involved in research with the areasof optimization, control, signal processing, wireless communication, andbiomedical engineering for more than 20 years.
Trung Q. Duong (S’05, M’12, SM’13) received hisPh.D. degree in Telecommunications Systems fromBlekinge Institute of Technology (BTH), Swedenin 2012. Currently, he is with Queen’s UniversityBelfast (UK), where he was a Lecturer (AssistantProfessor) from 2013 to 2017 and a Reader (As-sociate Professor) from 2018. His current researchinterests include Internet of Things (IoT), wirelesscommunications, molecular communications, andsignal processing. He is the author or co-author of350+ technical papers published in scientific journals
(210+ articles) and presented at international conferences (140+ papers).Dr. Duong currently serves as an Editor for the IEEE TRANSACTIONS
ON WIRELESS COMMUNICATIONS, IEEE TRANSACTIONS ON COMMUNI-CATIONS, and IET COMMUNICATIONS. He was awarded the Best PaperAward at the IEEE Vehicular Technology Conference (VTC-Spring) in 2013,IEEE International Conference on Communications (ICC) 2014, IEEE GlobalCommunications Conference (GLOBECOM) 2016 and 2019, and IEEEDigital Signal Processing Conference (DSP) 2017. He is the recipient ofprestigious Royal Academy of Engineering Research Fellowship (2016-2021)and has won a prestigious Newton Prize 2017.
15
H. Vincent Poor (S’72, M’77, SM’82, F’87) re-ceived the Ph.D. degree in EECS from PrincetonUniversity in 1977. From 1977 until 1990, he wason the faculty of the University of Illinois at Urbana-Champaign. Since 1990 he has been on the facultyat Princeton, where he is the Michael Henry StraterUniversity Professor of Electrical Engineering. From2006 until 2016, he served as Dean of Princeton’sSchool of Engineering and Applied Science. Hehas also held visiting appointments at several otherinstitutions, including most recently at Berkeley and
Cambridge. His research interests are in the areas of information theory andsignal processing, and their applications in wireless networks, energy systemsand related fields. Among his publications in these areas is the forthcomingbook Advanced Data Analytics for Power Systems (Cambridge UniversityPress, 2020).
Dr. Poor is a member of the National Academy of Engineering and theNational Academy of Sciences, and is a foreign member of the ChineseAcademy of Sciences, the Royal Society and other national and internationalacademies. He received the Technical Achievement and Society Awards ofthe IEEE Signal Processing Society in 2007 and 2011, respectively. Recentrecognition of his work includes the 2017 IEEE Alexander Graham BellMedal, the 2019 ASEE Benjamin Garver Lamme Award, a D.Sc. honoriscausa from Syracuse University, awarded in 2017, and a D.Eng. honoris causafrom the University of Waterloo, awarded in 2019.
Won-Joo Hwang (M’03, SM’17) received the Ph.D.Degree in Information Systems Engineering fromOsaka University Japan in 2002. He received hisB.S. and M.S. degree in Computer Engineeringfrom Pusan National University, Pusan, Republic ofKorea, in 1998 and 2000. He is now a full professorat Inje University, Gyeongnam, Republic of Korea.His research interests are in network optimizationand cross layer design. Dr. Hwang is a member ofthe IEICE and IEEE.