A New Class of Structured Beamforming for Content-Centric Fog Radio Access Networks

A multi-user fog radio access network (F-RAN) is designed for supporting content-centric services. The requested contents are partitioned into sub-contents, which are then 'beam- formed' by the remote radio heads (RRHs) for transmission to the users. Since a large number of beamformers must be designed, this poses a computational challenge. We tackle this challenge by proposing a new class of regularized zero forcing beamforming (RZFB) for directly mitigating the inter-content interferences, while the 'intra-content interference' is mitigated by successive interference cancellation at the user end. Thus each beamformer is decided by a single real variable (for proper Gaus- sian signaling) or by a pair of complex variables (for improper Gaussian signaling). Hence the total number of decision variables is substantially reduced to facilitate tractable computation. To address the problem of energy efficiency optimization subject to multiple constraints, such as individual user-rate requirement and the fronthauling constraint of the links between the RRHs and the centralized baseband signal processing unit, as well as the total transmit power budget, we develop low-complexity path- following algorithms. Finally, we actualize their performance by simulations.

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I Introduction

Fog radio access networks (F-RANs) [26, 6, 27, 14, 3], which place processing units at the network edge for reducing the backhaul’s latency and traffic load, have emerged for meeting the radical requirements of mission-critical cloud radio access networks (C-RANs), as exemplified by augmented reality/virtual reality (AR/VR), device-to-device (D2D) communications, smart living and smart cities, etc. (see e.g. [37, 17, 8, 12] and references therein). On one hand, the uniform quality of service (QoS) provision for cell-edge and cell-center users is satisfied by exploiting the strategic spatial distribution of the remote radio heads (RRHs) over the network [24, 2, 10, 39]. On the other hand, employing RRHs for proactively caching popular contents enables flawless low-latency service provision relying on a high network throughput at a high energy efficiency, because only a small fraction of the requested video clips must be fetched through the limited-capacity fronthaul links [36, 15, 38, 9].

Beamforming aided RRHs have been considered in [31, 23, 32, 20, 35]

and in the references therein. However, this design problem is high-dimensional, as it involves many beamforming vectors calculated for covering specific segments of contents. The resultant problem is nonconvex because the throughput function is neither convex nor concave, hence making the objection function constructed for maximizing the sub-contents’ throughput nonconcave, while the associated fronthaul constraints are nonconvex. The challenge of dimension is even further escalated in

[31, 23], which treat beamforming vectors of dimension ( is the number of RRH antennas), hence ultimately arriving at rank-one matrices of dimension . The resultant multiple rank-one constraints are then dropped for the sake of facilitating so-called difference of two convex functions based iterations [13]. Despite dropping these rank-one constraints, the problem still remains computationally complex, since it relies on logarithmic determinant function optimization, which has unknown computational complexity. In [31], no complexity analysis was provided, while the numerical examples in [23] are limited to the simplest and lowest-dimensional case of three single-antenna RRHs () serving three users. The path-following algorithms proposed in [20] handled more practical scenarios of five four-antenna RRHs serving five users, which became possible as a benefit of the structural exploitation of approximate zero-forcing beamforming. In contrast to the iterations used in [31, 23], the path-following algorithms of [20] invoke a convex fractional solver of polynomial complexity at each iteration. However, it still remained an open challenge to reduce the dimension of the approximate zero forcing beamformers, while maintaining the throughput.

Against the above background, this paper offers the following new contributions to the design of RRHs beamformers.

  • We propose a new class of regularized zero forcing beamformers (RZFB) operating in the presence of both ‘intra-content’ and ‘inter-content’ interferences, which allows us to recast the beamforming design into an optimization problem of moderate dimension.

  • We conceive a new convex quadratic approximation for the fronthaul constraints, which allows us to design new path-following algorithms for determining the beamforming vectors by relying on a low-complexity convex quadratic solver at each iteration for generating an improved feasible point.

The paper is organized as follows. Section II is devoted to the modeling of the constrained-backhaul FRAN. Sections III and IV constitute the main technical contribution of the paper, which are respectively devoted to RZFB-based proper Gaussian signaling (PGS) and improper Gaussian signaling (IGS). Our simulations are discussed in Section V, while our conclusions offered in Section VI. The Appendix provides fundamental inequalities, which support the derivation of the technical results.

Notation and most frequently used mathematics.   Only the optimization variables are boldfaced to emphasize their appearance in nonlinear functions; is the cardinality of the set ; and is the trace of the matrix ;

is the set of circular Gaussian random variables with zero mean and variance

(each is also called proper because ), while is the set of non-circular Gaussian random variables with zero means and variance ( for so is also called improper); for vector is a diagonal matrix with the entries of as its diagonal entries; is the mutual information between the random variable and random variable ; The dot product of the matrices and of appropriate size is defined as ; (, resp.) means is a Hermitian symmetric () and positive definite (positive semi-definite) matrix. Accordingly, merely means . One of the most fundamental properties of positive definite matrices is [11]. is the Frobenius norm, i.e. , which also implies whenever . Therefore, the function for is termed as convex quadratic, while is concave quadratic.

Ii FRAN modeling and signaling

As illustrated by Fig. 1, we consider a typical F-RAN consisting of a centralized baseband signal processing unit (BBU) and uniformly distributed RRHs indexed by , which provides content delivery for randomly localized users (UEs) indexed by . Each RRH is equipped by an -antenna array, while each user equipment (UE) has a single antenna. The RRHs are connected to the BBU through fronthaul links, each of which is of capacity .

Ii-a Edge caching

The file library consists of files labelled in order of their popularity, which is distributed according to Zipf’s distribution obeying [5] with the popularity exponent being . A larger results in a reduced number of extremely popular contents. Under the uncoded strategy, each file is split into subfiles , . Each RRH can store a fraction of each file in the pre-fetching phase [29], where is the library capacity. We define the binary-indicator function associated with such that if and only if is cached by RRH .111 is thus pre-determined by the caching strategy RRH used Then we have:

Fig. 1: F-RAN

Ii-B Content transmission and reception

Let us denote the set of requested files by :

where () is the number of requested files. For each , we define as the set of UEs requesting file :

(1)

where it is plausible that is equal to the cardinality of .

In what follows, for convenience of presentation, we use the notation to represent the above and .

Furthermore, the subfile requested by UE is transferred by the fronthaul links to those RRHs, which have the highest channel gains from them to UE among the RRHs, but do not have stored in their cache. Given , let us define as:

(2)

For , we denote by the specific set of subfiles that are either in the cache of RRH or are received by RRH from the BBU:

(3)

Let be the proper Gaussian information source that encodes the subfile . Each for is ‘beamformed’ into the signal for transmission.

Notation Description
a requested file
the -th subfile of file
the information source encoding
the set of users requesting file (defined by (1))
the number of users requesting file
user requests file
transmit beamformer for at RRH by the user ’s request (defined by (11))
optimization variable for allocating the power to in PGS (13)
optimization variable for allocating the power to in IGS (49)
quantization noise in BBU transmission to RRH
the introduced variable to control (defined by (54))
TABLE I: Nomenclature

Let . Under the soft-transfer fronthauling (STF) regime of [22], the BBU transfers the following quantization-contaminated version of the ‘beamformed’ subfiles to RRH :

(4)

with the independent quantization noise given by and

(5)

The fronthaul-rate constraint is constituted by the following reliably recovered rate-constraint of SFT [22]

(6)

It is plausible that a finer quantization results in a reduced error covariance , hence leading to a higher throughput, but the soft-transfer of the quantized signals is limited by the capacity of STF according to (6). As analysed in detail in [20], SFT is much more efficient than hard-transfer fronthauling (HFT) in the face of limited fronthauling capacity.

Furthermore, the signal transmitted by RRH is

(7)

which contains the quantization noise . This noise cannot be completely eliminated due to the limited fronthaul-rate constraint (6). The signal received by UE is

(8)

where is the channel spanning from RRH to UE , and is the background noise of covariance .

Iii Regularized zero-forcing beamforming for proper Gaussian signaling

To assist the reader, Table I provides a summary of the key notations at a glance.

Iii-a PGS Problem statement

It follows from (8) that the interference of to UE () is given by . Thus, for , we define the matrix of interfering channels by

(9)

and then the matrix of signal plus interfering channels as

(10)

Here the operation in (9) arranges row-vectors , , in a matrix of rows.

For and

(11)

associated with222 is the power budget

(12)

we propose the following class of RZFB

(13)
(14)

Let having all-zero entries except for the first entry, which is . Then we have

(15)
(16)

implying that the power of can be still amplified, while its interference inflicted upon UE () is approximately forced to zero. When the rank of is , which only occurs for , the perfect zero-interference condition of is achieved.

It should be noted that (14) represents a brand new class of RZFB specifically tailored for mitigating the inter-content interference only and as such it has quite a different structure compared to the traditional RZFB (see e.g. [21]), which aims for suppressing the multi-user interference.

Under the RZFB of (13), the signal defined in (4) and transferred from the BBU to RRH through the limited-rate backhaul link is reformulated as:

(17)

Then, the mutual information (MI) in the right-hand side (RHS) of (6) may be expressed as:

(18)

for , and

and

(19)

Then the fronthaul-rate constraint (6) becomes:

(20)

Observe that the unit in (18) is nat, while the unit of is bit, so the factor in (20) converts its right-hand side (RHS) to nats for consistency with its left-hand side (LHS).

The transmit signal at RRH is specialized to

(21)

For each let be the set of RRHs that do not have the entire file in their cache, i.e.

(22)

The power consumption of transmitting the signal carrying from RRH is expressed as:

(23)

for . The total power dissipated by delivering is given by the following convex quadratic function

(24)

where is the total ‘non-transmission’ power dissipation at the RRHs formulated as , where the antenna circuit power is .

Let us now rewrite equation (8) of the signal received by user as

(25)

for

(26)

and

(27)

with

Upon employing successive interference cancelation (SIC), user subtracts the detected and remodulated signal of

(28)

from the RHS of (25), yielding

(29)

for detecting by considering

(30)

in (III-A) as the signal of interest, and

(31)

as the interference-plus-noise term. As such, the throughput of at user is formulated as

(32)

for

(33)

which is a convex quadratic function.

The throughput of after SIC becomes:

(34)

The energy efficiency optimization related problem may then be formulated as333The final result should be divided by to express the energy efficiency in terms of bps/Hz/W:

(35a)
(35b)
(35c)

where is the maximum power budget of each RRH, and is the minimum rate required for guaranteeing the delivery of all files requested within a certain delivery deadline [1]. As such (35b

) constitutes the rate constraint to ensure a high-probability successful delivery according to Shannonian information theory.

Iii-B PGS Computation

Note that this problem is non-convex because the objective function (OF) in (35a) is not only non-smooth but also non-concave, while both the SFT constraint (20) and the rate-constraint (35b) are non-convex. We now develop a path-following algorithm, which iterates for finding ever better feasible points for (35) with the aid of convex solvers. To this end, we have to derive an inner convex approximation for the SFT constraint (20) and the rate-constraint (35b); and a lower-bounding concave approximation for the OF in (35a).

Let be the specific feasible point for (35) that is found by the th iteration.

Iii-B1 Inner convex approximation for the SFT constraint (20)

Using the inequality (86) in the appendix yields

(36)
(37)
(38)

for

The function is convex quadratic. The non-convex constraint (20) seen in (35) is then innerly approximated by the following convex constraint:

(39)

Iii-B2 Inner convex quadratic approximation for the rate-constraint (35b)

Furthermore, applying the inequality (Appendix: fundamental inequalities for convex quadratic approximations) of the Appendix yields

(40)
(41)

in conjunction with

and

as well as

Since the function is concave quadratic, the following function provides a concave lower-bounding approximation for :

leading to the following inner convex quadratic approximation for the non-convex constraint (35b) in (35)

(42)

Iii-B3 PGS Algorithm

We now solve the following convex optimization problem at the -th iteration for generating the next feasible point for (35)

(43)

which is equivalent to the following convex quadratic problem: