To support huge data rates in next-generation communication systems, mmWave technologies using wideband signals are widely considered as an attractive technology . From a research perspective, one of the challenges to overcome is the high path loss (PL) of the mmWave band compared to that of traditional bands. The channel PL in the mmWave bands is generally higher than that of traditional frequencies . In particular, the inherent propagation characteristics make the use of mmWave transmission sensitive to blockage. Thus, multiple-input multiple-output (MIMO) and beamforming (BF) techniques are adopted to compensate the severe PL conditions [3, 4]. Directional transmission is also known to be beneficial for reducing the interference in networks and for improving the spatial reuse of radio resources and the transmission range. However, with the densification of networks, directional transmission with narrow beams creates additional difficulties.
In contrast to traditional radio resource management (RRM) with physical (PHY) and medium access control (MAC) cross-layer approaches, where resources are usually managed in a time-frequency domain, mmWave communication systems also need to select appropriate transmit and receive beam directions and widths (beam configurations) of the network entities. As illustrated in Fig. 1, the large numbers of UE connected to different APs would have to share the frequency resources in the uplink, thus interference managament schemes coping with mutual interference are required. However, the design of interference management schemes that jointly optimizes the power, UE-AP assignments, and the beam configuration is difficult.
In this study, we consider a wireless mmWave access network where multiple low-mobility users in a hotspot area communicate with a set of APs in the uplink using optimized beam steering. We start by analyzing the possible interference cases in the considered multi-beam multi-user scenario. Next, for a given power budget of UEs and discrete beam configurations of APs in the network, we pose a weighted max-min problem involving the joint optimization of power and UE - AP assignments in the uplink. We show that this problem can be easily solved with a simple fixed point algorithm that is further combined with a heuristic based on simulated annealing  to search for an optimal beam configuration.
Our work builds upon a previous study on throughput and fairness trade-offs depending on beam width selection in multi-beam multi-user mmWave communication systems . Interference management via transmit beam width and direction for improving the system performance is one of the center topics in mmWave communications. E.g., the authors in  consider uplink mmWave cellular networks and minimize the interference by adapting only the transmit power of the UEs.  proposes a performance optimization approach for uplink mmWave communication systems based on a spatial modulation scheme. This scheme assumes an exact orthogonality between different beams, and such assumption is not valid for mmWave hotspot networks. Moreover, the impact of the transmit and receive beam widths to the system performance was not studied. Uplink inter-user interference in mmWave systems was considered in . The proposed scheme takes into consideration a single-cell scenario and assumes that the channel state information (CSI) is known perfectly at the AP.
In this study, we use the following standard definitions: scalars and variables are denoted by lowercase letters (e.g. and
). We use boldface letters to emphasize vectors (e.g.and ). The th element of a vector is denoted by . A vector inequality should be understood as an element-wise inequality. Sets are defined with calligraphic fonts (e.g. and
). Probability distributions are denoted with calligraphic letters. By, we denote the standard norm. Sets of non-negative and positive reals are denoted by and , respectively.
Ii-a Uplink Network Model
We consider a wireless network comprised of a set of transmitters (Tx), called user equipment (UE), and a set of receivers (Rx), called access points (APs). We assume fixed transmit beam widths . The transmit beam directions of the UEs
are uniformly distributed with. Furthermore, we assume a transmit power constraint for each UE given by . The transmit power vector takes values from a continuous power domain; i.e., , , . In contrast to the transmitter side, we assume that each receive beam width and direction can be adjusted by the AP. Let be the receive beamwidth vector, where takes values from a discrete set and , . Similarly, each receive beam can be steered by the AP in a specific angular direction and the vector of receive beam directions is denoted by , where takes values from a discrete set and , .
In this work, we assume that multiple UEs may be simultaneously connected to an AP, constituting a many-to-one association scenario. Hence, the AP is capable of processing several incoming uplink signals at the same time.
The CSI, which is needed to perform the beam and interference management, is assumed to be composed of the large-scale channel fading gains, based on the mmWave PL model (see Section V-A), for all users. That is, the instantaneous small-scale fading coefficients are assumed to be unknown, otherwise this would generate an excessively high amount of CSI feedback overhead, hardly implementable in mmWave systems.
Ii-B Directive Antenna Patterns
The beam width is one of the key variables that we will adjust in the proposed scheme in order to improve the system performance. We refer to an antenna model presented in . It uses the simplified and approximated beam gain pattern provided in Fig. 2 for both transmitters and receivers. An antenna with a gain pattern defined by beam width , gain in the mainlobe , and gain in the sidelobe with can be expressed by
Obviously, the beam gains in the mainlobe are increasing with smaller beamwidth. With we have an omnidirectional mode with unit gain.
Ii-C Interference Model
We adopt the interference model studied in [11, 12]. An UE is connected to a single AP , and the radiated power from other UEs is treated as the interference power at the AP . Hence, the overall interference at the receiver is expressed by
where is the transmit power of the interfering UE and is the power gain of the interference channel between UE and AP . The latter depends on Tx beamwidth , Tx beam direction , Rx beamwidth , Rx beam direction, and the distance from AP to UE . The interference power gain is expressed as follows:
where and are transmit and receive beam gains of UE and AP , respectively. The scalar denotes the path loss between UE and AP . As mentioned in Section II-A, we assume that the transmit beam width of all UEs is fixed , and UE is always in line-of-sight (LoS) with AP , if this is its serving access point.
We distinguish four interference scenarios, as shown in Fig. 3 (a)-(d). The respective transmit and receive beam gains are calculated as follows:
Above, and denote the mainlobe gains of UE and AP according to (1). For all four interference cases, expression (8) gives the combined transmit and receive beam gains that can be obtained in the network.
Iii Problem Statement
The objective of this study is to maximize the system utility in the network, which we define as a weighted rate allocation problem. The problem involves the optimization of the UE-AP assignments, the receive beam widths , the transmit beam directions , and the transmit power . In addition, the possible beam configurations are subject to discrete candidate sets and , and each component of the power vector cannot exceed the value .
Iii-a Uplink Data Rates
where is the transmit power of UE being connected to AP . The term is the interference power defined in (2), is the noise power at all APs (which we assume to be equal) and refers to the channel power gain between the serving AP and the UE, given by
where is the system bandwidth. For being fixed to the maximum transmit power budget , the maximum achievable rate, called interference-free rate, is given by
Iii-B The Weighted Rate Allocation Problem
As illustrated in Fig. 4, the objective of the optimization problem is to assign the user rates , , fairly, in the sense that every UE achieves the maximum common fraction of the interference-free rates . Formally, the proposed optimization problem is stated as the following mixed integer problem:
where is the transmit power vector, is the power budget, and is a set of all beam configurations of the APs.
Owing to the discrete parameters, it is hard to solve problem (10). However, if the tuple is fixed to a given beam configuration , one can optimally solve the weighted rate allocation problem (10). In this case, the objective reduces to the following problem:
where is the set of constraints (10a)-(10c) excluding the constraint on the beam configurations, which are fixed to . Problem (11) can be efficiently solved with an iterative fixed point algorithm that is described in the next section and also in the Appendix.
Problem (11) also enables us to define a function that maps an arbitrary beam configuration to a rate fraction as follows:
where is the component of the tuple () that solves (11) for a given beam configuration . In this study, we propose to maximize with a SA algorithm that adjusts the receive beam width and direction of the APs. Briefly, the proposed SA approach selects the parameters from a discrete beam configuration set such that
and the remaining optimal variables of problem (11) are obtained with the fixed point algorithm described next.
Iv Solution Framework
Or, more compactly,
for every . Note, that is a positive concave mapping with continuous extension at that fulfills the properties of def_2 given in the Appendix.
Consequently, for and a maximum power budget , the utility maximization problem in (11) can be stated as the power allocation problem:
The problem in (16) is a particular case of problem_1 in the Appendix. It can be solved with the simple iterative fixed point algorithm given in (22) in the Appendix. Its relation to problem (10) can summarized as follows. Suppose that is the optimal beam configuration to problem (10). If we solve (16) by fixing and , then the solution to (16) is also the optimal fraction and power to problem (10) . Furthermore, the optimal AP assignment to UE can be recovered from the equality
As shown above, if the optimal beam configuration is known, (10) can be solved optimally with a simple algorithm.
Iv-B Receive Beam Width and Direction Adjustment using Simulated Annealing Heuristics
Now, we propose a meta-heuristic based on SA  to obtain the optimal beam configuration. Recall that the SA algorithm works with a parameter called temperature , which is to be cooled down as the beam configurations change. The notion of cooling is interpreted as decreasing the probability of accepting solutions with worse utility as the search space is explored. We define the following main components of the SA that are relevant to our optimization problem:
Solution presentation: The solution presentation for determines that the utility (obtained by solving (16) with and ) is associated with the beam width and direction adjustment problem.
State transition mechanism (neighborhood search): The algorithm starts from the initial state . The state is chosen such that all APs select the largest beam width possible and the direction that points to the hotspot area. The main idea of the neighborhood search is that for a given temperature , we randomly select a new state , calculate the corresponding utility (12), and replace the current solution with if the utility is improved.
Cooling procedure: At the initial stage, the SA algorithm starts with the highest possible temperature, . Throughout an iterative procedure, the temperature is gradually decreased. In each iteration and for a given temperature , the algorithm determines and computes the acceptance probability of the new solution:
In case of the new solution is always accepted. For the new solution accepted if . This scheme aims to jump out from a temporary local minimum. The acceptance probability of the new solution decreases as the temperature decreases or as the utility of the new state is insufficient ( obtains a large negative value) as shown in (18).
Termination criteria: The SA algorithm terminates if no improvement on the utility is reached after a certain number of iterations. Otherwise, it continues the search procedure until the final temperature is reached.
The implemented steps of our SA scheme are given in Algorithm 2. After termination, we obtain a triplet that solves the problem in (10), and corresponding UE-AP assignment can be recovered from solution, as shown in (17).
V Numerical Evaluation
V-a Millimeter-wave Propagation Model
We use the mmWave path loss model proposed in . It assumes omni-directional antennas with unity gain for generality. In this work, the directional antenna patterns and gains are adapted to the PL model. The distance-dependent PL function in [dB] is given as follows:
where is the transmission distance in meters, is the free space path loss for carrier frequency in GHz at reference distance , is the path loss exponent andin dB (shadowing). It is a common assumption to set m. As described in , the above model can be parametrized for the so-called urban micro (UMi) open square LoS scenario. For an applicable range of GHz, we then obtain:
V-B Simulation Setup
For the performance evaluation of our proposed method, we consider a mmWave access network with UEs that are distributed uniformly at random within a hotspot area considering a separation distance of m. The size of the area is m and the AP locations are as shown in Fig. 5. Further system parameters are listed in Table I. The simulations are averaged over 500 random realizations of user positions.
In Fig. 8 to Fig. 8, we give the results from our proposed scheme using fixed point algorithms for a specified beam configuration , . For comparison, we also show the performance of a reference scheme, which assumes fully interfered transmissions with maximum power of each UE. It can be seen that, in the noise-limited power regime, both schemes perform similarly while our proposed approach outperforms the full power transmission in the interference-limited range. Not only is the worst network user (with minimum allocated fraction) made better off (Fig. 8), but the overall fairness in the network is also largely increased (Fig. 8). In addition, Fig. 8 shows that in mmWave networks, at a certain operation point in terms of , schemes that utilize orthogonal resources such as time-division multiple access (TDMA) may be preferable than schemes treating interference as noise (TIN), for the reason that TDMA guarantees the constant rates for all UEs. In TDMA, the fraction of the interference-free rate can be simply given as . As a general outcome from the study of our simulation setup it can be stated that interference cannot be ignored in our particular mmWave scenario.
Below, we show the improvements of the proposed weighted rate allocation scheme after running the simulated annealing (SA) heuristic in Algorithm 2. First, we exemplify the performance weighted rate allocation scheme over the whole beam configurations set in Fig. 9, i.e., for all possible receive beam configurations with the discrete candidate sets , . We use the small size problems since we compare the BF solution (in a large search space BF solution becomes an infeasible) to know the global optimum. In particular, we show the relative performance, called solution efficiency, compared to the best solution of problem (10) when a brute force (BF) search is applied (denoted by solution efficiency). The red path in Fig. 9 marks the neighborhood search with jumps in the states (beam configurations) when the utility is improved. Fig. 10 shows how the utility develops in the cooling procedure as parameter decreases over several iterations. Finally, we illustrate the solution efficiency in the simulated scenario when parameters in Algorithm 2 are changing. There is a trade-off between temperature and number of cycles per temperature which impact the utility. Hence, a certain parametrization can be obtained for a desired operational point.
In this work we have proposed an interference management and beamforming mechanism for uplink hotspot mmWave communication on shared resources. In particular, our centralized scheme jointly optimizes the uplink power, the UE-AP assignments, and the receive beam configurations of the APs. The proposed approach combines a simple fixed point algorithm with a heuristic based on SA, which is used to search for the optimal beam configurations. We showed that, if the SA heuristic is able to find the optimal beam configuration, then the fixed point algorithm provides us with the optimal power and the UE-AP assignments. Nevertheless, even if the beam configuration produced by the SA heuristic is a suboptimal beam configuration, then the fixed point algorithm is still optimal in the sense of maximizing the common fraction of interference-free rates for the given beam configuration.
The results in this study are related to properties of standard interference functions, which are defined as follows:
def_2 A function is said to be a standard interference function if the following properties hold:
If is given by , where are SIFs for every , then is said to be a standard interference mapping.
It is known that positive concave functions (e.g., the coordinate-wise functions used to construct in (16)) are a subclass of SIFs [14, 15]. Furthermore, if is a standard interference mapping, the following optimization problem, which is a generalization of that in (10), can be solved optimally with a simple normalized fixed point algorithm that we show below [16, 17].
Problem 1 (Canonical form of the utility maximization problem)
where is a design parameter (e.g., a power budget ), is an arbitrary monotone norm, and is an arbitrary continuous concave mapping in the class of standard interference mapping. In particular, the vector that solves problem_1 is the limit of the sequence generated by 
Once is known, we recover the optimal scalar of problem_1 by .
This work is partially supported by grant no. 01MD18008B of the German Federal Ministry for Economic Affairs and Energy (DigitalTWIN project).
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