I Introduction
Due to the increasing demand for wireless data, the radio frequency (RF) spectrum has become a very limited resource. Alternative approaches are investigated to support future growth in data traffic and nextgeneration highspeed wireless communication systems. Techniques such as massive multipleinput multipleoutput (MIMO), millimeter wave (mmWave) communications and LightFidelity (LiFi) are being explored. Among these technologies, LiFi is a novel bidirectional, highspeed and fully networked wireless communication technology. A typical LiFi system uses offtheshelf lowcost commercially available light emitting diodes (LEDs) and photodiodes (PDs) as front end devices [OBrienD]. Intensity modulation (IM) is used to encode the information in visible light communication (VLC) since the LED is an incoherent optical source. Direct detection (DD) is adopted at the receiver end. LiFi utilizes visible light as the propagation medium in the downlink for both illumination and communication purposes. It may use infrared light in the uplink in order to allow the illumination constraint of the room to be unaffected, and also to avoid interference with the visible light in the downlink [Haas]. The overall licensefree bandwidth of visible light is more than times greater than the whole RF spectrum [Haas]. Also, LiFi can provide enhanced security as the light does not penetrate through opaque objects [WuVLC5G]. In many large indoor environments, multiple light fixtures are installed, these luminaries can act as VLC access points (APs). A network consisting of multiple VLC APs is referred to as a LiFi attocell network [Haas]. Given the widespread use of LED lighting, a LiFi attocell network can use the existing lighting infrastructures to offer fully networked wireless connectivity. Moreover, LiFi attocells can be regarded as an additional network layer within the existing wireless networks because there is no interference to the RF counterparts such as femtocell networks [Haas]. These benefits of LiFi have made it favorable for recent and future research.
By improving the spatial reuse of the spectrum resources, cellular networks can achieve a higher area spectral efficiency [Alouini]. In comparison with RF femtocell networks, LiFi attocell networks use smaller cell sizes as the light beams from LEDs are intrinsically narrow [SpotlightingforVLC]. Thus, with the densely deployed optical APs, the LiFi attocell network can achieve a better bandwidth reuse and a higher area spectral efficiency. However, similar to other cellular systems, intercell interference (ICI) in LiFi attocell networks limits the system performance. This is because the signal transmitted to a user will interfere with other users who are receiving signals from the same frequency resource. Particularly, celledge users suffer from severe ICI. Despite the dense deployment of APs, due to ICI, LiFi may not provide a uniform coverage concerning data rate. Interference coordination mechanisms have been extensively investigated for VLC systems [Marsh, Cui, Chen, JointTrans_GlobCom_Cheng, ADT_JLT_Zhe]. The commonly used technique is static resource partitioning [Marsh]. By separating any two cells that reuse the same frequency resource with a minimum reuse distance, ICI is effectively mitigated. However, there is a significant loss in spectral efficiency. A combined wavelength division and code division multiple access scheme was proposed in [Cui]. Although this approach enhances the system bandwidth, it requires separate filters for each color band and thus creates additional cost. In [Chen], the fractional frequency reuse (FFR) technique is proposed to mitigate ICI. The FFR scheme is a costeffective approach to provide improvements both in celledge user performance and average spectral efficiency, but a low userdensity will decrease the average spectral efficiency significantly. Joint transmission (JT) has been proven to improve signal quality for celledge users [JointTrans_GlobCom_Cheng]. The downside of the JT systems is the extra signaling overhead. Moreover, the space division multiple access (SDMA) scheme using angle diversity transmitters proposed in [ADT_JLT_Zhe] can mitigate ICI by generating concentrated beams to users at different locations.
The angle diversity reception, first proposed in [KJM], allows the receiver to achieve a wide field of view (FOV) and high optical gain simultaneously. An angle diversity receiver (ADR) is composed of multiple narrowFOV PDs facing in different directions. In [ADR_NLOS_ICC, ADR_NLOS_TCOM, FlyEye_1992, ADR_JLT_Zhe, ADR_VTC_Zhe, DoubSource_GlobCom_Zhe, ADR_ICC_ZENG, ADR_WCNC_ZENG], the ADR is used to address the issue of ICI as well as frequency reuse in LiFi cellular systems, and different signal combining schemes are investigated. However, the proposed ADR structure is hard to implement and the optimum ADR design is not given. Moreover, the system is assumed to be interference limited instead of noise limited in [ADR_JLT_Zhe], which is not always true as the ADR can mitigate most of ICI with noise being the dominated part. Recently, due to the lower channel correlation achieved from the angle diversity scheme, ADRs are introduced to improve the performance of indoor MIMOVLC systems, and the pyramid receivers (PRs) are proposed [PR_ADR_MIMO]. The generalized structure of truncated pyramid receivers (TPRs) are given in [TPR_ADR] to reduce the signal to interference plus noise ratio (SINR) fluctuation. However, the optimum structures of the ADRs are not given and therefore the performance gain is not fully exploited. In addition, to obtain a more accurate evaluation of the system performance, the following three factors must be taken into consideration:
I1 User Device Orientation
Most of the studies on ADRs assume that the receiving device is pointed vertically upward. However, it has been shown in our previous works that the random orientation of mobile devices can significantly affect the direct current (DC) channel gain and thus the system performance [MDSorientation, ZhihongOrientation]. Therefore, the random orientation of the user equipment (UE) needs to be considered. A random device orientation model has been proposed in [MDSorientation]. This model will be applied in this study to evaluate the system more accurately.
I2 Diffuse Link Signal Propagation
The nonlineofsight (NLOS) link is neglected in most LiFi and VLC studies and only the lineofsight (LOS) channel is considered [Alouini, SpotlightingforVLC, Marsh, Cui, Chen, JointTrans_GlobCom_Cheng]. In [Downlink_Performance_OWC]
, it is shown that the LOS link is the dominate link and the effect of the reflected signal can be neglected. However, the UE is assumed to be positioned vertically upward, which is not realistic for mobile devices. In our study, we consider the effect of reflection when random device orientation is applied and the results show that the diffuse link cannot be ignored. A microscopic frequencydomain method for the simulation of the indoor VLC channel is presented in
[FrequencyDomain2016]. A closed form for the transfer function that contains all reflection orders is formulated. The method can be extended to multispot transmission without a significant increase in the computational complexity. Therefore, in this study, we will use the frequencydomain method to simulate the impact of the diffuse link.I3 Noise Power Spectral Density
The noise power spectral density of the PD has a huge impact on the analyses of system performance. For different levels of noise power spectral density, the system could be noiselimited, interferencelimited or noiseplusinterference limited, which could affect the choices of the signal combing schemes and the cell configurations.
The main contributions of this paper are summarized as follows:

The coverage area of ADRs is defined to differentiate from the coverage area of APs. Analytical expressions for the coverage area of both PRs and TPRs are given.

Based on the constraint set by the coverage area of ADRs, the lower bound of FOV of PDs on an ADR is given for the single source (SS) system. The performance of PRs and TPRs are compared, and optimized ADR structures are proposed to fully exploit the performance gain of ADRs. In addition, the joint effect of the receiver and transmitter bandwidth on the average data rate are analyzed.

The performance comparison between the select best combining (SBC) and maximum ratio combining (MRC) are given regarding different levels of noise power spectral density. It is the first time shown that under certain circumstances, the SBC can outperform the MRC.

The double source (DS) cell system is considered to further mitigate the NLOS interference. The lower bound of FOV of PDs on an ADR is derived and the optimized ADR structures are proposed for the DS system.

By comparing the average SINR between the DS system and the SS system under different levels of noise power spectral density, we present that, in a noisedominated scenario, the SS system should be applied, otherwise, the DS system is preferred.
The rest of this paper is organized as follows. The system model is introduced in Section . The generalized structures of ADRs are given in Section . Section presents the optimum FOV for PRs and TPRs. The concepts of the optical doublesource cell are proposed in Section . The simulation results and discussions are presented in Section . Finally, conclusions are drawn in Section .
Ii System Model
Iia Light Propagation Model
In indoor optical communications, the signal propagation consists of two components: the LOS link and the diffuse link. Therefore, the overall channel DC gain is the sum of the LOS and diffuse components:
(1) 
where represents the LOS DC channel gain between the transmitter (Tx) and receiver (Rx), and is the diffuse DC channel gain which is the superposition of all NLOS components that are caused by reflections from the surfaces of the walls.
IiA1 LOS link
It is typically assumed that the LED follows the Lambertian radiation pattern[KJM]. The LOS DC gain is thus given by [FrequencyDomain2016]:
(2) 
where is the Lambertian order, which is given as , and denotes the halfpower semiangle of the LED; is the distance between the Tx and the Rx; denotes the physical area of the PD; represents the signal transmission gain of the optical filter; represents the internal refractive index of the concentrator; denotes the FOV of the PD with concentrator; The irradiance angle of the transmitter is denoted as and the incidence angle of the receiving PD is denoted as . Note that can be obtained by , where
defines the distance vector between the Tx and the Rx. The dot product is denoted as
and denotes the Euclidean distance. Furthermore, is the normal vector of the PD. The visibility factor is denoted as , and if or [FrequencyDomain2016].IiA2 NLOS link
The diffuse link is due to the reflection from the walls. As mentioned earlier, the frequencydomain method in [FrequencyDomain2016] is used to obtain the diffuse link DC channel gain. We assume that all the wall surfaces are purely diffuse Lambertian reflectors with . All of the surfaces are divided into a number of small surface elements numbered by , with areas and reflective coefficients . To calculate the diffuse link DC channel gain, the propagation of light is divided into the following three parts. The first part of the diffuse link propagation is the light path between the and all the reflective surface elements of the room. The LOS DC channel gain between the and the surface element is defined as . The transmitter transfer vector, , is defined as , where defines the transpose of vectors. The second part of the diffuse link is the LOS link from all the surface elements to all the surface elements. The LOS DC channel gain between the surface elements and the surface element is given as . To describe the LOS links between all surfaces inside the room, the roomintrinsic transfer matrix, , is defined by its elements . In order to include the reflective coefficient of the surface elements, the reflectivity matrix is defined as [FrequencyDomain2016]. In the third part of the diffuse link, the light propagates from all the surfaces of the room to the . Similarly, we denote the LOS DC channel gain between the surface element and the as . The LOS DC channel gain between all the reflective elements of the room and the receiver are grouped to give the receiver transfer vector which is defined by its transpose
IiA3 The total diffuse DC channel gain
According to [FrequencyDomain2016], the total diffuse DC channel gain with infinite reflection can be calculated by the matrix product:
(3) 
where denotes the unity matrix.
IiB Signal Combining Schemes for ADR
An indoor LiFi network is studied and it is assumed that the total number of UE and LiFi APs are and , respectively. The set of APs is denoted by . The set of users is denoted as . The ADR is used as the Rx and the set of PDs on an ADR is denoted as , where denotes the total number of PDs on the ADR. In order to achieve high data rates, the direct current biased optical (DCO)OFDM is used in this study. The number of OFDM subcarriers is denoted as , where is an even and positive integer, and the sequence number of OFDM subcarriers is denoted by . Two constraints should be satisfied to ensure real and positive signals: i) , and ii) the Hermitian symmetry constraint, i.e., , for , where denotes the complex conjugate operator [DPO]. Therefore, the effective subcarrier set bearing information data is defined as , where is the set of natural numbers.
For an ADR, multiple PDs are receiving signals simultaneously. Thus, attention should be paid to the selection of the signal combing schemes. There are different combining schemes such as equal gain combining (EGC), SBC and MRC. An important metric to evaluate the link quality and capacity is the SINR. The SINR of user on subcarrier can be obtained based on [ADR_JLT_Zhe] and [DownlinkPerformanceChen]:
(4) 
where is the opticaltoelectrical conversion efficiency; is the transmitted optical power of the AP; denotes the combining weight of PD ; is the overall DC channel gain between the PD of user and the serving AP ; is the ratio of DC optical power to the square root of electrical signal power; represents the noise power spectral density of the additive white Gaussian noise and is the baseband modulation bandwidth; is the overall DC channel gain between the PD of user and the interfering LiFi AP . The serving AP for user is selected based on the signal strength strategy (SSS) where the UEs are connected to the APs providing the best received signal strength. Hence, the serving AP for user can be expressed as:
(5) 
When the EGC scheme is adopted, the signals received by the PDs are simply combined with equal weights, which can be described as:
(6) 
In terms of the SBC scheme, a switch circuit is required to output the information from the PD with the highest SINR. Hence, the weight of each PD is given as:
(7) 
where can be obtained by:
(8) 
On the subject of the MRC schemes, the weight for each PD is denoted as [ADR_JLT_Zhe]:
(9) 
Based on the Shannon capacity, assuming electrical signals after optical to electrical conversion, the data rate of the th UE on subcarrier can be expressed as [Dimitrov2013]:
(10) 
Hence, the data rate of the th UE can be obtained by .
Iii ADR Structure
The ADR is composed of multiple PDs facing in different directions. By using a PD in conjunction with a compound parabolic concentrator (CPC), a narrow FOV and high optical gain can be achieved [KJM]. However, the narrow FOV is achieved at the expense of the longer length of the CPC. Therefore, the number of PDs on the ADR should be limited due to the size limitation on the mobile devices and smartphones. In this study, the TPR [TPR_ADR] and the PR [PR_ADR_MIMO] are considered as they are both suitable for handheld devices. The number of PDs on the TPR and PR are separately denoted as and . The structure of the TPR with and the PR with are presented in Fig. 0(b) and 0(a), respectively. The ADR designs are analyzed in the following parts.
Iiia TPR Design
The TPR is composed of a central PD and a ring of equally separated side PDs. The side PDs are arranged uniformly in a circle of radius on the horizontal plane. Thus, the coordinate of the th PD on a TPR is represented as [ADR_WCNC_ZENG]:
(11) 
where is the UE position, denoted as . As the distance between the AP and the UE is much larger than , the distances between the AP and all PDs on a TPR are approximately the same. The normal vector of each PD can be described by two angles: the azimuth angle of a PD, , and the elevation angle of a PD, [ADR_ICC_ZENG]. When the UE is pointing vertically upward, the TPR has one vertically orientated central PD and inclined side PDs with identical elevation angles . In other words, the elevation angle of the th PD on a TPR can be expressed as:
(12) 
The azimuth angle of the th PD is given by:
(13) 
IiiB PR Design
The PR can be regarded as a TPR without the central PD. Therefore, the coordinate of the th PD on a PR is given by:
(14) 
When the UE is vertically orientated, the elevation angle and the azimuth angle of the th PD are separately expressed as:
(15) 
IiiC Random Orientation Model
The orientation of a UE has a great impact on the channel DC gain according to (2). In [MDSorientation], a model for the random orientation of mobile devices based on experiments is proposed so that the system performance of LiFi attocell networks can be evaluated more accurately. The random orientation model can be described by two angles: the elevation angle of a UE, , and the azimuth angle of a UE, . The geometrical representation of and is manifested in Fig. 2
. The probability density function (PDF) of
can be modeled as the truncated Laplace distribution and it can be simplified as [MDSorientation]:(16) 
where . The mean and scale parameters are set as and [MDSorientation]. In addition, the PDF of the azimuth angle of a UE,
, is modeled as a uniform distribution. It is assumed that the UE is initially pointing vertically upward and
. The normal vector of the UE after rotation becomes . The rotation can be simplified as rotating around the yaxis with and then rotating around zaxis with , which can be described by rotation matrices and separately [LSM]. Thus, is given by:(17) 
IiiD Normal Vector of the ADR
When the UE is pointing vertically upward, for both PRs and TPRs, the normal vector of the th PD is obtained as:
(18) 
However, the normal vector of the UE will change due to the random rotation. The random orientation model is described in Section IIIC. Thus, the normal vector of the th PD after the random rotation is obtained by:
(19) 
where and . Based on (19), after the random rotation, the elevation angle of the th PD can be obtained as:
(20) 
and the azimuth angle of the th PD can be expressed as:
(21) 
Therefore, the incidence angle of the th PD, , can be obtained based on and as .
IiiE Receiver Bandwidth vs PD Area
The bandwidth of a PD is affected by its physical area, , and the PD thickness, . The capacitance of the each PD is denoted as , where and are the permittivity of vacuum and and the relative permittivity of silicon, respectively. The load resistance is defined as while the hole velocity is denoted as . Therefore, the receiver bandwidth can be written as [ReceiverDesign_1997]:
(22) 
By solving , the optimum can be denoted as .
IiiF Visibility of an ADR
The visibility of an ADR was first defined in [ADR_ICC_ZENG]. An AP is visible to a PD when the AP is within the FOV of the PD. Hence, at the location and the orientation (), the visibility factor between the th PD on the ADR and the th AP can be expressed as:
(23) 
where = is the distance vector between the AP and the UE. The dot product is denoted as and is the norm operator. In terms of ADR, an AP is visible to an ADR if and only if the AP is visible to at least one of the PDs on the ADR. Hence, for a given UE position and orientation, the visibility of the ADR can be written as [ADR_ICC_ZENG]:
(24) 
It is assumed that both and
follow a uniform distribution. The probability of visibility of an ADR is defined as the probability that there is at least one AP within the visible area of the ADR for all UE positions and orientations, and it can be expressed as follows:
(25) 
where , and are the range of , and , respectively. Hence, it can be obtained that , and .
Iv The Optimum Field of View
Iva Optimization Problem
In (2), the LOS channel gain is a convex function of and decreases monotonically. Hence, the smaller the , the higher the channel gain. However, when the of the PD is too small, there is a high chance that no APs are visible to the ADR and the LOS link cannot be constructed. Thus, there is a trade off between the LOS channel gain and visibility. The optimization problem is formulated as maximizing the LOS channel gain based on the constraint that the ADR should provide visibility for all UE locations. Thus, the optimization function is written as:
(26) 
The solution set of is denoted as and is the minimum value in . As is a monotonically decreasing function, the maximum is achieved when . Hence, the optimization problem can be solved by finding the minimum value of , , which satisfies . Based on (25), cannot be solved in a closed form. Therefore, in the following parts, we will study the ADRs’ visible area on the ceilings to solve the solution set and find a closed form for .
IvB Coverage Area of ADR on the Ceiling
The coverage area of a PR for a verticalorientated UE is studied in [ADR_ICC_ZENG].
Fig. 2(a) demonstrates that the visible area of the PD mounted on the PR is an ellipse on the ceiling. Hence, the visible area of the 1st PD, where and , is given by [ADR_ICC_ZENG]:
(27) 
where
(28) 
and
(29) 
where is the vertical distance between the AP and UE. The detailed proof is given in Appendix A. Fig. 2(b) depicts that the shape of the visible area of the th PD can be obtained by rotating the 1st PD around with an angle of , which can be represented as:
(30) 
The TPR can be seen as the combination of a PR, where , and a central PD. When the UE is facing vertically upward, the visible area of the central PD is a circle. Therefore, the shape of the visible area of the th PD on a TPR is given by:
(31) 
where .
IvC Lower Bound of FOV
For a fixed UE location, the ADR has the smallest coverage area on the ceiling when vertically orientated. In other words, we will investigate the worst condition, i.e. the situation that an ADR is positioned vertically upward which provides the smallest coverage area on the ceiling. Under other orientation scenarios, the coverage area is larger. Based on (30) and (31), Fig. 4 illustrates the visible area of 4 different types of ADRs when the UE is at the cell corner, that is to say, the crosspoint of four LiFi cells. The blue curve is the outer boundary of the visible area. On the outer boundary, the points that have the shortest distance to the UE are defined as critical points, . denotes the horizontal distance between and the UE. To ensure , there are two constraints and the detailed explanation of these constraints are given as follows:
IvC1 Constraint
The central area above the ADR should be visible to the ADR.
As shown in Fig. 4(a), the total FOV of an ADR is represented as , which can be written as:
(32) 
In terms of PRs, if , then the central part is not covered by the visible area of the ADR as manifested in Fig. 4(a). If the UE is in the cell center, then no APs will be visible to the ADR. Hence, the condition should be satisfied so that the area directly above the UE is covered by the visible area of the ADR. Based on this constraint and (32), the lower bound of can be obtained as:
(33) 
With respect to the TPR, the area directly above the UE is covered by the central PD orientating vertically upwards as illustrated in Fig. 4(b). The concern should be the central coverage gap between the central PD and the side PDs. Therefore, is required to ensure there is no gap between them. By substituting this constraint into (32), it can be derived that:
(34) 
IvC2 Constraint
The outer boundary of the visible area should be large enough. The side length of a square cell is denoted as as shown in Fig. 4. The horizontal distance between the UE and the th AP is denoted as . When the UE is at the cell corner, = for any . With the decrease of , the outer boundary of the visible area will decrease, which means will decrease. If is smaller than the horizontal distance from the AP to the cell corner, which is , there will be no APs within the visible area of the ADR for cellcorner users. Therefore, to ensure that at least one AP is visible to the cellcorner UE, should be larger than the horizontal distance from the AP to the cell corner. By moving towards any direction, due to the symmetry, the cellcorner UE will get closer to at least one AP. In other words,
(35) 
That is to say, if there is at least one AP inside the outer boundary of the visible area for the cellcorner UE, then, when the UE moves to other locations, the AP will still be inside the outer boundary of the visible area. Hence, to meet the condition , it is required that . Also, it can be seen from Fig. 4 that is always inside the green reference circle, which has a radius of . Hence, , where .
Fig. 6 presents the geometrical relationship in a spherical coordinate system. The coordinate of is represented as and for both PRs and TPRs. The geometrical relationships among and , illustrated in Fig. 6, can be represented as:
(36) 
(37) 
(38) 
According to (37), (36) and (38), the lower bound of set by Constraint is derived in Appendix B and is represented as:
(39) 
where
(40) 
and
(41) 
IvC3 Summary
V Double Source Cell Configuration
In the conventional SS cell configuration, each cell is equipped with a single AP in the cell center. The double source (DS) cell configuration is proposed to further exploit the spatial diversity of the ADR in [DoubSource_GlobCom_Zhe]. As demonstrated in Fig. 7, each LiFi AP consists of two sources which transmit the same information signals but with opposite polarity. These two sources are termed as the positive source and the negative source, which transmit the time domain signal and respectively. In a single optical cell, the received optical signal at a PD is denoted as [DoubSource_GlobCom_Zhe]:
(43) 
where is the channel gain between the positive source and the PD; is the channel gain between the negative source and the PD. For a fair comparison, the total transmitting power for the SS system and DS system should be the same. Hence, the transmit power of each source is halved when the DS configuration is applied and the received optical power at the PD can be written as [DoubSource_GlobCom_Zhe]:
(44) 
Generally, the receiver is closer to the desired AP than the interfering AP. For the desired AP, due to the narrow FOV of ADRs, one PD can hardly receive LOS signals from both the positive source and negative source simultaneously, and only one appears as the LOS channel gain. In respect of the interfering AP, the channel gains and are both NLOS. Hence, the difference between and is small and the interference is attenuated accordingly. Therefore, the double source cell configuration can suppress the signal power from interfering APs [DoubSource_GlobCom_Zhe]. As the LOS interference can be mitigated by the narrow FOV of the ADR and the NLOS interference can be mitigated due to the adoption of the DS configuration, the SINR of user on subcarrier can be approximated by:
(45) 
where is the overall DC channel gain between the PD of user and the serving AP in the DS system. As manifested in Fig. 7, will vary according to the distance between the positive and negative sources, which can be represented as:
(46) 
Therefore, the lower bound of for the double source cell system can be calculated based on (39)  (42).
Parameter  Symbol  Value 

Transmitted optical power per AP  10 W  
Modulated bandwidth for LED  20 MHz  
Physical are of the single PD receiver  1  
FOV of the single PD receiver  
The total FOV of an ADR  
Halfintensity radiation angle  
PD responsivity  0.5 A/W  
Noise power spectral density  A/Hz  
Vertical distance between APs and UEs  m  
Wall reflectivity  0.8  
Ceiling reflectivity 