Stochastic Geometric Coverage Analysis in mmWave Cellular Networks with Realistic Channel and Antenna Radiation Models

Stochastic Geometric Coverage Analysis in mmWave Cellular Networks with Realistic Channel and Antenna Radiation Models

Abstract

Millimeter-wave (mmWave) bands will play an important role in 5G wireless systems. The system performance can be assessed by using models from stochastic geometry that cater for the directivity in the desired signal transmissions as well as the interference, and by calculating the signal-to-interference-plus-noise ratio () coverage. Nonetheless, the correctness of the existing coverage expressions derived through stochastic geometry may be questioned, as it is not clear whether they capture the impact of the detailed mmWave channel and antenna features. In this study, we propose an coverage analysis framework that includes realistic channel model (from NYU) and antenna element radiation patterns (with isotropic/directional radiation). We first introduce two parameters, aligned gain and misaligned gain, associated with the desired signal beam and the interfering signal beam, respectively. We provide the distributions of the aligned and misaligned gains through curve fitting of system-simulation results. The distribution of these gains is used to determine the distribution of the . We compare the obtained analytical coverage with the corresponding coverage calculated via system-level simulations. The results show that both aligned and misaligned gains can be modeled as exponential-logarithmically distributed random variables with the highest accuracy, and can further be approximated as exponentially distributed random variables with reasonable accuracy. These approximations are thus expected to be useful to evaluate the system performance under ultra-reliable and low-latency communication (URLLC) and evolved mobile broadband (eMBB) scenarios, respectively.

Millimeter-wave, channel model, antenna radiation pattern, large-scale cellular networks, stochastic geometry.

I Introduction

Millimeter-wave (mmWave) frequencies can provide - times larger bandwidth than current cellular systems. To enjoy this benefit in the 5G cellular systems, the significant distance attenuation of the desired mmWave signals needs to be compensated by means of sharpened transmit/receive beams [roh2014, mmwave3gpp]. The directionality of mmWave transmissions can induce intermittent but strong interference to the neighboring receivers. The signal-to-interference-plus-noise ratio () is affected as follows: sharpened directional beams reduce the interference probability from the mainlobe, while increasing the signal strength of the interfering mainlobe at the same time.

In this paper, we propose a tractable-yet-accurate downlink mmWave coverage analysis framework that incorporates measurement-based mmWave channel and antenna radiation models into the stochastic geometric analysis of mmWave networks, as visualized in Fig. 1. The novelty compared to the existing works on mmWave coverage analysis is summarized in the following subsections.

I-a Background and Related Works

The coverage of a mmWave cellular network has been investigated in [bai15, direnzo2015, park2016, li16, Gupta2016, Andrews:17, Kim:18] using stochastic geometry, a mathematical tool able to capture the random interference behavior in a large-scale network. Compared to traditional cellular systems using sub GHz frequencies, the major technical difficulty of mmWave coverage analysis comes from incorporating their unique channel propagation and antenna radiation characteristics in a tractable way, as detailed next.

Channel gain model

mmWave signals are vulnerable to physical blockages, which can lead to significant distance attenuation under non-line-of-sight (NLoS) channel conditions as opposed to under line-of-sight (LoS) conditions. This is incorporated in the mmWave path loss models by using different path loss exponents for LoS and NLoS conditions. Besides this large-scale channel gain, there exists a small-scale fading due to reflections and occlusions by human bodies. In order to capture this while maximizing the mathematical tractability, one can introduce an exponentially distributed gain as done in [park2016, li16, Gupta2016, Kim:18]. This implies assuming Rayleigh small-scale fading, which is not always realistic, particularly when modeling the sparse scattering characteristics of mmWave signals.

At the cost of making analytical tractability more difficult, several works have detoured this problem by considering generalized small-scale channel gains that follow a gamma distribution (i.e., Nakagami- fading)  [bai15, Andrews:17] or a log-normal distribution [direnzo2015]. Nevertheless, such generic fading models have not been compared with real mmWave channel measurements, and may therefore either overestimate or underestimate the actual channel behaviors.

(a) With the element pattern [rebato17].
(b) With the element pattern [mmWave_3gpp_channel].
Fig. 1: An illustration of our mmWave network model (top) and the channel model of each link with the transmitter/receiver antenna radiation model (bottom):
(a) With the element pattern, antenna gain parameters come from our previous work [rebato17]; (b) With the element pattern, antenna gain parameters follow from the specifications [mmWave_3gpp_channel]. For both element radiation patterns, channel parameters are obtained from a measurement-based mmWave channel model provided by the NYU Wireless Group [akdeniz14].

Antenna gain model

Both base stations (BSs) and user equipments (UEs) in 5G mmWave systems are envisaged to employ planar antenna arrays that enable directional transmissions and receptions. A planar antenna array comprises a set of patch antenna elements placed in a two-dimensional plane. The radiation pattern of each antenna element is either isotropic or directional, which are hereafter denoted as and element patterns, respectively. By superimposing the radiation from the antenna elements, a planar antenna array is able to enhance radiation in a target direction while suppressing radiation in other directions.

The element pattern is incorporated in the antenna gain model provided by the 3GPP [mmWave_3gpp_channel]. Compared to the element pattern, the directional antenna elements in the element pattern enables element-wise beam steering, thereby yielding the higher mainlobe gain and lower sidelobe gains, i.e., increase in front-back ratio1, as visualized in Fig. 0(b). Such benefit diminishes as the beam steering direction becomes close to the horizontal direction to the antenna array plane. In order to resolve this problem, the 3GPP suggests to equip each BS with -sectored antenna arrays [mmWave_3gpp_channel], restricting the beam steering angle to .

The said radiation characteristics and antenna structure of the element pattern complicate antenna gain analysis. For this reason, most of the existing stochastic geometric approaches [rebato17, bai15, direnzo2015, Andrews:17, park2016, li16, Gupta2016, Kim:18, Andrews:17bis] still resort to the element pattern. This underestimates the front-back ratio of the actual cellular system, degrading the accuracy in the mmWave coverage analysis. Furthermore, the antenna gains are commonly approximated by using two constants obtained from the maximum and the second maximum lobe gains [bai15, direnzo2015, Gupta2016, Andrews:17, park2016, li16, Kim:18]. It is unclear whether such an approximation is still applicable for the mmWave coverage analysis with realistic radiation patterns.

Recently, a few studies [Haenggi17] and [Haenggi18] incorporate the impact of directional antenna elements on the stochastic geometric coverage analysis, by approximating the element radiation pattern as a cosine-shaped curve under a one-dimensional linear array structure. Compared to these works, we consider two-dimensional planar arrays, and approximate the combined array-and-channel gain as a single term, as detailed in the following subsection.

Aligned/misaligned gain model

In order to resolve the aforementioned issue brought by inaccurate channel gains, one can use measurement-based channel gain models, such as the models provided by the New York University (NYU) Wireless Group [akdeniz14], which are operating at 28 GHz as described in [akdeniz14, samimi15, mezzavilla15, ford16]. However, this NYU channel gain model requires a numer of parameters, and is thus applicable only to system-level simulators with high complexity, as done in our previous study [rebato16].

In our preliminary work [rebato17], we simplified the NYU channel gain model via the following procedure so as to allow stochastic geometric coverage analysis.

  1. We separated the path loss gains from the original channel gains, and treated them independently using stochastic geometry. The channel term can be considered as channel gain per single link, which is independent of the link distance.

  2. For each downlink communication link, we combined the channel gain and the antenna gain into an aggregate gain. The aggregate gain is defined for the desired communication link as aligned gain and for an interfering link as misaligned gain, respectively.

  3. We applied a curve fitting method to derive the distributions of the aligned/misaligned gains.

  4. Finally, we derived the distribution of a reference user’s , which is a function of path loss gains and aligned/misaligned gains, by applying a stochastic geometric technique to the path loss gains and then by exploiting the aligned/misaligned gain distributions.

The limitation of our previous work [rebato17] is its use of the element pattern in step (ii). This induces excessive sidelobe gains, particularly including backward propagations, which are unrealistic. To fix this problem, in this study we apply the element pattern to the aforementioned aligned/misaligned gain model, thereby yielding a tractable mmWave coverage expression that ensures high accuracy, comparable to the results obtained from a system-level simulator. Moreover, instead of signal-to-interference ratio () as considered in [rebato17], we focus on by incorporating the impact of the noise power.

A recent work [Andrews:17bis] is relevant to this study. While neglecting interference, it firstly considers a simplified keyhole channel, and then introduces a correction factor. The aggregate channel gain thereby approximates the channel gain under the mmWave channel model provided by the  [3gppChannel:17]. Compared to this, using the NYU channel model [akdeniz14], we additionally consider a realistic antenna radiation pattern provided by the  [mmWave_3gpp_channel]. Besides, we explicitly provide the coverage probability expression using these realistic channel and antenna models, as well as its simplified expression.

I-B Contributions and Organization

The contribution of this paper is summarized below.

  • The distributions of aligned and misaligned gains are provided (see Remarks 1-4), which reflect the NYU mmWave channel model [akdeniz14] and the 3GPP mmWave antenna radiation model [mmWave_3gpp_channel].

  • With the element pattern, following from our preliminary study [rebato17], it is observed that the aligned gain follows an exponential distribution, despite the scarce multipaths in mmWave channels (Remark 1). On the other hand, the misaligned gain can be approximated with a log-logistic distribution (Remark 3) having a heavier tail than an exponential distribution’s, which can be lower and upper bounded by a Burr distribution and a log-normal distribution, respectively.

  • With the element pattern, by contrast, it is observed that both aligned and misaligned gains independently follow an exponential-logarithmic distribution (Remarks 2 and 4), which has a lighter tail compared to the exponential distribution.

  • Applying these aligned and misaligned gain distributions, the downlink mmWave coverage probabilities with the and element patterns are derived using stochastic geometry (Propositions 1 and 2).

  • In spite of the exponential-logarithmically distributed aligned/misaligned gains of the element pattern, it is viable for the calculation to approximate both gains independently as an exponential random variable with a proper mean value adjustment (Remark 5 and Fig. 10), yielding a further simplified coverage probability expression (Proposition 3).

The feasibility of the exponential approximation under the element pattern comes from the identical tail behaviors of both aligned/misaligned gains, that cancel each other out during the calculation. Following the same reasoning, this approach provides a similar approximation under the element pattern that leads to the different tail behaviors of both the aligned/misaligned gains due to the low front-back ratio obtained with isotropic elements (see Fig. 10 in Sect.V).

The remainder of this paper is organized as follows. Section II describes the channel model and antenna radiation patterns. Section III proposes the approximated distributions of aligned and misaligned gains. Section IV derives the coverage probability. Section V validates the proposed approximations and the resulting coverage probabilities by simulation, followed by our conclusion in Section VI.

Ii System Model

In this study, we consider a downlink mmWave cellular network where both BSs and UEs are independently and uniformly distributed in a two-dimensional Euclidean plane. Each UE associates with the BS that provides the maximum average received power, i.e., minimum path loss association. The UE density is assumed to be sufficiently large such that each BS has at least one associated UEs. Multiple UEs can be associated with a single BS, while the BS serves only a single UE per unit time slot according to a uniformly random scheduler, as assumed in [park2016, rebato17, Kim:18] under stochastic goemetric settings. Out of these serving users in the network, we hereafter focus on a reference user that is uniformly randomly selected, denoted as the typical UE. This typical UE’s is affected by the antenna array radiation patterns and channel gains, as described in the following subsections.

Ii-a Antenna Gain

Each antenna array at both BS and UE sides contributes to the received signal power, according to the radiation patterns of the antenna elements that comprise the antenna array. The amount is affected also by the vertical steering angle , horizontal steering angle , and polarization slant angle , as described next.

Element radiation pattern

For each antenna element in an antenna array, we consider two different radiation patterns: isotropic radiation and the radiation provided by the  [antenna_3gpp], respectively identified as and . Denoting as and the horizontal and vertical steering angles of the antenna array, respectively, the element radiation pattern (dB) for superscript specifies how much power is radiated from each antenna element towards the direction .

Following our preliminary study [rebato17], with the element pattern, each antenna element radiates signals isotropically with equal transmission power. Hence, for all and , the element radiation pattern is given as

(1)
(a) Element radiation gain.
(b) Array radiation gain with antenna elements.
Fig. 2: Illustrations of the element radiation gains and the array radiation gains for the and element patterns, with respect to the horizontal steering angle while the vertical steering angle is fixed at .

The element pattern is realized according to the specifications in [mmWave_3gpp_channel, antenna_3gpp] and [3d_channel_3gpp]. First, differently from the previous configuration, it implies the use of three sectors, thus three arrays, placed as in traditional mobile networks. Second, the single element radiation pattern presents high directivity with a maximum gain in the main-lobe direction of about 8 dBi. The of each single antenna element is composed of horizontal and vertical radiation patterns. Specifically, this last pattern is obtained as

(2)

where is the vertical 3 dB beamwidth, and is the side-lobe level limit. Similarly, the horizontal pattern is computed as

(3)

where is the horizontal 3 dB beamwidth, and  dB is the front-back ratio. Bringing together the previously computed vertical and horizontal patterns we can obtain the 3D antenna element gain for each pair of angles as

(4)

where  dBi is the maximum directional gain of the antenna element [mmWave_3gpp_channel]. The expression in (4) provides the dB gain experienced by a ray with angle pair due to the effect of the element radiation pattern.

Array radiation pattern

The antenna array radiation pattern determines how much power is radiated from an antenna array towards the steering direction . Following [antenna_3gpp], the array radiation pattern with a given element radiation pattern is provided as

(5)

The last term is the array factor with the number of antenna elements, given as

(6)

where is the correlation coefficient, set to unity by assuming the same correlation level between signals in all the transceiver paths [antenna_3gpp]. Since it represents the physical specifications of the array, is equally computed for both and models. The term is the amplitude vector, set as a constant while assuming that all the antenna elements have equal amplitude. The term is the beamforming vector, which includes the mainlobe steering direction, to be specified in Section II-B2. Further explanation of the relation between array and element patterns can be found in [rebato18] and [antenna_3gpp].

In Fig. 1(a) we report a comparison of the two continuous element radiation patterns (i.e., ). The figure permits to understand the difference between the element pattern showing a fixed gain and the element pattern providing dBi directivity. As a consequence of the element pattern used, we can see the respective shape of the array radiation pattern (i.e., ) in Fig. 1(b). The plot permits to see the reduction of undesired sidelobes and backward propagation when considering the curve with respect to the element pattern. Furthermore, shape and position of the main and undesired lobes vary as a function of the steerable direction. Further definitions and accurate examples for these concepts can be found in [rebato18].

Field pattern (i.e., antenna gain)

Finally, applying the given antenna array pattern , we obtain the antenna gain at the received signal power. This gain consists of a vertical field pattern and a horizontal field pattern , with the polarization slant angle . For simplicity, in this study we consider a purely vertically polarized antenna, i.e., . Following [3d_channel_3gpp], the vertical and horizontal field patterns are thereby given as follows

(7)
(8)

Ii-B Channel gain

Following the system-level simulator settings [akdeniz14], we divide the channel gains in independently into two parts: (i) path loss that depends on the link distance; and (ii) the channel gain multiplicative component. The latter gain is affected not only by the channel randomness but also by the antenna array directions. The following channel gain computation aspects are independent of the different radiation pattern considered, thus they are valid for both and .

The antenna array direction is determined by the BS-UE association. To elaborate, for each associated BS-UE link, denoted as the desired link, their beam directions are aligned, pointing their main-lobe centers towards each other. As a consequence, for all non-associated BS-UE links, denoted as interfering links, the beam directions can be misaligned. In order to distinguish them in (ii), we define aligned gain and misaligned gain as the channel gain for the desired link and for an interfering link, respectively. The definitions of path loss and aligned/misaligned gains are specified as follows.

Path loss

By definition, the set of BS locations follows a homogeneous Poisson point process (HPPP) with density . At the typical UE, the desired/interfering links can be in either LoS or NLoS state. To be precise, from the perspective of the typical UE, the set of all the BSs is partitioned into a set of LoS BSs and a set of NLoS BSs . According to the minimum path loss association rule, the desired link can be either LoS or NLoS, specified by using the subscript . Likewise, the LoS/NLoS state of each interfering link is identified by using the subscript .

For a given link distance , the LoS and NLoS state probabilities are and . Here, compared to the system-level simulator settings in [akdeniz14, samimi15], we neglect the outage link state induced by severe distance attenuation. This assumption does not incur a loss of generality for our analysis, since the received signal powers that correspond to outage are typically negligibly small.

When a connection link has distance and is in state , transmitted signals passing through this link experience the following path loss attenuation

(9)

where indicates the path loss exponent and is the path loss gain at unit distance [akdeniz14, Rappaport:17].

Aligned and misaligned gains

In both and element patterns, for a given link, a random channel gain is determined by the NYU channel model that follows mmWave channel specific parameters [akdeniz14, samimi15] based on the WINNER II model [winner2]. These parameters are summarized in Tab. I, and discussed in the following subsections. In this model, each link comprises clusters that correspond to macro-level scattering paths. For cluster , there exist subpaths, as visualized in Fig. 1. Moreover, the first cluster angle (i.e., ) exactly matches the LOS direction between transmitter and receiver in the simulated link.

Notation Meaning: Parameters
Carrier frequency: 28 GHz
BS locations following a HPPP with density
LoS state probability at distance :
Serving and interfering BSs or their coordinates
Path loss exponent, with : ,
Path loss gain at unit distance: ,
Path loss at distance in LoS/NLoS state
# of antennas of a BS and a UE
Aligned and misaligned gains, with
Aligned and misaligned gain PDFs
# of clusters
# of subpaths in the -th cluster
, Angular spread of subpath in cluster  [akdeniz14]:
Small-scale fading gain:
Delay spread induced by different subpath distances.
Power gain of subpath in cluster  [samimi15]:
, , ,
, ,
Tab. I: List of notations and channel parameters considered in the NYU mmWave network simulator [mezzavilla15].

Given a set of clusters and subpaths, the channel matrix of each link is represented as

(10)

where is the small-scale fading gain of subpath in cluster , and and are the 3D spatial signature vectors of the receiver and transmitter, respectively. Note that stands for the complex conjugate of vector . Furthermore, for brevity, we use subscript or superscript TX (RX), referring to a transmitter (receiver) related term. Moreover, is the angular spread of horizontal angles of arrival (AoA) and is the angular spread of horizontal angles of departure (AoD), both for subpath in cluster  [akdeniz14]. Note that, for ease of computation, we consider a planar network and channel, i.e., we neglect vertical signatures by setting their angles to 90 (i.e., radian). Finally, and are the field factor terms of transmitter and receiver antennas, respectively and they are computed as in (7) with .

We consider directional beamforming where the mainlobe center of a BS’s transmit beam points at its associated UE (we recall that is the mainlobe center angle as shown in the channel illustration of Fig. 1), while the mainlobe center of a UE’s receive beam aims at the serving BS. We assume that both beams can be steered in any directions. Therefore, considering the element pattern, we can generate a beamforming vector for any possible angle in . Instead, with the three-sectors consideration adopted in the element pattern, the beamforming vectors for any possible angles are mapped within one of the three sectors, thus using an angle in the interval .

At the typical UE, the aligned gain is its beamforming gain towards the serving BS at . With a slight abuse of notation for the subscript , is represented as

(11)
(12)

where is the transmitter beamforming vector and is the transposed receiver beamforming vector computed as in [tse_book, rebato18]. Their values contain information about the mainlobe steering direction and both are computed using the first cluster angle as

(13)

where , for all , , and . The terms and are the spacing distances between the vertical and horizontal elements of the array, respectively. Then, angles and are the steering angles and is kept fixed to . We assume all elements to be evenly spaced on a two-dimensional plane, thus it equals . The same expression can be used to compute the receiver beamforming vector with the exception that its dimension is .

Similarly, the typical UE’s misaligned gain is its beamforming gain with an interfering BS at

(14)

where and respectively are the transmitter and receiver beamforming vectors. It is noted that both and incorporate the effects not only of the mainlobes but also of all the other sidelobes. We highlight that even if both aligned and misaligned gain definitions are valid for both the and configurations, the gains will have a different distribution in the two radiation patterns.

Ii-C definition

The typical UE is regarded as being located at the origin, which does not affect its behaviors thanks to Slivnyak’s theorem [HaenggiSG] under the HPPP modeling of the BS locations. At the typical UE, let and respectively indicate the associated and interfering BSs as well as their coordinates.

Using equations (9), (12), and (14), we can represent as the received at the typical UE associated with , which is given by

(15)

Here, we assume that each BS transmits signals with the maximum power through the bandwidth . In (15), is normalized by . The term denotes the normalized noise power that equals where is the noise spectral density per unit bandwidth.

Iii Aligned and Misaligned Gain Distributions

In this section we focus on the aligned gain in (12) and the misaligned gain in (14) with and element patterns, and aim at deriving their distributions.

Following the definitions in Sect. II-B, the aligned gain is obtained for the desired received signal when the angles of the beamforming vectors and are aligned with the AoA and AoD of the spatial signatures and in the channel matrix . The misaligned gain is calculated for each interfering link with the beamforming vectors and spatial signatures that are not aligned.2 Fig. 1 shows an example of misalignment between the beam of the desired signal (yellow or green colored beam) and the interfering BSs beams (red colored beams).

In the following subsections, using curve fitting with the system-level simulation, we derive the distributions of the aligned gain and the misaligned gain .

Iii-a Aligned gain distribution

Running a large number of independent runs of the NYU simulator we empirically evaluated the distribution of the aligned gain . From the obtained data samples we have noticed that is roughly exponentially distributed when an element pattern is used. Indeed, the signal’s real and imaginary parts are approximately independent and identically distributed zero-mean Gaussian random variables. This exponential behavior finds an explanation in the small-scale fading effect implemented in the channel model using the power gain term computed as reported in Tab. I. We report in Fig. 3 an example of the exponential fit of the simulated distribution. The fit has been obtained using the curve fitting toolbox of MATLAB.

For the purpose of deriving an analytical expression, it is also interesting to evaluate the behavior of as a function of the number of antenna elements at both receiver and transmitter sides. For this reason, in our analysis we consider the term as a function of the number of elements. We show in Fig. 4 the trend of the parameter versus the number of antenna elements at the transmitter side and at the receiver side . Again, using the MATLAB curve fitting toolbox, we have obtained a two-dimensional power fit where the value of can be obtained as in the following remark.

Remark 1.

(Aligned Gain, ) At the typical UE, under the antenna model, the aligned gain can be approximated by an exponential distribution with probability density function (PDF)

(16)

where .

This result provides a fast tool for future calculations. Indeed, the expression found for the gain permits to avoid running a detailed simulation every time. We note that from a mathematical point of view the surface of the term is symmetric. In fact, the gain does not depend individually on the number of antennas at the transmitter or receiver sides, but rather on their product, so we can trade the complexity at the BS for that at the UE if needed.

Fig. 3: Fitting of the aligned gain with the element pattern. The empirical PDF of is obtained by the NYU mmWave network simulator [mezzavilla15], and is fit with the exponential distribution in Remark 1 (, ).
Fig. 4: Fitting of the aligned gain distribution parameter with the element pattern, with respect to the number of antenna elements and .

By contrast, using the element pattern, we have noticed that the data samples of can no longer be approximated as an exponentially distributed random variable. Instead, an exponential-logarithmic distribution provides the most accurate fitting result with the simulated desired gain, validated by simulation as shown in Fig. 5.

Remark 2.

(Aligned Gain, ) At the typical UE, and adopting the element pattern, the aligned gain can be approximated by an exponential-logarithmic distribution with PDF

(17)

where the parameters and are specified in Tab. II.

Exponential-logarithmic distributions are often used in the field of reliability engineering, particularly for describing the lifetime of a device with a decreasing failure rate over time [Tahmasbi:08]. Its tail is lighter than that of the exponential distribution, which is explained by the element pattern’s high directivity and sidelobe attenuation that mostly yield a higher aligned gain than the element pattern’s aligned gain.

Fig. 5: Fitting of the aligned gain with the element pattern. The empirical PDF of fits with the exponential-logarithmic distribution in Remark 2. It no longer fits with an exponential distribution, as opposed to the element pattern’s (, ).

An exponential-logarithmic distribution is determined by using two parameters and , as opposed to the element pattern’s exponential distribution with a single parameter . Precisely, the distribution is given by a random variable that is the minimum of independent realizations from , while is a realization from a logarithmic distribution with parameter . Due to its concatenated generation procedure, the relationship of the two parameters and the number of antenna elements is difficult to be generalized as done in Remark 1 for the element pattern. Instead, for practically possible combinations of and , the appropriate values of and are provided in Tab. II by curve-fitting of the system-level simulation results.

4 16 64 256
4
16
64
256
Tab. II: Aligned gain’s exponential-logarithmic distribution parameters with the element pattern for different and . The table is symmetric, so we hereafter report only the upper triangular part.

Iii-B Misaligned gain distribution

Following the same procedure as used for the aligned gain with the NYU simulator, we extract the distribution of the misaligned gain under the and element patterns. With the element pattern, we found that the PDF has a steep decreasing slope in the vicinity of zero, while showing a heavier tail than the exponential distribution. This implies that the occurrence of strong interference is not frequent thanks to the sharpened mainlobe beams, yet is still non-negligible due to the interference from sidelobes that include the backward propagation. We examined possible distributions satisfying the aforementioned two characteristics, and conclude that a log-logistic distribution provides the most accurate fitting result with the simulated misaligned gain.

Remark 3.

(Misaligned Gain, ) At the typical UE, and using antenna elements, the misaligned gain can be approximated by a log-logistic distribution with PDF

(18)

where the values of and are provided in Tab. III.

A log-logistic distribution is given by a random variable whose logarithm has a logistic distribution. The shape is similar to a log-normal distribution, but has a heavier tail [Bennett:83]. For a similar reason addressed after Remark 2, a log-logistic distribution is determined by two parameters and , and their relationship with the number of antenna elements is difficult to be generalized. We instead report the appropriate values of and for 16 combinations of and in Tab. III.

4 16 64 256
4
16
64
256
Tab. III: Misaligned gain’s log-logistic distribution parameters with the element patterns for different and .

Next, with the element pattern, we identified that the PDF has a lighter tail than the exponential distribution, which is far different from the heavy-tailed distribution. In this case, we found that the misaligned gain fits well an exponential-logarithmic distribution, as also used for the aligned gain in Remark 2.

Remark 4.

(Misaligned Gain, ) At the typical UE, and adopting the element pattern, the misaligned gain can be approximated by an exponential-logarithmic distribution with PDF

(19)

where the values of parameters and are provided in Tab. IV.

Although both and can be described by using exponential-logarithmic distributions, these two results come from different reasons, respectively. For , it follows from the higher mainlobe gains than under the element pattern that yields the exponentially distributed . For , on the contrary, its light-tailed distribution originates from attenuating sidelobes, reducing the interfering probability. For these distinct reasons, the distribution parameters for and for are different, as shown in Tab. II and Tab. IV.

Note that is often considered as a Nakagami- or a log-normal distributed random variable [bai15, Andrews:17, direnzo2015]. In Sect. V, we will thus compare our proposed distributions for with them. For a fair comparison, for a Nakagami- distribution with and , we will use its best-fit distribution parameters obtained by curve-fitting with the system-level simulation, which are given with the PDF as follows.

(20)

With this PDF, we will observe in Sec. V that a Nakagami- distribution underestimates the tail behavior of too much, thereby leading to a loose empirical upper bound for the coverage probability.

Likewise, for a log-normal distribution with and , we will consider the following PDF with the parameters.

(21)

Under the element pattern, it will be shown in Sec. V that a log-normal distribution is a better fit than a Nakagami- distribution, yet it still underestimates the interference, yielding an empirical upper bound to the coverage probability.

As an auxiliary result, we will also provide the result with a Burr distribution [burr1942]. This overestimates the tail behavior of , leading to the empirical lower bound of the coverage probability. For this, we will consider the following PDF under and .

(22)
4 16 64 256
4
16
64
256
Tab. IV: Misaligned gain’s exponential-logarithmic distribution parameters with the element patterns for different and .

Iv mmWave Coverage Probability

In this section, we aim at deriving the closed-form expression of the coverage probability , defined as the probability that the typical UE’s is no smaller than a target threshold , i.e., . In the first subsection, utilizing the aligned/misaligned gains provided in Sect. III, we derive the exact coverage expressions under and element patterns. In the following subsection, applying a first-moment approximation to aligned/misaligned gains, we further simplify the coverage expressions.

 

(28)

Iv-a Coverage

Let denote the association distance of the typical UE associating with . By using the law of total probability, at the typical UE can be represented as

(23)
(24)

In (24), two expectations are taken over the typical UE’s association distance . The PDF of is given by [Gupta2016] as

(25)
(26)

where , and indicates the opposite LoS/NLoS state with respect to .

For the element pattern, the typical UE’s coverage probability in (24) is then derived by exploiting while applying Campbell’s theorem [HaenggiSG] and the distribution in Remark 1.

Proposition 1.

(Coverage, ) At the typical UE, and considering arrays with radiation elements, the coverage probability for a target threshold is given as

(27)

where is the Laplace transform of the interference from BSs , for , to the typical UE and is given in (28) with at the bottom of this page.
Proof:  See Appendix.

Note that is the mean aligned gain in Remark 1. The misaligned gain PDF and its corresponding parameters are provided in Remarks 3 and 4 as well as in Tab. III. Then, the term is the LoS/NLoS channel state probability defined in Sec. II-B.

For the element pattern, following the same procedure and distribution in Remark 2, we obtain as shown in the following proposition.

Proposition 2.

(Coverage, ) At the typical UE, and considering arrays with radiation elements, the coverage probability for a target threshold is given as

(29)
(30)

where the Laplace transform for is given in (28) with at the bottom of this page.

Proof: Replacing the exponentially distributed in the proof of Proposition 1 by the following an exponential-logarithmic distribution with the CCDF completes the proof.

It is worth noting that the Laplace transform expression in (28) is used for both and element patterns, i.e., in Propositions 1 and 2. Here, the element pattern is differentiated only by the distribution of the misaligned gain contained therein. For different element patterns and their fitting results, we can thus change accordingly while keeping the rest of the terms, thereby allowing us to quickly compare the resulting s. This is an advantage of the analysis, that avoids redundant calculations.

 

(39)

where