On Beamforming Gain Models for Performance Evaluation and Analysis of Narrowband and Wideband Wireless Networks

Directional antennas and beamforming techniques that bring promising transmission gain to the wireless links are widely incorporated in the system-level analysis of wireless networks. In many existing studies, the beamforming gain model to calculate the beamforming gain considers beamforming pattern and aligned as well as misaligned cases. However, the channel properties, e.g., the K factor and the spatial distribution of multi-paths, are neglected, which could signiﬁcantly inﬂuence the beamforming gain. In this paper, a general beamforming gain model is appropriately deﬁned, while the traditional beamforming gain model is proved to be only a special case in the proposed general model by considering an oversimpliﬁed channel with no angular spread. In light of this, expressions of the received signal amplitude and the beamforming gain are rigorously derived for narrowband fading and wideband statistical mmWave channels, respectively. Thorough comparison between the proposed beamforming gain model and the traditional beamforming gain is provided, which demonstrates and validates that the traditional model incorrectly captures the beamforming gain and thereby, leads to inaccurate system-level network analysis. To this end, the effectiveness and importance of the proposed general beamforming gain model are revealed, particularly for millimeter-wave wideband systems.


I. INTRODUCTION
System-level performance evaluation and analysis are widely recognized to be critical in design of wireless networks [1]. In the past few decades, stochastic geometry has emerged as a powerful approach due to its analytical tractability in the performance evaluation and analysis of wireless networks for ad hoc networks, cellular networks, and local and personal area networks [2]- [5].
With directional antennas and MIMO techniques, directional beam demonstrates the potential of combating fading, increasing spectral efficiency and potentially reducing interference. Recently, it has been incorporated in the stochastic-geometry-based network evaluation and analysis as an additional beamforming gain [2]- [16]. For instance, the authors in [16] propose use finite homogeneous Poisson point process, which is suitable for finite regions with a random number of nodes, and comprehensively address the modeling and analysis of finite mmWave wireless networks using the properties of the PPP. Moreover, the transmitter selection strategy that chooses maximum received power among LoS and NLoS transmitters is considered. In particular, the beamforming gain model utilized in those studies to calculate the statistics of the beamforming gain takes the beamforming pattern and beamforming case into account. However, the critical channel properties, e.g., the K factor and direction distribution of the multi-paths, are missing in the traditional beamforming gain model that we refer to in the rest of this paper. Specifically, the traditional model is based on a sectored beamforming pattern and developed by assuming two cases of the beamforming, i.e., the aligned beamforming and the misaligned beamforming case. In the aligned case, the beamforming gain is assumed as the gain of the main lobe of the beamforming pattern. In the misaligned beamforming case, the main lobe is randomly directed, and thus the beamforming gain is a discrete random variable, taking the value of the gain of each lobe with the probability of the normalized beamwidth of the lobe.
In line with this, the authors in [12] recently attempt to address the inappropriate modeling of the channel gain and beamforming gain by considering the realistic channel model and the realistic antenna radiation model in the antenna array for mmWave network analysis. They focus on replacing the isotropic antenna radiation model with a realistic antenna radiation model (3GPP model) and adopting a measurement-based channel model to generate a wireless channel. Unfortunately, they do not provide any analytical derivation on the channel gain and the beamforming gain. Instead, they use a curve-fitting method to approximate the statistics of the channel gain and the beamforming gain based on the simulation results.
The objectives of this paper are twofold. First, we address the incorrect modeling of the beamforming gain in the system-level analysis of the wireless networks in the literature [2]- [16]. Second, we demonstrate the traditional beamforming gain model leads to an incorrect evaluation of the signal power and hence system performance. To reach these goals, we first introduce the system model for wireless communications with beamforming and traditional beamforming gain model utilized in the system-level analysis. Moreover, we present the general definition of beamforming gain deriving from the perspective of signal power, which accounts for the beamforming pattern, beamforming case as well as channel parameters. Furthermore, we rigorously derive the true beamforming gain models for the most common narrowband fading channel models, i.e., Rayleigh fading and Rician fading model, in sub-6 GHz band, as well as the wideband statistical channel models in mmWave band under aligned and misaligned beamforming cases. Finally, through extensive numerical analysis, we investigate the impact of the beamforming gain model on the distribution of the received power, under different channel models and beamforming cases. To further show the impact of the proposed beamforming gain model on the system-level performance analysis and evaluation of wireless networks, we analyze the distributions of signal-to-interference-noise-ratio (SINR) and the consequent coverage probability based on the stochastic geometry method, with varying beamforming gain models under the wideband NYU channel model. The contributions of this paper are summarized as follows. • We present a proper definition of the beamforming gain that incorporates the beamforming pattern, beamforming case as well as spatial channel properties. Indeed, it is proved that the traditional beamforming gain model in [2]- [16] is a special case of the proposed beamforming gain model when the channel energy is directed in one direction. • We develop the beamforming gain model in a narrowband fading channel, i.e., Rayleigh fading and Rician fading channels in a scattering environment. We further derive the closed-form expressions of the received signal amplitude with the beamforming gain in the two fading channels in comparison with the traditional beamforming model (Remarks 1-4). • We develop the semi-closed-form expression of the beamforming gain in the wideband statistical channel models. The boundaries of the beamforming gain and the probabilities of the boundaries are analytically investigated (Remarks 5-6).
• Extensive comparison between the proposed beamforming gain model and the traditional beamforming gain model is carried out in both narrowband and wideband channels. We validate and show that the traditional beamforming gain model incorrectly evaluates the beamforming gain. The inaccuracy of the traditional model increases with a large number of antennas and the low K factor.
• The impact of the channel parameters and antenna array parameters on the received signal amplitude and the beamforming gain are thoroughly analyzed. We reveal that the channel parameters significantly influence the boundaries and the distribution of beamforming gain, which is not captured in the traditional beamforming gain model. Numerical results demonstrate that the impact of the channel parameters as well as the number of antennas are inappropriately estimated under the traditional beamforming gain model.
• System-level analysis by the stochastic geometry method with different beamforming gain models is studied. We observe that the traditional model overestimates the SINR as well as ergodic capacity with massive antennas and a small channel K factor, since it underestimates the interference links. In addition, the larger number of antennas steadily brings significant improvement of SINR in the traditional model, which is however incorrect.
Instead, the proposed models show that the increment of SINR decreases as the number of antennas increases.
The remainder of this paper is organized as follows. In Sec. II, the traditional beamforming gain model is introduced. Then the general definition of the beamforming gain is provided and compared with the traditional model. The beamforming gain in the narrowband fading channel is analyzed in Sec. III. The beamforming gain in the wideband statistical channel is investigated in Sec. IV. Extensive numerical results and analysis are presented in Sec. V. Finally, this paper is concluded in Sec. VI.

II. SYSTEM MODEL AND BEAMFORMING GAIN MODEL
In this section, we introduce a system model for wireless communication networks with beamforming techniques. In wireless networks, a base station (BS) is the transmitter (Tx) side that is equipped with multiple antennas and conducts beamforming and beam-tracking technique, while the receiver (Rx) is the user equipment (UE), as shown in

A. System Model for System-level Analysis
In the system-level analysis of the wireless communication networks, the received power, P r , is regarded as a superposition of the beamforming gain at Tx, G t , channel gain, H, transmit power, P t with additional noise power, N 0 , given by, where channel gain H is determined by the channel model. In the system-level analysis of wireless communication networks, a narrowband fading model is commonly used, in which the channel gain consists of two parts, i.e., large-scale fading and small-scale fading [17].
On one hand, the large-scale fading usually comprises of path loss and shadowing. On the other hand, the small-scale fading is described by a complex random variable. The distribution of the complex random variable is different in different narrowband small-scale fading models.
Commonly-used small-scale fading models are Rayleigh fading, Rician fading, and Nakagami fading models [9], [18]. For example, in a Rayleigh fading model, one assumes the real and imaginary parts are two Gaussian with zero mean and same variance, and thereby, the envelope of the channel is Rayleigh distributed, and its phase is uniformly distributed. This model is usually utilized in the rich scattering environment with no dominant path, e.g., a blockage of the line-of-sight (LoS) path.

B. Beamforming Pattern
Before we introduce the beamforming gain model utilized in the literature, we need to introduce the beamforming pattern. The beamforming pattern, A p (θ), describes the amplitude gain of the transmitted waves at the angular direction θ, which is determined by the configuration of the antenna array or the directional antenna at the base station. Without loss of generality, we assume that the max amplitude gain is at the direction of θ = 0 • , i.e., A max = A p (0). The beamforming pattern after beamforming and beam-tracking that directs the main lobe to the direction of θ main , A(θ; θ main ), can be expressed by, A(θ; θ main ) = A p (θ − θ main ).
As the beamforming pattern usually consists of a main lobe and multiple side lobes, the beamforming pattern is generally modeled by a multi-cone model, A MC p (θ), as illustrated in Fig. 2(a), given by [13], where N c stands for the number of all the beam lobes, and g i refers to the amplitude gains of the i th beam lobe respectively, which are all non-negative real values. In addition, Θ i denotes the angular interval of the i th beam lobe, and |Θ i | describes the width of the i th beam lobe. We emphasize that the multi-cone model is a general model for beamforming pattern and it can be regarded error-free if N c is large enough. Therefore, we adopt multi-cone model in the rest of this paper for derivations.
Specifically, as demonstrated in Fig. 2(b), when N c equals to 2, the multi-cone model boils down to a cone-plus-sphere (CPS) model, A CPS p (θ). We denote g 1 as the amplitude gain of the main lobe, while g 2 describes the amplitude gain of the side lobes. By further defining the parameter k as the ratio between the energy of the main lobe and the one leaked to side lobes, we obtain g 2 1 = 2[(1 − cos(|Θ 1 |/2)) + k(1 + cos(|Θ 1 |/2))], and g 2 2 = kg 2 1 .

C. Traditional Beamforming Gain Models
In the traditional beamforming gain models adopted in [2]- [14], the beamforming gain is regarded to be dependent on the beamforming pattern and the beamforming cases, however, independent on the spatial channel properties. Next, we elaborate on the beamforming gain models in aligned and misaligned beamforming cases, respectively.
1) Aligned Beamforming Case: When we consider the link between the BS and target UE (redpoint and black point in Fig. 1), the main lobe of the base station with the highest amplitude gain is directed to the LoS path with the strongest power, i.e., θ main is a determined constant.
The resulting beamforming gain in the traditional beamforming gain model is then regarded as a constant and equals to the square of the maximum amplitude gain, as 2) Misaligned Beamforming Case: The misaligned beamforming case occurs in the interference links. The main lobe of the interference BS (blue point in Fig. 1) is assumed to be uniformly distributed in [−π, π]. This can be explained that BS directs its main lobe to its target UE so that the main lobe direction follows uniform distribution from the sight of the other UEs since the location of the target UE follows a infinite HPPP [6]. It should be noted that in some studies [11], the link between BS and its target UE is regarded to be a misaligned beamforming case when the link is non-line-of-sight (NLoS). The reason is that as there is no significantly strong LoS path, the beamforming strategy decides to randomly direct the main lobe. As a result, the beamforming gain, G t , is a random variable as, where θ u is a random variable following uniform distribution in [−π, π]. When applying the multi-cone model according to (2), the PDF of G t , P G t (x), is given by, where δ(·) is the Dirac delta function.
Remarkably, it should be noted that in the work that considers finite point process [15], the direction of the main lobes of the transmitter, θ u , is uniformly distributed in a range which depends on the location of the transmitter instead of the entire [−π, π] with infinite HPPP. This difference does not violate the proof of Lemma 1, which is the key contribution of this work.
In the remainder of this paper, we adopt infinite HPPP without loss of generality.

D. True Beamforming Gain Models
We first denote the transmit signal u(t) that is a baseband complex signal, and the basebandequivalent complex channel impulse response as h(t, θ), respectively. Then, the received signals with beamforming and an isotropic beam pattern, r BF (t) and r ISO (t), are given by, r BF (t) = π −π A(θ; θ main )(h(t, θ) * u(t))dθ, and r ISO (t) = π −π h(t, θ) * u(t)dθ where * denotes a convolution operator with respect to time t.
Based on the above system model, we define the beamforming gain G t from the perspective of power, which is obtained as a ratio between the received signal power with beamforming, P BF r , and the received signal power with an isotropic beam pattern, P ISO r , as From (6), we notice that the beamforming gain is dependent on the beamforming pattern, beamforming case as well as the spatial channel properties. A remark is drawn here that the channel property, h(t, θ), is incorrectly ignored in the traditional beamforming gain models in Sec. II-C.
Lemma 1: The traditional model is equivalent to a special case of the general beamforming gain model in (6), if the channel has not angular spread (the energy of the channel is concentrated to one direction).
Proof: We consider a channel consisting of N p (N p ≥ 1) paths. As a result, the spatial channel impulse response, h(t, θ) = h R (t, θ), is described as, where h k , τ k , θ k denote the complex amplitude gain, delay and direction of departure (DoD) of the k th path, respectively, which are independent. Without loss of generality, we consider that the paths are sorted in an ascending order with respective to the amplitude gain. Next, we proceed with the proof by separating the aligned and misaligned beamforming cases.

1) Aligned Beamforming Case:
The main lobe is directed to the DoD of the first path, i.e., θ main = θ 1 .
By substituting (7) into (6), we have If all the DoDs equal to the DoD of the first path, i.e., θ k = θ 1 , then we have, which is the same as (3).

2) Misaligned Beamforming Case:
The main lobe is directed to a random direction, i.e., θ main = θ u , which leads to If all the DoDs equal to the DoD of the first path, i.e., θ k = θ 1 , the above expression boils down to which demonstrates the same expression as (4). Note that (a) holds since θ u is uniformly distributed in the circular domain.
Furthermore, we observe that the distributions of h k , θ k and τ k as well as the transmit signal u(t) have no impact on the aforementioned results. Therefore, this proof holds under any channel condition as long as the energy of the channel is concentrated in one direction.

III. BEAMFORMING GAIN MODELS IN NARROWBAND FADING CHANNEL
In this section, we analyze the beamforming gain based on the narrowband channel models, i.e., Rayleigh and Rician fading.

A. Rayleigh Fading Model
In the Rayleigh fading model, the spatial channel impulse response, h(t, θ) = h RL (t, θ), is given by [19], where N p indicates the number of multi-path components (MPCs) and f c is the carrier frequency.
Additionally, h k represents the amplitude gain of k th MPC, which is an i.i.d. random variable satisfying E{ |h k | 2 } = 2σ 2 h . φ k is the phase shift caused by reflection, scattering, etc, which is also an i.i.d random variable following a uniform distribution. Moreover, τ k and θ k denote the delay and DoD of the k th MPC, respectively. As f c is large, a small variation of τ k can cause very large shift of the phase term of 2πf c τ k , which leads that Φ k = 2πf c τ k + φ k is regarded to be i.i.d. uniform distributed. Without loss of generality, θ k is assumed to be i.i.d von Mises distribution with PDF [20]- [22], where I 0 (·) stands for the zero-order modified Bessel function of the first kind, µ k accounts for the mean value and k v is a real parameter controlling angle spread. For k v = 0, the von Mises distribution is equivalent to a uniform distribution that denotes isotropic scattering environment.
By contrast, k v > 0 describes a non-isotropic scattering environment.
Moreover, in the narrowband channel, the transmit signal, u(t), is a time-invariant random variable with normalized power, i.e., E{|u(t)| 2 } = 1. For simplicity, we remove (t) in the rest of this section. Hence, the received signal with isotropic beam pattern under the Rayleigh fading channel, r ISO (t), is expressed as, where the time dependence is omitted since the narrowband channel is time-invariant. Moreover, is expressed as, where r BF I and r BF Q are the in-phase and quadrature components of the received signal r BF , given by Next, we investigate the received signals in the Rayleigh fading channel with aligned and misaligned beamforming separately.

1) Aligned Beamforming Case:
The main lobe is directed to a deterministic direction θ d , i.e., θ main = θ d , since we do not know the optimal direction of the main beam. Being aware that the terms in r BF I and r BF Q are i.i.d., by invoking the CLT, when N p is sufficiently large, r BF I and r BF Q are two independent Gaussian variables with zero mean and variance of E{A 2 (θ k ; θ d )}σ 2 h . Hence, the received signal with beamforming also follows a Rayleigh distribution with a scaling factor of E{A 2 (θ k ; θ d )}σ h .
By using the property of the Rayleigh distribution, we can derive that the beamforming gain G t defined in (6) is a constant and equals to E{A 2 (θ k ; θ d )} in this case, which is given by, It can be seen that E{A 2 (θ k ; θ d )} is a function of θ d with given parameters k v and µ v , if k v > 0. There actually exists an optimal direction θ opt in non-isotropic scattering environment to maximize the beamforming gain. This optimal direction is determined by the beam pattern A(θ) and the distribution of DoD, θ k .

Remark 1:
In the aligned beamforming case under Rayleigh fading channel, the beamforming gain G t is equal to max{E{A 2 (θ k ; θ d )}}, instead of the result G t = A 2 max in traditional beamforming gain models. Gaussian variables with zero mean and variance of E{A 2 p (θ u )}σ 2 h where the expectation operator is with respect to θ u . We observe that the received signal with beamforming also follows a Rayleigh distribution with a scaling factor E{A 2 p (θ u )}σ h , given by Remark 2: In the misaligned beamforming case under Rayleigh fading channel, the beamforming gain G t using the definition in (6) is equal to E{A 2 p (θ u )} that yields a constant. However, in the traditional beamforming gain models, G t = A 2 p (θ u ) is a random variable.

B. Rician Fading Model
In a Rician fading model, the channel can be regarded as a Rayleigh fading channel plus a dominant path. Then the spatial channel impulse response, h(t, θ) = h Ri (t, θ), boils down to [23], where h 0 , Φ 0 , τ 0 and θ d denote the amplitude, phase, delay and direction of departure (DoD) of the dominant path, respectively. According to the CLT, the in-phase and quadrature components are independent Gaussian random variables with the same mean and variance, i.e., r ISO Hence, the received signal with isotropic beam pattern follows a Rician distribution, of which the resulting PDF of the amplitude is given by, where the K-factor K = K ISO = h 2 0 /2σ 2 h is the ratio between the power in the direct path and total received power, and Ω = Ω ISO = h 2 0 + 2σ 2 h denotes the total received power. The in-phase and quadrature components of the received signal with beamforming under the Rician fading channel are given by, Next, we investigate the received signals in the Rician fading channel with aligned and misaligned beamforming separately.

1) Aligned Beamforming Case:
The main lobe is directed to the DoD of the dominant path θ 0 , i.e., θ main = θ 0 . Then the received signals are, By observing (21), we find that the received signal r(t) consists of a Rayleigh variable with a scaling factor E{A 2 (θ k ; θ 0 )}σ h and a LoS component constant as θ 0 is fixed, the received signal r(t) also follows a Rician distribution with parameters Remark 3: In the aligned beamforming case under Rician fading channel, the beamforming gain G t in this case is a random variable that is expressed as a ratio of two Rician random variables with different parameters. Inaccurately, the traditional beamforming gain G t is regarded as a constant |G max | 2 , which makes the received signal as a Rician random variable with K = 2) Misaligned Beamforming Case: The main lobe is randomly directed, i.e. θ main = θ u . Then the received signals are We observe that r BF I and r BF Q are not Gaussian random variables since the LoS component A 0 (θ u )h 0 is also a random variable rather than a constant. Therefore, when performing beamforming and beam-tracking, the received signal cannot be considered as a Rician random variable, although the channel is under Rician fading. By denoting the PDF of A(θ u ) as p A (x), and the PDF of Gaussian random variable with zero mean and variance of σ 2 r = E{|A(θ u )| 2 }σ 2 h as p G (x), the PDFs of r BF I and r BF Q are given by p Then the joint PDF of r BF I and r BF Q is calculated as, p I,Q (x, y) = p I (x)p Q (y). By using a Jacobi transform, the joint PDF of the amplitude |r BF | = (r BF I ) 2 + (r BF Q ) 2 and the phase Φ = arctan Finally, the PDF of the amplitude |r BF | is derived as, Remark 4: In the misaligned case under Rician fading channel, the beamforming gain under random beamforming is a random variable, which is the ratio between a random variable whose PDF given in (24) and a Rician random variable. This brings a noticeable difference from the beamforming gain in the traditional beamforming as a random variable with PDF of p A (x).
When applying the multi-cone model according to (2), we let Φ 0 = 0 for further derivation, which yields the PDF of the amplitude |r BF | under the Rician fading channel and random beamforming strategy, as By comparison, the amplitude |r BF | in a traditional beamforming gain model is |A(θ u )|r ISO and its PDF is calculated as,

C. Summary of Beamforming Gain in Narrowband Channels
In light of the received signal amplitude, the comparison between the proposed beamforming gain model and the traditional beamforming gain model in narrowband channels are summarized in Tab. I. To put in a nutshell, the traditional beamforming gain model is independent of the channel, while the beamforming gain is highly related to the channel fading and direction distribution of multi-paths. To be concrete, the traditional beamforming gain model assumes that the beamforming gain is a constant in aligned beamforming case and a random variable in the misaligned beamforming case. Instead, the proposed true model suggests that the beamforming gain is constant in the Rayleigh fading channel, while the beamforming gain is a random variable in Rician fading channel, in both aligned and misaligned cases. Therefore, the mistreated beamforming gain models in both aligned and misaligned cases make substantial impact on the distribution of the received signal and consequently, leading significantly inaccurate system-level analysis.  Rician fading with parameters given in (25) PDF given in (29) IV. BEAMFORMING GAIN MODELS FOR WIDEBAND STATISTICAL CHANNEL In this section, we proceed the analysis on beamforming gain models in wideband channels, which are common in the mmWave and THz bands.

A. Wideband Channel Framework
In a general wideband space-time statistical model, the channel consists of a LoS path and several NLoS paths. The resulting spatial channel impulse response with the signal power of P s is given by, where h LoS and h m refer to the path gain amplitudes for the LoS path and m th path, respectively.
Moreover, τ LoS and τ m are the ToA for the LoS path and m th path. θ LoS and θ m are the DoD for the LoS path and m th path. P s is the received power. Additionally, K is the power ratio between the LoS path and NLoS paths, which is sensitive to the frequency and propagation environment.
It is reported to have a mean value of 3.8 dB in the indoor environment and 6 dB in the urban environment at 28 GHz [24], [25]. Nokia Bell Lab shows that 80% of the positions have the K factor less than 0 dB at 28 GHz in outdoor small cell environment [26]. Also, its mean value ranges from -5 dB to 2.75 dB at 5.2 GHz in urban macro-cell scenario [27]. And at 800 MHz and 2.6 GHz in urban environment, the K factor decreases to 0.22 and 0.68, respectively [28].
We adopt the NYU channel model framework proposed for mmWave communication networks [29], in which the assumptions about the statistics of the channel parameters are commonly accepted in many mmWave and THz channel models [30]- [34]. The signal power P s is calculated by α − β path loss model as, where P t is the transmitted power. α is simply taken as the free-space path loss at distance of 1 m while β is taken as 2 for LoS channel given in [29]. Moreover, ξ describes the shadowing fading and is Gaussian distributed with standard deviation of σ SF .
In the NYU channel model framework, the number of NLoS paths, N m , follows a cut-off  (6) is rearranged as, In particular, the distribution of the received power with the true beamforming gain model is calculated as, where p Ps is the PDF of P s that follows 10-based log-normal distribution while p Gt (x) is the PDF of the true beamforming gain model that will be given in Sec. IV-B and Sec. IV-C.
In comparison, the distribution of the received power with traditional beamforming gain model for aligned beamforming case is given as, Moreover, the distribution of the received power with traditional beamforming gain model for misaligned beamforming case is stated as,

B. Aligned Beamforming Case
In the aligned beamforming case, the main lobe is directed to the LoS path, i.e., θ main = θ LoS .
Therefore, the beamforming gain G t is expressed as, Since, G t is a function of N m which is a discrete random variable, the PDF of G t is calculated as, where P (N m = l) = where If we consider the multi-cone model for beamforming pattern, p G (m) Ali is derived by introducing We reveal that some conditions where the denominator of (37) is zero that deserves special considerations. If g 2 I 1 − (K + 1)x = 0, we have x = g 2 I 1 /(K + 1), which corresponds to G t = lobe of the beamforming pattern, which thereby cause non-continuous components in the PDF of G t , p Gt (x). Therefore, the probability of these conditions are given as, where I 1 = 1, 2, . . . , N c . The lower bound and upper bound of the beamforming gain are obtained by taking I 1 as N c and 1, which yields G lower = g 2 Nc + Kg 2 1 /(K + 1) and G upper = g 2 1 . Remark 5: In the aligned beamforming case under wideband statistical channel, the boundaries of the beamforming gain is determined by the K value, and the PDF of beamforming gain consists of continuous and discrete components. By contrast, the beamforming gain in the traditional model is inaccurately assumed as a constant g 2 1 .

C. Misaligned Beamforming Case
In the misaligned beamforming case, the main lobe is uniformly and randomly directed, i.e., θ main = θ u . Therefore, the beamforming gain G t is expressed as, As the number of NLoS paths, N m , follows a cut-off Poisson distribution, the PDF of G t , p Gt , is expressed as, where p G (l) Mis is the PDF of the beamforming gain in (39) when the total number of NLoS paths is l. If we consider the multi-cone model for beamforming pattern, p G (m) Mis is given by, We observe that some conditions where the denominator of (41) is zero need special considerations, given by, G t = lobe. The probability of these conditions are, where I 0 , I 1 = 1, 2, . . . , N c . The boundaries of the beamforming gain are obtained by taking I 0 and I 1 as N c and 1, which yields G lower = g 2 Nc and G upper = g 2 1 . Remark 6: In the misaligned beamforming case under wideband statistical channel, the PDF of beamforming gain consists of continuous and discrete components. On the contrary, the beamforming gain in the traditional model is assumed as a discrete random variable with PDF given in (5).

V. NUMERICAL RESULTS AND ANALYSIS
In this section, we validate our theoretical beamforming gain models with simulations for        2) Misaligned Beamforming Case: In Fig. 4(a), we observe that the received signal amplitude does not vary with different k v , which is consistent with (18), since the beamforming gain only depends on the beamforming pattern A p (θ). Moreover, the deviation between the received signal amplitude in the true beamforming gain model and the traditional model can easily reach 10 dB.
Specifically, at a low signal amplitude, the traditional model shows a much higher cumulative probability than the true model, while at a high signal amplitude, the traditional model yields a slightly smaller cumulative probability.
In Fig. 4(b), the increase of the number of antennas tremendously degrades the accuracy of the traditional models in the misaligned beamforming case, similarly in the aligned beamforming case. The difference between the signal amplitude exceeds 15 dB with 32 antennas. Additionally, the traditional beamforming gain model underestimates the impact of the number of antennas (or equivalently the main lobe gain g 1 ) on the signal amplitude, which is opposite to the aligned beamforming case. To be concrete, the differences among the dashed lines are fairly small, especially at low signal amplitudes. However, the average difference among the solid lines is 3 dB, which suggests doubling the number of antennas results in doubled signal power.  Therefore, u v = 0 makes the power distributed in a narrow direction, which is consistent with the assumption made in the traditional beamforming gain model. In Fig. 5(b), we further investigate the impact of K factor of the Rician channel on the received signal amplitude. First, the traditional model underestimates the impact of K. We notice that the differences among solid lines (proposed model) reach 5 dB, which are much larger than that among the dashed lines (traditional model). This is due to the fact that the traditional model assumes that the power of all the path components is enhanced by the main lobe. However, in reality, only the LoS path and a small fraction of the Rayleigh scattering components benefit from the power improvement of the main lobe. Therefore, K, which denotes the power ratio between the direct path components and Rayleigh scattering components, demonstrates a sound impact on the received signal amplitude. Second, the traditional model wrongly estimates the distribution behavior of the signal amplitude, as discussed in Sec. III-B. We notice that the CDF of the signal amplitude with large K is almost vertical, which implies that the fading vanishes after beamforming, and the channel tends to be deterministic. However, this phenomenon is not The impact of the number of antennas N ant is evaluated in Fig. 5(c). First, more antennas cause a steeper CDF of the signal amplitude in the proposed model, compared to that of the traditional model. Second, a more severe inaccuracy of the traditional model is observed with massive antennas. For example, the difference between the two models is 7 dB with 4 antennas, which, however, reaches 12.5 dB with 32 antennas. Third, at a low signal amplitude, the cumulative probability of the traditional model is higher than the proposed model. By contrast, at a high signal amplitude, the cumulative probability of the proposed model becomes higher in turn.
2) Misaligned Beamforming Case: In this case, the main lobe is randomly directed. Fig. 6 depicts the impact of the channel parameters and the number of antennas on the distribution of the signal amplitude. In Fig. 6(a), we observe that firstly, there are generally two cross-points of the CDFs of the signal amplitude under the proposed model and traditional model. Therefore, in the range of low amplitudes, the cumulative probability in the traditional model is lower than that in the proposed model. In the middle amplitude range, the cumulative probability in the traditional model is slightly higher. Then, when the signal amplitude rises up, while the cumulative probability based on the traditional model is below that of the proposed model.
Second, the traditional model underestimates the impact of K as in the aligned beamforming case, for the same reason, as explained in Sec. IV-B- (2). Third, a low K value can lead to indispensable inaccuracy of the traditional model as the solid black line, and the black dashed line show the largest difference, which reaches 7 dB.

C. Beamforming gain in Wideband Statistical Channel
Next, we adopt the wideband statistical channel model to analyze the beamforming gain, based on the proposed model in Sec. IV, in contrast with the traditional beamforming gain model.

1) Aligned Beamforming Case:
In this case, the main lobe points towards the LoS path, which has the lowest path loss among all MPCs of the channel. In Fig. 7, we evaluate the beamforming gain with varying parameters, including K, λ m , and the number of antennas, N ant .
We observe that the beamforming gain is bounded in the region [G lower , G upper ]. Specifically, the lower bound, G lower , satisfies with the probability, In addition, the upper bound G upper is equal to g 2 1 = N 2 ant with the probability calculated as, The explanations of the boundaries of G t are as follows. The LoS path component contains K/(K + 1) portion of the received power, which is enhanced by a factor of |g 1 | 2 . Therefore, the beamforming gain is at least K/(K + 1) · |g 1 | 2 . However, the beamforming gain cannot exceed |g 1 | 2 , since the maximum beamforming gain in any direction is |g 1 | 2 . By contrast, the traditional beamforming gain model superficially assumes that the beamforming gain is equal to a constant value |g 1 | 2 , which is generally overestimated. As the beamwidth of the main lobe |Θ 1 | is generally smaller than the beamwidth of the side lobe |Θ 2 |, P [G t = G lower ] is larger than Therefore, the traditional model by considering the beamforming gain stays at the upper bound, which indeed incorrectly estimates the beamforming gain.
As demonstrated in Fig. 7(a), we observe that the beamforming gain is highly related to K value. Moreover, the CDF at the lower bound of beamforming gain is very high, and is larger than 0.6 for any K values in our proposed model, which is consistent with our calculations.
We further demonstrate that the number of NLoS paths, λ m , affects the beamforming gain in Fig. 7(b). In particular, an increasing value of λ m causes a large difference between the proposed model and the traditional model. The reason is that a large λ m stands for a large number of NLoS paths and consequently, the NLoS power turns to be dispersive in the angular domain.
By incorrectly assuming all the power concentrates in a certain direction, the traditional model degrades the accuracy for a large λ m . Fig. 7(c) presents that massive antennas make a large difference between the beamforming gain in the proposed model and the traditional model. Still, we emphasize that the beamforming gain lies in the region (K/(K + 1)N 2 ant , N 2 ant ]. Therefore, the ratio between the upper bound and the lower bound is determined only by K value. If we depict the x-axis in a log scale, the difference between the two models maintains at 10 * log 10 (K/(K + 1)) dB constantly, which is independent of the number of antennas.    2) Misaligned Beamforming Case: In this case, the main lobe is directed to a random direction. Fig. 8 illustrates the comparison of the beamforming gain in our proposed model and the traditional model. It can be seen that the traditional model regards that the beamforming gain is a discrete random variable with the probabilities, P [G t = g 2 2 ] = |Θ 2 | 2π , and P [G t = g 2 1 ] = |Θ 1 | 2π . In comparison, the CDF of the proposed beamforming model is a continuous function along with some discrete components with the beamforming gain bounded by [|g 2 | 2 , |g 1 | 2 ]. This is consistent with the bounds of that of the traditional model, although being different from the bounds in the aligned case. The reason is that the power in any direction provides a beamforming gain over the range between |g 1 | 2 and |g 2 | 2 . The probabilities of the lower bound and the upper bound in the proposed beamforming gain model are calculated according to (42) as, We verify that the result in the aligned beamforming case, still holds.
As shown in Fig. 8(a), we observe that a higher K value leads to a larger difference between the proposed model and the traditional model. The reason is the same as we discussed in the narrowband fading channels in Sec. V-B. A high K indicates the most of the power concentrates in a direction, which owes to the assumption of the traditional beamforming gain model. Moreover, a sudden rise of the CDF is observed near the position of beamforming gain of K/(K + 1)|g 1 | 2 , or equivalently, |g 1 | 2 = 16 as the number of antennas is 4. For example, the sudden rise occurs at the beamforming gain of 8 and 13.3, for K = 1 and K = 5, respectively, which can be explained as follows. As the LoS path contains K/(K + 1) energy while the NLoS paths carry the rest 1/(K + 1), if the LoS path is within the main lobe and the NLoS paths are located in the side lobe, the beamforming gain equals to Kg 2 1 +g 2 2 K+1 . As g 2 is much smaller than g 1 , the beamforming gain approximates to K/(K + 1)|g 1 | 2 .
Similar to the aligned beamforming case, we further analyze that a higher λ m value with severe power dispersion in the angular domain leads to smaller accuracy of the traditional model, as demonstrated in Fig. 8(b). Moreover, the sudden rise of the CDF does not change with varying λ m , which shows that the sudden rise depends only on K and g 1 . In Fig. 8(c), we can observe that a large number of antennas cause severe inaccuracy of the traditional model. For example, the largest difference between the two models exceeds 20 dB with N ant = 16, while the difference is 10 dB with N ant = 4. In addition, we notice that the sudden rise vanishes as the number of antennas increases. The explanation is that the vertical length of the sudden rise is proportional to the beamwidth of the main lobe as, P [G t = ≤ |Θ 1 | 2π . Therefore, the vertical length decreases with the number of antennas.

D. System-level SINR Analysis with Wideband NYU channel model
To further evaluate our analysis of the beamforming gain model, we investigate the distribution of SINR by a stochastic-geometry method with the wideband NYU channel model [29], at 60 GHz with 1 GHz bandwidth and 15 dBm transmit power. The The system geometry for simulation is the same as described in our previous work [35].
We first choose an UE and assign it as the receiver of interest, Rx 0 . Then, we consider random   without loss of generality, which is a standard approach using the stochastic geometry method.
The associated BS with which Rx 0 communicates is the nearest one to Rx 0 , while the other BSs are regarded as interference source. The channel between the associated BS and UE is the aligned beamforming case, while the channel between the interference BS and the UE is the misaligned beamforming case. The probability of coverage is interpreted as the probability that an arbitrarily UE can achieve a target SINR, T , as which boils down to the CCDF of SINR as, P c (T ) = 1 − P (SIN R ≤ T ) = CCDF SIN R (T ).
The probability of coverage is computed in Fig. 9, as a function the target SINR, T (i.e., CCDF of SINR) with varying K and N ant values. First, we observe from Fig. 9(a) and 9(b) that the traditional model significantly overestimates SINR with larger N ant and smaller K values. We take the target SINR T corresponding to 90% coverage probability, i.e., T 0.9 (P c (T 0.9 ) = 90%), is that the traditional model overestimates the beamforming gain in the aligned case while underestimates the beamforming gain in the misaligned case. More importantly, the traditional model shows that increasing N ant improves SINR without limit in 9(b), which can be also observed in [5]. On average, doubled N ant leads to 3.3 dB increment of T 0.9 in the traditional model. However, the increment of SINR indeed decreases as N ant increases in our proposed model. This upper bound is decided by K value as shown in Fig. 9(c), where a larger K value leads to a higher level of SINR. To be concrete, T 0.9 is 0.75 dB with K = 1 and 1.41 dB with K = 10 and N ant = 128, respectively. The reason is given in our previous explanation in Sec. V-A that the traditional model underestimates the impact of increasing large N ant on beamforming gain in the misaligned beamforming case and thereby the interference link power.

E. Complexity of the beamforming gain models
As a benchmark, one realization of the traditional beamforming gain model costs 0.5 ns and 25 ns for aligned and misaligned beamforming cases, respectively. Then, we evaluate the ratio of the computation times of the proposed beamforming gain model and the traditional beamforming gain model for different beamforming cases and channel models in Table II.

VI. CONCLUSION
In this paper, we first point out that the traditional beamforming gain model in the literature ignores the channel properties. To be concrete, in the narrowband fading channel, the channel property is the spatial distribution of the scattering path and Rician K factor. In the wideband statistical channel, the ignored channel properties include K factor and the parameters of the number of NLoS paths. Instead, we properly present the general definition of the beamforming gain, and thereby propose a new beamforming gain model that incorporates the beamforming pattern, beamforming cases, and spatial channel properties. We further prove that the traditional beamforming gain model is indeed a special case of the proposed model when the energy of the channel is directed to one direction.
Furthermore, we derive the beamforming gain and the distribution of the signal amplitude in both aligned and misaligned beamforming cases for narrowband fading channels, i.e., Rayleigh fading channel and Rician fading channel by using the proposed beamforming gain model.
By comparison, we reveal that the traditional model wrongly estimates the distribution of the signal amplitude in the narrowband fading channels. Moreover, we derive the beamforming gain model in the wideband statistical channel model. The PDF of the proposed model is a continuous function with some discrete components, while that of the traditional model is a discrete function.
In addition, the boundaries and distribution of the beamforming gain are highly related to the channel parameters, which are ignored in the traditional model. However, it underestimates the impact of the number of antennas in the misaligned beamforming case in both the narrowband fading and wideband statistical channels. In addition, the traditional model underestimates the effect of K factor in both aligned and misaligned beamforming cases for narrowband fading channels and wideband statistical channels.
The lessons for the system-level performance analysis of the wireless networks are as follows.
First, the beamforming gain is highly related to the K factor according to the proposed model, and the traditional model is less accurate with a small K factor. Therefore, in the analysis of those environments with a low K factor, the traditional model is not suggested. Second, the traditional mode greatly differs from the proposed model with a narrower beam. The traditional model is not suggested in the cases of massive antennas or high-gain directional antenna. Third, we notice that in the misaligned beamforming case, which is mostly the case of interference links, the traditional model underestimates the beamforming gain as well as the impact of the increasing number of antennas on the beamforming gain.
Remarkably, the traditional model overestimates the SINR as well as ergodic capacity with massive antennas, which is validated in our analysis by the stochastic geometry approach. In addition, the larger number of antennas steadily brings significant improvement of SINR and ergodic capacity in the traditional model, which is however incorrect. Instead, the increment of SINR and ergodic capacity in our proposed model decreases as the number of antennas increases.
We conclude that missing consideration of the channel characteristics in the beamforming gain model leads to incorrect system-level performance analysis of wireless networks.