A Statistical Characterization of Localization Performance in Millimeter-Wave Cellular Networks

We consider downlink transmission in a mmWave cellular network where the locations of BSs are modeled using a homogeneous PPP[5]. As illustrated in Fig. 1, we assume that the target is located at the origin O and the BSs are randomly distributed over the $\mathbb{R}^{2}$ plane. The red triangle represents the nearest BS to the target that is located inside the disk, whereas the green triangle indicates the furthest BS from the target residing in the disk. Furthermore, blue and yellow triangles represent the BSs that are located inside and outside the disk, respectively. Let us denote the locations of the BS as $\bm{\psi}_{l}=[x_{l},y_{l}]\in\mathbb{R}^{2}$ and the distance between the $l$-th BS and target as $r_{l}=||\bm{\psi}_{l}||$. Based on the system model, the probability density function (PDF) and cumulative distribution function (CDF) of the $L$-th nearest BS are given by [21]

$$\begin{split}f_{r_{L}}(r)&=\frac{2(\lambda\pi r^{2})^{L}}{r(L-1)!}e^{-\lambda\pi r^{2}},\\ F_{r_{L}}(r)&=1-\sum_{n=0}^{L-1}\frac{1}{n!}e^{-2\pi\lambda r^{2}}(2\pi\lambda r^{2})^{n},\end{split}$$

(1)

where $\lambda$ represents the BS density. Conditioned on the distance of the $L$-th BS from O, the remaining BSs closer to the origin than the $L$-th BS form a uniform binomial point process (BPP) on $\bm{b}(\textit{O},R_{L})$ [22], where the PDF and CDF of $r_{l}$ are given by

$$f_{r_{l}}(r)=\frac{2r}{r_{L}^{2}-r_{1}^{2}},\quad F_{r_{l}}(r)=\frac{r^{2}}{r_{L}^{2}-r_{1}^{2}},$$

(2)

with $r_{1}\leq r_{l}\leq r_{L}$. In Section IV, we applied order statistic to obtain the distribution of the ordered distances.

We assume that each BS is equipped with a directional antenna array composed of $N_{t}$ elements and all BSs operate at a constant power $P_{t}$. In the mmWave channel, the non-line-of-sight (NLOS) interference is negligible since the channel gains of NLOS paths are typically 20 dB weaker than those from the line-of-sight (LOS) [23]. The effect of path-loss can be reduced due to the utilization of the directional antenna arrays, and it is also applied to provide highly directional beams. The received signal from the $l$-th BS to the origin is given by

$$\begin{split}y(t)&=\sqrt{P_{t}\beta}\bm{h}_{\psi_{l}}\bm{w}_{\psi_{l}}r_{l}^{-\frac{\alpha}{2}}s_{\psi_{l}}(t)+n(t)\\ &+\sum_{\bm{\psi}\in\bm{\psi}^{\prime}}\sqrt{P_{t}\beta}\bm{h}_{\bm{\psi}}\bm{w}_{\bm{\psi}}||\bm{\psi}||^{-\frac{\alpha}{2}}s_{\bm{\psi}}(t),\quad t\in\left[0,T\right],\end{split}$$

(3)

where $s_{\bm{\psi}}(t)$ is the transmit signal, $\bm{h}_{\bm{\psi}}$ is the channel vector, $\alpha$ and $\beta$ respectively represent the path-loss exponent and path-loss intercept, $\bm{w}_{\bm{\psi}}$ denotes the beamforming vector of the node at location $\bm{\psi}$, and $n(t)$ represents the additive white Gaussian noise (AWGN) with variance $\sigma^{2}$. Note that the locations of the interfering transmitters are denoted as $\bm{\psi}^{\prime}$.

Due to high free-space path-loss, the mmWave propagation environment is well characterized by a clustered channel model, known as the Saleh-Valenzuela model[24]:

$$\bm{h}_{\bm{\psi}}=\sqrt{N_{t}}\sum_{n=1}^{N}\rho_{\bm{\psi},{n}}\bm{a}_{t}^{H}(\theta_{\bm{\psi},{n}}),$$

(4)

where $N$ is the number of clusters and $\rho_{\bm{\psi},{n}}$ represents the complex small-scale fading coefficient of the $n$-th cluster. We assume that the fading channel power gain follows a gamma distribution, i.e., $|\rho_{\bm{\psi}}|^{2}\sim\Gamma(M,\frac{1}{M})$, with Nakagami parameter $M$. In this paper, we focus on LOS paths, i.e., $N=1$, and adopt a uniformly random single path (UR-SP) channel model that is commonly used in the mmWave network analysis [25]. The $\bm{a}_{t}(\theta_{\bm{\psi}})$ represents the transmit array response vector corresponding to the AOA $\theta_{\bm{\psi}}$. We consider a uniform linear array (ULA) with $N_{t}$ antenna elements, where the transimt array response vectors are given by

$$\bm{a}_{t}(\theta_{\bm{\bm{\psi}}})=\frac{1}{\sqrt{N_{t}}}\left[1,\ldots,e^{j2\pi k\theta_{\bm{\psi}_{1}}},\ldots,e^{j2\pi(N_{t}-1)\theta_{\bm{\psi}}}\right]^{T},$$

(5)

where $d$ is the antenna spacing, $\lambda_{w}$ represents the wavelength, $\phi_{\bm{\psi}}$ denotes the AOA, $k\in\left[0,N_{t}\right]$ is the antenna index, and $\theta_{\bm{\psi}}=\frac{d}{\lambda_{w}}\sin\phi_{\bm{\psi}}$ is uniformly distributed over $\left[-\frac{d}{\lambda_{w}},\frac{d}{\lambda_{w}}\right]$.

Once the AOA measurements are obtained, we compute the location of the target, where we assume LOS propagation. As shown in Fig. 2, we denote the AOA between the target and $l$-th BS as $\theta_{l}$ and the location of target as $\bm{\psi}_{t}=[x_{t},y_{t}]$,

$$\tan(\theta_{l})=\frac{y_{l}-y_{t}}{x_{l}-x_{t}},\quad l=\{1,\ldots,L\}.$$

(6)

The AOA measurement at the $l$-th BS is modeled as follows

$$r_{{\rm{AOA}},l}=\theta_{l}+n_{{\rm{AOA}},l}=\tan^{-1}\left(\frac{y_{l}-y_{t}}{x_{l}-x_{t}}\right)+n_{{\rm{AOA}},l},$$

(7)

where $n_{{\rm{AOA}},l}$ is the AWGN with variance $\sigma_{{\rm{AOA}},l}^{2}$. The AOA measurements in (7) can be represented by a vector form

$${\bm{r}}_{\rm{AOA}}=\bm{f}_{AOA}(\bm{\psi})+\bm{n}_{{\rm{AOA}},l},$$

(8)

where ${\bm{r}}_{\rm{AOA}}$, ${\bm{n}}_{\rm{AOA}}$, and $\bm{f}_{AOA}(\bm{\psi})$ are respectively defined by

$$\begin{split}{\bm{r}}_{\rm{AOA}}&=\left[r_{{\rm{AOA}},1},r_{{\rm{AOA}},2},\ldots,r_{{\rm{AOA}},L}\right]^{T},\\ {\bm{n}}_{\rm{AOA}}&=\left[n_{{\rm{AOA}},1},n_{{\rm{AOA}},2},\ldots,n_{{\rm{AOA}},L}\right]^{T},\\ \bm{f}_{AOA}(\bm{\psi})&=\left[\tan^{-1}\left(\frac{y_{1}-y_{t}}{x_{1}-x_{t}}\right),\ldots,\tan^{-1}\left(\frac{y_{L}-y_{t}}{x_{L}-x_{t}}\right)\right]^{T}.\end{split}$$

(9)

Assuming that the AOA of the channel between the BS at location $\bm{\psi}_{l}$ and its serving user at location $\bm{\psi}_{t}$ is $\theta_{\bm{\psi}_{l}}$, the beamforming vector is given by

$$\bm{w}_{\bm{\psi}_{l}}=\bm{a}_{t}(\theta_{\bm{\psi}_{l}}),$$

(10)

which means that the BS should align the beam direction exactly with the propagation channel to obtain the maximum power gain. However, the beam direction cannot always align with the transmit signal. Hence, we consider the single main-lobe and single side-lobe at antennas of both BS and mobile user, and all lobes are approximated by a flat-top antenna pattern[26]. That is, the single main-lobe with beam-width $\theta_{1}$ has antenna gain $G_{1}$ and each side lobe with identical beam-width $\theta_{2}$ has antenna gain $G_{2}$. We assume that the power spectrum density (PSD) of the main-lobe and side-lobe at a distance $r$ are denoted as $P_{m}$ and $P_{s}$. Hence, the total transmit power $P_{T}$ consists of the main-lobe and side-lobe radiation powers, which is given by [26]

$$P_{T}=P_{m}2\pi r^{2}\left[1-\cos\frac{\theta_{1}}{2}\right]+N_{t}P_{s}2\pi r^{2}\left[1-\cos\frac{\theta_{2}}{2}\right],$$

(11)

where $P_{m}=G_{1}P_{T}/4\pi r^{2}$ and $P_{s}=G_{2}P_{T}/4\pi r^{2}$. Let us denote $k=G_{2}/G_{1}$ where $k\in(0,1)$, i.e., $G_{2}=kG_{1}$.

For the associated signal transmission, we assume perfect alignment where both the BS and user utilize the main-lobe, achieving the squared gain $G_{1}^{2}$. For the interfering signal, the interfering BSs are randomly distributed in $[0,2\pi)$. The transmit antenna gain $G_{Tx}$ at the transmitter and the receive antenna gain $G_{Rx}$ at the receiver are randomly chosen from a discrete set $\left\{G_{1},G_{2}\right\}$ with probability $p_{a}=\frac{\theta_{1}}{2\pi}$ and $p_{b}=1-p_{a}$, respectively. Let $G_{TRx}=G_{Tx}G_{Rx}$, we have

$$G_{TRx}=\left\{\begin{array}[]{ll}G_{1}^{2},&p_{1}=p_{a}^{2}\\ G_{1}G_{2},&p_{2}=2p_{a}p_{b}\\ G_{2}^{2},&p_{3}=p_{b}^{2}.\end{array}\right.$$

(12)

Based on the antenna radiation pattern, the product of small-scale fading gain and beamforming gain of the BS at location $\bm{\psi}$ is computed as:

$$|\bm{h}_{\bm{\psi}}\bm{w}_{\bm{\psi}}|^{2}=N_{t}|\rho_{\bm{\psi}}|^{2}G_{TRx}.$$

(13)

A target is localizable if there are a sufficient number of participating BSs such that the localization procedure can be conducted. We introduce L-Localizability, which is a probability to have $L$ localizable BSs within the network [7]. Conventionally, commonly-accepted minimum values of $L$ for the unambiguous operation of a localization system are $2$, $3$, $3$ and $3$ for AOA, TOA or RSS, and TDOA, respectively [4]. If we treat the interference originated from outside of the circular disk with radius $R_{L}$ as a noise, the signal-to-interference-plus-noise ratio (SINR) of the link from the $k$-th BSs to the target can be expressed as a function of $L$ as

$${\rm{SINR}}_{k}(L)=\frac{G_{1}^{2}|\rho_{\bm{\psi}_{k}}|^{2}r_{k}^{-\alpha}}{\sigma_{n}^{2}+J},$$

(14)

where $\sigma_{n}^{2}=\frac{\sigma_{T}^{2}+\sigma_{{\rm{out}}}^{2}}{\beta P_{t}N_{t}}$ is the normalized noise power, including the thermal noise power $\sigma_{T}^{2}$ and interference power $\sigma_{{\rm{out}}}^{2}$ outside the circular disk. The interference from nodes inside the disk, denoted by $J$, is expressed as:

$$J=\sum_{i=1,i\neq k}^{L-1}a_{i}G_{TRx,i}|\rho_{\bm{\psi}_{i}}|^{2}||\bm{\psi}_{i}||^{-\alpha},$$

(15)

where $a_{i}\in\{0,1\}$ is utilized to simulate the network load. The probability of $a_{i}=1$ equals $q$ which is the probability of a BS inside the disk to be activated. The $a_{i}$ represents whether the BS is activated in the localization procedure and we assume that the activation probability $P\left(a_{i}=1\right)=q$ is fixed throughout the localization procedure.

For a given ${\bm{\psi}}\in\mathbb{R}^{2}$, a mobile device is said to be $L$-localizable if at least $L$ BSs participate in the localization procedure. Let us denote the SINR threshold as $\tau$ and the maximum number of BSs that can participate in the localization procedure as $\gamma$, defined as

$$\gamma={\mathop{\arg\max}_{L}}\left(L\cdot\prod_{k=1}^{L}\mathbb{I}\left({\rm{SINR}}_{k}(l)\geq\tau\right)\right),$$

(16)

where $\mathbb{I}(.)$ is the indicator function. Then, the $L$-localizability, denoted by $P_{L}$, is derived as:

$$P_{L}=P\left(\gamma\geq L\right)=\mathbb{E}\left[\prod_{k=1}^{L}\mathbb{I}\left({\rm{SINR}}_{k}(l)\geq\tau\right)\right].$$

(17)

Since the SINR from a BS farther from the mobile device is lower than that of the closer BS, the following inequality holds: $\mathbb{I}({\rm{SINR}}_{k}(L)\geq\tau)\geq\mathbb{I}({\rm{SINR}}_{l}(L)\geq\tau)$ for all $k\geq l\geq L$. Then, the $L$-localizability $P_{L}$ can be computed as follows

$$\begin{split}P_{L}&=\mathbb{E}\left[\mathbb{I}({\rm{SINR}}_{L}(L)\geq\tau\right]=P\left({\rm{SINR}}_{L}(L)\geq\tau\right)\\ &=1-P\left(|\rho_{\psi_{L}}|^{2}\leq\frac{\tau\leavevmode\nobreak\ r_{L}^{\alpha}}{G_{1}^{2}}\left(\sigma_{n}^{2}+J\right)\right)\\ &\overset{a}{\simeq}1-\mathbb{E}_{r_{L}}\left[\left(1-e^{-\nu\frac{\tau}{G_{1}^{2}}r_{L}^{\alpha}\left(\sigma_{n}^{2}+J\right)}\right)^{M}\right]\\ &\overset{b}{=}\mathbb{E}_{r_{L}}\left[\sum_{i=1}^{M}(-1)^{i+1}\binom{M}{i}e^{-s\sigma_{n}^{2}}\mathcal{L}_{I}(s)\right]\\ &=\int_{0}^{\infty}f_{r_{L}}(r)\sum_{i=1}^{M}(-1)^{i+1}\binom{M}{i}e^{-s\sigma_{n}^{2}}\leavevmode\nobreak\ \mathcal{L}_{I}(s)dr\\ \end{split}$$

(18)

where $\nu=M(M!)^{-\frac{1}{M}}$, $s=i\nu\frac{\tau}{G_{1}^{2}}r_{L}^{\alpha}$ and $\mathcal{L}_{I}(s)=\mathbb{E}_{I}[e^{-sJ}]$ is the Laplace transform of the interference. Step $(a)$ follows by the Alzer’s inequality and $(b)$ is obtained based on the binomial expansion. The Laplace transform of the interference is [27]

$$\begin{split}\mathcal{L}_{I}(s)&=\mathbb{E}_{I}[e^{-sJ}]\\ &=\exp\bigg{[}-2\pi\lambda q\underbrace{\int_{r_{1}}^{r_{L}}(1-\mathbb{E}_{g_{{\bm{\psi}}}}[e^{-sJ}]){r}dr}_{\triangleq\Lambda}\bigg{]},\\ \end{split}$$

(19)

where $g_{{\bm{\psi}}}=G_{TRx}|\rho_{{\bm{\psi}}}|^{2}$ represents the combined effect of antenna gain and channel gain at the location ${\bm{\psi}}$. The term $\Lambda$ is computed as:

$$\begin{split}\Lambda&=\int_{r_{1}}^{r_{L}}\left(1-\mathbb{E}_{g_{{\bm{\psi}}}}[e^{-sJ}]\right)rdr\\ &=-\frac{1}{2}\Big{[}r_{L}^{2}-\delta r_{L}^{2}\mathbb{E}_{g_{{\bm{\psi}}}}\left[E_{1+\delta}\left(sg_{{\bm{\psi}}}r_{L}^{-\alpha}\right)\right]\\ &\quad-r_{1}^{2}+\delta r_{1}^{2}\mathbb{E}_{g_{{\bm{\psi}}}}\left[E_{1+\delta}\left(sg_{{\bm{\psi}}}r_{1}^{-\alpha}\right)\right]\Big{]},\end{split}$$

(20)

where $\delta=\frac{2}{\alpha}$ and $E_{1+\delta}(.)$ is the generalized exponential integral [28]. The term $\mathbb{E}_{g_{{\bm{\psi}}}}[E_{1+\delta}(sg_{{\bm{\psi}}}r^{-\alpha})]$ is given by

$$\begin{split}&\mathbb{E}_{g_{{\bm{\psi}}}}[E_{1+\delta}(sg_{{\bm{\psi}}}r^{-\alpha})]\\ =&\frac{s^{\delta}\Gamma\left(-\delta\right)}{r^{2}}\left[\mathbb{E}_{g_{\psi}}\left[g_{\psi}^{\delta}\right]+\frac{\alpha}{2}-\sum_{p=1}^{\infty}\frac{(-s)^{p}\cdot\mathbb{E}_{g_{\psi}}\big{[}g_{\psi}^{\delta}\big{]}}{r^{\alpha p}\cdot p!\cdot(p-\delta)}\right],\end{split}$$

(21)

and the fractional moment of $g_{{\bm{\psi}}}$ is derived as:

$$\begin{split}&\mathbb{E}_{g_{{{\psi}}}}[g_{{{\psi}}}^{\delta}]=\mathbb{E}_{|\rho_{{\bm{\psi}}}|^{2},G_{TRx}}\left[\left(|\rho_{{\bm{\psi}}}|^{2}G_{TRx}\right)^{\delta}\right]\\ &=\frac{\Gamma(M+\delta)}{\Gamma(M)M^{\delta}}\cdot\mathbb{E}_{G_{TRx}}(G_{TRx}^{\delta})\\ &=\frac{\Gamma(M+\delta)}{\Gamma(M)M^{\delta}}\cdot\left[G_{1}^{2\delta}p_{a}^{2}+2(G_{1}G_{2})^{\delta}p_{a}p_{b}+G_{2}^{2\delta}p_{b}^{2}\right].\end{split}$$

(22)

Hence, the $L$-localizability can be numerically evaluated by substituting (19)-(22) into (18).

We derive the AOA-based random CRLB using the Fisher information matrix (FIM), denoted by $\bm{{\rm{I}}}(\bm{\psi})$ [4]

$$\bm{{\rm{I}}}(\bm{\psi})=\left(\frac{\partial{\bm{f}}_{\rm{AOA}}(\bm{\psi})}{\partial\bm{\psi}}\right)^{T}\bm{C}_{\rm{AOA}}^{-1}\frac{\partial{\bm{f}}_{\rm{AOA}}(\bm{\psi})}{\partial\bm{\psi}},$$

(23)

where $\bm{C}_{\rm{AOA}}^{-1}$ represents the inverse of the noise covariance matrix and the derivative of ${\bm{f}}_{\rm{AOA}}(\bm{\psi})$ which is the angle vector with respect to $\bm{\psi}$ are given by

$$\begin{split}\bm{C}_{\rm{AOA}}^{-1}&=\text{diag}\left(\frac{1}{\sigma_{{\rm{AOA}},1}^{2}},\frac{1}{\sigma_{{\rm{AOA}},2}^{2}},\cdots,\frac{1}{\sigma_{{\rm{AOA}},L}^{2}}\right),\\ \newline \\ \frac{\partial{\bm{f}}_{\rm{AOA}}(\bm{\psi})}{\partial\bm{\psi}}&=-\begin{bmatrix}\frac{y-y_{1}}{(x-x_{1})^{2}+(y-y_{1})^{2}}&\frac{x-x_{1}}{(x-x_{1})^{2}+(y-y_{1})^{2}}\\ \frac{y-y_{2}}{(x-x_{2})^{2}+(y-y_{2})^{2}}&\frac{x-x_{2}}{(x-x_{2})^{2}+(y-y_{2})^{2}}\\ \vdots&\vdots\\ \frac{y-y_{L}}{(x-x_{L})^{2}+(y-y_{L})^{2}}&\frac{x-x_{L}}{(x-x_{L})^{2}+(y-y_{L})^{2}}\end{bmatrix}.\end{split}$$

(24)

Without loss of generality, $\sigma_{\rm{AOA}}$ is considered to be a known quantity and assumed to be identical for each BSs, i.e., $\sigma_{{\rm{AOA}},1}=\sigma_{{\rm{AOA}},2}=\cdots=\sigma_{{\rm{AOA}},L}$[8]. The numerical value of $\sigma_{\rm{AOA}}$ depends on the average SNR of the mmWave networks, denoted by $\rm{\overline{SNR}}$, as follows

$${\rm{\overline{SNR}}}=\frac{G_{c}P_{t}}{N_{0}W_{\rm{TOT}}}$$

(25)

where $G_{c}$ is the average channel gain, $N_{0}$ is the spectral density of the WGN, and $W_{\rm{TOT}}$ is the total system bandwidth [29, 30]. Hence, $\bm{{\rm{I}}}(\bm{\psi})$ is

$$\begin{split}&\bm{{\rm{I}}}_{\rm{AOA}}(\bm{\psi})\\ =&\sigma_{\rm{AOA}}^{2}\begin{bmatrix}\sum_{i=1}^{L}\frac{(y-y_{i})^{2}}{r_{i}^{4}}&-\sum_{i=1}^{L}\frac{(x-x_{i})(y-y_{i})}{r_{i}^{4}}\\ -\sum_{i=1}^{L}\frac{(x-x_{i})(y-y_{i})}{r_{i}^{4}}&\sum_{i=1}^{L}\frac{(x-x_{i})^{2}}{r_{i}^{4}}\end{bmatrix}.\end{split}$$

(26)

To assess the distribution of the CRLB, we introduce the position error bound (PEB), which is the square root of the CRLB [31]. We will denote the PEB by $S$ and its closed-form expression can be obtained by using (26)

$$\begin{split}S&\triangleq\sqrt{\rm{CRLB}}=\sqrt{{\rm{tr}}(\bm{I}_{\rm{AOA}}^{-1}(\bm{\psi}))}=\sigma_{\rm{AOA}}\frac{\sqrt{L}}{\sqrt{Q_{1}-Q_{2}}},\end{split}$$

(27)

where $Q_{1}$ and $Q_{2}$ are

$$\begin{split}Q_{1}&=\sum_{i=1}^{L}\frac{(y_{i}-y_{t})^{2}}{r_{i}^{4}}\sum_{j=1}^{L}\frac{(x_{j}-x_{t})^{2}}{r_{j}^{4}},\\ Q_{2}&=\sum_{i=1}^{L}\frac{(x_{i}-x_{t})^{2}(y_{i}-y_{t})^{2}}{r_{i}^{8}}.\end{split}$$

(28)

Since (27) and (28) are functions of $2L$ random variables, i.e., $\left(x_{i},y_{i}\right)$ for $1\leq i\leq L$, we need to simplify (28) using its asymptotic bounds, which will enable us to characterize the distribution of (27). In the following proposition, we derived a tight upper bound for $Q_{1}-Q_{2}$ and through simulation, we verified that approximation error is less than $5\%$ for $L\geq 8$.

Based on Proposition 1, the PEB $S$ is lower bounded by

$$\begin{split}S&\geq\frac{2\sigma_{\rm{AOA}}\cdot\sqrt{L}}{\sqrt{\left(\sum_{i=1}^{L}\frac{1}{r_{i}^{2}}\right)^{2}-\sum_{i=1}^{L}\frac{1}{r_{i}^{4}}}}=\frac{2\sigma_{\rm{AOA}}\cdot\sqrt{L}}{\sqrt{\sum\limits_{\begin{subarray}{c}\scriptstyle i,j=1\\ \scriptstyle i\neq j\end{subarray}}^{L}\frac{1}{r_{i}^{2}r_{j}^{2}}}}.\end{split}$$

(36)

In the following assumption, we introduced an approximation of (36), which provides a tractable asymptotic bound of $S$. Through simulation, we justified the approximation accuracy.

Based on (36) and (37), we can approximate $S$ by

$$S\approx\frac{2\sigma_{\rm{AOA}}}{\sqrt{L-1}}\leavevmode\nobreak\ r_{\lceil L/4\rceil}^{2},$$

(42)

where $S$ is a function of $r_{\lceil L/4\rceil}$. In the following proposition, we derived the distribution of $S$ using order statistic.

In Figs. 11-11, we investigate how the network parameters, including network loads, path-loss exponent and Nakagami fading parameter, affect the performance of $L$-localizability. Fig. 11 compares the $L$-localizability $P_{L}$ versus the SINR threshold for different network loads $q$. The simulation results are plotted in dotted curves, where as the analytical results are represented by solid curves with a marker. All of the numerical results indicate that the analytical results accurately match the simulation results, justifying the analytical derivation. We observed that increasing the network load leads to a decrease in $P_{L}$. It means that network design should be optimized so that there is a sufficient number of BSs to meet the localization requirement. Fig. 11 demonstrates the impact of path-loss on the $L$-localizability. As the path-loss exponent increases, the transmitted power across the mmWave link will significantly decline, causing a significant drop in $P_{L}$. In Fig. 11, we observe the impact of Nakagami fading parameter $M$ on $P_{L}$. Since the Nakagami channel becomes deterministic as the $M$ parameter increases, the $L$-localizability escalates with higher $M$ values.

In Figs. 11-11, we evaluate the distribution of $S$ for various network parameter configurations. Since the approximation of CRLB using $r_{\lceil L/4\rceil}$ provides an accurate approximation to the original CRLB, we used the approximation based on the $\lceil L/4\rceil$-th distance across Figs. 11-11. In Fig. 11, we examine the impact of the number of participating BSs $L$ on the distribution of $S$. This is accomplished by varying the number of activated BSs transmitting during a localization procedure. It is observed that the value of $P[S\leq abscissa]$ increases for a larger $L$. Since the localization error reduces as the number of BSs increases, a network designer looking to improve the localization accuracy may aim to optimize the network environment to ensure a sufficient number of BSs participate in the localization procedure.