Jittering Effects Analysis and Beam Training Design for UAV Millimeter Wave Communications

We assume that a UPA with $N_{B}=N_{B,x}\times N_{B,z}$ element antennas is vertically mounted on the 2-D plane $y=0$ at BS side, which is shown in Figure 1. We denote the location of BS as $\mathbf{p}_{B}=(0,0,0)^{T}$, and thus the location of the $i$-th element antenna at BS side is represented as

$\displaystyle\mathbf{p}_{B,\iota}=\mathbf{p}_{B}+\mathbf{a}_{B,\iota}$

(1)

where $\mathbf{a}_{B,\iota}$ is the relative position of the $\iota$-th element antenna with reference to the $1$-st element antenna (reference antenna)

$\displaystyle\mathbf{a}_{B,\iota}=\frac{\lambda}{2}[x_{\iota},0,z_{\iota}]^{T}$

(2)

where $\lambda$ is the wavelength of the carrier, $\frac{\lambda}{2}$ is the inter antenna spacing, the antenna indexes $x_{\iota}\in\{0,1,\cdots,N_{B,x}-1\}$ and $z_{\iota}\in\{0,1,\cdots,N_{B,z}-1\}$, and the relative position of the reference antenna is $\mathbf{a}_{B,1}=[0,0,0]^{T}$.

Similarly, we assume that there is a UPA with $N_{U}=N_{U,x}\times N_{U,y}$ element antennas mounted on the bottom of UAV, which is shown in Figure 2. We denote the location of UAV as $\mathbf{p}_{U}$, and thus the location of the $\kappa$-th element antenna at UAV side is

$\displaystyle\mathbf{p}_{U,\kappa}=\mathbf{p}_{U}+\underbrace{\mathbf{R}\mathbf{a}_{U,\kappa}}_{\bar{\mathbf{a}}_{U,\kappa}}$

(3)

where $\mathbf{R}$ is the rotation matrix that depicts the attitude of UAV,

Note that the rotation matrix $\mathbf{R}$ is characterized by Euler angles, i.e., yaw, pitch and roll [27], which is shown in Figure 2, and it is written by

$\displaystyle\mathbf{R}=\mathbf{R}_{Yaw}(\alpha)\mathbf{R}_{Pitch}(\beta)\mathbf{R}_{Roll}(\gamma)$

(4)

where

	$\displaystyle\mathbf{R}_{Yaw}(\alpha)$	$\displaystyle=\left(\begin{array}[]{ccc}\cos\alpha&-\sin\alpha&0\\ \sin\alpha&\cos\alpha&0\\ 0&0&1\\ \end{array}\right)$		(8)
	$\displaystyle\mathbf{R}_{Pitch}(\beta)$	$\displaystyle=\left(\begin{array}[]{ccc}\cos\beta&0&\sin\beta\\ 0&1&0\\ -\sin\beta&0&\cos\beta\\ \end{array}\right)$		(12)
	$\displaystyle\mathbf{R}_{Roll}(\gamma)$	$\displaystyle=\left(\begin{array}[]{ccc}1&0&0\\ 0&\cos\gamma&-\sin\gamma\\ 0&\sin\gamma&\cos\gamma\\ \end{array}\right)$		(16)

and $\mathbf{R}$ can be further expressed as (20).

$\mathbf{a}_{U,\kappa}$ is the relative position of the $\kappa$-th element antenna with reference to the $1$-st element antenna (reference antenna) when UPA is on the 2-D plane $z=0$, which is represented as

$\displaystyle\mathbf{a}_{U,\kappa}=\frac{\lambda}{2}\left[x_{\kappa}-\frac{N_{U,x}-1}{2},y_{\kappa}-\frac{N_{U,y}-1}{2},0\right]^{T}$

(21)

where the antenna indexes $x_{\kappa}\in\{0,1,\cdots,N_{U,x}-1\}$ and $y_{\kappa}\in\{0,1,\cdots,N_{U,y}-1\}$, and the relative position of the reference antenna is $\mathbf{a}_{U,1}=\frac{\lambda}{2}[-\frac{N_{U,x}-1}{2},-\frac{N_{U,y}-1}{2},0]^{T}$.

The channel response for mmWave MIMO with array antenna can be depicted using the geometry dependent model [28, 1], in which each propagation path is characterized by the array shape of Tx/Rx antenna and the spatial domain parameters, i.e., AoA/AoD. However, the exact relationship between AoA/AoD and the placement of antenna arrays (i.e., relative position and antenna orientation) is rarely explored in wireless communication systems. The relationship is the prerequisite for modeling UAV jittering effects on mmWave channel. In order to unravel the relationship, we start channel modeling from the element antennas at BS side and UAV side. According to the free-space path loss model in Eq. (2.7) and Eq. (2.8) of [29], the complex coefficient of the LoS path between the $\kappa$-th element antenna in UAV side and the $\iota$-th element antenna at BS side is expressed as

$\displaystyle h_{\kappa,\iota}=\frac{\lambda}{4\pi d_{\kappa,\iota}}e^{j\frac{-2\pi f_{c}d_{\kappa,\iota}}{c}}=\frac{\lambda}{4\pi d_{\kappa,\iota}}e^{j\frac{-2\pi d_{\kappa,\iota}}{\lambda}}$

(22)

where $c$ is the speed of light, $f_{c}$ is the carrier frequency, $\lambda=\frac{c}{f_{c}}$ is the signal wavelength and $d_{\kappa,\iota}$ is the distance between the $\kappa$-th antenna of UAV and the $\iota$-th antenna of BS, which is represented as

$\displaystyle d_{\kappa,\iota}$

$\displaystyle=\|\mathbf{p}_{B,\iota}-\mathbf{p}_{U,\kappa}\|_{2}=\|\mathbf{p}_{B}+\mathbf{a}_{B,\iota}-\mathbf{p}_{U}-\bar{\mathbf{a}}_{U,\kappa}\|_{2}$

(23)

In (22), the term $\frac{1}{d_{\kappa,\iota}}$ determines the magnitude of $h_{\kappa,\iota}$, and the term $d_{\kappa,\iota}$ determines the phase of $h_{\kappa,\iota}$.

Through Taylor series expansion, we obtain the approximations of $\frac{1}{d_{\kappa,\iota}}$ and $d_{\kappa,\iota}$ as follows.

	$\displaystyle\frac{1}{d_{\kappa,\iota}}\approx$	$\displaystyle\frac{1}{\|\mathbf{p}_{B}-\mathbf{p}_{U}\|_{2}}$		(24a)
	$\displaystyle d_{\kappa,\iota}\approx$	$\displaystyle\|\mathbf{p}_{B}-\mathbf{p}_{U}\|_{2}+(\mathbf{a}_{B,\iota}-\bar{\mathbf{a}}_{U,\kappa})^{T}\frac{\mathbf{p}_{B}-\mathbf{p}_{U}}{\|\mathbf{p}_{B}-\mathbf{p}_{U}\|_{2}}$		(24b)

In (24a), only the zeroth order component is conserved because $\mathbf{a}_{B,\iota}-\bar{\mathbf{a}}_{U,\kappa}$ is of millimeter-level and by contrast $\mathbf{p}_{B}-\mathbf{p}_{U}$ is of kilometer-level; In (24b), as $\frac{d_{\kappa,\iota}}{\lambda}$ determines phase of $h_{\kappa,\iota}$, both zeroth and first order components are conserved, and second and higher order components are omitted due to their negligible impact on phase.

Substituting (24) into (22), we have

$\displaystyle h_{\kappa,\iota}\approx\frac{\lambda}{4\pi d_{BU}}e^{j\frac{-2\pi d_{BU}}{\lambda}}e^{j\frac{-2\pi(\mathbf{a}_{B,\iota}-\bar{\mathbf{a}}_{U,\kappa})^{T}\mathbf{e}_{BU}}{\lambda}}$

(25)

where

	$\displaystyle d_{BU}$	$\displaystyle\triangleq\|\mathbf{p}_{B}-\mathbf{p}_{U}\|_{2}$		(26a)
	$\displaystyle\mathbf{e}_{BU}$	$\displaystyle\triangleq\frac{\mathbf{p}_{B}-\mathbf{p}_{U}}{\|\mathbf{p}_{B}-\mathbf{p}_{U}\|_{2}}$		(26b)

Using spherical coordinates, the direction vector $\mathbf{e}_{BU}$ can be represented as

$\displaystyle\mathbf{e}_{BU}=[\cos\phi\cos\theta,\;\cos\phi\sin\theta,\;\sin\phi]^{T}$

(27)

where $\phi$ is elevation angle and $\theta$ is azimuth angle of the LoS path.

Owing to the poor scattering property of mmWave frequency band [30, 31], LoS propagation dominates mmWave communications, and it is especially true for the link between UAV and BS. Following [21, 22, 19], we assume that the channel of UAV-to-BS mmWave communications is LoS-channel. Thus, the channel matrix $\mathbf{H}$ between BS and UAV is written as

	$\displaystyle\mathbf{H}=$	$\displaystyle\left(\begin{array}[]{cccc}h_{1,1}&h_{1,2}&\cdots&h_{1,N_{B}}\\ h_{2,1}&h_{2,2}&\cdots&h_{2,N_{B}}\\ \vdots&\vdots&\ddots&\vdots\\ h_{N_{U},1}&h_{N_{U},2}&\cdots&h_{N_{U},N_{B}}\\ \end{array}\right)$		(32)
	$\displaystyle\approx$	$\displaystyle\frac{\lambda}{4\pi d_{BU}}e^{j\frac{-2\pi d_{BU}}{\lambda}}\left(\begin{array}[]{cccc}v_{1,1}&v_{1,2}&\cdots&v_{1,N_{B}}\\ v_{2,1}&v_{2,2}&\cdots&v_{2,N_{B}}\\ \vdots&\vdots&\ddots&\vdots\\ v_{N_{U},1}&v_{N_{U},2}&\cdots&v_{N_{U},N_{B}}\\ \end{array}\right)$		(37)

where $v_{\kappa,\iota}=e^{j\frac{-2\pi(\mathbf{a}_{B,\iota}-\bar{\mathbf{a}}_{U,\kappa})^{T}\mathbf{e}_{BU}}{\lambda}}$. In accordance to the geometry dependent channel model in [1], [28], Eq. (37) can be written as

$\displaystyle\mathbf{H}\approx$

$\displaystyle\frac{\lambda}{4\pi d_{BU}}e^{j\frac{-2\pi d_{BU}}{\lambda}}\mathbf{v}_{U}\mathbf{v}_{B}^{H}$

(38)

where the array response vectors $\mathbf{v}_{B}$ and $\mathbf{v}_{U}$ are represented as

	$\displaystyle\mathbf{v}_{B}=\left[e^{j\frac{2\pi\mathbf{a}_{B,1}^{T}\mathbf{e}_{BU}}{\lambda}},e^{j\frac{2\pi\mathbf{a}_{B,2}^{T}\mathbf{e}_{BU}}{\lambda}},\cdots,e^{j\frac{2\pi\mathbf{a}_{B,N_{B}}^{T}\mathbf{e}_{BU}}{\lambda}}\right]^{T}$		(39a)
	$\displaystyle\mathbf{v}_{U}=\left[e^{j\frac{2\pi\bar{\mathbf{a}}_{U,1}^{T}\mathbf{e}_{BU}}{\lambda}},e^{j\frac{2\pi\bar{\mathbf{a}}_{U,2}^{T}\mathbf{e}_{BU}}{\lambda}},\cdots,e^{j\frac{2\pi\bar{\mathbf{a}}_{U,N_{U}}^{T}\mathbf{e}_{BU}}{\lambda}}\right]^{T}$		(39b)

To parameterize the array response vectors, we decompose them as follows.
(1) Decomposition of the array response vector $\mathbf{v}_{B}$ at BS side.

With $(\Psi_{B},\Omega_{B})$, the array response vector $\mathbf{v}_{B}$ can be further decomposed as

$\displaystyle\mathbf{v}_{B}=\mathbf{v}(\Psi_{B},N_{B,x})\otimes\mathbf{v}(\Omega_{B},N_{B,z})$

(42)

where

	$\displaystyle\mathbf{v}(\Psi_{B},N_{B,x})$	$\displaystyle=\left[1,\;e^{j\pi\Psi_{B}},\;\cdots,\;e^{j(N_{B,x}-1)\pi\Psi_{B}}\right]^{T}$		(43a)
	$\displaystyle\mathbf{v}(\Omega_{B},N_{B,z})$	$\displaystyle=\left[1,\;e^{j\pi\Omega_{B}},\;\cdots,\;e^{j(N_{B,z}-1)\pi\Omega_{B}}\right]^{T}$		(43b)

In (39b), $\bar{\mathbf{a}}_{U,\kappa}^{T}\mathbf{e}_{BU}$ can be represented as

		$\displaystyle\bar{\mathbf{a}}_{U,\kappa}^{T}\mathbf{e}_{BU}=\mathbf{a}_{U,\kappa}^{T}\mathbf{R}^{T}\mathbf{e}_{BU}$
	$\displaystyle=$	$\displaystyle\frac{\lambda}{2}\left[x_{\kappa}-\frac{N_{U,x}-1}{2},y_{\kappa}-\frac{N_{U,y}-1}{2},0\right]\mathbf{R}^{T}\mathbf{e}_{BU}$
	$\displaystyle=$	$\displaystyle\frac{\lambda}{2}\left(x_{\kappa}[1,0,0]\mathbf{R}^{T}\mathbf{e}_{BU}+y_{\kappa}[0,1,0]\mathbf{R}^{T}\mathbf{e}_{BU}\right)$
		$\displaystyle\qquad-\frac{\lambda}{2}\left[\frac{N_{U,x}-1}{2},\frac{N_{U,y}-1}{2},0\right]\mathbf{R}^{T}\mathbf{e}_{BU}$		(44)

Let

	$\displaystyle\Psi_{U}$	$\displaystyle\triangleq[1,0,0]\mathbf{R}^{T}\mathbf{e}_{BU}$		(45a)
	$\displaystyle\Omega_{U}$	$\displaystyle\triangleq[0,1,0]\mathbf{R}^{T}\mathbf{e}_{BU}$		(45b)

Then, (44) is rewritten as

		$\displaystyle\quad\bar{\mathbf{a}}_{U,\kappa}^{T}\mathbf{e}_{BU}$
	$\displaystyle=$	$\displaystyle\frac{\lambda}{2}(x_{\kappa}\Psi_{U}+y_{\kappa}\Omega_{U})-\frac{\lambda}{2}\left(\frac{N_{U,x}-1}{2}\Psi_{U}+\frac{N_{U,y}-1}{2}\Omega_{U}\right)$		(46)

Therefore, the array response vector $\mathbf{v}_{U}$ can be decomposed as

$\displaystyle\qquad\mathbf{v}_{U}=\mathbf{v}(\Psi_{U},N_{U,x})\otimes\mathbf{v}(\Omega_{U},N_{U,y})$

(47)

where

		$\displaystyle\quad\mathbf{v}(\Psi_{U},N_{U,x})$
	$\displaystyle=$	$\displaystyle e^{-j\left(\frac{N_{U,x}-1}{2}\pi\Psi_{U}\right)}\left[1,\;e^{j\pi\Psi_{U}},\;\cdots,\;e^{j(N_{U,x}-1)\pi\Psi_{U}}\right]^{T}$		(48a)
		$\displaystyle\quad\mathbf{v}(\Omega_{U},N_{U,y})$
	$\displaystyle=$	$\displaystyle e^{-j\left(\frac{N_{U,y}-1}{2}\pi\Omega_{U}\right)}\left[1,\;e^{j\pi\Omega_{U}},\;\cdots,\;e^{j(N_{U,y}-1)\pi\Omega_{U}}\right]^{T}$		(48b)

From (41) and (50), it can be seen that $(\Psi_{B},\Omega_{B})$ is only determined by UAV position, and $(\Psi_{U},\Omega_{U})$ is determined by both UAV position and UAV attitude. For illustrative purposes, the dependency of the two-dimensional AoA/AoD on UAV position and UAV attitude is shown in Table. II.

The derived channel response in (1) is in accordance with the widely used far-field mmWave channel model for UPA [32, 33], which is characterized by the two dimensional AoA/AoD $(\Psi_{B},\Omega_{B})$ at BS side and $(\Psi_{U},\Omega_{U})$ at UAV side. A notable difference from the existing work, which simply treat the two dimensional AoA/AoD as unknown variables to be estimated, is that (50) extracts the relationship between the two dimensional AoA/AoD $(\Psi_{B},\Omega_{B})$ (or $(\Psi_{U},\Omega_{U})$) and UAV position & attitude related angles.

To obtain insights into jittering effects, we firstly explore the physical meaning of the two dimensional AoA/AoD $\Psi_{U},\Omega_{U}$.

The UPA can be decomposed into a set of vertical/horizontal uniform linear arrays (ULAs). For the purpose of illustration, the spatial relationship of the vertical/horizontal ULA and incident/emergent electromagnetic (EM) wave is shown in Figure 3. When UAV is in the plane $z=0$ and facing towards positive x axis (left side of Figure 3), the direction vector of the vertical ULA is $\mathbf{e}_{U,v}\triangleq[1,0,0]^{T}$, and the direction vector of the horizontal ULA is $\mathbf{e}_{U,h}\triangleq[0,1,0]^{T}$. While, the direction vector of the EM wave between UAV and BS is $\mathbf{e}_{BU}$. Thus, the dot product of $\mathbf{e}_{U,v}$ and $\mathbf{e}_{BU}$ is

$\displaystyle\mathbf{e}_{U,v}\cdot\mathbf{e}_{BU}=\mathbf{e}_{U,v}^{T}\mathbf{e}_{BU}=\cos\psi_{U}=\Psi_{U}$

(51)

where $\psi_{U}$ is the angle between $\mathbf{e}_{U,v}$ and $\mathbf{e}_{BU}$, and the dot product of $\mathbf{e}_{U,H}$ and $\mathbf{e}_{BU}$ is

$\displaystyle\mathbf{e}_{U,h}\cdot\mathbf{e}_{BU}=\mathbf{e}_{U,h}^{T}\mathbf{e}_{BU}=\cos\omega_{U}=\Omega_{U}$

(52)

where $\omega_{U}$ is the angle between $\mathbf{e}_{U,h}$ and $\mathbf{e}_{BU}$.

When the UPA is rotated by $\mathbf{R}$ (right side of Figure 3), the direction of the vertical ULA turns to be $\mathbf{R}\mathbf{e}_{U,v}$, and the direction of the horizontal ULA turns to be $\mathbf{R}\mathbf{e}_{U,h}$, thus the dot products of the direction vectors of ULAs and EM wave become

	$\displaystyle\mathbf{R}\mathbf{e}_{U,v}\cdot\mathbf{e}_{BU}=\mathbf{e}_{U,v}^{T}\mathbf{R}^{T}\mathbf{e}_{BU}=\cos\psi_{U}=\Psi_{U}$		(53a)
	$\displaystyle\mathbf{R}\mathbf{e}_{U,h}\cdot\mathbf{e}_{BU}=\mathbf{e}_{U,h}^{T}\mathbf{R}^{T}\mathbf{e}_{BU}=\cos\omega_{U}=\Omega_{U}$		(53b)

To summarize, $\Psi_{U}$ and $\Omega_{U}$ are the cosine of the angles between the decomposed ULAs (vertical/horizontal) and incident/emergent EM wave.

The most direct reflection of jittering effects on UAV is the fluctuation of attitude angles around the desired attitude [34]. To investigate jittering effects on UAV mmWave channel, we should firstly model the jittering effects on UAV attitude. To this end, we represent the instantaneous attitude angles of the UAV as

	$\displaystyle\alpha=\bar{\alpha}+\Delta\alpha$
	$\displaystyle\beta=\bar{\beta}+\Delta\beta$
	$\displaystyle\gamma=\bar{\gamma}+\Delta\gamma$

where $\bar{\alpha},\bar{\beta},\bar{\gamma}$ are the desired attitude angles which are configured by the flight control system, and $\Delta\alpha,\Delta\beta,\Delta\gamma$ are the fluctuations about the desired attitude angles caused by jittering effects. Owing to the small-scale fluctuation of the attitude angles, the two dimensional AoA/AoD $\Psi_{U},\Omega_{U}$ can be approximated using first-order Taylor polynomial, i.e.,

$\displaystyle\left[\begin{array}[]{c}\Psi_{U}(\alpha,\beta,\gamma)\\ \Omega_{U}(\alpha,\beta,\gamma)\\ \end{array}\right]\approx\left[\begin{array}[]{c}{\Psi_{U}}(\bar{\alpha},\bar{\beta},\bar{\gamma})\\ {\Omega}_{U}(\bar{\alpha},\bar{\beta},\bar{\gamma})\\ \end{array}\right]+\mathbf{J}(\bar{\alpha},\bar{\beta},\bar{\gamma})\left[\begin{array}[]{c}\Delta\alpha\\ \Delta\beta\\ \Delta\gamma\\ \end{array}\right]$

(61)

where the Jacobian matrix $\mathbf{J}(\alpha,\beta,\gamma)$ is represented as

$\displaystyle\mathbf{J}(\alpha,\beta,\gamma)=\left[\begin{array}[]{ccc}\frac{\partial\Psi_{U}}{\partial\alpha}&\frac{\partial\Psi_{U}}{\partial\beta}&\frac{\partial\Psi_{U}}{\partial\gamma}\\ \frac{\partial\Omega_{U}}{\partial\alpha}&\frac{\partial\Omega_{U}}{\partial\beta}&\frac{\partial\Omega_{U}}{\partial\gamma}\\ \end{array}\right]$

(64)

and

$\displaystyle\begin{split}\frac{\partial\Psi_{U}}{\partial\alpha}&=-\sin\alpha\cos\beta\cos\phi\cos\theta+\cos\alpha\cos\beta\cos\phi\sin\theta\\ \frac{\partial\Psi_{U}}{\partial\beta}&=-\cos\alpha\sin\beta\cos\phi\cos\theta-\sin\alpha\sin\beta\cos\phi\sin\theta\\ &-\cos\beta\sin\phi\\ \frac{\partial\Psi_{U}}{\partial\gamma}&=0\\ \frac{\partial\Omega_{U}}{\partial\alpha}&=(-\sin\alpha\sin\beta\sin\gamma-\cos\alpha\cos\gamma)\cos\phi\cos\theta\\ &+(\cos\alpha\sin\beta\sin\gamma-\sin\alpha\cos\gamma)\cos\phi\sin\theta\\ \frac{\partial\Omega_{U}}{\partial\beta}&=(\cos\alpha\cos\beta\sin\gamma)\cos\phi\cos\theta\\ &+(\sin\alpha\cos\beta\sin\gamma)\cos\phi\sin\theta-\sin\beta\sin\gamma\sin\phi\\ \frac{\partial\Omega_{U}}{\partial\gamma}&=(\cos\alpha\sin\beta\cos\gamma+\sin\alpha\sin\gamma)\cos\phi\cos\theta\\ &+(\sin\alpha\sin\beta\cos\gamma-\cos\alpha\sin\gamma)\cos\phi\sin\theta\\ &+\cos\beta\cos\gamma\sin\phi\end{split}$

Following [19, 25, 35], we model the attitude angles under jittering effects by Gaussian distribution, i.e., $\Delta\alpha\sim\mathcal{N}(0,\sigma_{\alpha}^{2})$, $\Delta\beta\sim\mathcal{N}(0,\sigma_{\beta}^{2})$, and $\Delta\gamma\sim\mathcal{N}(0,\sigma_{\gamma}^{2})$ and $\Delta\alpha,\Delta\beta,\Delta\gamma$ are independent of each other. Then, the following theorem is obtained about the distribution of $[{\Psi_{U}}(\alpha,\beta,\gamma),{\Omega}_{U}(\alpha,\beta,\gamma)]^{T}$ under jittering effects.

According to the properties of the marginal distribution of a bivariate Gaussian distribution [36], the marginal distributions of ${\Psi_{U}}(\alpha,\beta,\gamma)$ and ${\Omega_{U}}(\alpha,\beta,\gamma)$ are obtained as follows.

To provide further insights into jittering effects on UAV mmWave channel, we carry out numerical simulations in the following three scenarios.
Scenario 1. UAV is hovering at the position $\mathbf{p}_{U}=[-100,100,50]^{T}$ (Cartesian coordinates) with its desired flight attitude being $\bar{\alpha}=\bar{\beta}=\bar{\gamma}=0$ rad;
Scenario 2. UAV is hovering at the position $\mathbf{p}_{U}=[-100,100,50]^{T}$ with its desired flight attitude being $\bar{\alpha}=1$ rad, $\bar{\beta}=\bar{\gamma}=0$ rad;
Scenario 3. UAV is hovering at the position $\mathbf{p}_{U}=[0,100,50]^{T}$ with its desired flight attitude being $\bar{\alpha}=\bar{\beta}=\bar{\gamma}=0$ rad;

In all the three scenarios, we set $N_{B,x}=N_{B,z}=N_{U,x}=N_{U,y}=16$ and the operating frequency as $28$GHz. We assume that the angles of yaw, pitch and roll follow Gaussian distribution with the standard deviation $\sigma_{\alpha}=\sigma_{\beta}=\sigma_{\gamma}=0.05$, which means $99.73\%$ of the values of $\alpha$, $\beta$, $\gamma$ are less than $0.15$ rad (equivalently $8.59$ degree) biased from their expected values.

From Figure 4, Figure 4, and Figure 4, we can find that UAV jitter does not cause fluctuation to $\Omega_{B}$ and $\Psi_{B}$. With respect to $\Omega_{U}$ and $\Psi_{U}$, we firstly derive the mean vector and covariance matrix according to Theorem 2 and then based on which we will study the three examples respectively.

In Scenario 1, the mean vector and covariance matrix are

$\displaystyle\bm{\mu}$

$\displaystyle=[0.6667,-0.6667]^{T},\;\bm{\Sigma}$

$\displaystyle=\left[\begin{array}[]{cc}0.0014&0.0011\\ 0.0011&0.0014\\ \end{array}\right]$

(76)

According to Lemma 1, the standard deviation of the marginal distribution of $\Psi_{U}$ is $0.0374$ and the standard deviation of the marginal distribution $\Omega_{U}$ is $0.0374$. Thus, the $99.73\%$ confidence interval (three-sigma interval) of $\Psi_{U}$ is $(0.5545,0.7789)$, and the $99.73\%$ confidence interval of $\Omega_{U}$ is $(-0.7789,-0.5545)$. From Figure 4 and Figure 4, $\Omega_{U}$ and $\Psi_{U}$ both fluctuate around $0.6667$ and $-0.6667$, respectively, and the degrees of the fluctuations are almost the same, which is in accordance to our theoretical expectation.

In Scenario 2, UAV hovers in the same position as Scenario 1 but changes its facing direction from $\bar{\alpha}=0$ to $\bar{\alpha}=1$. The mean vector and covariance matrix are

$\displaystyle\bm{\mu}=[-0.2008,-0.9212]^{T},\;\bm{\Sigma}=\left[\begin{array}[]{cc}0.0024&-0.0005\\ -0.0005&0.0004\\ \end{array}\right]$

(79)

It is noteworthy that the mean vector and covariance matrix are different from Scenario 1. According to Lemma 1, the standard deviation of the marginal distribution of $\Psi_{U}$ is $0.0489$ and the standard deviation of the marginal distribution $\Omega_{U}$ is $0.02$. Thus, the $99.73\%$ confidence interval of $\Psi_{U}$ is $(-0.3475,-0.0541)$, and the $99.73\%$ confidence interval of $\Omega_{U}$ is $(-0.9812,-0.8612)$. From Figure 4 and Figure 4, we can see that (1) the expected values of $\Omega_{U}$ and $\Psi_{U}$ are different from Scenario 1, and (2) $\Omega_{U}$ experiences mild fluctuation, but $\Psi_{U}$ experiences intense fluctuation, which validates our conclusion in Theorem 2 that UAV attitude is a determinant factor to the jittering effects on UAV mmWave channel.

In Scenario 3, UAV’s attitude is the same as Scenario 1, but UAV’s position changes to $\mathbf{p}_{U}=[0,100,50]^{T}$. The mean vector and covariance matrix are

$\displaystyle\bm{\mu}=[0,-0.8944]^{T},\;\bm{\Sigma}=\left[\begin{array}[]{cc}0.0025&0\\ 0&0.0005\\ \end{array}\right]$

(82)

It is noteworthy that the mean vector and covariance matrix are different from Scenario 1. According to Lemma 1, the standard deviation of the marginal distribution of $\Psi_{U}$ is $0.05$ and the standard deviation of the marginal distribution $\Omega_{U}$ is $0.0224$. Thus, the $99.73\%$ confidence interval of $\Psi_{U}$ is $(-0.15,0.15)$, and the $99.73\%$ confidence interval of $\Omega_{U}$ is $(-0.9616,-0.8272)$. From Figure 4 and Figure 4, we can see that (1) the expected values of $\Omega_{U}$ and $\Psi_{U}$ are different from Scenario 1, and (2) $\Omega_{U}$ experiences mild fluctuation, but $\Psi_{U}$ experiences intense fluctuation, which validates our conclusion in Theorem 2 that UAV position is also a determinant factor to the jittering effects on UAV mmWave channel.