It is assumed that the linear array is placed in the *z*-axis, thus having a unique elevation angle in the main direction of its steering beam, which facilitates the formulation of channel modeling and algorithm design when using the steering beam. As shown in Fig. 1, the main direction of the steering beam of the linear array is donut-shaped and can be divided into multiple pieces by vertical slicing, each piece being approximated by one steering main beam of the same virtual planar array in the *x*-*y* plane. Moreover, due to the limitations of scatterer size in the channel and beamwidth at the transmitter side, the scattering path of the mmWave channel in practical cellular communication scenarios is more directional and can generally be described by the array response of a planar array. Accordingly, the channel path can be specified by the array response of the virtual planar array corresponding to the linear array at the BS side, which is thus mathematically coherent from the transmitter, channel, and receiver side. In addition, it is important to ensure that the power in one direction of the donut-shaped main beam is not excessive by the angular separation of the horizontal azimuth *ϕ* of the two adjacent slices. Furthermore, it is not necessary for this virtual ring main beam to be complete. If there is no scattering path in some horizontal azimuth range, the slices in the corresponding range can be removed to give a clearer picture of the connection between the transmitter beam and the channel scattering path. If there are multiple AoDs within the half-power beamwidth (HPBW) of the virtual planar array corresponding to the same donut-shaped main beam, the relationship to these channel paths can be represented by a slice in the main beam instead.

### 2.1 Number of elements in the virtual planar array

As shown in Fig. 1, the normalized array factor for a linear array of *N* uniformly spaced array elements placed on the *z*-axis and with the origin set as the reference point can be written as

$$ \overline{\mathrm{A\!F}}_{\text{Linear}}=\frac{1}{N}\sum_{n=1}^{N}e^{j(n-1)\psi}, $$

(1)

where

$$ \psi=kd\cos \theta-\beta, $$

(2)

where *k* is the wave number, *d* is the array element spacing, *θ* is the elevation angle, and *β* is the phase difference of the emitted signals from adjacent array elements. After adjusting the reference point to the array midpoint [6], the array factor can be approximated as

$$ \overline{\mathrm{A\!F}}_{\text{Linear}}\approx{\frac{\sin{\left({\frac{N}{2}}\psi\right)}}{\frac{N}{2}\psi}}. $$

(3)

In (2), by setting *ψ*=0, we have

$$ \beta = kd\cos \theta_{m}, $$

(4)

where *θ*_{m} is the elevation angle corresponding to any horizontal azimuth angle in the donut-shaped steering main beam direction of the linear array, which can be kept to a uniform value of *θ*_{ring} by selecting a suitable coordinate system. Since it is possible for the donut-shaped main beam to be irradiating several scatterers in the physical channel, the connection between the transmitting main beam and the channel path can be described by a set of steering main beams of a planar array with disjoint HPBW ranges when assuming that the energy radiated by the side lobes in the transmitting beam at the receiving end is highly attenuated and negligible due to atmospheric absorption of mmWave. Thus, by approximating the part of the array factor of the linear array in the first quadrant through the combination of the normalized array factors of a virtual planar array in the *x*-*y* plane in Fig. 1, and more precisely, by approximating the main lobe of the steering beam of the linear array by the main lobes of several steering beams of the planar array and neglecting the effect of the side lobes, we have

$$ \overline{\mathrm{A\!F}}_{\text{linear}}(\theta_{m})\approx \sum_{k=1}^{K}{\overline{\mathrm{A\!F}}_{\text{planar}}(\phi_{m_{k}},\theta_{m})}. $$

(5)

The half-power direction *θ*_{h} of the main beam of the linear array can be determined by \(\overline {\text {AF}}=\sqrt {0.5}\). When \(\frac {\sin x}{x}=\sqrt {0.5}, x\! \approx \! \pm 1.391, x=N\psi /2\) from (3), which is substituted into (2) to give^{Footnote 1}

$$ \theta_{{hi}}\approx\arccos\left[\frac{1}{kd}\left(\beta\pm\frac{2.782}{N}\right)\right]. $$

(6)

Therefore, the HPBW of the linear array can be approximated as

$$ \Theta_{h}\left(f_{i}\right)\approx2\left|\theta_{{mi}}-\theta_{{hi}}\right|. $$

(7)

On the other hand, the HPBW of a planar array in the *x*-*y* plane can be expressed as [6]

$$ \Theta_{h}\left(f_{i}\right)=\left.1\middle/\sqrt{\cos^{2}\!\theta_{{mi}}\left[\Theta_{x0}^{-2}\cos^{2}\!\phi_{m}+\Theta_{y0}^{-2}\sin^{2}\!\phi_{m}\right]}\right., $$

(8)

$$ \Psi_{h}\left(f_{i}\right)=\left.1\middle/\sqrt{\Theta_{x0}^{-2}\sin^{2}\!\phi_{m}+\Theta_{y0}^{-2}\cos^{2}\!\phi_{m}}\right., $$

(9)

where *Θ*_{x0} and *Θ*_{y0} are the HPBWs of linear arrays equally spaced on the *x*- and *y*-axes with the number of array elements *M*_{x} and *M*_{y}, respectively, with the beam direction toward the *z*-axis, with

$$ \Theta_{x0}\left(f_{i}\right)\approx2\arcsin\left(\frac{2.782c}{2\pi {dM}_{x}f_{i}}\right), $$

(10)

in which *c* is the velocity of electromagnetic wave. Similarly, replacing *M*_{x} with *M*_{y} in (10) gives *Θ*_{y0}. When a square planar array is employed, such that *M*=*M*_{x}=*M*_{y}, (8) and (9) can be further simplified to

$$\begin{array}{*{20}l} \Theta_{h}\left(f_{i}\right)\!=\Theta_{x0}/\cos\theta_{{mi}}, \end{array} $$

(11)

$$\begin{array}{*{20}l} \Psi_{h}\left(f_{i}\right)\!=\Theta_{x0}. \end{array} $$

(12)

To approximate the steering main beam of the linear array by steering main beams of the planar array, let (7) and (11) have equal HPBWs, which gives

$$ \arccos\left(\frac{\beta}{kd}-\frac{2.782}{kdN} \right)-\theta_{{mi}}=\left.\arcsin\left(\frac{2.782c}{2\pi. {dMf}_{i}}\right)\right/\cos\theta_{{mi}} $$

(13)

Since the HPBW is not perfectly symmetrical on both sides in the main beam direction, the relationship in (13) can be written exactly as

$$ \arccos\!\left(\! \frac{\beta}{kd}-\frac{2.782}{kdN} \!\right)-\arccos\!\left(\! \frac{\beta}{kd}+\frac{2.782}{kdN} \!\right)\!=2\!\left.\arcsin\!\left(\!\frac{2.782c}{2\pi {dMf}_{i}}\!\right)\!\right/\!\cos\theta_{{mi}}. $$

(14)

Substituting (4) into (14) yields

$$ \begin{aligned} M&=\frac{2.782c}{2\pi f_{i} d}\\ &\quad\cdot\frac{1}{\sin\left\{ \!{0.5\cos\theta_{{mi}}\! \left[ \arccos\!\left(\!\cos\theta_{{mi}} -\frac{2.782c}{2\pi f_{i} d N} \right) - \arccos\!\left(\!\cos\theta_{{mi}}+\frac{2.782c}{2\pi f_{i} d N} \right) \right]} \right\} }\\ &\stackrel{(a)}{=}\frac{2.782c}{2\pi f_{i} d \sin\left\{ {0.5\cos\theta_{{mi}} \left[ \theta_{{hi}}^{-} - \theta_{{hi}}^{+} \right]} \right\} }\\ &\stackrel{(b)}{=}\frac{2.782c}{2\pi f_{i} d \sin\left\{ {0.5\cos\theta_{{mi}}\Theta_{h}{\left(f_{i}\right)}} \right\} }\\ &\stackrel{(c)}{=}\frac{2.782c}{2\pi f_{i} d \sin\left(\Theta_{x0}/2 \right) }, \end{aligned} $$

(15)

where (a) is based on the assumption that the two terms in square brackets in the denominator are the half-power direction of the main beam, (b) is based on the definition of the HPBW, and (c) is due to substituting (11). If (10) is substituted into (c) again, we get *M*, i.e., deriving (15) from the top down and from the bottom up, for the assumption to hold sufficient necessary for the half-power direction \(\theta _{{hi}}^{\pm }\) and satisfies \(\theta _{{hi}}^{-} > \theta _{{hi}}^{+}\). Therefore, the half-power direction of the frequency squint main beam in elevation can also be noted as

$$ \theta_{{hi}}^{\pm} = \arccos\left(\cos\theta_{{mi}} \pm \frac{2.782c}{2\pi f_{i} d N} \right). $$

(16)

Alternatively, the approximate expression of the above equation can also be obtained by substituting *β*=*kd* cos*θ*_{m} into (13) and by a simple derivation, namely

$$ M\approx\frac{2.782c}{2\pi f_{i} d \sin\left\{ {\cos\theta_{{mi}}\arccos\left(\cos\theta_{{mi}} -\frac{2.782c}{2\pi f_{i} d N} \right) - \theta_{{mi}}\cos\theta_{{mi}}} \right\} }. $$

(17)

Since the number of elements on each side of a planar antenna array is an integer, simply rounding to the nearest whole number, we have

$$ \widetilde{M}=\lfloor M \rceil. $$

(18)

First, by investigating the relationship between the main beam direction *θ*_{m} and *M* for the linear array operating at the half-wavelength spacing frequency *f*_{c}, we have

$$ M\approx\frac{2.782}{\pi \sin\left\{ {\cos\theta_{m}\arccos\left(\cos\theta_{m} -\frac{2.782}{\pi N} \right) - \theta_{m}\cos\theta_{m}} \right\} }. $$

(19)

To investigate the relationship between the half-power beam direction and the virtual planar array size, rewrite (19) as

$$\begin{aligned} M&(f_{c},\theta_{m})=\\ &\frac{2.782}{\pi \sin\left\{ 0.5 {\cos\theta_{m} \left[ \arccos\left(\cos\theta_{m} -\frac{2.782}{\pi N} \right) - \arccos\left(\cos\theta_{m} +\frac{2.782}{\pi N} \right) \right]} \right\} }, \end{aligned} $$

(20)

and since from (16) it follows that

$$ \theta_{h}^{\pm}=\arccos\!\left(\cos\theta_{m}\pm\frac{2.782}{ \pi N}\right). $$

(21)

Substituting (21) into (20), eventually we get

$$ M(f_{c},\theta_{h}^{\pm})= \frac{\pm 2.782}{\pi \sin\!\left\{ \!0.5 { \left(\! \cos\theta_{m} \pm \frac{2.782}{\pi N} \right) \!\left[ \theta_{m} - \arccos\!\left(\! \cos\theta_{m} \pm \frac{2.782}{\pi N}\times 2 \!\right) \!\right]} \!\right\} }. $$

(22)

Note that the half-power direction \(\theta _{h}^{\pm }\) in (22) needs to be kept within [0^{∘},90^{∘}]. And since the derivation in (22) is more complicated, there is no analytical comparison for the moment. By simulation experiments, as shown in Fig. 2, in the corresponding main beam angle range satisfying \(M(f_{c},\theta _{h}^{-})>M(f_{c},\theta _{h}^{+})\). Furthermore, since the definition of *Θ*_{x0} in (c) of (15) is the HPBW of the linear array in the vertical direction, which is independent of the actual beam direction *θ*_{mi}, it is related to the number of antenna elements. Therefore, it can be assumed that changing the beam direction will cause a change in the number of elements of the virtual planar array. Due to the small angular spread of the mmWave band and the narrow HPBW due to the high number of elements in the array, we use the same virtual planar array to approximate paths in the HPBW range of the same scattering cluster for ease of handling.

Next, we study the relationship between the direction of the frequency squint main beam *θ*_{mi} and the parameter *M* of the virtual planar antenna array once the main beam direction *θ*_{m} at frequency *f*_{c} has been determined. Since a linear array on the *z*-axis is considered, there is

$$ \theta_{{mi}}=\arccos\left(\frac{c \beta}{2\pi {df}_{i}}\right). $$

(23)

As the main direction of the beam at frequency *f*_{c} is *θ*_{m}, substituting into (23) gives *β*=*π* cos*θ*_{m}; again substituting into (23) gives

$$ \theta_{{mi}}=\arccos\left(\frac{c \cos\theta_{m}}{2 {df}_{i}}\right). $$

(24)

By substituting (24) into (17), we get

$$\begin{aligned} M&\left(f_{i}\right)=\frac{2.782c}{2\pi f_{i} d}\\ \cdot&\frac{1}{\sin\left\{ \frac{c\cos\theta_{m}}{2{df}_{i}}\!\left[ \arccos\!\left(\frac{c}{2df_{i}}\left(\cos\theta_{m}-\frac{2.782}{\pi N} \right) \right)-\arccos\left(\frac{c}{2df_{i}}\cos\theta_{m} \right) \right] \right\} }. \end{aligned} $$

(25)

By fixing the array spacing of the virtual planar array to half the wavelength corresponding to the central frequency *f*_{c} and rewriting (25), we get

$$\begin{aligned} M&\left(f_{i}\right)=\\ &\frac{2.782f_{c}}{\pi f_{i} \sin\left\{ \frac{f_{c}\cos\theta_{m}}{f_{i}}\!\left[ \arccos\!\left(\frac{f_{c}}{f_{i}}\left(\cos\theta_{m}-\frac{2.782}{\pi N} \right) \right)-\arccos\left(\frac{f_{c}}{f_{i}}\cos\theta_{m} \right) \right] \right\} }. \end{aligned} $$

(26)

Figure 3 shows the quantized value *M* in (26) versus the operating frequency *f*_{i} for a central frequency of 45 GHz. In the lower left corner of the curve for the main beam direction *θ*_{m}=45^{∘}, there is a part of the frequency with a zero value of *M*, that is because according to (24), when the squint beam is pointing at *θ*_{mi}=0^{∘}, its frequency is about 31.92 GHz, and when the frequency is any lower, it is out of the operating frequency range.

According to (55) in Appendix 1, when the maximum angle of the main beam at central frequency *f*_{c} is within 60^{∘}, i.e., *θ*_{m}≤60^{∘}, if \(f_{i}<\frac {3}{2}f_{c} \), no grating lobes will appear in the resulting squint beams. When *θ*_{m}=60^{∘}, if the operating frequency range is chosen to be \([\frac {1}{2}f_{c},\;\frac {3}{2}f_{c})\), the coverage angle range is approximately *θ*_{mi}∈[0^{∘},70.53^{∘}).

### 2.2 System model

Following the multi-path scattering Saleh-Valenzuela channel model [5], the channel of a particular user can generally be represented as

$$ \mathbf{H}=\sum_{c=1}^{N_{c}}\sum_{l=1}^{N_{{pc}}}\alpha_{{cl}}\mathbf{a}_{r}\left(\phi^{r}_{{cl}},\theta^{r}_{{cl}}\right)\mathbf{a}_{t}\left(\phi_{{cl}}^{t},\theta_{{cl}}^{t}\right)^{H}, $$

(27)

where *N*_{c} is the number of all clusters from the BS to that user, *N*_{pc} is the number of paths within the *c*^{th} cluster, and \(\mathbf {a}_{r}(\phi ^{r}_{{cl}},\theta ^{r}_{{cl}})\) and \(\mathbf {a}_{t}(\phi _{{cl}}^{t},\theta _{{cl}}^{t})\) are the steering beam vectors represented by array responses at the receiving and transmitting ends for the *l*^{th} path in the *c*^{th} channel cluster, respectively.

For convenience, due to the small angular spread of the beam in the actual mmWave sparse scattering environment, it is assumed that the paths within each cluster can be represented by the array response of the same virtual planar array; if the user has multiple scattering clusters on the same donut-shaped main beam, we still use the array response of the same virtual planar array to approximate the paths in multiple scattering clusters; when the difference in elevation angle between clusters is large, corresponding to several steering donut-shaped beams at the transmitting side, the difference in size of the virtual planar array between clusters is more pronounced, and we assume that the donut-shaped main beam affects a range within the HPBW of the transmitting elevation angle \(\theta _{\text {ring}}^{t}\) in the real channel environment. According to the approximation of the array factor in (5), the actual channel caused by a particular steering beam of the linear array can then be equivalently expressed as

$$ \widetilde{\mathbf{H}}(\theta_{\text{ring}}^{t})\approx\sum_{c=1}^{\bar{N}_{c}}\sum_{l=1}^{\bar{N}_{{pc}}}\alpha_{{cl}}\mathbf{a}_{r}\left(\phi^{r}_{{cl}},\theta^{r}_{{cl}}\right)\tilde{\mathbf{a}}_{t}\left({\phi}_{{cl}}^{t},\bar{\theta}_{{cl}}^{t}\right)^{H}, $$

(28)

where \(\bar {N}_{c}\) and \(\bar {N}_{{pc}}\) are the number of clusters and paths within the HPBW range of \(\theta _{\text {ring}}^{t}\), respectively; the elevation angle \(\bar {\theta }_{{cl}}^{t}\) in the AoD of each path in the channel needs to be within the HPBW range of \(\theta _{\text {ring}}^{t}\) at the transmitter side, i.e., satisfying \(|\theta _{\text {ring}}^{t}-\bar {\theta }_{{cl}}^{t}|\leq \Theta _{h}\left (f_{i}\right)/2, \bar {\theta }_{{cl}}^{t}\in \{\theta _{{cl}}^{t}\}\); \(\tilde {\mathbf {a}}_{t}\) is the vector form of the virtual planar array response corresponding to a certain steering beam of the linear array.

When the user at the receiving end uses a square antenna array with *M*_{r} elements on each side and placed in the *x*-*y* plane, its array response at frequency *f*_{s} with the main beam orientated to \((\phi _{s}^{r},\theta _{s}^{r})\) is

$$ \begin{aligned} \mathbf{a}_{r}&\left(\phi_{s}^{r},\theta_{s}^{r}\right)=\frac{1}{M_{r}}\\ \cdot&\left[1,e^{-j\pi\sin\theta_{s}^{r}\cos\phi_{s}^{r}f_{s}/f_{c}},\dots,e^{-j\pi(M_{r}-1)\sin\theta_{s}^{r}\cos\phi_{s}^{r}f_{s}/f_{c}},\dots,\right.\\ &\left.\dots,e^{-j\pi(M_{r}-1)\sin\theta_{s}^{r}\sin\phi_{s}^{r}f_{s}/f_{c}},\dots,e^{-j\pi(M_{r}-1)\left(\sin\theta_{s}^{r}\cos\phi_{s}^{r}+ \sin\theta_{s}^{r}\sin\phi_{s}^{r} \right)f_{s}/f_{c}}\right]^{T}. \end{aligned} $$

(29)

And the array response of the virtual planar array at the transmitter side can be written as

$$ \begin{aligned} \tilde{\mathbf{a}}_{t}&\left({\phi}_{s}^{t},\theta_{s}^{t}\right)=\frac{1}{\widetilde{M}_{t}}\\ \cdot&\left[1,e^{-j\pi\sin\theta_{s}^{t}\cos\phi_{s}^{t}f_{s}/f_{c}},\dots,e^{-j\pi(\widetilde{M}_{t}-1)\sin\theta_{s}^{t}\cos\phi_{s}^{t}f_{s}/f_{c}},\dots,\right.\\ &\left.\dots,e^{-j\pi(\widetilde{M}_{t}-1)\sin\theta_{s}^{r}\sin\phi_{s}^{t}f_{s}/f_{c}},\dots,e^{-j\pi(\widetilde{M}_{t}-1)\left(\sin\theta_{s}^{t}\cos\phi_{s}^{t}+ \sin\theta_{s}^{t}\sin\phi_{s}^{t} \right)f_{s}/f_{c}}\right]^{T}, \end{aligned} $$

(30)

where \(\widetilde {M}_{t}\) is the number of elements per side of the virtual array obtained from (18), *f*_{s} is the operating frequency, and \((\phi _{s}^{t},\theta _{s}^{t})\) is the virtual main beam direction.

Consequently, the equivalent baseband signal at the user receiver side can be represented as

$$ \begin{aligned} y_{i}=&\sqrt{P}\mathbf{w}_{i}^{H} \mathbf{H}_{i}(f)\mathbf{f}s_{i}+\mathbf{w}_{i}^{H} n_{i}(f)\\ \approx & \sqrt{P}\mathbf{a}_{r}\left(\phi^{r}_{i},\theta^{r}_{i}\right)^{H} \widetilde{\mathbf{H}}\left(\theta_{\text{ring}}^{t}\right) \sum_{j=1}^{K}\tilde{\mathbf{a}}_{t}\left({\phi}_{j}^{t},\bar{\theta}_{\text{ring}}^{t}\right)s_{i}+\mathbf{a}_{r}\left(\phi^{r}_{i},\theta^{r}_{i}\right)^{H} n_{i}(f), \end{aligned} $$

(31)

where *P* is the transmitted power of the individual steering beams.

Hence, the rate for this user can be expressed as

$$ \begin{aligned} \mathrm{R}_{i}= &\mathrm{log_{2}}\left(1+ \frac{P\left|\mathbf{w}_{i}^{H} \mathbf{H}_{i}(f) \mathbf{f} \right|^{2}}{\sigma_{n}^{2} \mathbf{w}_{i}^{H}\mathbf{w}_{i} }\right)\\ \stackrel{(d)}{\approx}& \mathrm{log_{2}} \bigg(1+\frac{P|\alpha_{l}(f)|^{2}}{ \sigma_{n}^{2}\mathbf{w}_{i}^{H}\mathbf{w}_{i}} \left| \mathbf{a}_{r}\left(\phi_{i}^{r},\theta_{i}^{r}\right)^{H} \mathbf{a}_{r}\left(\phi_{l}^{r},\theta_{l}^{r}\right) \tilde{\mathbf{a}}_{t}\left({\phi}_{l}^{t},\bar{\theta}_{l}^{t}\right)^{H} \tilde{\mathbf{a}}_{t}\left({\phi}_{l}^{t},\theta_{\text{ring}}^{t}\right) \right|^{2} \bigg) \\ \stackrel{(e)}{\approx}& \mathrm{log_{2}}\left(1+ \frac{ P\left|\alpha_{l}(f) \tilde{\mathbf{a}}_{t}\left({\phi}_{l}^{t},\bar{\theta}_{l}^{t}\right)^{H} \tilde{\mathbf{a}}_{t}\left({\phi}_{l}^{t},\theta_{\text{ring}}^{t}\right) \right|^{2} }{\sigma_{n}^{2}}\right), \end{aligned} $$

(32)

where (d) is obtained by representing the channel according to (28) as the actual channel generated by a single donut-shaped transmitting main beam in the scattering environment and assuming that the path *l* with the highest received energy in the actual channel is taken through the steering beam at the receiver end; (e) is assumed that the quantization resolution of the phase shifter of the wideband analog array at the receiver side is high enough to satisfy \({\phi }_{i}^{r}={\phi }_{l}^{r}\) and \({\theta }_{i}^{r}={\theta }_{l}^{r}\).