In this section, the system model will be presented at first, followed by the investigation of the available codebook in the beamspace when the maximum elevation angle of the transmitting beam is limited, and finally, the effect of multi-subarray on the orthogonality of the beam domain vectors and the matching problem between the phase shifter quantization and the beam domain codebook will be discussed.

### 2.1 System model

We assume a single-cell downlink topology containing one BS and a number of UEs, where the BS side consists of multi-subarray and the UE end is all equipped with a single antenna. As shown in Fig. 1, all subarrays at the BS end are composed of wideband phased planar arrays of the same size, with the set of subarrays as \({\mathcal {N}}_{\mathrm{sub}}=\{1,2,\ldots ,N_{\mathrm{sub}}\}\), and each subarray consists of a rectangular antenna array with the elements number of \(M_{\mathrm{sub}}=M_x\times M_y\), whose spacing between adjacent elements is one half of the wavelength corresponding to the center working frequency. The set of UEs in the system is \({\mathcal {N}}_u=\{1,2,\ldots ,N_{u}\}\). Considering the long distance between the transceiver and the small size of the mmWave subarray used, it is assumed that the difference in angle-of-departure (AoD) of the same channel path between different subarrays can be neglected. Without loss of generality, the horizontal angle \(\phi _m\) and the elevation angle \(\theta _m\) of the steering beam emitted by subarrays are all within the interval \([0,\pi /2]\).

In a downlink LoS scenario where each subarray at the BS end serves the UE individually, the equivalent baseband received signal of its *i*-th UE (\(i\in {\mathcal {N}}_u\)) can be expressed as

$$\begin{aligned} r_{i}=\underbrace{\sqrt{\rho _i}{\mathbf {h}}_{i}^H{\mathbf {b}}_is_i}_{\mathrm{signal}} \quad +\quad \underbrace{\sum ^{l\in {\mathcal {N}}_u^i} \sqrt{\rho _l}{\mathbf {h}}_i^H{\mathbf {b}}_l s_l}_{\mathrm{interference}} \quad +\quad \omega _{i}, \end{aligned}$$

(1)

where \(\rho _i\) is the power of transmitted steering beam for this UE, \({\mathbf {h}}_{i}\) is its channel vector when the subarray serves this UE, \({\mathbf {b}}_{i}\) is the precoding vector of this subarray, and \(s_i\) is the user signal which obeys the \(\mathcal {CN}(0,1)\) distribution. \({\mathcal {N}}_u^i\) is the set of co-channel UEs of this UE in the system, which satisfies \(\lambda _l=\lambda _i\) when \(l\in {\mathcal {N}}_u^i\), and \(\lambda _{i}\) denotes the working wavelength of this UE. \(\omega _i \sim \mathcal {CN}(0,\sigma _{i}^2)\) is the noise at this UE end.

$$\begin{aligned} {\mathbf {h}}_{i}=\alpha _{i}\sqrt{M_{\mathrm{sub}}}{\mathbf {a}}(\phi _i,\theta _i,\lambda _i), \end{aligned}$$

(2)

where \({\mathbf {a}}(\phi _i,\theta _i,\lambda _i)\) is the normalized array response^{Footnote 1} at the transmitting subarray end, \(\alpha _ {i}=|\alpha _{i}|e^{j\psi _{i}}\) is the channel coefficient, and according to the Friis transmission formula [15, 16], we have

$$\begin{aligned} |\alpha _{i}|=\frac{ \lambda _{i} \sqrt{D_0} }{ 4\pi d_{i} }, \end{aligned}$$

(3)

where \(d_{i}\) is the distance between this UE end and the BS side, and \(D_0\) is the directivity of the array factor, which can be approximated as [17]

$$\begin{aligned} D_0\approx \frac{\pi ^2}{\Theta _h^i \Psi _h^i}, \end{aligned}$$

(4)

where \(\Theta _h^i\), \(\Psi _h^i\) are the elevation and horizontal orientation half-power beam width (HPBW) of the transmitting main beam, respectively, and according to (6-98) and (6-99) in [17], we get

$$\begin{aligned} \Theta _h^i= \left. 1\bigg /\sqrt{\cos ^2\theta _i\left[ \Theta _{x0}^{-2}\cos ^2\phi _i+\Theta _{y0}^{-2}\sin ^2\phi _i\right] }\right. , \end{aligned}$$

(5)

$$\begin{aligned} \Psi _h^i= \left. 1\bigg /\sqrt{\Theta _{x0}^{-2}\sin ^2\phi _i+\Theta _{y0}^{-2}\cos ^2\phi _i}\right. , \end{aligned}$$

(6)

where \(\Theta _{x0}\approx 2\arcsin \left( \frac{2.782}{\pi M_x}\right)\) and \(\Theta _{y0}\approx 2\arcsin \left( \frac{2.782}{\pi M_y}\right)\).

Following (1), the signal-to-noise ratio of this UE can be obtained as

$$\begin{aligned} \mathrm{SINR}_i=\frac{\rho _i |{\mathbf {h}}_i^H{\mathbf {b}}_i|^2 }{ \sum _{l\in {\mathcal {N}}_u^i} \rho _l|{\mathbf {h}}_i^H{\mathbf {b}}_l|^2+\sigma _i^2}. \end{aligned}$$

(7)

Thus the sum rate of the system can be written as

$$\begin{aligned} R=\sum _{i\in {\mathcal {N}}_u'}^{}\log _2(1+\mathrm{SINR}_i) \quad \text {bits/s/Hz}, \end{aligned}$$

(8)

where \({\mathcal {N}}_u'\) is the set of users served simultaneously.

### 2.2 Beamspace on center frequency

When considering the LoS scenario, the beam domain is divided into \(M_x\times M_y\) small zones, each corresponding to an orthogonal beam in the beamspace, so the BS can distinguish the UEs in different small zones by the orthogonal beams of the beamspace [9]. We take the set of orthogonal beams in the beamspace as the codebook of the precoding vector, and when the AoD of a UE appears in a certain zone, the vector corresponding to that zone in the beamspace is used as the transmitting precoding vector to ensure the orthogonality among multiple transmitting beams.

From the perspective of the phase shifter group [6, 17] the so-called beamspace [9], that is, the DFT matrix column space formed by a set of orthogonal steering vectors constructed by taking values of uniform quantization points in the range of one phase period \([0,2\pi )\) as the phase difference \(\beta\) of adjacent elements in the array. For a uniformly spaced linear array consisting of *M* elements, let \(\beta ^i=2\pi i/M,\, i\in \{0,1,\ldots ,M-1\}\) such that \({\mathbf {a}}(\beta ^i)^H{\mathbf {a}}(\beta ^j)=0,\forall i\ne j\), where \({\mathbf {a}}(\beta ,M) =\frac{1}{\sqrt{M}}[1,e^{-j\beta },\ldots ,e^{-j(M-1)\beta }]^T\) is the steering vector. And for a \(M_x\times M_y\)-dimensional rectangular planar array, with one of its vertices as the origin and making the *x*-axis and *y*-axis extend through the two sides of the antenna array respectively to establish the coordinate system, its steering vector may be expressed as \({\mathbf {a}}(\beta _x,\beta _y)={\mathbf {a}}(\beta _x,M_x)\otimes {\mathbf {a}}(\beta _y,M_y)\) [15], where \(\beta _x\) and \(\beta _y\) are the emission phase difference between adjacent elements along the *x*-axis and *y*-axis of the antenna array, respectively. By the nature of the matrix Kronecker product \(({\mathbf {A}}\otimes {\mathbf {B}})({\mathbf {C}}\otimes {\mathbf {D}})=({\mathbf {A}}{\mathbf {C}})\otimes ({\mathbf {B}}{\mathbf {D}})\), we have \(({\mathbf {a}}(\beta _x^i,M_x)\otimes {\mathbf {a}}(\beta _y^l,M_y))^H({\mathbf {a}}(\beta _x^j,M_x)\otimes {\mathbf {a}}(\beta _y^k,M_y))=({\mathbf {a}}(\beta _x^i,M_x)^H{\mathbf {a}}(\beta _x^j,M_x))\otimes ({\mathbf {a}}(\beta _y^l,M_y)^H{\mathbf {a}}(\beta _y^k,M_y))=0\), \(\forall i\ne j\) or \(l\ne k\), which means the beamspace of the two-dimensional planar array can be constructed by following the way of the steering vector in the beamspace of the linear array.

As shown in Fig. 1, the phase difference [17] between its adjacent array elements along the *x*-axis and *y*-axis can be obtained when the transmitting main beam direction of a subarray is \((\phi _m,\theta _m)\), namely

$$\begin{aligned} \beta _x= kd_x\cos \phi _m\sin \theta _m, \end{aligned}$$

(9)

$$\begin{aligned} \beta _y= kd_y\sin \phi _m\sin \theta _m. \end{aligned}$$

(10)

It can be seen that (9) and (10) also establish the connection between the beamspace and the main beam direction for the planar array.

Consider the main beam in the right half of Fig. 1 which is active only in the first quadrant, with \(\phi _m,\theta _m \in [0,\pi /2]\) in this case. Assume that the elevation angle \(\theta _m\) of the main beam at the center frequency is chosen in the interval \([0,\theta _{m}^{\mathrm{max}}]\) and the elevation angle in the range \([0,\pi /2]\) can be covered by the chosen frequency. Substituting \(\theta _m^{\mathrm{max}}\) into (9) and (10) yields the corresponding \(\beta _x\) and \(\beta _y\) selection ranges of both \([0,\pi \sin \theta _ {m}^{\mathrm{max}}]\), the corresponding set of available vectors in this beamspace is

$$\begin{array}{*{20}l} {{\mathbb{B}} = \{ {\mathbf{b}}_{{{\text{card}}({\mathcal{M}}_{y} )i + l}} = {\mathbf{a}}(\beta _{x}^{i} ,M_{x} ) \otimes {\mathbf{a}}(\beta _{y}^{l} ,M_{y} ):} \\ {\beta _{x}^{i} + \beta _{y}^{l} \le \sqrt 2 \pi \sin \theta _{m}^{{{\text{max}}}} ,\theta _{m}^{{{\text{card}}({\mathcal{M}}_{y} )i + l}} \le \theta _{m}^{{{\text{max}}}} ,} \\ {\beta _{x}^{i} = 2i\pi /M_{x} ,\beta _{y}^{l} = 2l\pi /M_{y} ,i \in {\mathcal{M}}_{x} ,l \in {\mathcal{M}}_{y} \} ,} \\ \end{array}$$

(11)

where \({\mathcal {M}}_x=\left\{ 0,1,\ldots ,\left\lfloor \frac{\pi \sin \theta _{m}^{\mathrm{max}}}{2\pi /M_x} \right\rfloor \right\} and {\mathcal {M}}_y=\left\{ 0,1,\ldots ,\left\lfloor \frac{\pi \sin \theta _{m}^{\mathrm{max}}}{2\pi /M_y} \right\rfloor \right\}\). When \(\mathrm{card}({\mathcal {M}}_x)>1\), the phase difference of adjacent elements along the *x*-axis can be divided as

$$\begin{aligned} \left\{ \begin{array}{ll} {[}0, \pi /M_x), &{}\quad i=0,\\ {[}\frac{2\pi }{M_x}(\left\lfloor \frac{M_x}{2} \right\rfloor -0.5),\pi \sin \theta _{m}^{\mathrm{max}}], &{}\quad i=\left\lfloor \frac{\pi \sin \theta _{m}^{\mathrm{max}}}{2\pi /M_x} \right\rfloor ,\\ {[} \frac{2\pi }{M_x}(i-0.5),\frac{2\pi }{M_x}(i+0.5)), i \in &{}\quad \{1,\ldots ,\left\lfloor \frac{\pi \sin \theta _{m}^{\mathrm{max}}}{2\pi /M_x} \right\rfloor -1\}. \end{array} \right. \end{aligned}$$

(12)

Similarly, the phase difference division of adjacent array elements along the *y*-axis can be obtained by rewriting \(M_x\) in (12) as \(M_y\), namely

$$\begin{aligned} \left\{ \begin{array}{ll} {[}0, \pi /M_y), &{}\quad l=0,\\ {[} \frac{2\pi }{M_y}(\left\lfloor \frac{M_y}{2} \right\rfloor -0.5),\pi \sin \theta _{m}^{\mathrm{max}}], &{}\quad l=\left\lfloor \frac{\pi \sin \theta _{m}^{\mathrm{max}}}{2\pi /M_y} \right\rfloor ,\\ {[} \frac{2\pi }{M_y}(l-0.5),\frac{2\pi }{M_y}(l+0.5)), l \in &{}\quad \{1,\ldots ,\left\lfloor \frac{\pi \sin \theta _{m}^{\mathrm{max}}}{2\pi /M_y} \right\rfloor -1\}. \end{array} \right. \end{aligned}$$

(13)

(12) and (13) jointly present an interval division of the two-dimensional continuous beam domain. In addition, \(\beta _x^i+\beta _y^l \le \sqrt{2}\pi \sin \theta _{m}^{\mathrm{max}}\) and \(\theta _m^{\mathrm{card}({\mathcal {M}}_y)i+l}\le \theta _m^{\mathrm{max}}\) in (11) are the restrictions on the effective range of the beam interval obtained by summing up (9) and (10) as well as the maximum elevation angle of the main beam at the center frequency, respectively. The AoD of the actual LoS channel path is mapped to a certain zone in the beam domain by (12) and (13), and the codebook corresponding to this zone in (11) is taken as the precoding vector, which is the beamspace codebook-based precoding method we adopt.

### 2.3 The effect of subarray on orthogonality in beamspace

Assuming that the adjacent elements of a planar array composed of multi-subarray all maintain a half-wavelength spacing corresponding to the center frequency, in order to investigate the orthogonality of the beams between the subarrays, from the perspective of a large array it may be useful to rewrite the steering beam vector of the *k*-th subarray as

$$\begin{aligned} {\mathbf {a}}_k(\beta _x^i,\beta _y^j)= {[}\underbrace{0,\ldots ,0}_{M_{\mathrm{sub}}\times (k-1)},{\mathbf {a}}(\beta _x^i,\beta _y^j)^T,\underbrace{0,\ldots ,0}_{M_{\mathrm{sub}}\times (N_{\mathrm{sub}}-k)}]^T. \end{aligned}$$

(14)

Then the following two equations are valid, namely

$$\begin{aligned} &{\mathbf {a}}_k(\beta _x^i,\beta _y^j)^H {\mathbf {a}}_s(\beta _x^m,\beta _y^n)=0,\\&k,s \in {\mathcal {N}}_{\mathrm{sub}} \, \text {and} \, k \ne s , \\&i,m \in {\mathcal {M}}_x,\\&j,n \in {\mathcal {M}}_y, \end{aligned}$$

(15)

and

$$\begin{aligned} &{\mathbf {a}}_k(\beta _x^i,\beta _y^j)^H {\mathbf {a}}_k(\beta _x^m,\beta _y^n)=0,\\&k \in {\mathcal {N}}_{\mathrm{sub}}, \\&i,m \in {\mathcal {M}}_x, \\&j,n \in {\mathcal {M}}_y,\\&i \ne j \,\,\text{or}\,\, m \ne n. \end{aligned}$$

(16)

When each subarray precoding vector is selected from the beamspace shown in (11), (15) guarantees the orthogonality of the transmitted beams between subarrays, (16) guarantees the orthogonality between different beam vectors of the same subarray. Therefore, the orthogonality of the precoding vectors between subarrays and the orthogonality of multiple transmitted beams within the same subarray can be guaranteed when the same beamspace codebook is used for the precoding vectors of all subarrays.

### 2.4 The effect of phase shifter quantization on beamspace

For a linear array, the matrix composed of the phase shift vectors in its beamspace is actually a DFT matrix, so all the elements in it can be written in the form of a rotation factor power. And when all phase shifters in the phase shifter group of the antenna array are *Q* bits uniformly quantized, let its individual quantized phase shift cells and the rotation factor have the same phase shift value as \(\beta _{\mathrm{cell}}=2\pi /M\), so according to the steering vector formula of the linear array and \(\beta ^i=i\beta _{\mathrm{cell}}\), when satisfying \(2^Q \beta _{\mathrm{cell}} \ge (M-1)\beta ^{(M-1)}\), that is, \(2^Q \ge (M-1)^2\), all the beams in the beamspace codebook can be emitted, at which time the minimum number of quantized bits is \(Q_{\mathrm{min}}=2\lceil \log _2(M-1) \rceil\).

For a planar array whose phase shifter group also adopts the above quantization, according to Eq. (37) in [6], it needs to satisfy \((M_x-1)\beta _x^i+(M_y-1)\beta _y^l\le 2^Q\beta _{\mathrm{cell}}\) when transmitting the beam through the beamspace in (11), therefore leading to

$$\begin{aligned} Q_{\mathrm{min}} = \left\lceil \log _2[ (M_x-1)(\mathrm{card}({\mathcal {M}}_x)-1)+ (M_y-1)(\mathrm{card}({\mathcal {M}}_y)-1) ] \right\rceil . \end{aligned}$$

(17)

Since the steering beams in the beamspace matrix can all be represented by a group of uniformly quantized phase shifters, we can realize the beamspace vector of the antenna array when the phase shifter group is unquantized by a uniformly quantized phase shifter group and its maximum transmit elevation angle is no greater than \(\theta _m^{\mathrm{max}}\) once (17) is satisfied.