In this section, we propose multi-user beamforming and precoding methods that utilize the angular-domain CSI feedback described in the previous section. We consider first multi-user beamforming with a single stream per user in Section 4.1 and extend to multiple streams per user in Section 4.2.

### Leakage-bounded angular beamforming

In this section, we assume that the base station transmits only a single stream *L*_{
u
}=1 to each served user \(u \in \mathcal {S}[n]\), i.e., the precoding matrices **F**_{
u
}[*n*] reduce to beamforming vectors **f**_{
u
}[*n*]. From the angular CSI feedback provided by the users, the base station has at time instant *n* delayed knowledge of the channel expansion vectors \(\mathbf {d}_{u}^{s}[n-m]\) and the norm of the corresponding expansion coefficient vectors \(\left \|\mathbf {c}_{u}^{s}[n-m]\right \|\), where *m* represent the delay of the CSI feedback path. To calculate the transmit beamformers, we interpret the expansion vectors \(\mathbf {d}_{u}^{s}[n-m]\) as specular channel contributions and propose a beamformer optimization that maximizes the expected received signal power of user *u* over its specular components, while restricting the interference leakage caused to the other users over their respective specular components. To shorten notation, we omit the time indices [*n*] and [*n*−*m*] in the following derivations.

Consider the signal power received over the *S* specular components corresponding to the available CSIT:

$$ \begin{aligned} P_{u} &= \mathbb{E}\left(\left\|\left(\sum\limits_{s = 1}^{S} \mathbf{c}_{u}^{s} \left(\mathbf{d}_{u}^{s}\right)^{\mathrm{T}}\right)\mathbf{f}_{u} s_{u}\right\|^{2}\right) \\ &= \text{tr}\left(\sum_{s = 1}^{S} \sum_{k = 1}^{S} \mathbb{E}({(\mathbf{c}_{u}^{k})}^{\mathrm{H}} \mathbf{c}_{u}^{s}) (\mathbf{d}_{u}^{s})^{\mathrm{T}} \mathbf{f}_{u} \mathbf{f}_{u}^{\mathrm{H}} (\mathbf{d}_{u}^{k})^{*}\right). \end{aligned} $$

(19)

The value of the inner product \((\mathbf {c}_{u}^{k})^{\mathrm {H}} \mathbf {c}_{u}^{s} = r_{u}^{k,s} e^{j \varphi _{u}^{k,s}}\) depends on the relative phase-shift \(\varphi _{u}^{k,s}\) between these expansion coefficient vectors. Since we do not provide feedback information about these relative phase-shifts, we assume \(\varphi _{u}^{k,s} \sim \mathcal {U}\left (0,2\pi \right)\) such that \(\mathbb {E}(r_{u}^{k,s} e^{j \varphi _{u}^{k,s}}) = 0\). Notice, this assumption also makes sense for our targeted application scenario of high user mobility. In such scenarios, the angular CSI feedback corresponding to the expansion vectors \(\mathbf {d}_{u}^{s}\) is relatively stable, whereas the phase of the expansion coefficient vectors \(\mathbf {c}_{u}^{s}\) varies significantly over time. Hence, relying on such phase information at high mobility is not recommended. With this assumption of uncorrelated phases, the signal power is:

$$ P_{u} = \sum\limits_{s = 1}^{S} \left\|\mathbf{c}_{u}^{s}\right\|^{2} (\mathbf{d}_{u}^{s})^{\mathrm{T}} \mathbf{f}_{u} \mathbf{f}_{u}^{\mathrm{H}} (\mathbf{d}_{u}^{s})^{*} $$

(20)

and the transmitter has sufficient CSIT to calculate this value. Similarly, we can determine the interference leakage power caused by the transmission to user *u* and received over specular component *k* of user *j* as:

$$ L_{j,u}^{k} = \left\|\mathbf{c}_{j}^{k}\right\|^{2} (\mathbf{d}_{j}^{k})^{\mathrm{T}} \mathbf{f}_{u} \mathbf{f}_{u}^{\mathrm{H}} (\mathbf{d}_{j}^{k})^{*}. $$

(21)

With these definitions, we now proceed with the formulation of our optimization problem. We consider independent optimization of the transmit beamformers of the individual users according to the following optimization problem:

$$\begin{array}{*{20}l} & \max_{\mathbf{f}_{u} \in \mathbb{C}^{N_{t} \times 1}} P_{u} \text{P1} \\ \text{s. t.:} & \left\|\mathbf{f}_{u}\right\|^{2} \leq 1/\left|\mathcal{S}\right|, \\ & L_{j,u}^{k} \leq L_{\text{max}},\quad \forall j \in \mathcal{S}, j \neq u,\ \forall k \in \{1,\ldots,S\}. \end{array} $$

(P1)

where the parameter *L*_{max} denotes the maximum tolerable interference level. Replacing \(\max _{\mathbf {f}_{u} \in \mathbb {C}^{N_{t} \times 1}} P_{u}\) equivalently by \(\min _{\mathbf {f}_{u} \in \mathbb {C}^{N_{t} \times 1}} (-P_{u})\), this optimization problem can be formulated as a quadratically constrained quadratic program; however, since the matrix \(-\sum _{s = 1}^{S} \left \|\mathbf {c}_{u}^{s}\right \|^{2} (\mathbf {d}_{u}^{s})^{*} (\mathbf {d}_{u}^{s})^{\mathrm {T}}\), which determines the quadratic objective function, is negative semidefinite, the problem is in general non-convex and non-deterministic polynomial-time (NP)-hard. An approximate solution is possible by applying a semidefinite programming relaxation (SDR) to the optimization variable **f**_{
u
} and by recovering a feasible suboptimal rank-one beamforming solution through randomization [56]. The approximation performance of this approach in general deteriorates with increasing dimension *N*_{
t
} of the optimization variable as well as with growing number of constraints in (P1) [57, 58]. Hence, for our envisioned use case of FD-MIMO systems with large *N*_{
t
} and \(\left |\mathcal {S}\right |\), the achieved approximation performance might not be satisfactory.

We therefore consider a modification of (P1) to avoid the SDR: Instead of maximizing the sum signal power over all *S* specular components of user *u*, we rather maximize the power received only over the strongest specular component *s*=1, while still considering the interference leakage caused over all specular components:

$$\begin{array}{*{20}l} \text{P2} & \max_{\mathbf{f}_{u} \in \mathbb{C}^{N_{t} \times 1}} \left\|\mathbf{c}_{u}^{1}\right\|^{2} (\mathbf{d}_{u}^{1})^{\mathrm{T}} \mathbf{f}_{u} \mathbf{f}_{u}^{\mathrm{H}} (\mathbf{d}_{u}^{1})^{*} \\ \text{s. t.:} & \left\|\mathbf{f}_{u}\right\|^{2} \leq 1/\left|\mathcal{S}\right|, \\ & L_{j,u}^{k} \leq L_{\text{max}},\quad \forall j \in \mathcal{S}, j \neq u,\ \forall k \in \{1,\ldots,S\}. \end{array} $$

(P2)

This problem can be transformed to a convex problem by recognizing that (P2) is invariant w.r.t. the absolute phase of **f**_{
u
}. Hence, we can apply the approach considered, e.g., in [59], and restrict \((\mathbf {d}_{u}^{1})^{\mathrm {T}} \ \mathbf {f}_{u}\) to be real valued to obtain the following equivalent convex second-order cone program:

$$\begin{array}{*{20}l} \text{P3} & \max_{\mathbf{f}_{u} \in \mathbb{C}^{N_{t} \times 1},\, p_{u} \in \mathbb{R}}\ \ p_{u} \\ \text{s. t.:} & \left\|\mathbf{f}_{u}\right\|^{2} \leq 1/\left|\mathcal{S}\right|, \\ & \Re\left((\mathbf{d}_{u}^{1})^{\mathrm{T}} \mathbf{f}_{u} \right) \geq p_{u}, \\ & \Im\left((\mathbf{d}_{u}^{1})^{\mathrm{T}} \mathbf{f}_{u} \right) = 0, \\ & L_{j,u}^{k} \leq L_{\text{max}},\quad \forall j \in \mathcal{S}, j \neq u,\ \forall k \in \{1,\ldots,S\}. \end{array} $$

(P3)

We denote the optimal solution of this optimization problem as the *non-robust leakage-bounded angular beamformer*. As this is a second-order cone program, its complexity scales with the square root of the number of cone constraints [60, 61].

The solution to problem (P3) allows to efficiently control the interference leakage caused to other users, provided the angular CSIT is accurate. However, since this CSIT is obtained by limited feedback, angular quantization is unavoidable which impairs the quality of the CSIT. Additionally, in mobile situations, the azimuth and elevation angles representing the specular components change over time, causing a mismatch between the angular CSI feedback and the actual angles, due to the processing delay of the feedback link. These effects increase the inter-user interference and thus deteriorate the performance of the system. To improve the robustness of our beamformer solution w.r.t. angular uncertainty, we consider a robust problem formulation in the following.

Given the angular feedback \(\left (\phi _{u}^{s},\theta _{u}^{s}\right)\), we propose to incorporate angular uncertainty regions into the beamformer optimization problem. We denote the discrete angular uncertainty region corresponding to specular component *s* of user *u* as \(T_{u}^{s}\subseteq \left [-\frac {\pi }{2},\frac {\pi }{2}\right ] \times [0,\pi ]\), such that \((\phi _{u}^{s},\theta _{u}^{s}) \in T_{u}^{s}\). In our investigations, we utilize symmetric rectangular regions around \((\phi _{u}^{s},\theta _{u}^{s})\). The sizes of these regions can, e.g., be set according to the angular variation observed over the previous CSI feedback interval, i.e., \(2 \left |\phi _{u}^{s}[n-m] - \phi _{u}^{s}[n-2m]\right |\) and \(2 \left |\theta _{u}^{s}[n-m] - \theta _{u}^{s}[n-2m]\right |\), where the factor 2 accounts for the unknown direction of movement. The density of the lattice points considered within these regions, i.e., the angular resolution of \(T_{u}^{s}\), must be chosen sufficiently large to ensure that no side-lobes of the radiation pattern are missed. As a guideline, consider a UPA with equal gain beamforming: here, zeros in the radiation pattern in azimuth occur at angles \(\sin \phi = \pm \frac {n \lambda }{d_{h} N_{h}}\); hence, the angular resolution in azimuth must be chosen to assure that the peaks in-between such zeros are resolved with sufficient accuracy. Notice, though, that the complexity of the proposed optimization problems, more specifically the number of constraints, scales linearly with the size of each set \(T_{u}^{s}\); hence, one should avoid unnecessary oversampling of \(T_{u}^{s}\) to reduce computational complexity.

Utilizing these angular uncertainty regions, we now formulate the optimization problem for the *robust leakage-bounded angular beamformer*:

$$\begin{array}{*{20}l} \text{P4} & \max_{\mathbf{f}_{u} \in \mathbb{C}^{N_{t} \times 1}} \left\|\mathbf{c}_{u}^{1}\right\|^{2} (\mathbf{d}_{u}^{1})^{\mathrm{T}} \mathbf{f}_{u} \mathbf{f}_{u}^{\mathrm{H}} (\mathbf{d}_{u}^{1})^{*} \\ \text{s. t.:} & \left\|\mathbf{f}_{u}\right\|^{2} \leq 1/\left|\mathcal{S}\right|, \\ & \left\|\mathbf{c}_{j}^{k}\right\|^{2} (\mathbf{a}_{t}(\boldsymbol{\Psi}_{i,j}^{k}))^{\mathrm{T}} \mathbf{f}_{u} \mathbf{f}_{u}^{\mathrm{H}} (\mathbf{a}_{t}(\boldsymbol{\Psi}_{i,j}^{k}))^{*} \leq L_{\text{max}}, \\ & \forall j \in \mathcal{S}, j \neq u,\ \ \forall k \in \{1,\ldots,S\},\ \ \forall \boldsymbol{\Psi}_{i,j}^{k} \in T_{j}^{k}. \end{array} $$

(P4)

In this problem, the vector \(\mathbf {a}_{t}(\boldsymbol {\Psi }_{i,j}^{k})\) denotes the transmit antenna array response evaluated at angle \(\boldsymbol {\Psi }_{i,j}^{k} \in T_{j}^{k}\); here, the same antenna array response as in the CSI feedback dictionary (10) is used. Notice, we account for the angular uncertainty only in the interference terms, but not in the signal term. This is because the system reacts much more sensitive w.r.t uncertainty in the interference directions as compared to the signal direction, since nulls in the beamforming pattern are commonly very narrow whereas peaks are comparatively broad in the angular domain (see also the example in Fig. 2). Another reason for neglecting uncertainty in the angular direction of the intended signal is to keep the problem convex.^{Footnote 3} Problem (P4) can be brought into the form of a second-order cone program, similar to Problem (P3).

### Leakage-bounded angular precoding

In this section, we extend the robust beamformer design to multi-user precoding with multiple streams per user. When considering multi-stream precoding, it is not sufficient to maximize the sum received power over all *L*_{
u
} streams of user *u* because this will generally lead to a rank one beamforming solution that steers all signal energy over the largest singular value of the channel. Hence, to obtain a multi-stream precoding solution of rank *L*_{
u
}, we have to consider the achievable rate of a user in terms of the mutual information. With the available CSIT, however, we can only obtain a coarse estimate of the achievable rate as detailed below.

Let us start by considering the achievable rate of user *u* in terms of the mutual information:

$$\begin{array}{*{20}l} R_{u} = \mathbb{E} \text{ log}\left|{\mathbf{I}_{N_{r}} + \left(\sigma_{n,u}^{2} \mathbf{I}_{N_{r}} + \sum_{\substack{j \in \mathcal{S} \\ j\neq u}} \mathbf{R}_{u,j} \right)^{-1} \mathbf{R}_{u}}\right| \end{array} $$

(22)

$$\begin{array}{*{20}l} \mathbf{R}_{u} = \mathbf{H}_{u} \mathbf{F}_{u}(\mathbf{H}_{u} \mathbf{F}_{u})^{\mathrm{H}},\ \ \mathbf{R}_{u,j} = \mathbf{H}_{u} \mathbf{F}_{j} (\mathbf{H}_{u} \mathbf{F}_{j})^{\mathrm{H}}, \end{array} $$

(23)

where **R**_{
u
} and **R**_{u,j} denote the signal and interference covariance matrices, respectively. We consider individual optimization of the achievable rate of each user w.r.t. its precoder, putting additional interference constraints on the signal leakage caused to the other users. This means that we do not attempt to jointly optimize the sum rate of all users; notice, though, that for single-stream beamforming, it has been shown in [62, 63] that such an approach can still achieve the maximal sum-rate, provided the leakage constraints are appropriately selected. When optimizing *R*_{
u
} w.r.t. **F**_{
u
}, we can then focus on

$$ \tilde{R}_{u} = \mathbb{E} \text{ log}\left|\sigma_{n,u}^{2} \mathbf{I}_{N_{r}} + \sum_{\substack{j \in \mathcal{S} \\ j\neq u}} \mathbf{R}_{u,j} + \mathbf{R}_{u}\right|, $$

(24)

since the remaining terms are independent of **F**_{
u
}. Because the CSIT contains no information about the channel estimation error in (8) and the channel decomposition error in (9), the transmitter can only calculate approximations \(\hat {\mathbf {R}}_{u}\) and \(\hat {\mathbf {R}}_{u,j}\) of the covariance matrices utilizing the angular-domain CSI feedback; this gives a corresponding approximation \(\hat {\tilde {R}}_{u}\) of \(\tilde {R}_{u}\). In the Appendix, we derive the following upper bound on \(\hat {\tilde {R}}_{u}\):

$$ {}\hat{\tilde{R}}_{u} \leq \text{ log} \left|\sigma_{n,u}^{2} \mathbf{I}_{S} + \left(|\mathcal{S}|-1\right) \frac{S L_{\text{max}}}{\text{tr }(\boldsymbol{\Sigma}_{u}^{2})} \boldsymbol{\Sigma}_{u}^{2} + \boldsymbol{\Sigma}_{u}^{2} \mathbf{U}_{u}^{\mathrm{H}} \mathbf{C}_{u} \mathbf{U}_{u}\right|, $$

(25)

where the matrices \(\boldsymbol {\Sigma }_{u}^{2}\) and **U**_{
u
}, as defined in (36), can be calculated from the available CSIT and **C**_{
u
}=**F**_{
u
}(**F**_{
u
})^{H} denotes the transmit covariance matrix associated to user *u*.

We now utilize this upper bound to formulate the optimization problem for the *non-robust leakage-bounded angular precoder*:

$$\begin{array}{*{20}l}\text{P5} {}\max_{\mathbf{C}_{u} \in \mathbb{C}^{N_{t} \times N_{t}}} \left|\sigma_{n,u}^{2} \mathbf{I}_{S} + \left(|\mathcal{S}|-1\right) \frac{S L_{\text{max}}}{\text{tr }(\boldsymbol{\Sigma}_{u}^{2})} \boldsymbol{\Sigma}_{u}^{2} + \boldsymbol{\Sigma}_{u}^{2} {\mathbf{U}_{u}^{\mathrm{H}}} \mathbf{C}_{u} \mathbf{U}_{u} \right| \\ \text{s. t.:} \quad \text{tr }(\mathbf{C}_{u}) \leq 1/\left|\mathcal{S}\right|,\ \mathbf{C}_{u} \succeq \mathbf{0}, \\ L_{j,u}^{k} \leq L_{\text{max}}, \ \ \forall j \in \mathcal{S}, j \neq u,\ \forall k \in \{1,\ldots,S\}, \end{array} $$

(P5)

with \(L_{j,u}^{k}\) as defined in (38). Notice, this actually represents an SDR of the optimization w.r.t. the precoder \(\mathbf {F}_{u} \in \mathbb {C}^{N_{t} \times L_{u}}\). In general, the optimal solution \(\mathbf {C}_{u}^{\text {(opt)}}\) of (P5) is not guaranteed to be of rank *L*_{
u
}. Due to the definition of \(\boldsymbol {\Sigma }_{u}^{2}\) and **U**_{
u
} in (36), \(\text {rank }{\mathbf {C}_{u}^{\text {(opt)}}} \leq S\). This is because a solution of rank greater than *S* would steer part of the transmit energy into the null space of **U**_{
u
}, which cannot maximize our objective function. The restriction \(\text {rank }{\mathbf {C}_{u}^{\text {(opt)}}} \leq S\) also implies that we have to feed back at least as many specular components *S* as the number of data streams *L*_{
u
} to make sure that the solution \(\mathbf {C}_{u}^{\text {(opt)}}\) can potentially support the transmission of *L*_{
u
} streams. Still, even with *S*≥*L*_{
u
}, it can happen that the optimal solution \(\mathbf {C}_{u}^{\text {(opt)}}\) has \(\text {rank }{\mathbf {C}_{u}^{\text {(opt)}}} < L_{u}\); this is likely to be the case at low signal-to-noise ratio SNR where the transmission of less than *L*_{
u
} streams can provide an advantageous beamforming gain. If \(\text {rank }{\mathbf {C}_{u}^{\text {(opt)}}} < L_{u}\), we simply transmit less than *L*_{
u
} streams over the eigenvectors corresponding to the non-zero eigenvalues of \(\mathbf {C}_{u}^{\text {(opt)}}\), since this maximizes our objective function. If \(\text {rank }{\mathbf {C}_{u}^{\text {(opt)}}} > L_{u}\), we apply Gaussian randomization to derive a feasible precoder of rank *L*_{
u
}, by appropriately modifying the randomization approaches described in more detail in [56, 64]. Notice, though, if we set *S*=*L*_{
u
}, i.e., we feed back as many specular components as the number of data streams, then the rank of \(\mathbf {C}_{u}^{\text {(opt)}}\) is upper bounded by *L*_{
u
} and we therefore do not have to apply randomization at all. Thus, by setting *S*=*L*_{
u
}, we can guarantee that the SDR provides a globally optimal solution for the precoder **F**_{
u
}.

Similar to (P4), we can also implement a *robust leakage-bounded angular precoder* by accounting for the angular uncertainty regions \(T_{j}^{k}\):

$$\begin{array}{*{20}l}\text{P6} {}\max_{\mathbf{C}_{u} \in \mathbb{C}^{N_{t} \times N_{t}}} \left|\sigma_{n,u}^{2} \mathbf{I}_{S} + \left(|\mathcal{S}|-1\right) \frac{S L_{\text{max}}}{\text{tr }(\boldsymbol{\Sigma}_{u}^{2})} \boldsymbol{\Sigma}_{u}^{2} + \boldsymbol{\Sigma}_{u}^{2} {\mathbf{U}_{u}^{\mathrm{H}}} \mathbf{C}_{u} \mathbf{U}_{u} \right| \\ \text{s. t.:} \quad \text{tr }(\mathbf{C}_{u}) \leq 1/\left|\mathcal{S}\right|,\ \mathbf{C}_{u} \succeq \mathbf{0}, \\ \left\|\mathbf{c}_{j}^{k}\right\|^{2} \text{tr }{(\mathbf{a}_{t}(\boldsymbol{\Psi}_{i,j}^{k}))^{*} (\mathbf{a}_{t}(\boldsymbol{\Psi}_{i,j}^{k}))^{\mathrm{T}} \mathbf{C}_{u}} \leq L_{\text{max}}, \\ \forall j \in \mathcal{S}, j \neq u,\ \ \forall k \in \{1,\ldots,S\},\ \ \forall \boldsymbol{\Psi}_{i,j}^{k} \in T_{j}^{k}. \end{array} $$

(P6)

Applying a similar step as from (P2) to (P3), both problems (P5) and (P6) can be brought into convex form by adding an extra optimization variable \(p_{u} \in \mathbb {R}\) and maximizing *p*_{
u
} subject to the logarithmic term in (P5), (P6) being greater than or equal to *p*_{
u
}. However, such a determinant constraint is substantially more complex than the corresponding linear constraint in (P3). Hence, we consider in our simulations in Section 6.3 also as an alternative to determinant optimization a simple *per stream optimization*: we apply (P3) or (P4) *L*_{
u
} times to obtain beamformers that maximize the received power over the *L*_{
u
} strongest specular components and we concatenate these beamformers to obtain the *L*_{
u
} dimensional precoding matrix.

In our derivation, we assumed that the precoders of the other \(|\mathcal {S}|-1\) users are unknown when calculating the precoder of user *u*. Yet, in principle when optimizing the *u*-th user, the base station could already utilize the knowledge of the precoders calculated for those *u*−1 users that have been optimized before. Similarly, an alternating optimization of precoders with a proper termination criterion could be applied to obtain an iterative approach that might provide better performance. For complexity reasons, however, we have not implemented such an approach in our simulations.

### Implementation issues

#### Computational complexity

Our derivation in Section 4.1 shows that leakage-bounded transmit optimization as proposed in (P1) is in general an NP-hard quadratically constrained quadratic optimization problem, which cannot be solved efficiently. Approximate solution by means of an SDR is a convex optimization problem, which can be solved efficiently; yet, it is still computationally demanding, since the optimization variable of the SDR is of dimension *N*_{
t
}×*N*_{
t
} and hence grows quadratically with the size of the FD-MIMO antenna array. This complexity issue is relaxed by problem (P2) and its convex second-order cone programming reformulation (P3), in which the dimension of the optimization variable **f**_{
u
} grows only linearly with the size of the FD-MIMO antenna array. Since (P3) is a convex problem, it can be solved in polynomial time, e.g., by means of an interior point method; more details on solving second-order cone programs can for example be found in [65]. In problem (P4), we extend the number of constraints of our optimization problem by accounting for the angular uncertainty regions; this directly impacts the computational complexity of the problem. However, since the worst-case complexity of a second-order cone program scales with the square root of the number of constraints [60, 61], the increase in complexity is moderate. Considering the two multi-stream optimization problems (P5) and (P6), they both admit solution by means of an interior point method or a Newton conjugate-gradient approach as proposed in [66]. However, as our computer experiments have shown, this approach can be computationally demanding and slow. Therefore, the per-stream optimization approach proposed in Section 4.2 appears practically more relevant, since its complexity only grows linearly with the number of data streams per user.

#### Combination with hybrid architectures

As mentioned in the introduction of this paper, much research work on FD-MIMO systems is currently devoted to reducing system complexity through so-called hybrid architectures, which divide the precoding operation into an analog RF domain part and a digital base band processing part. The RF domain precoder thereby reduces the dimension of the effective base band channel, i.e., the product of channel matrix and RF domain precoder, which simplifies channel estimation, alleviates the CSI feedback overhead burden, and reduces the number of RF chains required. The RF domain precoder optimization is commonly required to provide a precoder solution with unit modulus matrix entries, to enable implementation with simple phase-shifter elements, and to be constant over the entire system bandwidth, to facilitate application in the analog domain. Both these constraints are not fulfilled by the leakage-bounded precoding optimization problem proposed in this paper. Nevertheless, we consider an extension of our leakage-bounded precoding optimization to hybrid architectures as promising future work.

Specifically, our leakage-bounded angular beamforming approach might be well suited for the design of the RF domain precoder, by applying the per stream optimization mentioned in Section 4.2 for each user more than *L*_{
u
} times in order to obtain a precoder of dimension equal to the number of available RF chains. However, the corresponding optimization problem (P3) or (P4) will in general not output a unit-modulus solution. It is well-known that the unit modulus precoder constraint of RF precoding is a non-convex constraint. A common solution to deal with this issue is to relax the constraint in the precoder optimization and to orthogonally project the resulting solution onto the set of unit modulus matrices [14]; this approach can also be applied to our problem at hand. Notice also, by spending twice the amount of phase-shifters it is actually possible to avoid the unit-modulus constraint all together [67].

A second issue that needs to be addressed is that the channel decomposition (11) is frequency selective due to the multipath delay spread introduced by the channel, which implies that the precoder also needs to be frequency selective and can thus not be implemented in the analog domain. However, in LOS situations, the frequency selectivity is not very distinct and can potentially be neglected for the cost of a slight frequency-dependent mismatch of the decomposition.

In principle, the so obtained analog RF domain precoder can then be combined with any base band precoding approach that is designed for the effective base band channel, similar to the hybrid BD precoding scheme described in Section 5.2 below.