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Robust fulldimension MIMO transmission based on limited feedback angulardomain CSIT
EURASIP Journal on Wireless Communications and Networking volume 2018, Article number: 58 (2018)
Abstract
In this paper, we propose robust multiuser beamforming and precoding techniques for fulldimension MIMO transmission based on limited feedback. We propose to employ an overcomplete basis decomposition in the angular domain to approximate the channel matrices as sums of few dominant specular components, facilitating efficient channel state information (CSI) quantization at the users. The selected expansion vectors of such a sparse approximation, parametrized by azimuth and elevation angles, are relatively robust with respect to channel estimation errors as well as channel variations over time. Based on this CSI feedback, we propose incoherent beamforming/precoding methods that make use only of the azimuth and elevation angles as well as the norm of the expansion coefficients and do not rely on coherent multipath interference to eliminate interuser interference. Our optimization aims at maximizing the signal power or the achievable rate of a user, while limiting the amount of interference leakage caused to other users. To further improve robustness, we account for uncertainty in the angular channel decomposition in the proposed precoder optimization.
Introduction
Fulldimension multipleinput multipleoutput (FDMIMO) transmission refers to wireless transmission systems that support twodimensional antenna arrays with a large number of antenna elements. This enables highresolution beamforming in both, the elevation and the azimuth domain, in order to achieve spacedivision multiple access gains through spatial separation of users, as well as to enhance the energy efficiency of wireless data transmission by concentrating the radiated energy towards intended users [1–3]. The great potential of FDMIMO systems has been recognized by standardization bodies, such as the third generation partnership project (3GPP); correspondingly, development and standardization efforts are ongoing within longterm evolution (LTE) Release 14 and 5G new radio (NR) to further enhance the initial realization of FDMIMO in Release 13. Currently, most implementation proposals of FDMIMO transceivers are based on hybrid architectures, where part of the signal processing is performed in base band and part in the analog radio frequency (RF) domain in order to limit the number of required RF chains. This approach is especially important in the millimeter wave band, where hardware complexity is a limiting factor [4–8]. When designing hybrid precoding systems, several challenges need to be addressed [9–11]: the design freedom of RF precoders is commonly restricted by hardware limitations, such as discrete angular resolution and constant modulus of the phaseshifting elements, as well as partial connectivity of the phaseshift network to limit insertion loss. An alternative to hybrid architectures, which potentially provides even higher spectral and energy efficiency in certain operating regimes, is to employ cheap low (even single bit) resolution analog to digital (AD)/digital to analog (DA) converters in FDMIMO transceivers. At the receiveside of most existing MIMO transceivers, highresolution AD converters are employed to minimize the signal distortions introduced by the receiver. Such highresolution AD converters are, however, among the most power hungry devices of the receiver chain. This motivates the use of lowresolution energy efficient AD converters in FDMIMO systems to achieve energyefficient operation [12–15]. At the transmit side, on the other hand, power expenditure is dominated by power amplifiers, which are usually required to operate within the high linearity regime to avoid signal distortion. Employing lowresolution DA converters relaxes these linearity requirements, allowing the amplifiers to operate closer to saturation thereby increasing their efficiency [16, 17].
In this work, our focus is on an enhancing the robustness of downlink threedimensional FDMIMO point to multipoint transmission with respect to imperfections of the channel state information at the transmitter (CSIT), especially due to high user mobility, without specifically addressing the restrictions of hybrid precoding or lowresolution AD/DA implementations. In MIMO transmission in general, and especially when considering multipoint communication, CSIT is an essential ingredient. CSIT for downlink transmission is commonly obtained either from uplink measurements using dedicated uplink pilot signals [18, 19] or from limited channel state information (CSI) feedback from the users [20–22]. We consider frequency division duplex (FDD) transmission in this paper, where CSIT is provided via feedback from the users over dedicated limited capacity feedback links. For this purpose, the users have to quantize the required CSI utilizing a proper quantization codebook. Two approaches are most common in current mobile communication systems [23]: (1) Implicit CSI feedback, where the users select preferred transmit beamformers/precoders from a predefined codebook given the current channel conditions. This has the advantage that the users can accurately estimate the expected performance during transmission, provided the channel conditions are quasistatic. (2) Explicit CSI feedback, where the users directly quantize the MIMO channel matrix. This approach is better suited for multiuser MIMO transmission because it facilitates calculation of efficient multiuser precoders at the transmitter onthefly. Either way, providing CSI via limited feedback is always prone to inaccuracies, on the one hand, caused by the quantization process, which is inevitable due to the limited capacity of the feedback links. On the other hand, CSI inaccuracies also result from the delay of the feedback path, leading to a mismatch of the CSI experienced during the quantization phase and the CSI valid during transmission. Several theoretical studies exist that evaluate the performance degradation due to imperfect and outdated CSIT for a variety of transmission scenarios [24–27].
Popular coherent multiuser beamforming/precoding methods, such as zero forcing (ZF) beamforming [28], block diagonalization (BD) precoding [29] and interference alignment (IA) [30, 31], utilize the phase difference between the multipath components contributing to the channel between the transmitter and the receiver to achieve constructive/destructive signal interference. The phases of multipath components, however, can change very quickly over time, especially for mobile users, and hence coherent beamforming/precoding methods react very sensitive to nonstatic scenarios [32, 33]. The corresponding performance degradation can partly be mitigated by robust coherent beamforming/precoding techniques that account for the CSIT imperfection. One prominent example of robust beamforming techniques is minimum variance robust adaptive beamforming [34], which accounts for imperfections in the estimation of signal covariance matrices. In [35], the authors achieve robustness w.r.t. CSI quantization by utilizing multiple receive antennas to improve system performance without the need for full channel feedback information; this idea is in essence similar to [36], where subspaceselection strategies for CSI feedback are discussed that enable mitigating the impact of CSI quantization. In [37], robust singleuser MIMO precoding methods are proposed that maximize the worstcase received signaltonoise ratio or minimize the worstcase error probability with imperfect CSIT. Furthermore, hybrid transceiver architectures that incorporate robustness against channel estimation errors are proposed in [38].
Contribution In this paper, we take a different approach to achieve robustness of multiuser beamforming/precoding w.r.t. CSIT imperfections due to limited feedback. Specifically, we employ an overcomplete basis decomposition of the MIMO channel matrix at each receiver to obtain a sparse approximation of the CSI, which can efficiently be quantized and fed back to the transmitter. The overcomplete basis decomposition is based on an angulardomain quantization codebook (also called dictionary following the nomenclature of sparse approximation literature), which is parametrized by azimuth and elevation signal departure angles. Under line of sight (LOS) conditions, the selected expansion vectors of the sparse approximation are robust w.r.t. channel estimation errors as well as channel variations over time. We demonstrate this robustness by means of numerical simulations employing stateoftheart threedimensional stochasticgeometric channel models. However, the complexvalued coefficients of the overcomplete basis expansion are still significantly impacted by channel estimation errors and channel variations over time, which can impair the performance of coherent precoding techniques, such as, as ZF beamforming, BD precoding and IA, that rely on destructive multipath interference to eliminate interuser interference. We therefore propose to apply incoherent beamforming/precoding techniques that are inherently robust w.r.t. phase uncertainties of multipath components, similar to spacedivision multiple access (SDMA). Specifically, we propose leakagebased angular beamforming/precoding methods that make use only of the azimuth and elevation angles as well as the norm of the expansion coefficients and do not rely on the phase of the expansion coefficients to achieve destructive multipath interference. This approach also improves robustness of beamforming/precoding solutions at high mobility, where the relative phases of multipath components change very quickly over time. To further improve robustness we additionally account for uncertainty in the angular channel decomposition in the proposed precoder optimization. Naturally, this approach cannot compete with coherent precoding methods in case of highly accurate CSIT; however, our numerical simulations demonstrate significant gains in terms of achievable rate and symbol error ratio (SER) in case of imperfect CSIT under LOS conditions. The present work is an extension of our two conference publications [39, 40] to multiuser beamforming and multistream multiuser precoding. Specifically, in [39] we propose and approximately solve the FDMIMO transmit beam pattern optimization problem of worstcase signal to leakage ratio (SLR) maximization, where we maximize the ratio of signal power radiated into an angular region of interest to leakage power radiated into an unintended angular region. A similar design and optimization principle guides us also in the present paper, yet extended to adaptive operation with limited feedback from the users and transmission of multiple data streams per user. In [40], we consider socalled doublesided FDMIMO systems, where both transmitter and receiver are equipped with FDMIMO antenna arrays, and we apply a similar overcomplete basis decomposition as in the present paper to achieve efficient limited feedback operation. However, we do not consider robust multiuser beamforming/precoding in [40]. Hence, the present paper combines results of [39, 40] and extends them to adaptive limitedfeedback based multiuser precoding with multistream transmission per user.
Organization In Section 2, we introduce our system model and state the underlying assumptions of this work. In Section 3, we present the proposed CSI feedback approach that is based on a sparse approximation of the channel utilizing an overcomplete basis decomposition. Based on this CSI feedback, we then propose in Section 4 our robust leakagebounded angular beamforming and precoding methods. We briefly summarize for the readers convenience in Section 5 the benchmark methods employed in our simulations, in order to clarify how these methods are applied in our considered scenario. Finally, in Section 6, we investigate the performance of the proposed beamformers and precoders by means of numerical simulations and we provide concluding remarks in Section 7.
Notation We denote scalars by upper and lower case italic letters (e.g., N, n), vectors by lower case bold letters (e.g., x), and matrices by upper case bold letters (e.g., H). To address the element in the ith row and the jth column of matrix H, we use the notation [H]_{i,j}. We write the trace of a matrix as tr(H), the transpose as H^{T}, the conjugatetranspose as H^{H}, the Frobenius norm as ∥H∥, the pseudoinverse as pinv(H), and the vector obtained by stacking the columns of the matrix below each other as vec(H). We denote the logarithm of the determinant of a quadratic matrix H as logH. We denote sets by calligraphic letters (e.g., \(\mathcal {A}\)); the size of set \(\mathcal {A}\) is \(\left \mathcal {A}\right \). To calculate the expected value of a random variable r, we employ the notation \(\mathbb {E}{r}\). The Dirac delta function is δ(t); the discrete Kronecker delta is δ_{ ij }, i.e., δ(t)→∞ifft=0 and δ_{ ij }=1iffi=j. We denote the real part of a complex number z as ℜ(z), the imaginary part as I(z), and the complex conjugate as z^{∗}. We denote the uniform distribution on the interval [a,b] as \(\mathcal {U}(a,b)\) and the vectorvalued complexGaussian distribution with mean vector μ and covariance matrix C as \(\mathcal {CN}\left (\boldsymbol {\mu },\mathbf {C}\right)\).
System model
Channel model
In wireless communications, the channel between transmitter and receiver can be characterized by a doubledirectional timevariant channel impulse response [41]:
with N_{ L } denoting the number of multipath components and α_{ ℓ },τ_{ ℓ }, respectively, representing the complexvalued amplitude and the propagation delay associated to path ℓ. The vectors Ψ_{t,ℓ}=[ϕ_{t,ℓ},θ_{t,ℓ}]^{T} and Ψ_{r,ℓ}=[ϕ_{r,ℓ},θ_{r,ℓ}]^{T} denote the azimuth and elevation angles of departure and arrival. Although not explicitly shown in (1), the parameters on the righthand side may all implicitly depend on the absolute time t due to movement of users and scattering objects. Incorporating the responses of the transmit and receive antenna elements w.r.t. a plane wave from angular direction Ψ, i.e., a_{ t }(Ψ) and a_{ r }(Ψ), we obtain the timevariant channel impulse response as:
If we furthermore incorporate the overall impulse response of the transmit and receive filters f(τ), we obtain the transceiverdependent timevariant channel impulse response as:
where again the time dependence of the parameters on the righthand side is implicit. Under the wellknown planar wave and narrowband array assumptions [42], we can obtain the timevariant impulse response of a MIMO system simply by concatenating the individual impulse responses of antenna element pairs:
with \(\mathbf {a}_{r}(\boldsymbol {\Psi }) \in \mathbb {C}^{N_{r} \times 1}, \mathbf {a}_{t}(\boldsymbol {\Psi }) \in \mathbb {C}^{N_{t} \times 1}\) denoting the receive and transmit antenna array response vectors. For example, consider a uniform planar array (UPA) with N_{ v } rows and N_{ h } columns at the transmitter, with vertical antenna element spacing d_{ v } and horizontal antenna element spacing d_{ h } in multiples of the wavelength λ. Then, the response of the antenna element in the ℓth row and the kth column with respect to a plane wave from direction Ψ=[ϕ,θ]^{T} is [43]:
where angles are measured as illustrated in Fig. 1 in relation to the antenna array and the individual antenna element responses are arranged rowbyrow in the vector \(\mathbf {a}_{t}^{\textrm {UPA}}(\phi,\theta) \in \mathbb {C}^{N_{t} \times 1}\). In (5), the function \(g_{e}(\phi,\theta) \in \mathbb {C}\) denotes the complexvalued angledependent antenna element gain pattern, e.g., a Hertzian dipole pattern. We do not focus on a specific antenna array geometry in this work nor do we restrict the antenna element gain pattern g_{ e }(ϕ,θ). Notice, in (4), we do not account for coupling inbetween antenna elements, which may be incorporated with additional coupling matrices [43].
Transmission model
We consider downlink transmission from a base station that is equipped with an antenna array of N_{ t } antenna elements, e.g., a UPA of size N_{ t }=N_{ h }·N_{ v } as illustrated in Fig. 1. The base station serves U users. The users are each equipped with N_{ r } receive antennas; we assume N_{ t }≥N_{ r }. We consider orthogonal frequency division multiplexing (OFDM) transmission; the MIMOOFDM channel matrix \(\mathbf {H}[n,k] \in \mathbb {C}^{N_{r} \times N_{t}}\) at OFDM symbol n and subcarrier k is obtained by sampling the Fourier transform of the timevariant impulse response \(\tilde {\mathbf {H}}(t,\tau)\) of (4). We assume that time variations within OFDM symbols are small enough such that the intercarrier interference due to Doppler shifts is negligible. In the following, we focus on a single subcarrier and omit the subcarrier index k for brevity. To refer to the channel matrix associated to a specific user u, we employ a subscript, such as H_{ u }[n].
The base station selects at each time instant a set \(\mathcal {S}[n] \subseteq \{1,\ldots, U\}\) of users to be served in parallel, employing spatial multiplexing by means of linear beamforming/precoding. Assuming transmission of L_{ u } streams to user \(u \in \mathcal {S}[n]\), the base station applies the precoding matrix \(\mathbf {F}_{u}[n] \in \mathbb {C}^{N_{t} \times L_{u}}\) to transmit the data symbol vector \(\mathbf {s}_{u}[n] \in \mathbb {C}^{L_{u} \times 1}\) to user u. We assume \(\mathbb {E}(\mathbf {s}_{u}[n] {\mathbf {s}_{u}[n]^{\mathrm {H}}}) = \mathbf {I}_{L_{u}}\) and account for the power allocation in F_{ u }[n], such that \(\text {tr}\,(\mathbf {F}_{u}[n]\mathbf {F}_{u}[n]^{\mathrm {H}}) = 1/\left \mathcal {S}[n]\right \). The inputoutput relationship of user u hence is
with n_{ u }[n] denoting zeromean complex Gaussian receiver noise of variance \(\sigma _{n,u}^{2}\).
Targeted deployment scenarios
The beamforming and precoding methods proposed below are basically enhanced interferenceaware SDMA techniques. As such, the methods require a certain degree of spatial separability of users in the angular domain. Such separability can mainly be guaranteed in limited scattering environments with a small number of dominant specular components per user and relatively weak diffuse background scattering, as common in LOS conditions. Even though this might sound restrictive for mobile communications, many scenarios envisioned for deployment of FDMIMO systems are expected to exhibit dominant LOS transmission. Example scenarios where LOS propagation could dominate are the urban micro and macro scenarios illustrated in ([2], Fig. 2) and ([44], Fig. 2), where users on different floors of a building are served in parallel by means of spatial beamforming from base stations mounted on neighboring buildings. Another interesting deployment scenario is multiuser multiplexing in large indoor open spaces, e.g., within stadiums, airport terminals, shopping malls, where highceilingmounted small cells and remote radio heads can provide LOS connectivity to users. Furthermore of interest are vehicular communication scenarios on highways and motorways, where FDMIMO beamforming can be employed for multiuser transmission between roadside access points and vehicles as described in [45]. Notice also that channel fading models that contain only two dominant specular components in addition to weak diffuse background scattering, such as the twowave diffuse power (TWDP), generalized tworay (GRT), and fluctuating tworay (FRT) fading models [46–50], are well suited to capture the behavior of channel measurements conducted in the millimeter wave band [49, 51], suggesting that millimeter wave transmissions are in many cases well characterized by few specular components. Importantly, in the millimeter wave band, even relatively low mobility of users (pedestrians) causes significant time selectivity of the channel due to the small wavelength; hence, CSIT becomes outdated even faster.
Channel state information feedback
We assume that the transmitter inserts pilot symbols in the OFDM timefrequency grid to support channel estimation for CSI feedback calculation at the receivers. Notice, these CSI pilots may be provided sparsely in the timefrequency grid to facilitate coarse channel estimation only for CSI calculation; additional more densely populated precoded pilot symbols can be employed to support accurate channel estimation for data detection, similar to LTE’s CSI and UEspecific pilot symbols [52]. We consider channel quantization and feedback in the frequency domain; alternatively, it is also possible to quantize the channel impulse response [53]. We write the MIMOOFDM channel matrix H_{ u }[n] as a function of the estimated matrix \(\hat {\mathbf {H}}_{u}[n]\) as:
with \(\mathbb {E}\left (\text {vec}(\mathbf {E}_{u}[n])\,\text {vec}(\mathbf {E}_{u}[n])^{\mathrm {H}}\right) = \sigma _{e,u}^{2} \mathbf {I}_{N_{r}N_{t}}\). If we assume that downlink CSI pilot signals are provided relatively sparse in the timefrequency domain, to keep the pilot signal overhead small, the channel estimation error variance \(\sigma _{e,u}^{2}\) will be comparatively large.
In order to attain more robust behavior w.r.t. channel estimation errors and outdated CSIT, we propose to employ an overcomplete basis decomposition of \(\hat {\mathbf {H}}_{u}[n]\) to obtain a sparse approximation of H_{ u }[n].^{Footnote 1} Choosing a proper quantization codebook/dictionary \(\mathcal {D}\) for the decomposition, we can achieve a signal denoising gain that provides the wanted robustness. More specifically, we apply the following decomposition:
with \(\mathbf {d}_{u}^{s}[n] \in \mathcal {D}\) representing the sth selected entry/atom from the dictionary \(\mathcal {D}\) and \(\mathbf {c}_{u}^{s}[n] \in \mathbb {C}^{N_{r} \times 1}\) being the corresponding expansion coefficient vector. Matrix \(\mathbf {R}_{u}[n] \in \mathbb {C}^{N_{r} \times N_{t}}\) is the residual decomposition error. The dictionary \(\mathcal {D} = \{\mathbf {d}_{1},\ldots,\mathbf {d}_{\mathcal {D}}\},\, \mathbf {d}_{i} \in \mathbb {C}^{N_{t} \times 1}, \left \mathcal {D}\right  \geq N_{t}\) is chosen to span the entire N_{ t }dimensional complex Euclidean space, in order to be able to represent in principle any matrix errorfree with sufficiently many expansion coefficients. Our goal is to obtain a good sparse approximation of \(\hat {\mathbf {H}}_{u}[n]\) with a very small number S of expansion terms by means of a dictionary that supports easy parametrization and therefore efficient CSI quantization/feedback. This can be achieved if the channel matrix is compressible in the chosen dictionary [54], which basically means that the norm of the expansion coefficient vectors \(\left \\mathbf {c}_{u}^{s}[n]\right \\) decreases exponentially with the index s (assuming sorting in decreasing order).
If we assume transmission under LOS conditions, the channel impulse response \(\tilde {\mathbf {H}}(t,\tau)\) is dominated by the LOS specular component. This dominant component is then also present in the Fourier transform H_{ u }[n]. Correspondingly, we propose to utilize a dictionary that is matched to the transmit antenna array response vector a_{ t }(Ψ), since this will result in a compressible decomposition of the channel matrix.^{Footnote 2} We therefore propose to set the dictionary as:
With \(\mathcal {D} = N_{\phi } N_{\theta } \geq N_{t}\), this dictionary spans the N_{ t }dimensional complex Euclidean space and can thus be used to decompose in principle any matrix; however, for a general unstructured matrix, such decomposition will not be compressible. Notice, as the codebook is matched to the transmit antenna array geometry, the users must be aware of this geometry which requires additional downlink signaling. However, since mainly UPAs are employed in practice, we expect this to be a minor issue. The size of the quantization codebook in the azimuth and elevation domains N_{ ϕ } and N_{ θ } should be chosen according to the spatial resolution of the antenna array, i.e., with a larger number of antenna elements in the horizontal/vertical dimension also the resolution of the quantization codebook in the corresponding angular domain should be increased. In order to be able to closely match the angular directions of the dominant specular components, we employ a dictionary with \(\left \mathcal {D}\right  \gg N_{t}\) in our simulations.
To obtain a sparse channel decomposition, we consider the following optimization problem:
We determine an approximate solution of this optimization problem by applying a variation of the orthogonal matching pursuit (OMP) algorithm [55], as detailed in Algorithm 1. Notice, to approximately solve (11), we would actually have to adapt the number S of expansion terms in Algorithm 1 according to the desired approximation accuracy ε^{2}; however, in our simulations, we employ a fixed S. Notice the objective function in (11) can also be written as an ℓ1norm ∥[∥c_{1}∥,…,∥c_{ S }∥]∥_{1} justifying its use for sparse decomposition. Given the sparse approximation, the CSI feedback is set equal to the corresponding codebook indices of the optimal azimuth and elevation angles \(\left (\phi _{u}^{s}[n],\theta _{u}^{s}[n]\right)\) of the expansion vectors \(\mathbf {d}_{u}^{s}[n]\). Additionally, we provide the norms of the expansion coefficient vectors \(\left \\mathbf {c}_{u}^{s}[n]\right \\) as feedback information; the reason for this will become clear in Section 4. In our simulations in Section 6, we consider unquantized feedback of the expansion coefficient vector norms; we have investigated the impact of quantization of these norms for N_{ r }=1 in [39], showing that coarse quantization is possible without significant performance degradation.
Leakagebounded angular beamforming and precoding
In this section, we propose multiuser beamforming and precoding methods that utilize the angulardomain CSI feedback described in the previous section. We consider first multiuser beamforming with a single stream per user in Section 4.1 and extend to multiple streams per user in Section 4.2.
Leakagebounded 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 \(\mathbf {d}_{u}^{s}[nm]\) and the norm of the corresponding expansion coefficient vectors \(\left \\mathbf {c}_{u}^{s}[nm]\right \\), where m represent the delay of the CSI feedback path. To calculate the transmit beamformers, we interpret the expansion vectors \(\mathbf {d}_{u}^{s}[nm]\) 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:
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 phaseshift \(\varphi _{u}^{k,s}\) between these expansion coefficient vectors. Since we do not provide feedback information about these relative phaseshifts, 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:
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:
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:
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 nonconvex and nondeterministic polynomialtime (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 rankone 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 FDMIMO 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:
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 secondorder cone program:
We denote the optimal solution of this optimization problem as the nonrobust leakagebounded angular beamformer. As this is a secondorder 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 interuser 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., \(2 \left \phi _{u}^{s}[nm]  \phi _{u}^{s}[n2m]\right \) and \(2 \left \theta _{u}^{s}[nm]  \theta _{u}^{s}[n2m]\right \), 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 sidelobes 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 inbetween 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 leakagebounded angular beamformer:
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 secondorder cone program, similar to Problem (P3).
Leakagebounded angular precoding
In this section, we extend the robust beamformer design to multiuser precoding with multiple streams per user. When considering multistream 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 multistream 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:
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 singlestream beamforming, it has been shown in [62, 63] that such an approach can still achieve the maximal sumrate, provided the leakage constraints are appropriately selected. When optimizing R_{ u } w.r.t. F_{ u }, we can then focus on
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 angulardomain 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 \(\hat {\tilde {R}}_{u}\):
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 nonrobust leakagebounded angular precoder:
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 signaltonoise 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 nonzero 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 leakagebounded angular precoder by accounting for the angular uncertainty regions \(T_{j}^{k}\):
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 uth 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 leakagebounded transmit optimization as proposed in (P1) is in general an NPhard 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 FDMIMO antenna array. This complexity issue is relaxed by problem (P2) and its convex secondorder cone programming reformulation (P3), in which the dimension of the optimization variable f_{ u } grows only linearly with the size of the FDMIMO 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 secondorder 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 worstcase complexity of a secondorder cone program scales with the square root of the number of constraints [60, 61], the increase in complexity is moderate. Considering the two multistream optimization problems (P5) and (P6), they both admit solution by means of an interior point method or a Newton conjugategradient approach as proposed in [66]. However, as our computer experiments have shown, this approach can be computationally demanding and slow. Therefore, the perstream 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 FDMIMO systems is currently devoted to reducing system complexity through socalled 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 phaseshifter 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 leakagebounded precoding optimization problem proposed in this paper. Nevertheless, we consider an extension of our leakagebounded precoding optimization to hybrid architectures as promising future work.
Specifically, our leakagebounded 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 unitmodulus solution. It is wellknown that the unit modulus precoder constraint of RF precoding is a nonconvex 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 phaseshifters it is actually possible to avoid the unitmodulus 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 frequencydependent 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.
Benchmark methods
In this section, we summarize several wellknown beamforming and precoding techniques that we employ as benchmarks for our proposed methods. We provide this section as a reference for the interested reader as well as to clarify how these benchmark methods are applied in our considered setup. Additionally, we propose in Section 5.2.4 a modification of the popular signal to leakage and noise ratio SLNR metric to enable its application with the angulardomain CSI feedback proposed in this paper.
Interferenceunaware beamsteering
The following methods do not explicitly account for multiuser interference in the beamformer design. This methods are pure angleofdeparturebased beamsteering techniques, which can be viewed as typical SDMA implementations. As is well known, the performance of SDMA strongly depends on the selection of coscheduled users [68], i.e., the transmitter should coschedule users that experience close to orthogonal channels to avoid extensive multiuser interference. In our simulations in Section 6.2, we avoid the impact of imperfect user scheduling by employing an exhaustive search over all possibilities of coscheduled users.
Maxgain beamsteering
The simplest beamforming/precoding technique is to steer the transmit signal along the L_{ u } strongest angular directions provided by the CSI feedback:
This technique does provide large signal gain; yet, its major weakness is that it does not account for interference among users. Hence, it only works properly if the base station serves users that are a priori well separated in the angular domain.
Chebyshevtapered beamsteering
Maxgain beamsteering, as considered above, in general, produces large side lobe levels and thus potentially strong interuser interference. Such large side lobe levels can be reduced by array weight tapering methods [69]. A popular weighting technique is Chebyshev weighting, which achieves the smallest main lobe width for a given side lobe level [70].
In order to steer the main lobe in an intended direction, array tapering is combined with the beamsteering approach of the previous section; i.e., the actual beamformer is obtained as the product of the array taper and the maxgain beamsteering vectors.
Interferenceaware precoding
The following methods take multiuser interference into account and attempt to mitigate it.
Blockdiagonalization precoding
BD precoding enables interuser interferencefree transmission of a total number of at most N_{ t } data streams [29]. This is achieved by steering the transmit signal of a user into the nullspace of the channel matrices of all other users. In our simulations in Section 6, we do not employ BD precoding in combination with our CSI feedback methods, but we rather assume perfect knowledge of H_{ u } or its estimate \(\hat {\mathbf {H}}_{u}\). We do this since BD precoding requires phase information which is not provided by our CSI feedback; see, e.g., [33], for an efficient CSI feedback method that can be applied with BD precoding.
Angle of departure blockdiagonalization precoding
BD precoding is very sensitive to CSI imperfections since it relies on destructive multipath interference to eliminate multiuser interference [71–73]. It also has heavy CSI requirements as it requires knowledge of the entire channel matrix H_{ u } of each user. To reduce the CSI requirements, the authors of [74] propose to calculate the precoder not for the entire channel matrix H_{ u } but rather only for the dominant propagation paths between transmitter and receiver. Specifically, the authors of [74] consider single stream transmission per user and propose to calculate the corresponding ZF beamformer for the single strongest multipath component of each coscheduled user. Rather than decomposing the frequency response channel matrix as we do in the present paper, the authors of [74] directly consider the time domain channel impulse response (4) and select the strongest multipath component according to the magnitude of α_{ ℓ }; the multiuser ZF beamformer is then calculated from the corresponding transmit antenna array response vectors a_{ t }(Ψ_{t,ℓ,u}). This can be extended to multistream transmission per user by means of BD precoding in a straightforward way, by calculating the BD precoder from the L_{ u } transmit antenna array response vectors corresponding to the angles of departure of the L_{ u } strongest multipath components.
Hybrid blockdiagonalization precoding
The angle of departure BD precoding approach described above does not achieve full multiuser interference cancelation, since it mitigates interference only over the strongest multipath component of each user. The residual multiuser interference leads to a significant performance loss compared to regular BD precoding in case of perfect CSI. To reduce this performance gap, while still keeping CSI requirements at a reasonable level, the authors of [74] propose hybrid BD precoding. Here, the hybrid precoding principle is applied to first reduce the dimension of the effective base band channel matrix of each user, by employing a RF domain precoder. This RF domain precoder \(\mathbf {F}_{\text {RF}} \in \mathbb {C}^{N_{t} \times \sum _{u \in \mathcal {S}[n]} L_{u}}\) is obtained as the matched filter to the L_{ u } strongest multipath components of each scheduled user. The base band BD precoder is then calculated for the product of channel matrix and RF domain precoder, which is called the effective base band channel. CSI requirements for estimating this effective base band channel are reduced compared to the full channel matrix, since the dimension of the effective channel matrix is smaller.
Incoherent SLNR precoding
SLNR beamforming is a popular coherent beamforming technique for multiuser broadcast and interference channels [75]. If full CSI is available at the base station, that is the full channel matrix H_{ u } is known for each user u, the SLNR beamformer is obtained by maximizing a corresponding Rayleigh quotient. If we apply the CSI feedback of Section 3, however, the transmitter only has knowledge of the channel decomposition expansion vectors \(\mathbf {d}_{u}^{s}\) and the norm of their corresponding expansion coefficients \(\left \\mathbf {c}_{u}^{s}\right \\). Hence, the transmitter does not know whether the multipath transmissions over the different specular components \(\mathbf {d}_{u}^{s}\) add up constructively or destructively, since it has no phase information available. We therefore modify the SLNR metric to consider the incoherent ratio of signal and leakage powers, utilizing the delayed angular CSI available at the transmitter:
If we restrict to semiunitary precoding with equal power allocation, such that \({\mathbf {F}_{u}^{\mathrm {H}}} \mathbf {F}_{u} = \frac {1}{L_{u} \left \mathcal {S}\right } \mathbf {I}_{L_{u}}\), we can pull the precoder in the denominator of (28) left and right out of the sum, taking into account that \(\text {tr }({\mathbf {F}_{u}^{\mathrm {H}}} \mathbf {F}_{u}) = 1/\left \mathcal {S}\right \). This leads to a generalized Rayleigh quotient that is maximized by the following semiunitary precoder:
where the operator \(\max \text {eig vec}_{L_{u}}\) calculates the L_{ u } orthonormal eigenvectors corresponding to the largest eigenvalues.
Simulations
In this section, we evaluate the performance of the proposed beamforming and precoding methods by means of numerical simulations. We first provide in Section 6.1 an example of the radiation beam pattern obtained by leakagebounded angular beamforming with and without robustness constraints. This facilitates the development of a basic understanding of the behavior of the beamforming solutions produced by the proposed optimization problem, specifically with respect to the influence of the robustness constraints. In Section 6.2, we compare the achievable transmission rate of the proposed leakagebounded angular beamforming methods to the classical interferenceunaware beamsteering methods summarized in Section 5.1. In these simulations we consider a simple Ricianfading channel model, which allows to easily control the compressibility of the CSI in the angular domain via the Rician Kfactor. In Section 6.3, we then utilize the more realistic QUADRIGA channel model [76] to compare the proposed leakagebounded angular precoding methods to the interferenceaware precoding methods summarized in Section 5.2^{Footnote 4}. We first of all evaluate the compressibility of the channel realizations obtained from QUADRIGA in the angular domain. Based on these results, we then investigate the performance of the different precoding methods in terms of the achievable transmission rate. We furthermore simulate the SER of uncoded transmission as a function of the variance of the channel estimation error \(\sigma _{e}^{2}\) in (8), demonstrating the robustness of the proposed leakagebounded beamformer. To find solutions to the proposed optimization problems, we apply CVX [77].
Radiation beam pattern
In our first example, we illustrate the radiation beam pattern obtained by leakagebounded angular beamforming with and without robustness constraints. We consider singlestream beamforming and assume a UPA of isotropic antenna elements at the transmitter of size N_{ t }=N_{ h }·N_{ v }=11·11=121. We assume that U=10 users are randomly placed within an angular area stretching in azimuth ϕ from 60° to 60° and in elevation θ from 70° to 110°^{Footnote 5}. In this example, we consider only a single specular component S=1 per user; in the robust leakagebounded optimization problem, we additionally assume an angular uncertainty region of width ψ_{max}=± 1.6° in azimuth and elevation around each specular component. This uncertainty represents, for example, LOS transmission to users that move on a radius of distance d=50 m from the base station with v=100 km/h when highresolution CSI feedback is provided every τ=50 ms (\(\psi _{\text {max}} = \frac {v \tau }{d})\).
In Fig. 2, we compare the radiation beam patterns of robust and nonrobust leakagebounded angular beamforming for a user that is located at approximately (ϕ,θ)=(22°,95°) (indicated with the green circle) and nine leakage areas corresponding to the other U−1 users (indicated in black). In this example, we set the maximum tolerable interference level to L_{max}=−40 dB relative to the unit transmit power. In Fig. 2 (left), we observe that the radiation pattern of nonrobust leakagebounded angular beamforming shows very narrow valleys wherever interference leakage constraints are active. Therefore, the transmission to the intended user does not cause substantial interference to the other users, provided the angular CSIT is accurate. However, due to the narrow width of the valleys at the interference leakage constraints, the residual interuser inference grows significantly if the angular CSIT is not accurate. To compensate for such angular uncertainty, we consider interference leakage uncertainty regions of width ψ_{max} in the robust leakagebounded angular beamformer. As shown in Fig. 2 (right), the interference leakage is then low over the entire uncertainty regions, thus improving the robustness with respect to imperfect angular CSIT. However, the maximum achievable gain of the intended user is slightly reduced, since it is not possible any more to perfectly steer the signal power towards the intended user while still satisfying all leakage constraints. Therefore, there exists a tradeoff between the achievable signal gain of the intended user and the robustness of the interference leakage constraints.
Comparison to interferenceunaware beamsteering
In our second example, we investigate multiuser beamforming with a single stream per user, comparing the interferenceunaware angle of departurebased SDMA beamsteering methods of Section 5.1 and our robust/nonrobust leakagebounded angular beamformers. We consider users with a single receive antenna N_{ r }=1 and a transmitter with a UPA of size N_{ t }=N_{ h }·N_{ v }=11·11. We employ a Ricianfading channel model to generate the channel vector \(\mathbf {h}_{u} \in \mathbb {C}^{1 \times N_{t}}\) according to:
with \(\beta _{u} \sim \mathcal {U}\left (\pi,\pi \right)\) denoting a random phaseshift, \(\phi _{u} \sim \mathcal {U}\)(60°,60°) and \(\theta _{u} \sim \mathcal {U}\)(70°,110°) representing the LOS directions, and \(\mathbf {h}_{u,d} \sim \mathcal {CN}\left (\mathbf {0},\mathbf {I}\right)\) being a diffuse background scattering component. The Rician Kfactor controls the relative strength of the LOS component. We do not utilize OMP in this example to decompose the channel vector, but we rather assume that the receivers employ a dictionary that is matched to the UPA and that the receivers are able to estimate the LOS angles with an accuracy of ±2°. Hence, the CSIT utilized for beamforming is \(\mathbf {a}_{t}^{\textrm {UPA}}(\hat {\phi }_{u},\hat {\theta }_{u})\) with \(\hat {\phi }_{u} \sim \mathcal {U}\left (\phi _{u} 2^{\circ },\phi _{u} + 2^{\circ }\right)\) and \(\hat {\theta }_{u} \sim \mathcal {U}\left (\theta _{u} 2^{\circ },\theta _{u} + 2^{\circ }\right)\) corresponding to a dictionary with 4^{∘} angular resolution. In our simulations, we consider a total number of U=10 users; out of these ten users we select the best subset \(\mathcal {S} \subseteq \{1,\ldots,U\}\) of users through exhaustive search with varying size \(\left \mathcal {S}\right  \in [1,U]\) to avoid an impact of imperfect user scheduling. We set the noise variance equal to \(\sigma _{n,u}^{2} = 10^{3}\). For Chebyshevtapered beamsteering, we set the sidelobe level equal to the maximum of noise power and residual interference caused by the diffuse component:
such that the multiuser interference caused by the sidelobes is in the order of the maximum of diffuse background interference and noise power.
In Fig. 3, we show the achievable rate performance of the interferenceunaware beamsteering methods and our robust/nonrobust leakagebounded angular beamformers for varying Kfactors. The performance of the interferenceunaware beamsteering methods (max. gain and Cheb. tapered beamsteering) is plotted for the case of perfect angular CSIT; however, their performance with imperfect CSIT of accuracy ± 2^{∘} is virtually the same. The performance of leakagebounded beamforming is shown for perfect angular CSIT as well as for imperfect angular CSIT with robust/nonrobust interference leakage constraints. We observe that our leakagebounded beamforming method can achieve better performance and can serve more users in parallel than the interferenceunaware beamsteering schemes (max. gain and Cheb. tapered beamsteering). With decreasing Rician Kfactor (left to right) the performance deteriorates since the residual interference over the diffuse background scattering component is not mitigated by our beamformer optimization. Still, even with the relatively low value of K=10 our proposed method outperforms the interferenceunaware beamsteering schemes. We furthermore observe in Fig. 3 that the performance loss due to imperfect angular CSIT can partly be compensated with our robust beamforming method as compared to the nonrobust approach.
Comparison to interferenceaware precoding
For the remaining simulations, we utilize the QUADRIGA channel model to test our proposed methods on more realistic channel realizations. We summarize the most important simulation parameters in Table 1, which have been selected to be LTE standard compliant. We consider users that are uniformly distributed within a sector of 120° with a maximum distance of 500 m from the transmitter and we assume that the users move perpendicular to the transmit antenna array boresight direction with 150 km/h. For such user placement, the variation of the elevation angle is in most cases very small and we therefore consider beamforming only in the azimuth domain utilizing a highresolution horizontal uniform linear array (ULA) of 50 antenna elements. In our simulations, we do not consider macroscopic pathloss even though it is considered in the output of the QUADRIGA channel model. We therefore normalize the channel matrices obtained from the QUADRIGA channel model to satisfy \(\mathbb {E}\left (\left \\mathbf {H}_{u}[n]\right \{~}^{2}\right) = N_{r} N_{t}\phantom {\dot {i}\!}\). We apply this normalization since we do not consider multiuser scheduling in this section, which would be necessary in case of strong received signal power disparities of different users. We employ a dictionary/quantization codebook that is matched to the horizontal ULA and we consider an azimuthal angular resolution for CSI quantization of 1°^{Footnote 6}. We consider a blockfading channel model with constant channel during each transmission time interval (TTI).
In our first simulation of this section, we investigate the compressibility of the persubcarrier channel matrices H_{ u }[n] in the angular domain. That is, we apply the OMP algorithm to decompose H_{ u }[n] in the angular domain using a dictionary that is matched to the employed ULA and we determine the average norm of the expansion coefficient vectors \(\left \\mathbf {c}_{u}^{s}[n]\right \^{2}\) of the decomposition. The corresponding results are shown in Fig. 4 for different simulation scenarios as specified in the QUADRIGA channel model. We observe that in the LOS situations, the first expansion coefficient contains the majority of the channel energy, especially in the 3GPP compliant scenarios; this expansion coefficient corresponds to the LOS direction. The second largest coefficient lies already approximately an order of magnitude below the first coefficient. In the nonline of sight (NLOS) scenarios, however, the channel energy distributes more equally over many expansion coefficients. Correspondingly, compressibility with the employed dictionary is worse. In the remaining simulations we therefore consider LOS transmission and employ the 3GPP compliant UMa scenario. We also conducted simulations in NLOS situations, where our proposed method however does not achieve a significant gain over the interferenceaware benchmark precoders.
In the next simulation, we compare the achievable transmission rate of the interferenceaware precoding schemes summarized in Section 5.2 to our robust leakagebounded precoding method. We consider transmission to U=5 users in parallel with L_{ u }=2 streams per user; hence, the total number of spatially multiplexed data streams is ten. We have seen above that in the LOS scenario most channel energy is contained in a single specular component. To efficiently support transmission of two spatial streams per user, we therefore employ a distributed antenna system (DAS) composed of two remote radio heads (RRHs) that are separated by 25 m. Each RRH radiates the same transmit signal; that is, the radio heads are fed with the same RF signal, e.g., via radiooverfiber connections. Correspondingly, the two RRHs do not appear as independently steerable antenna arrays since they are fed with the same transmit signal; hence, the effective number of actively steerable transmit antennas (and thus the size of the channel matrix that must be estimated) is still N_{ t }=50. In that way, we guarantee that each user observes two strong specular components of approximately equal strength corresponding to the two antenna arrays of the RRHs.
We assume that the receivers estimate and feed back the CSI only at the first TTI, to enable calculation of the precoders; these precoder are then utilized for the entire transmission duration of 150 TTIs. For BD precoding and hybrid BD precoding, we assume perfect unquantized CSIT of the (effective) base band channel at the first TTI. For angle of departure BD precoding, we assume perfect knowledge of the LOS angles w.r.t. the two RRHs and hence perfect knowledge of the corresponding transmit antenna array response vectors. For the other schemes, we utilize our OMPbased channel decomposition to determine the S=2 strongest specular components, in order to facilitate transmission of two spatial streams. We have observed that the performance does not improve if we consider S>L_{ u } expansion coefficients; when S is too large, the performance rather deteriorates since too many interference leakage constraints are active. For BD precoding and its hybrid variant, we assume perfect base band CSIT on each of the 600 subcarriers within the first TTI. For angular BD precoding, only broadband feedback of the angles of departure of the strongest multipath components needs to be provided, since this CSI is obtained from the time domain channel impulse response; notice, though, that this either requires estimation of the time domain channel impulse response or direction estimation of the dominant angles from uplink training signals [74]. For the remaining schemes, we consider a CSI feedback resolution of 60 subcarriers, i.e., the S=2 strongest specular components are provided for every 60th subcarrier. Therefore, to estimate the full channel matrix at the receiver for CSI feedback calculation, we require a total of 600/60·N_{ t }=500 orthogonal pilotsymbols within the first TTI, amounting to an overhead of 500/(14·600)≈6 % within the first TTI. However, since we provide CSI feedback only every 150th TTI the pilot overhead for CSI feedback calculation is negligible.
For our robust leakagebounded precoders, we determine the size of the angular uncertainty regions to match the uncertainty of the LOS direction caused by the user movement. To avoid unnecessarily large uncertainty regions, we recalculate the robust leakagebounded precoders every 25 TTIs with correspondingly increasing angular uncertainty regions. Notice, in this recalculation, only the size of the uncertainty regions is updated and not the angular CSIT itself; it does therefore not require renewed CSI feedback from the users. In Fig. 5, we show the achievable rate performance of the considered precoding schemes. We observe that regular BD precoding as well as hybrid BD precoding achieve very high data rate in the first TTI since at this time the CSIT is perfect and thus interferencefree transmission is achieved; notice, we assume a noise variance of \(\sigma _{n,u}^{2} = 10^{2}\) in this simulation. However, the performance of BD precoding and its hybrid variant deteriorates very quickly over time due to increasing residual interference caused by growing precoder mismatch, since the base band channel varies quickly over time. Our proposed leakagebounded precoding scheme cannot compete with BD precoding in the first TTI. This is because it does not achieve interferencefree transmission, since there is always residual interference caused by the channel decomposition error. Even with S very large and therefore a small channel decomposition error, our method does not achieve the multiplexing capabilities of BD precoding because it is an incoherent scheme that does not make use of coherent multipath interference, as BD precoding does to eliminate multiuser interference. Yet, our method is much less sensitive to the temporal variation of the channel matrices, since the employed angular channel decomposition is comparatively stable over time. This means that the angular directions of the largest channel expansion coefficients change only slowly over time and therefore the leakagebased precoding solution performs well even after many TTIs. Notice, the incoherent SLNR precoder as well as the angle of departure BD precoder achieve virtually the same performance as our scheme at the beginning of the transmission; however, their performance deteriorates faster because they do not consider angular uncertainty regions. Nevertheless, both methods provide a reasonably good tradeoff between robustness and computational complexity. We did attempt to incorporate angular uncertainty regions in the SLNR metric, but we were not successful. In Fig. 5, we also compare the performance of leakagebounded precoding utilizing the determinant optimization approach and the per stream optimization approach described in Section 4.2. We observe that per stream optimization exhibits only a small rate degradation compared to determinant optimization and therefore might be practically more relevant due to lower computational complexity.
In our final example, we investigate the sensitivity of leakagebounded beamforming and BD precoding with respect to channel estimation errors. For this purpose, we distort the output H_{ u }[n] of the QUADRIGA channel model to emulate a channel estimation error:
with \(\left [\mathbf {E}_{u}[n]\right ]_{i,j} \sim \mathcal {CN}(0,1)\) denoting the normalized channel estimation error and \(\sigma _{e}^{2}\) controlling its relative strength, such that \(\mathbb {E}\left (\left \\hat {\mathbf {H}}_{u}[n]\right \^{2}\right) = N_{r} N_{t}\)^{Footnote 7}. We consider uncoded 16 QAM transmission to a varying number of users U∈{3,5,7} with L_{ u }=1 stream per user. We use highresolution angular quantization of a single S=1 specular component, such that the CSI uncertainty of this single expansion vector is only due to the channel estimation error.
In Fig. 6, we show the obtained SER as a function of the channel estimation error variance \(\sigma _{e}^{2}\). We observe that leakagebounded beamforming exhibits an SER floor due to residual interference caused by the channel decomposition error, which grows with the number of users served in parallel. Yet, the performance of the method is hardly impacted by the channel estimation error; this robustness is achieved by the signal denoising properties of the employed angular basis decomposition. Only for very large channel estimation errors the decomposition fails due to the growing impact of E_{ u }[n]. BD precoding, on the other hand, is much more sensitive w.r.t. channel estimation errors, since such errors directly cause residual interuser interference; yet, for small error variance it outperforms our proposed method. We want to stress that the performance shown in (8) is for uncoded transmission; with channel coding, the SER can of course be substantially reduced.
Conclusions
Multiuser beamforming and precoding approaches are in general very sensitive to CSI imperfections at the transmitter. Most existing approaches are not efficiently applicable in highmobility situations, since channel matrices change so quickly over time that already small processing delays can lead to substantial interuser interference. In this paper, we provide a remedy to this problem by considering noncoherent beamforming/precoding in the angular domain with robustness w.r.t. angular uncertainty. In addition to improving the robustness of the beamformer/precoder, this has the further advantage that only a minimal amount of CSIT is required, namely the angles of departure of the most significant specular components together with their respective gains as obtained from a lowrank basis decomposition. We show that the relevant CSI can efficiently be quantized by applying a variant of the OMP algorithm. Naturally, under ideal circumstances of perfect CSIT, such a noncoherent approach cannot compete with coherent methods in terms of the achievable multiplexing gain. Also, it is not well suited for strong scattering environments with a large number of significant specular components because the lowrank channel decomposition has a large residual error in such cases. Nevertheless, the proposed technique can provide substantial improvements in situations with few dominant specular components, as for example expected in the millimeter wave regime, and it enables robust multiuser beamforming/precoding with low requirements on CSIT accuracy.
Appendix
Upper bound on the approximation \(\hat {\tilde {R}}_{u}\)
In this appendix we derive an upper bound on the approximation \(\hat {\tilde {R}}_{u}\) of (24) that can be calculated from the provided CSI feedback. We start by applying Sylvester’s determinant theorem and Jensen’s inequality to (24):
Let us next consider the approximation of a single interference term in detail:
with \(\mathbf {C}_{j} = \mathbf {F}_{j} {\mathbf {F}_{j}^{\mathrm {H}}}\) being the transmit covariance matrix. In this equation, we again utilized the phase independence of different specular components as in the steps from (19) to (20). We next apply an eigendecomposition to the sum of the specular components:
where we assume that S≤N_{ t } as our focus is on largescale antenna arrays and lowrank channel decomposition.
Let us next consider the interference leakage power caused by the transmission to user j w.r.t. all specular components of user u:
In the precoder optimization problem (P5), we restrict this interference leakage to satisfy \(L_{u,j}^{s} \leq L_{\text {max}}\) and hence L_{u,j}≤SL_{max}. From this upper bound on the interference leakage power, we can determine an upper bound on the interference leakage covariance (35). For that purpose, we decompose the precoding matrix in terms of the S dimensional basis U_{ u } and a basis \(\mathbf {U}_{u}^{\perp }\) of its N_{ t }−S dimensional orthogonal complement:
With this, we can reformulate the leakage power as:
We next assume \(\mathbf {F}_{j}^{\parallel } \in \mathbb {C}^{S \times L_{j}}\) to be isotropically distributed, such that in combination with (41) we get
Utilizing these results in (35), we obtain:
where ≼ means that the difference between the righthand side and lefthand side is positive semidefinite. This brings us to the desired upper bound on the approximation \(\hat {\tilde {R}}_{u}\):
where we again applied Sylvester’s determinant theorem.
Notes
 1.
A basis of a vector space is said to be overcomplete if it is complete even after the removal of a vector from the basis.
 2.
A related lowrank basis expansion model has recently also been investigated in [78]; however, in [78] the authors consider an orthogonal basis expansion model, whereas we consider an overcomplete basis set containing far more entries \(\left \mathcal {D}\right \) than the dimension N_{ t } of the vector space.
 3.
We investigate a related maxmin problem that accounts for uncertainty in the signal direction in [39]; it can only be approximately solved via SDR.
 4.
The simulation results were generated using QuaDRiGa Version 1.4.8571.
 5.
Notice, the antenna boresight direction is (ϕ,θ)= (0°,90°).
 6.
Notice, for multiuser beamforming with a single stream per user we investigate the impact of the angular CSI resolution in [40] and demonstrate that an angular resolution of 4° achieves close to optimal performance for a UPA of size 10·10. Since the spatial resolution of the 50 element ULA considered here is better, we increase the angular CSI resolution to 1°.
 7.
The definition of \(\sigma _{e}^{2}\) here is slightly different as compared to (8), in order to keep the energy of the estimated channel constant irrespective of \(\sigma _{e}^{2}\).
Abbreviations
 3GPP:

Third generation partnership project
 AD:

Analog to digital
 BD:

Block diagonalization
 BS:

Base station
 CoMP:

Coordinated multipoint transmission
 CSI:

Channel state information
 CSIT:

Channel state information at the transmitter
 DAS:

Distributed antenna system
 DA:

Digital to analog
 DoF:

Degrees of freedom
 FD:

Fulldimension
 FDD:

Frequency division duplex
 FDMIMO:

Fulldimension multipleinput multipleoutput
 FRT:

Fluctuating tworay
 GRT:

Generalized tworay
 IA:

Interference alignment
 i.i.d.:

Independent and identically distributed
 ITS:

Intelligent transport systems
 LOS:

Line of sight
 LTE:

Long term evolution
 MBMS:

Multimedia broadcast multicast service
 eMBMS:

Evolved MBMS
 MCIC:

Multicast interference channel
 MIMO:

Multipleinput multipleoutput
 MISO:

Multipleinput singleoutput
 NLOS:

Nonline of sight
 NP:

Nondeterministic polynomialtime
 NR:

New radio
 OFDM:

Orthogonal frequency division multiplexing
 OMP:

Orthogonal matching pursuit
 QAM:

Quadrature amplitude modulation
 RF:

Radio frequency
 RRH:

Remote radio head
 SDMA:

Spacedivision multiple access
 SDR:

Semidefinite programming relaxation
 SER:

Symbol error ratio
 SINR:

Signal to interference and noise ratio
 SLNR:

Signal to leakage and noise ratio
 SLR:

Signal to leakage ratio
 SNR:

Signal to noise ratio
 SVD:

Singular value decomposition
 TDD:

Time division duplex
 TTI:

Transmission time interval
 TWDP:

Twowave diffuse power
 UCIC:

Unicast interference channel
 UE:

User equipment
 UMTS:

Universal mobile telecommunications system
 ULA:

Uniform linear array
 UPA:

Uniform planar array
 ZF:

Zero forcing
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Acknowledgements
The financial support by the Austrian Federal Ministry of Science, Research and Economy and the National Foundation for Research, Technology and Development is gratefully acknowledged. The author acknowledges the TU Wien University Library for the financial support through its Open Access Funding Program.
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Keywords
 Fulldimension MIMO
 3D beamforming
 Robust precoding
 Interference leakage
 High mobility
 Limited feedback