A novel approach for beamforming based on adaptive combinations of vector projections

Digital Signal Processing 97 (2020) 102621

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Digital Signal Processing www.elsevier.com/locate/dsp

A novel approach for beamforming based on adaptive combinations of vector projections Ciro André Pitz a , Eduardo Luiz Ortiz Batista a , Rui Seara a,∗ , Dennis R. Morgan b,1 a

LINSE – Circuits and Signal Processing Laboratory, Department of Electrical and Electronics Engineering, Federal University of Santa Catarina, Florianópolis, Santa Catarina, 88040-900, Brazil b Morristown, NJ, USA

a r t i c l e

i n f o

Article history: Available online 25 November 2019 Keywords: Adaptive antenna arrays Beamforming Interference suppression Mobile communications Stochastic gradient method

a b s t r a c t Multi-antenna systems have emerged as a key technology to meet the growing demand for capacity in mobile communications. The spatial filtering capability of these systems can be exploited to enhance the signal-to-interference-plus-noise ratio (SINR) in wireless communication channels, allowing to reduce transmission power and increase data rates. However, this is not an easy task due to computational and spatial-selectivity limitations, requiring the use of effective beamforming algorithms to provide adequate balance between signal-of-interest maximization and interference minimization. In this context, a new framework for developing beamforming algorithms is proposed in this paper. Such a framework, termed adaptive combination of vector projections (ACVP), is derived from a geometric analysis of stochastic algorithms and is based on a linear combination of vectors belonging to the subspaces spanned by signals available at the array input. The proposed framework is used to devise a new beamforming algorithm, which applies a sigmoid function along with the stochastic gradient method for dynamically adjusting the linear combination of vector projections. The resulting algorithm, named sigmoid-based ACVP algorithm, exhibits low computational burden and provides higher SINR levels than competing techniques from the open literature. Numerical simulation results are shown aiming to confirm the effectiveness of the proposed approach. © 2019 Elsevier Inc. All rights reserved.

1. Introduction The growing demand for spectrum and energy efficiency in mobile communications has motivated a continuous research effort towards improving system capacity while keeping power consumption at low levels. In this context, adaptive beamforming has shown promise, since the real-time spatial filtering capability provided by this technique can be used for enhancing the signal-tointerference-plus-noise ratio (SINR) in both uplink and downlink channels [1–3]. Such an enhancement, in turn, allows improving system capacity by means of reducing frequency reuse [4], as well as through more efficient modulation and coding schemes [5,6]. Moreover, the enhanced SINR levels obtained by using beamforming algorithms can be exploited to develop power control schemes

*

Corresponding author. E-mail addresses: [email protected] (C.A. Pitz), [email protected] (E.L.O. Batista), [email protected] (R. Seara), [email protected] (D.R. Morgan). 1 Dennis R. Morgan was with Bell Laboratories, Alcatel-Lucent, Murray Hill, NJ, USA. https://doi.org/10.1016/j.dsp.2019.102621 1051-2004/© 2019 Elsevier Inc. All rights reserved.

that reduce the energy consumption while maintaining acceptable link quality [7–9]. Beamforming algorithms can be designed to operate in either the uplink or downlink channel of base stations. Some important aspects distinguish uplink beamforming from downlink beamforming. For instance, in the uplink case, the SINR of each user does not depend on the beamformer obtained for the other users (i.e., the SINR of all users are decoupled) [10]. As a consequence, the implementation of uplink beamforming algorithms can be carried out in a distributed manner. In contrast, the downlink beamformer of each user affects the crosstalk experienced by the remaining users [10,11]. Thereby, centralized processing is usually required to fully coordinate the downlink beamforming of multiple base stations [12–16], which entails high-capacity backhaul links [16] and constitutes a single point of failure for the system [12]. To cope with these problems, several approaches have been discussed in the open literature aiming to achieve global optimality without the need for inter-cell communication [12,14–20]. In order to implement downlink beamforming algorithms in either coordinated or distributed fashion, estimation of the channel state information (CSI) is usually required [4,21]. Obtaining this information is not a straightforward task, since it depends on

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C.A. Pitz et al. / Digital Signal Processing 97 (2020) 102621

the downlink signals arriving at the mobile terminals, which are not readily available at the base station. To circumvent this problem in time-division duplexing (TDD) systems, the downlink CSI is generally acquired considering the reciprocity assumption [22–24], which allows obtaining accurate estimates of downlink covariance matrices from uplink measurements. In contrast, frequencydivision duplexing (FDD) systems usually rely on feedback-based approaches [23–25] for obtaining the downlink CSI, particularly due to the uncorrelated nature of the frequency-separated uplink and downlink channels [25]. Thus, uplink pilot (training) signals [26–28], angle-of-arrival estimation [29–31], or feedback schemes [23–25] are required for obtaining the downlink CSI [32,33], implying a significant allocation of system resources specially in nonstationary scenarios. As shown in [34–36], the requirement for pilot signals or angle-of-arrival estimation can be avoided in codedivision-multiple-access (CDMA) systems by using both spread and despread signals to obtain covariance matrices that can be exploited to implement effective beamforming algorithms. Moreover, in non-CDMA systems, the space-time equalization structure discussed in [37] can be used for separating the signal of interest (SOI) from the interferences, which allows the development of beamforming algorithms that do not rely on pilot signals or angleof-arrival estimation. Examples of these algorithms are the constrained stochastic gradient (CSG) and the improved CSG (ICSG) from [17] and [18], respectively. Another example is the adaptiveprojection CSG (AP-CSG) from [38], which allows overcoming implementation challenges present in both the CSG and ICSG algorithms. Such an approach is also exploited in [39] to derive the adaptive-projection quadratically-constrained stochastic gradient (AP-QCSG) algorithm. In this paper, a new framework is discussed for developing adaptive beamforming algorithms for mobile communication systems. Such a framework, named adaptive combination of vector projections (ACVP), is based on a geometric interpretation of CSGtype algorithms [38,39], consisting of a linear combination of vectors belonging to subspaces spanned by the signals from the array input. The proposed ACVP framework paves the way for development of a new family of beamforming algorithms that can be used in mobile communication systems, requiring neither pilot signals nor angle-of-arrival estimation. An algorithm of this family, termed here sigmoid-based ACVP (SB-ACVP), is derived in this paper, representing a first practical outcome of the proposed framework. Numerical simulation results are shown, aiming to validate the proposed ACVP framework as well as to confirm the effectiveness of the proposed SB-ACVP algorithm. The main contributions of this paper can be summarized as follows:

• A new vector-projection-based framework is proposed for developing beamforming algorithms that neither requires estimating the angle-of-arrival of the involved signals nor relies on using pilot signals. • A novel effective adaptive beamforming algorithm is formulated by using the proposed ACVP framework. The remainder of this paper is organized as follows. Section 2 presents the system model and problem formulation considered for developing beamforming algorithms. Section 3 is dedicated to a review of CSG-type algorithms. Section 4 presents the main contributions of this research, namely a unifying view on the behavior of adaptive-projection CSG-type algorithms, the proposed ACVP framework, and the proposed SB-ACVP algorithm. Simulation results are shown in Section 5. Finally, Section 6 presents concluding remarks.

Fig. 1. Uplink scenario in which a SOI (dark line) and interfering signals (dashed lines) arrive at a given base station B i .

2. System model and problem statement The mobile communication scenario considered in this paper consists of M single-antenna mobile terminals (users) that share the same channel (co-channel users) and R multi-antenna base stations. The ith user is denoted T i and the base station assigned to such a user, denoted B i , is equipped with an array of K i antennas used for both transmission and reception. Note that B i and B j may be the same (i.e., B i = B j ) if both the ith and jth users (T i and T j , respectively) are assigned to the same base station. A multipath fading channel model is assumed for both direct and reverse channels (downlink and uplink, respectively), with L m,i representing the number of independent paths between the mth user T m and the station assigned to the ith user B i , giving rise to an angle (azimuth) spread of θm,i around the mean angle-of-arrival θm,i . 2.1. Uplink signal model At the uplink, the signals transmitted by all M users arrive at each one of the R multi-antenna base stations. Thus, for a given base station B i , one has the scenario illustrated in Fig. 1. In this scenario, the baseband input signal can be modeled as a K i -dimensional complex vector, given by

xˆ i (n) =

M 

xˆ m,i (n) + rˆ i (n)

(1)

m =1

with rˆ i (n) representing complex (circular) additive white Gaussian noise (AWGN) with average power σrˆ2 present in the antennas of B i , and

xˆ m,i (n) =

i



ˆ m,i sˆm,i (n) Pˆ m H

(2)

denoting the uplink signal vector related to T m , where the constituents are explained as follows. Variable Pˆ m is the uplink ˆ m,i = [hˆ (θm,i ,1 ) hˆ (θm,i ,2 ) . . . transmission power for T m ; matrix H

ˆ (θm,i , L )] is a K i × Lm,i matrix whose lth column hˆ (θm,i ,l ) deh m ,i notes the K i -dimensional steering vector [40] of the lth multipath signal from T m that arrives at B i with angle-of-arrival θm,i ,l ; and sˆm,i (n) is an L m,i -dimensional vector containing the complex envelope of each multipath signal for the pair T m , B i . Such a complex envelope characterizes normalized and independent fading, which implies H E[ˆsm,i (n)ˆsm ,i (n)] =

1 L m ,i

C

(3)

C.A. Pitz et al. / Digital Signal Processing 97 (2020) 102621

3

T i , B m . Moreover, rˇi (n) is complex AWGN with average power

σrˇ2 i

ˇ m is the K m -dimensional present in the single antenna of T i , and w downlink beamforming vector used in B m to transmit data to T m . Similarly to the derivation of (5), the mean power of yˇ i (n) can be expressed as E[| yˇ i (n)|2 ] =

M  m =1

Fig. 2. Downlink scenario in which a SOI (dark line) and interfering signals (dashed lines) arrive at the considered user (T i ).

with C representing an L m,i × L m,i diagonal matrix with Tr(C) = L m,i . Next, the input signal vector xˆ i (n) is processed at B i using beamforming vectors that are specific for each in-cell (local) user. In this context, the array output for user T i is given by

ˆH ˆ i (n) yˆ i (n) = w i x

(4)

ˆ i denotes the corresponding K i -dimensional uplink beamwhere w forming vector. The link improvement provided by the beamformer is typically measured in terms of SINR. Aiming to evaluate the SINR at the uplink channel, an expression for the mean power of yˆ i (n) needs to be obtained first. Then, considering (3), substituting (1) into the squared magnitude of (4), and taking the expected value of the resulting expression, the mean power of yˆ i (n) can be written as E[| yˆ i (n)|2 ] =

M  m =1

ˆ ˆ ˆ i + σ 2 ||w ˆ i ||2 ˆH w i P m Rm,i w rˆi

1 L m ,i

ˆ m,i CH ˆH H m ,i

(6)

denoting the uplink spatial covariance matrix for the pair T m , B i . From (5), one can individually access the terms related to both interference-plus-noise power and power of T i , resulting in the following expression for the uplink SINR:

γˆi =

ˆ w ˆH ˆ w Pˆ R i i i ,i i M  m=i

ˆ w ˆH ˆ w Pˆ R i m m ,i i

.

(7)

2 ˆ ||wi ||2 rˆi

+σ

To model the signals received in the downlink channel, we assume that the data symbols transmitted to each user are multiplied by the corresponding downlink beamforming vector. Thus, considering that the signals transmitted to all users inevitably arrive at the ith user T i , as depicted in Fig. 2, one can model the signal received by such a user as M 

ˇ H ˇ m + rˇi (n) sˇ H i ,m (n)Hi ,m w

(8)

m =1

ˇ i ,m represent, respectively, the downlink verwhere sˇ i ,m (n) and H ˆ m,i (see Section 2.1), but now for the pair sions of sˆm,i (n) and H

(9)

ˇ i ,m denoting the downlink spatial covariance matrix bewith R tween T i and B m . The analytical expression of such a matrix is ˆ m,i and Lm,i with H ˇ i ,m and L i ,m , respecgiven by (6), replacing H tively. Finally, the downlink SINR γˇi can be derived by isolating the power related to T i from the remaining terms of (9), resulting in

ˇ w ˇH ˇ w R i i ,i i

γˇi =

M  m=i

Hˇ ˇm ˇm w Ri ,m w

.

(10)

+σ

2 rˇi

In contrast to (7), the downlink SINR given by (10) depends on the beamforming vectors obtained for all users. Thus, the maximizaˇ i ∀i. tion of (10) requires joint processing of w 2.3. Uplink and downlink beamforming problems The uplink beamforming aims at maximizing the uplink SINR of all users, i.e., to maximize γˆi ∀i. From (7), one can notice that the uplink SINR between users is not interdependent, since γˆi deˆ i . Thus, uplink beamforming algorithms can be pends only on w ˆ i computed locally at implemented in a distributed manner, with w B i considering the following optimization problem: ˆi w

ˆ i || = 1 subject to ||w

(11)

where the norm constraint in (11) is imposed for uniqueness of ˆ i || [see (7)]. its solution, since γˆi is invariant with respect to ||w The solution of (11) is the generalized eigenvector corresponding ˆ in , Rˆ s to the smallest generalized eigenvalue related to the pair R [41], where

ˆ s = Pˆi Rˆ i ,i R

(12)

is the autocorrelation matrix of the SOI and

ˆ in = R

M  m=i

2.2. Downlink signal model

yˇ i (n) =

i

maximize γˆi (5)

with

ˆ m ,i = R

Hˇ ˇm ˇ m + σrˇ2 w Ri ,m w

ˆ m ,i + σ 2 I K Pˆ m R rˆi

(13)

is the autocorrelation matrix of the interference plus noise, with I K representing the K × K identity matrix. Regarding the downlink beamforming, an appropriate problem formulation must inevitably consider the interdependence between the downlink beamforming vectors, which is evident ˇ i with i = 0, 1, . . . , M. from (10), noticing that γˇi depends on w As a consequence, approaches considered in the open literature are often based on maximizing some arbitrary objective function f (γˇ1 , . . . , γˇM ) that is strictly increasing with γˇi ∀i (e.g., the sumrate capacity) subject to a transmission power constraint [42–45]. The resulting optimization problem can be expressed as

maximize f (γˇ1 , . . . , γˇM ) ˇ 1 ···w ˇM w

subject to

M  m =1

ˇ m ||2 ≤ Pˇ max ||w

(14)

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C.A. Pitz et al. / Digital Signal Processing 97 (2020) 102621

where Pˇ max is a limit on the total downlink transmit power [44]. The optimum solution of (14) is obtained by means of a centralized processing scheme, which increases the backhaul load and demand high computational resources at the mobile switching center. To cope with these problems, distributed beamforming schemes exploit channel reciprocity using the directions of the uplink beamˇ i locally at B i . More specififorming vectors [44] for obtaining w ˇ i is given by a scaled version of w ˆ i available at B i , i.e., cally, w



ˇi= w

Pˇ i

ˆi w ˆ i || ||w

(15)

ˇ i ||2 represents the transmit power allocated to the where Pˇ i = ||w ith user [46]. By considering fixed values of Pˇ i ∀i, the use of (15) at all R base stations results in maximization of the downlink SINR γˇi . Thus, power control algorithms can be implemented along with uplink beamforming algorithms, aiming to solve (14) [38,46]. It is important to mention that power control is not the focus of this paper, and thus the beamforming algorithms presented in the next sections consider distributed implementations with fixed transmission power values. 3. Review of CSG-type beamforming algorithms The aim of CSG-type algorithms is to iteratively maximize the SINR by using only instantaneous samples of the signals available at the uplink channel of cellular systems. These algorithms are developed by considering that a K -dimensional complex vector dˆ i (n), containing snapshots of the SOI (ith user) at each antenna of the array, can be estimated from xˆ i (n). Such an estimation can be carried out in non-CDMA systems by detecting data symbols at each antenna of the array [17,37]. After estimating dˆ i (n), a signal vector zˆ i (n), composed of snapshots of the interference-plus-noise, can be obtained by subtracting dˆ i (n) from xˆ i (n) [18]. For simplicity, the subscript i is omitted from now on without loss of generality, since all variables are related to the SOI (ith user). Thus, the instantaneous uplink SINR for the SOI can be written as

γ˜ (n) =

ˆ ˆ H dˆ (n)dˆ H (n)w w ˆ ˆ H zˆ (n)ˆzH (n)w w

(16)

which is the objective function used for deriving CSG-type beamforming algorithms for non-CDMA systems. For CDMA systems, dˆ (n) and zˆ (n) are replaced by the signal vectors available at the input and output of the two-dimensional RAKE receiver [34,35]. ˆ and its maximization, Note that (16) is a nonconcave function of w which can be attained by solving a generalized eigenvalue problem [41], is intricate and computationally costly. As a consequence, different strategies have been developed aiming to iteratively maximize (16) with focus on real-time applications, giving rise to the CSG-type algorithms revisited in this section. The strategy used in [17] and [18] to develop the CSG and ICSG algorithms is based on the maximization of (16) in two stages [17]: in the first, the denominator is kept fixed while the numerator is maximized; in the second, the denominator is minimized, keeping the numerator fixed. As shown in [17] and [18], the CSG and ICSG algorithms reduce the computational complexity and provide high SINR levels under Rayleigh fading channels, with the ICSG outperforming the CSG for cases in which the SOI lies in the interference subspace. Regarding the AP-CSG and AP-QCSG algorithms, the denominator of (16) is minimized while an adaptive-projection ˆ towards the SOI subspace [38,39]. constraint is used for steering w In contrast to the standard CSG and ICSG algorithms, AP-CSG and AP-QCSG do not depend on individual estimation of the interferences, but rather depend on the estimation of the sum of interfering signals, making such algorithms simpler to implement and

Table 1 Summary of the CSG algorithm [17].

ˆ (0) and w ˇ (0) Initialize w Do for n = 0, 1, 2, 3 . ..

 zˆ 2 (n)ˆzH 2 (n) ˆ (n) − dˆ (n)dˆ H (n)w ||ˆz1 (n)||2 ||ˆz2 (n)||2    dˆ (n)dˆ H (n)  ˆ  (n) = w ˆ  (n) − μ I K − ˆ 2 (n)ˆzH ˆ (n) w zˆ 1 (n)ˆzH 1 (n) + z 2 (n) w 2 ˆ ||d(n)||  ˆ (n) w ˆ (n + 1) = w ˆ  (n)|| ||w ˆ (n + 1) ˇ (n + 1) = Pˇ w w ˆ (n) + μ I K − ˆ  (n) = w w

zˆ 1 (n)ˆzH 1 (n)

Table 2 Summary of the ICSG algorithm [18].

ˆ (0) and w ˇ (0) Initialize w Do for n = 0, 1, 2, 3 . ..

ˆ zH zˆ 1 (n)ˆzH 1 (n) z2 (n)ˆ 2 (n) + ||ˆz2 (n)||2 ||ˆz1 (n)||2 ||ˆz2 (n)||2  ˆ zˆ 2 (n)ˆzH zH 2 (n) z1 (n)ˆ 1 (n) ˆ (n) + dˆ (n)dˆ H (n)w ||ˆz2 (n)||2 ||ˆz1 (n)||2    dˆ (n)dˆ H (n)  ˆ  (n) − μ I K − ˆ 2 (n)ˆzH ˆ  (n) = w ˆ (n) w zˆ 1 (n)ˆzH 1 (n) + z 2 (n) w ||dˆ (n)||2 ˆ  (n) w ˆ (n + 1) = w ˆ  (n)|| ||w ˇ (n + 1) = Pˇ w ˆ (n + 1) w ˆ  (n) = w ˆ (n) + μ I K − w

zˆ 1 (n)ˆzH 1 (n)

||ˆz1 (n)||2

−

zˆ 2 (n)ˆzH 2 (n)

Table 3 Summary of the AP-CSG algorithm [38].

ˆ (0) and w ˇ (0) Initialize w Do for n = 0, 1, 2, 3 .  ..

ˆ (n) − μ1 I K − ˆ  (n) = w w ˆ (n + 1) = w

ˆ  (n) w

dˆ (n)dˆ H (n)

||dˆ (n)||2

 ˆ (n) + μ2 zˆ (n)ˆzH (n)w

dˆ (n)dˆ H (n)

||dˆ (n)||2

ˆ (n) w

ˆ  (n)|| ||w ˇ (n + 1) = Pˇ w ˆ (n + 1) w Table 4 Summary of the AP-QCSG algorithm [39].

ˆ (0) and w ˇ (0) Initialize w Do for n = 0, 1, 2, 3 . . .

ˆ H (n)dˆ (n)|2 G d (n) = |w ˆ H (n)dˆ (n)dˆ H (n)ˆz(n)ˆzH (n)w ˆ (n) G dz (n) = w 2

(n) = (1 + μ2 )G 2d (n) − μ21 Im[G dz (n)]

 1 1 χ (n) = μ1 G ∗dz (n) + (n) − 1 G d (n) 2   dˆ (n)dˆ H (n) dˆ (n)dˆ H (n) ˆ (n) − μ1 I K − ˆ (n) + χ (n) ˆ  (n) = w ˆ (n) w zˆ (n)ˆzH (n)w w 2 ˆ 2||d(n)|| ||dˆ (n)||2  ˆ (n) w ˆ (n + 1) = w ˆ  (n)|| ||w ˇ (n + 1) = Pˇ w ˆ (n + 1) w

easily applicable to CDMA systems [38,39]. In comparison with the AP-CSG, the AP-QCSG provides higher SINR levels, especially in realistic scenarios where the signals are transmitted by spatially distributed sources [39]. The iterative processes of the CSG and ICSG algorithms are summarized in Tables 1 and 2, respectively, considering a scenario with two interfering users. In these tables, ˆ  (n) and w ˆ  (n) μ and μ are the step-size parameters, whereas w denote the beamforming vectors obtained, respectively, in the first and second stages. Tables 3 and 4 summarize, respectively, the APCSG and AP-QCSG algorithms, with μ1 and μ2 denoting step-size parameters.

C.A. Pitz et al. / Digital Signal Processing 97 (2020) 102621

E[χ (n)] ∼ =

4. Proposed approach In this section, a new framework for the development of adaptive beamforming algorithms is introduced. The starting point for deriving the proposed framework is the analysis of the mean weight behavior of CSG-type algorithms based on adaptive projections [38], [39]. This analysis gives rise to a unifying view of the mean weight behavior of such algorithms, which constitutes the foundation upon which the proposed framework is then derived. The first practical algorithm developed by using the proposed framework is also introduced in this section.

Stochastic models have been used for the study and development of CSG-type algorithms since the work of Kolodziej et al. [18], where expressions describing the SINR and mean weight behavior of the CSG algorithm were derived. From these expressions, it has been observed that the standard CSG algorithm leads to an unbalance between maximization of the SOI power and minimization of interference power for certain practical scenarios [18]. To overcome this problem, the ICSG algorithm has been developed (also in [18]) by means of first adjusting the model expressions to obtain the desired behavior and then deriving the corresponding weight update expressions. Another example of statistical analysis for a CSG-type algorithm is found in [38], where a stochastic model for the AP-CSG algorithm (see Table 3) has been presented. According to this model, the mean weight behavior of the AP-CSG can be described as

ˆ  (n)] = E[w ˆ (n)] − μ1 (I K − Rˆ s )Rˆ in E[w ˆ (n)] + μ2 Rˆ s E[w ˆ (n)] E[w (17) and

ˆ  (n)] E[w

(18)

ˆ  (n)]|| ||E[w

ˆ s denoting the normalized autocorrelation matrix of the SOI, with R i.e.,



ˆs = E R

dˆ (n)dˆ H (n)

||dˆ (n)||2



(19)

.

4.2. Mean weight behavior for the AP-QCSG algorithm Aiming to obtain an expression to describe the mean weight behavior of the AP-QCSG algorithm, we consider here the approximations described in [18] and [38] for modeling the CSG and the AP-CSG, respectively. Thus, taking the statistical expectation of the weight-update equation of the AP-QCSG algorithm (next to last expression in Table 4) and manipulating the resulting expression, we obtain



1  ˆ ˆ in E[w ˆ ˆ ˆ (n)] E[w (n)] = E[w(n)] − μ1 I K − Rs R 2

(20)

ˆ (n)] + E[χ (n)]Rˆ s E[w with

E[χ (n)] =

1 2



μ1 E

G ∗dz (n) G d (n)

√

 +E

 (n) − 1. G d (n)

2

μ1

E[G ∗dz (n)] E[G d (n)]

√ +

E[(n)]

E[G d (n)]

−1

(22)

with

E[(n)] ∼ = (1 + μ2 ){E[G d (n)]}2 − μ21 Im {E[G dz (n)]} 2

(23)

ˆ H (n)]Rˆ s E[w ˆ (n)] where E[G d (n)] = E[w and E[G dz (n)] = ˆ H (n)]Rˆ s Rˆ in E[w ˆ (n)]. Finally, considering again the small disperE[w ˆ the expected value of the normalization expression (last sion of w, one in Table 4) can be obtained by using (18). 4.3. A unifying view on the behavior of the AP-CSG and AP-QCSG algorithms

4.1. On the mean weight behavior of CSG-type algorithms

ˆ (n + 1)] ∼ E[w =

1

5

(21)

Now, for the sake of mathematical simplicity, we assume small values for μ1 and μ2 , resulting in a small dispersion in the evolution ˆ [18,38]. In this case, (21) can be approximated by of w

From (17) and (20), one can notice that the mean-weight behavior of both AP-CSG and AP-QCSG algorithms share important similarities. To make such similarities more evident, (17) and (20) can be rewritten in a unified form as

ˆ  (n)] = E[w ˆ (n)] − μ1 v1 (n) + μ2 v2 (n) + μ3 v3 (n) E[w

(24)

with

ˆ in E[w ˆ (n)] v1 (n) = R

(25)

ˆ s E[w ˆ (n)] v2 (n) = R

(26)

and

ˆ s Rˆ in E[w ˆ (n)]. v3 (n) = R

(27)

Expression (17) (mean weight behavior of the AP-CSG) can be obtained by making μ3 = μ1 in (24), whereas (20) (mean weight behavior of the AP-QCSG) can be obtained by making μ2 = E[χ (n)] and μ3 = μ1 /2. From (24)-(27), it is easy to observe that the weight update of both AP-CSG and AP-QCSG algorithms behaves according to a linear combination of vectors v1 (n), v2 (n), and v3 (n). In the case of the AP-CSG, the linear combination is carried out with fixed coefficients given by −μ1 for v1 (n), μ2 for v2 (n), and μ1 for v3 (n). In contrast, the better performance of the AP-QCSG is attained by using −μ1 and μ2 /2 as fixed coefficients for v1 (n) and v3 (n), respectively, whereas the time-varying coefficient corresponding to v2 (n) is E[χ (n)]. In this context, from the difference of performance between the AP-CSG and AP-QCSG algorithms [39], we conclude that the strategy used to combine v1 (n), v2 (n), and v3 (n) plays a key role on the behavior and performance of the corresponding algorithm. 4.4. Proposed framework for adaptive combination of vector projections The objective now is to develop a framework for deriving CSGtype beamforming algorithms that allows exploiting the aforementioned influence of the combination of vectors v1 (n), v2 (n), and v3 (n) on the algorithm behavior. More specifically, the idea is to dynamically adjust the linear combination of these vectors for the sake of performance enhancement. In this context, the first difficulty lies in the fact that the individual contributions of v1 (n), v2 (n), and v3 (n) to the summation on the right-hand side of (24) are very unbalanced. Such an unbalance is evidenced by the difference in the upper bounds of the Euclidean norms of v1 (n), v2 (n), and v3 (n). For instance, considering (25) and also the eigendecomˆ in (i.e., Rˆ in = Qin 2 QH ), the square of the Euclidean position of R in in norm of v1 (n) can be written as

ˆ H (n)]Qin 2in QH ˆ (n)] ||v1 (n)||2 = E[w in E[w

(28)

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C.A. Pitz et al. / Digital Signal Processing 97 (2020) 102621

ˆ in where in is a diagonal matrix composed of the eigenvalues of R and Qin denotes an orthonormal matrix whose columns are the ˆ in . Next, considering (28) along with the Cauchyeigenvectors of R Schwarz inequality, one can conclude that the square of the Euclidean norm of v1 (n) is bounded according to

ˆ (n)]||2 . ||v1 (n)||2 ≤ Tr(2in )||E[w

(29)

An analysis similar to (28)-(29), but now considering (26), leads to

ˆ (n)]||2 = ||E[w ˆ (n)]||2 ||v2 (n)||2 ≤ Tr(2s )||E[w

(30)

where Tr(s ) = 1, since s is a diagonal matrix composed of the ˆ s . From (29) and eigenvalues of the normalized correlation matrix R (30), one can observe that, while the bound for ||v2 (n)||2 depends ˆ (n)]||2 , the bound for ||v1 (n)||2 depends also exclusively on ||E[w ˆ in and consequently on the volatile (highlyon the eigenvalues of R varying) characteristics of the interference plus noise. This fact is also true for the bound of ||v3 (n)||2 , since v3 (n) is also a function ˆ in . As a result, one observes that the relative importance of the of R terms that depend on v1 (n) and v3 (n) in the summation (24) tend to constantly change, making the development of algorithms based on dynamic combinations of v1 (n), v2 (n), and v3 (n) a cumbersome ˆ in by task. To circumvent this problem, the idea here is to replace R its normalized version

 H ˆ ˆRin = E z(n)ˆz (n) ||ˆz(n)||2

(31)

in both (25) and (27), which can be carried out without loss of ˆ in and Rˆ in share the same eigenvectors, and the generality, since R ˆ in are scaled versions of the eigenvalues of Rˆ in eigenvalues of R [18]. Thus, one obtains

ˆ in E[w ˆ (n)] v¯ 1 (n) = R

(32)

and

ˆ s Rˆ in E[w ˆ (n)] v¯ 3 (n) = R

(33)

which are normalized versions of v1 (n) and v3 (n), respectively, ˆ (n)]|| (i.e., ||¯v1 (n)||2 ≤ with magnitudes bounded above by ||E[w ˆ (n)]||2 and ||¯v3 (n)||2 ≤ ||E[w ˆ (n)]||2 ). Now, considering v¯ 2 (n) = ||E[w v2 (n) along with (32) and (33), and also replacing the step-size parameters μ1 , μ2 , and μ3 by time-varying combination coefficients denoted as β1 (n), β2 (n), and β3 (n), (24) can be rewritten as

ˆ  (n)] = E[w ˆ (n)] + E[w

3 

βi (n)¯vi (n).

(34)

i =1

Note that by properly choosing the values of β1 (n), β2 (n), and β3 (n), one can obtain (24), and consequently (17) and (20) (i.e., mean-weight behaviors of AP-CSG and QAP-CSG algorithms, respectively). Moreover, by efficiently choosing the values of β1 (n), β2 (n), and β3 (n), mean-weight behaviors of algorithms that outperform the AP-CSG and QAP-CSG can be obtained. The focus now is on deriving an iterative update expression that defines a general algorithm whose mean-weight behavior is described by (34). To this end, vectors v¯ 1 (n), v¯ 2 (n), and v¯ 3 (n) are replaced by instantaneous estimates vˆ 1 (n), vˆ 2 (n), and vˆ 3 (n), which ˆ (n) along with instantaneous estimates of are computed by using w ˆ s and Rˆ in given, respectively, by R

˜ s (n) = R and

dˆ (n)dˆ H (n)

||dˆ (n)||2

Fig. 3. Geometric representation of the vector projections considered in the ACVP framework.

˜ in (n) = R

zˆ (n)ˆzH (n)

||ˆz(n)||2

˜ in (n)w ˆ (n), vˆ 2 (n) = R˜ s (n)w ˆ (n), and Then, considering vˆ 1 (n) = R ˆ (n), the following general update expression ˆv3 (n) = R˜ s (n)R˜ in (n)w is obtained:

ˆ (n) + ˆ  (n) = w w

3 

βi (n)ˆvi (n).

(37)

i =1

ˆ that may arise Finally, aiming to prevent an indefinite growth of w ˆ (n + 1) is obtained from (37), the a posteriori beamforming vector w ˆ  (n), i.e., after normalizing w

ˆ (n + 1) = w

ˆ  (n) w ˆ  (n)|| ||w

.

(38)

The characteristics of the coefficient update process described by (37) can be better understood by using a geometric representation of the vectors involved in such an expression, as shown in ˜ s (n) and R˜ in (n) are projection matrices [see (35) and Fig. 3. Since R (36)], one can verify that vˆ 1 (n) corresponds to the vector projecˆ  (n) onto zˆ (n) (interference-plus-noise vector), whereas tion of w ˆ  (n) onto dˆ (n) (SOI vˆ 2 (n) corresponds to the vector projection of w vector). As a consequence, the coefficients β1 (n) and β2 (n) can be ˆ  (n) onto properly adjusted in order to increase the projection of w the SOI subspace, denoted here S , and reduce its projection onto the interference-plus-noise subspace, denoted I . However, in cases ˆ  (n) onto I may lead in which S ⊂ I , to reduce the projection of w to a beamforming vector that lies in the null space of S , thus impairing the SINR performance. In this context, the vector projection of vˆ 1 (n) onto dˆ (n), given by vˆ 3 (n), plays a key role in (37). More specifically, vˆ 3 (n) becomes more significant in (37) as the projection of dˆ (n) onto zˆ (n) increases and, therefore, β3 (n) can be used ˆ from being steered toin conjunction with β2 (n) to prevent w wards the null space of S . From the discussion above, it becomes clear that by effectively adapting the values of the combination coefficients β1 (n), β2 (n), and β3 (n), one can improve the algorithm performance by means of a proper adjustment of the vector projections represented by vˆ 1 (n), vˆ 2 (n), and vˆ 3 (n). This is the main idea behind the framework for the development of beamforming algorithms proposed in this paper. The effective adjustment of β1 (n), β2 (n), and β3 (n) required in the ACVP framework is attained considering the following optimization problem:

maximize

(35)

(36)

.

β1 (n),β2 (n),β3 (n)

f (γˆ )

(39)

where f (γˆ ) denotes some arbitrary utility function that is strictly increasing with SINR γˆ .

C.A. Pitz et al. / Digital Signal Processing 97 (2020) 102621

4.5. Sigmoid-based ACVP algorithm

Table 5 Summary of the SB-ACVP algorithm.

This section is dedicated to the derivation of the first algorithm based on the ACVP framework. Such an algorithm, called sigmoidbased (SB-) ACVP, is developed by using a sigmoid function to support the adjustment of the combination coefficients β1 (n), β2 (n), and β3 (n). The choice of such a function is due to its limited range (from 0 to 1), making easier to control the excursion of the combination coefficients, and also due to its small gradient for values close to the range limits. Now, aiming to define an expression for β1 (n), we first recall ˆ  (n) onto the that such a coefficient weighs the projection of w interference-plus-noise vector zˆ (n), as described in the previous section. Since such a projection tends to contribute negatively to the SINR performance, we define β1 (n) as a negative coefficient, resulting in

β1 (n) = −μφ[α1 (n)]

(40)

where 0 < μ ≤ 1 is a parameter that determines the range of β1 (n) and

φ[α1 (n)] =

1

(41)

1 + e−α1 (n)

denotes the chosen sigmoid function with −∞ < α1 (n) < ∞. In contrast to β1 (n), β2 (n) is set as a positive coefficient since ˆ  (n) onto the SOI it weighs the contribution of the projection of w ˆ vector d(n), which is usually positive for SINR enhancement. Thus, one has

β2 (n) = μφ[α2 (n)]

β3 (n) = μ − β2 (n).

(43)

As can be seen from (42) and (43), β3 (n) reaches its maximum when β2 (n) → 0, ensuring that a vector projection lying in S is always considered in (37) for S ⊂ I . Thus, substituting (42) into (43), one gets

β3 (n) = μ{1 − φ[α2 (n)]}.

(44)

Now, regarding the utility function f (γˆ ) required in (39), the idea here is to use the natural logarithm function of the SINR, i.e.,



ˆ (0), w ˇ (0), Initialize w

ˆ (n + 1) ˆ H (n + 1)Rˆ s w w ˆ H (n + 1)Rˆ in w ˆ (n + 1) w

 .

(45)

This function meets the requirement of being strictly increasing in γˆ and also allows easy calculation of the partial derivatives of f (γˆ ) with respect to αi (n). As a consequence, the steepest ascent method can be used for maximizing (45), resulting in the following updating rule for αi (n) with i = 1, 2:

   ˆ (n + 1) ˆ H (n + 1)Rˆ s w ∂ w αi (n + 1) = αi (n) + · log 2 ∂ αi (n) ˆ H (n + 1)Rˆ in w ˆ (n + 1) w

μα

(46)

α1 (0) and α2 (0)

Do for n = 0, 1, 2, 3 . . . 1 1 , φ[α2 (n)] = φ[α1 (n)] = 1 + e−α1 (n) 1 + e−α2 (n) H zˆ (n)ˆz (n) dˆ (n)dˆ H (n) ˆ (n) − μφ[α1 (n)] ˆ  (n) = w ˆ (n) + μφ[α2 (n)] ˆ (n) w w w ||ˆz(n)||2 ||dˆ (n)||2 dˆ (n)dˆ H (n) zˆ (n)ˆzH (n)

ˆ (n) w ||dˆ (n)||2 ||ˆz(n)||2 zˆ (n)ˆzH (n) ˆ (n) q1 (n) = −μ{1 − φ[α1 (n)]}φ[α1 (n)] w ||ˆz(n)||2  ˆd(n)dˆ H (n)

zˆ (n)ˆzH (n) ˆ (n) q2 (n) = μ{1 − φ[α2 (n)]}φ[α2 (n)] IK − w 2 2 ˆ ||ˆ z ( n )|| ||d(n)|| Do for i = 1, 2 ˆ  (n)] ˆ  (n)] Re[qH (n)dˆ (n)dˆ H (n)w Re[qH (n)ˆz(n)ˆzH (n)w i i αi (n + 1) = αi (n) + μα − μα  H (n)ˆ H (n)w  (n)  H H  ˆ ˆ ˆ ˆ w z ( n )ˆ z ˆ (n) ˆ (n)d(n)d (n)w w ˆ  (n) w ˆ (n + 1) = w ˆ  (n)|| ||w ˇ (n + 1) = Pˇ w ˆ (n + 1) w + μ{1 − φ[α2 (n)]}

where μα denotes the step-size parameter for the steepest ascent method and a factor 1/2 is included in (46) for canceling out a factor of 2 that appears after computing the partial derivatives. Next, substituting (38) into (46) and considering the chain rule along ˆ s and Rˆ in [given, respectively, with the instantaneous estimates of R by dˆ (n)dˆ H (n) and zˆ (n)ˆzH (n)], we get

αi (n + 1) = αi (n) + μα − μα

(42)

where α2 (n) is defined in the same way as α1 (n). With respect to β3 (n), the strategy proposed here aims to ensure that the beamforming vector is not steered towards the null space of S during the adaptive process. This situation may occur when both β2 (n) and β3 (n) are very small and S ⊂ I . Then, to avoid very small values for these coefficients simultaneously, β3 (n) is bound to β2 (n) as

f (γˆ ) = log

7

ˆ ˆH ˆ Re[qH i (n)d(n)d (n)w (n)] ˆ  (n) ˆ  H (n)dˆ (n)dˆ H (n)w w

ˆ  (n)] Re[qH (n)ˆz(n)ˆzH (n)w i

(47)

ˆ  (n) ˆ  H (n)ˆz(n)ˆzH (n)w w

with

qi (n) =

ˆ  (n) ∂w . ∂ αi (n)

(48)

Finally, in order to obtain q1 (n) and q2 (n), we substitute (40), (42), and (44) into (37), and the resulting expression into (48). Then, considering that ∂φ[αi (n)]/∂ αi (n) = {1 − φ[αi (n)]}φ[αi (n)], we obtain

q1 (n) = −μ{1 − φ[α1 (n)]}φ[α1 (n)]

zˆ (n)ˆzH (n)

||ˆz(n)||2

ˆ (n) w

(49)

and

q2 (n) = μ{1 − φ[α2 (n)]}φ[α2 (n)]

× IK −

zˆ (n)ˆz (n) H

||ˆz(n)||2

dˆ (n)dˆ H (n)



||dˆ (n)||2

(50)

ˆ (n). w

The iterative process of the SB-ACVP algorithm is detailed in ˆ  (n) presented in such Table 5. From the update expression for w a table, the convergence of the proposed algorithm can be analyzed in terms of the contributions of φ[α1 (n)] and φ[α2 (n)]. For ˆ  (n) is completely steered toφ[α1 (n)] = 0 and φ[α2 (n)] > 0 ∀n, w wards S as n → ∞ (see Fig. 3). On the other hand, any positive ˆ  (n) value of φ[α1 (n)] contributes to reduce the projection of w  ˆ onto I . Thus, one can conclude that w (n) invariably converges to the subspace spanned by the basis of both S and NI , with NI denoting the null space of I . Table 6 presents the resulting computational load of the algorithms considered here. Regarding the number of operations required to compute the sigmoid function given by (41), we consider the first two terms of the Taylor series expansion. From Table 6, we

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Table 6 Computational load for K antennas. Number of real operations per iteration

Algorithm

Multiplications CSG [17] ICSG [18] AP-CSG [38] AP-QCSG [39] SB-ACVP (proposed)

54K 66K 28K 30K 46K

+ 22 + 34 +6 + 37 + 77

Additions 52K 64K 24K 28K 38K

−8 −6 −4 +5 + 22

Divisions

Square roots

5 6 2 3 5

1 1 1 2 1

verify that the load required by the proposed SB-ACVP algorithm is on the same order O ( K ) as the other CSG-type algorithms considered in this paper. More specifically, the computational burden of the proposed algorithm is only moderately larger than that of the AP-CSG and AP-QCSG algorithms and below the cost of the CSG and ICSG algorithms. Moreover, similarly to both the AP-CSG and AP-QCSG, the proposed SB-ACVP algorithm does not depend on individually estimating the interfering signals, which makes its implementation easier than those of both the CSG and ICSG. It is also important to point out that the choice of the step-size parameter, which is a recurring problem for the algorithms derived by using gradient methods (such as all CSG-type algorithms), is significantly easier in the case of the proposed SB-ACVP. This is ˆ  (n) solely based on due to the fact that the SB-ACVP updates w vector projections whose magnitudes are invariant with respect to the norms of dˆ (n) and zˆ (n) (see Table 5). Thus, the powers of the involved signals have no influence on the selection of parameters for the SB-ACVP algorithm, contrasting with the other CSG-type algorithms for which the step size tuning requires considering the power of both SOI and interfering signals. In this context, a default value of μ = 0.3 is suggested for the SB-ACVP (which worked well in several tests performed by considering a range of different scenarios), whereas μα = 1 can be adopted as a default value for updating αi (n), since the norm of qi (n) is automatically reduced as φ[αi (n)] approaches either 0 or 1. 5. Simulation results In this section, results of Monte Carlo simulations (200 independent runs) are presented aiming to assess the performance of the proposed SB-ACVP algorithm and make comparisons with other competing algorithms that also do not require estimating the angle-of-arrival neither rely on pilot signals, namely the CSG [17], ICSG [18], AP-CSG [38], and AP-QCSG [39]. Scenarios involving uplink and downlink beamforming in a TDD system are considered in these simulations. For the uplink cases, we consider a singleuser scenario formed by a SOI (in-cell user) and M − 1 interfering signals coming from nearby co-channel cells. In contrast, the downlink beamforming is evaluated considering multiuser scenarios due to the interdependence between the beamforming vectors of all users [see (10)]. In all simulations, we assume that the base stations are equipped with a linear array having K = 8 omnidirectional antennas uniformly spaced by a half wavelength. Each of the signals transmitted by all users and base stations are assumed to travel through 12 independent fading paths (i.e., L m,i = 12, ∀m, i), generating 5◦ of azimuth spread around the mean angle-ofarrival. Such spatially distributed sources lead to a full-rank matrix  

M

m=i

ˆ m,i Pˆ m R

and thus the beamformers will not be able to

completely suppress the interferers. A fast Rayleigh fading channel model is considered in the simulations, which is obtained by multiplying the steering vector corresponding to each multipath by a complex Gaussian random variable with zero mean and variance 1/ L m,i . The same noise power is considered for all users and base stations, leading to a signal-to-noise ratio [(7) and (10) with zero interference] of 40 dB. Moreover, the step-size parameters are cho-

Fig. 4. Example 1. (a) SINR comparison. (b) Radiation pattern comparison. (c) Mean behavior of the sigmoid functions φ[α1 (n)] and φ[α2 (n)] used by the proposed SBACVP algorithm.

sen aiming to provide approximately the same initial convergence rate for all algorithms, and the beamforming vectors are initialized ˇ (0) = [1 0 · · · 0]T , which corresponds to an omnidiˆ (0) = w as w rectional radiation pattern over the azimuth plane. 5.1. Interference rejection in the uplink channel 5.1.1. Example 1 The uplink scenario considered in the first example is formed by a SOI located at 30◦ and two interfering signals located at 60◦ and −60◦ . All signals are transmitted with the same power and the interfering signals (coming from co-channel cells) are attenuated by 10 dB due to free space loss. Fig. 4 illustrates the results obtained in this example. One observes from Fig. 4(a) that the proposed SB-ACVP algorithm attains a SINR level considerably higher than those obtained by the other competing algorithms (around 12 dB of improvement). The obtained radiation patterns, displayed in Fig. 4(b), show that the proposed algorithm is capable of imposing lower gains in the interference directions (indicated by ×) while maintaining a similar gain in the SOI direction (indicated by ◦ ), which explains its better performance. The evolution of φ[α1 (n)] and φ[α2 (n)], presented in Fig. 4(c), also gives interesting insights on the performance of the proposed algorithm. Notice that, at the initial transient phase, the value of φ[α2 (n)] grows, increasing the ˆ towards the direction of the SOI value of β2 (n), and steering w subspace S [see Section 4.4 and Fig. 4(b)]. Afterwards, φ[α2 (n)] is

C.A. Pitz et al. / Digital Signal Processing 97 (2020) 102621

9

noise levels (SNR less than 10 dB), which is expected since in this ˆ i ||2 dominates the denominator of (7), limiting the SINR case σrˆ2 ||w i

improvement achieved by suppressing directional interferers. On the other hand, as the input SNR increases, the proposed SB-ACVP significantly outperforms the other algorithms considered, which is due to the lower gains obtained in the interference directions.

Fig. 5. Example 1. Steady-state SINR at the beamformer output versus input SNR.

Fig. 6. Example 2. (a) SINR comparison. (b) Radiation pattern comparison. (c) Mean behavior of the sigmoid functions φ[α1 (n)] and φ[α2 (n)] used by the proposed SBACVP algorithm.

reduced towards zero while φ[α1 (n)] is increased, which in turn ˆ towards the null space of the interference subspace I . steers w Such behavior of φ[α1 (n)] and φ[α2 (n)] illustrates the effectiveness of the strategy for obtaining the linear combination coefficients used by the SB-ACVP algorithm. Additional results are presented in this example aiming to assess the performance of the considered algorithms under different levels of noise power. More specifically, the steady-state SINR provided by each algorithm is evaluated as a function of the input SNR. One can notice, from the resulting curves shown in Fig. 5, that all algorithms achieve similar SINR performance under high

5.1.2. Example 2 Now, we consider a scenario where the SOI and one of the interfering signals arrive at the base station with nearby anglesof-arrival. More specifically, the two interfering signals are now located at −25◦ and 25◦ , the SOI at 30◦ , and the remaining scenario characteristics are the same as in Example 1. Fig. 6 presents the results of this example. One notices from Fig. 6(a) that the SINR levels attained by all algorithms are lower than in Example 1, which is expected due to the proximity of the SOI and one interferer. Despite this, the proposed SB-ACVP algorithm again leads to significantly higher SINR levels as compared with the other algorithms assessed (more than 7-dB improvement in this case). The radiation patterns presented in Fig. 6(b) reveal that the proposed SB-ACVP algorithm again imposes lower gains in the interference directions (indicated by ×), while maintaining the gain in the SOI direction (indicated by ◦ ) at an acceptable level. In contrast, the competing algorithms are ineffective on either attenuating the interferences (ICSG, AP-CSG, and AP-QCSG) or producing an acceptable gain in the SOI direction (CSG). In addition, one observes from Fig. 6(c) that, due to the SOI-interferer proximity, φ[α2 (n)] takes longer to reach zero as compared with the results obtained in Example 1. This behavior confirms the proper operation of the proposed algorithm, since a higher contribution of vˆ 2 (n) is required in the case considered in Example 2 aiming to avoid SOI suppression. 5.1.3. Example 3 In this example, we consider a scenario with higher interference levels, in which five interfering signals arrive at the base station with the same power as the SOI. The angles-of-arrival of all signals are randomly modified at each of the 200 simulation runs, considering a uniform probability distribution over the range (−90◦ , 90◦ ). The CSG and ICSG algorithms are not considered in this example, since they have been originally formulated for scenarios with two interfering signals and also require individual estimation of each signal received at the base station. Results of this example are depicted in Fig. 7. The performance of the considered algorithms is compared in Fig. 7(a) in terms of the steady-state SINR obtained at each simulation run. For better visualization, the SINR values provided by the SB-ACVP algorithm are used as a reference in Fig. 7(a) for sorting the simulation runs. From this figure, one notices that the proposed SB-ACVP outperforms both AP-CSG and AP-QCSG algorithms in most of the runs. Moreover, in those runs in which the SB-ACVP is outperformed by the other algorithms, the difference of SINR is very small. The better performance of the proposed algorithm is confirmed by curves of the empirical complementary cumulative distribution function (CCDF) given in Fig. 7(b), which show that the SB-ACVP increases the probability of higher SINR levels as compared with AP-CSG and AP-QCSG algorithms. 5.2. Interference rejection in the downlink channel Aiming to assess the performance of the beamforming algorithms for downlink channels, we consider a multiuser scenario with fixed transmission powers and the same propagation geometry for uplink as was used for the downlink channels. Thus, the algorithms can be implemented in a distributed manner and (15) can be used for obtaining the beamforming coefficients. The scenario is formed by three mobile terminals (M = 3), each one as-

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C.A. Pitz et al. / Digital Signal Processing 97 (2020) 102621

Table 7 Example 4. Angles of arrival of the signals received at each base station from each mobile terminal.

T1 T2 T3

B1

B2

B3

0◦ −60◦ 60◦

75◦ 30◦ 70◦

30◦ −30◦ 60◦

Fig. 7. Example 3. (a) Steady-state SINR obtained at each simulation run. (b) Empirical CCDF of the steady-state SINR.

Fig. 9. Example 4. Radiation pattern comparison in the downlink channel of (a) [ B 1 , T 1 ], (b) [ B 2 , T 2 ], and (c) [ B 3 , T 3 ].

signed to a certain base station, resulting in [user, base station] pairs given by [T 1 , B 1 ], [T 2 , B 2 ], and [T 3 , B 3 ]. Moreover, we assume that each mobile terminal receives the signal transmitted by all base stations, and the signals coming from the non-assigned base stations are attenuated by 10 dB due to free space loss.

Fig. 8. Example 4. Comparison of the SINR curves obtained for the downlink channel of (a) [ B 1 , T 1 ], (b) [ B 2 , T 2 ], and (c) [ B 3 , T 3 ].

5.2.1. Example 4 This first downlink beamforming example involves three different base stations and therefore B 1 = B 2 = B 3 . The angles-of-arrival of the signals coming from each of the three different users (T 1 , T 2 , and T 3 ) and received at each base station are given in Table 7. From the downlink SINR curves shown in Fig. 8, we notice that the proposed SB-ACVP algorithm is capable of obtaining significantly higher SINR levels for all users as compared with the other algorithms considered here. The better performance of the proposed algorithm is corroborated by the radiation patterns presented in Fig. 9, showing that the SB-ACVP considerably reduces the power

C.A. Pitz et al. / Digital Signal Processing 97 (2020) 102621

Fig. 10. Example 5. Comparison of the SINR curves obtained for the downlink channel of (a) [ B 1 , T 1 ], (b) [ B 1 , T 2 ], and (c) [ B 1 , T 3 ].

radiated in the direction of the co-channel mobile terminals (indicated by ×) and provide the same power in the direction of the in-cell mobile terminal (indicated by ◦ ). 5.2.2. Example 5 Now, a single cell is considered, i.e., B 1 = B 2 = B 3 , referring to the same base station. The signal transmitted by such a base station is assumed to arrive at all mobile terminals (T 1 , T 2 , and T 3 ) with the same power, since they are located in the same cell. The angles-of-arrival of T 1 , T 2 , and T 3 are −25◦ , 60◦ , and 80◦ , respectively. As can be seen in Fig. 10, the SB-ACVP outperforms all the other evaluated algorithms by a significant margin in terms of steady-state SINR. In addition, Fig. 11 shows the resulting radiation patterns, confirming the better performance of the proposed SB-ACVP algorithm. 6. Concluding remarks The main focus of this research work was on the development of a new framework for deriving adaptive beamforming algorithms. Such a framework, named adaptive combination of vector projections (ACVP), was developed considering a unifying formulation for the mean-weight behavior of adaptive-projection CSG-type algorithms. This formulation makes it evident that a linear combination of vectors belonging to the subspaces spanned by the signals available at the array input can be exploited for developing ef-

11

Fig. 11. Example 5. Radiation pattern comparison in the downlink channel of (a) [ B 1 , T 1 ], (b) [ B 1 , T 2 ], and (c) [ B 1 , T 3 ].

fective beamforming algorithms. In this context, a first algorithm based on the proposed ACVP framework (termed SB-ACVP) was proposed, which outperforms competing algorithms from the open literature, presenting the best tradeoff between performance and computational cost. Numerical simulation results were shown confirming the validity of the ACVP framework, as well as corroborating the effectiveness of the SB-ACVP algorithm. Declaration of competing interest The authors declare that there is no conflict of interest concerning the manuscript submitted to Elsevier Digital Signal Processing. Acknowledgments The authors would like to thank the Handling Editor and the Reviewers, whose constructive comments and valuable suggestions have significantly benefited the quality of this paper. This research work was supported in part by the Brazilian National Council for Scientific and Technological Development (CNPq). References [1] Y. Zhang, C. Tepedelenlioglu, Transmit beamforming with power adaptation in downlink multi-user systems, IEEE Trans. Wirel. Commun. 9 (8) (Aug. 2010) 2424–2429.

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Ciro André Pitz received the B.S. degree in telecommunication engineering and the M.Sc. degree in electrical engineering from the Regional University of Blumenau, Brazil, in 2008 and 2010, respectively. In 2015, he received the Ph.D. degree in electrical engineering from Federal University of Santa Catarina, Brazil. From 2015 to 2017, he was a postdoctoral researcher at the LINSE–Circuits and Signal Processing Laboratory, Federal University of Santa Catarina, Brazil. In 2018, he joined the Department of Control, Automation and Computational Engineering at the Federal University of Santa Catarina, Brazil, where he is currently a Professor. His present research interests include adaptive signal processing theory and its application in communication systems. Eduardo Luiz Ortiz Batista received the B.S., M.Sc., and Ph.D. degrees from the Federal University of Santa Catarina, Florianópolis, Brazil, in 2002, 2004, and 2009, respectively, all in electrical engineering. Since 2010, he has been with the Federal University of Santa Catarina, where he is currently a Professor. He is also with the LINSE–Circuits and Signal Processing Laboratory, Federal University of Santa Catarina. His current research interests include nonlinear adaptive filtering, reduced-complexity adaptive algorithms, beamforming algorithms, active vibration control, and statistical analysis of adaptive filters. Rui Seara received the B.S. and M.Sc. degrees in electrical engineering from Federal University of Santa Catarina, Brazil, in 1975 and 1980, respectively. In 1984, he received the Doctoral degree in Electrical Engineering from the Paris-Sud University, Paris, France. He joined the Electrical Engineering Department at the Federal University of Santa Catarina, Brazil, in 1976, where he is currently a Professor of Electrical Engineering, and Director of LINSE–Circuits and Signal Processing Laboratory. His research interests include digital and analog filtering, adaptive signal processing algorithms, image and speech processing, and digital communications.

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Dennis R. Morgan received the B.S. degree, in 1965, from the University of Cincinnati, OH, and the M.S. and Ph.D. degrees from Syracuse University, Syracuse, NY, in 1968 and 1970, respectively, all in electrical engineering. From 1965 to 1984, he was with the General Electric Company, Electronics Laboratory, Syracuse, NY. From 1984 to 2014, he was a Distinguished Member of Technical Staff with Bell Laboratories, AlcatelLucent (formerly Lucent Technologies, formerly AT&T): from 1984 to 1990, he was with the Special Systems Analysis Department, Whippany NJ; from

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1990 to 2002, he was with the Acoustics Research Department, Murray Hill NJ; from 2002 to 2014, he was with Wireless Research, Murray Hill NJ. Since 2014, he has been a signal processing consultant, Morristown NJ. He has authored numerous journal publications and is coauthor of Active Noise Control Systems: Algorithms and DSP Implementations (New York: Wiley, 1996).