# Butler Matrix Frequency Diverse Retrodirective Array Beamforming: An Energy-Efficient Technique for mmWave Networks

Authors: Shaddrack Yaw Nusenu; Emmanuel Asare
Journal: Wireless Communications and Mobile Computing
 (2020)

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## Abstract

Millimeter-wave (mmWave) networks with the frequency spectrum ranging from 30 GHz (with wavelength 10 mm) to 300 GHz (with wavelength 1 mm), can support massive wireless data in fifth-generation (5G) systems. Importantly, large colocated antenna elements can be exploited at the base station (BS) to facilitate beam-steering synthesis along the users’ directions. This paper proposes the Butler matrix (BM) frequency diverse retrodirective array (BM-FDRA) beamforming network at the BS for multiuser communications in mmWave networks. We utilize the orthogonal feature of theM×M BM with M elements of the FDRA to create directional beams for concurrent transmission towards different users in range-angle locations. The proposed scheme has the following merits: (a) there is beam-steering orthogonality without beam interferences and (b) there is automatic tracking functionality, i.e., the prior knowledge on the user location is not required by the proposed BM-FDRA at the BS. The proposed method can serve multiple users concurrently using the beam-steering orthogonality property. Furthermore, performance metrics such as the multiaccess secrecy sum rate (SSR) model, bit error rate (BER), system capacity, and energy efficiency are examined. The proposed BM-FDRA scheme achievements are highlighted via simulation results.


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## Body

## 1. Introduction

The recent growth in mobile devices as well as the use of Internet mobile applications has called for higher data capacity. In particular, mmWave networks offered an attractive approach to meet this challenge due to their high-frequency spectrum from 30 GHz (with wavelength 10 mm) to 300 GHz (with wavelength 1 mm) [1–5]. In the fifth generation (5G), the employment of the large-frequency spectrum (i.e., mmWave bands) can be a realistic solution for communications [6–11]. However, since the mmWave has high attenuation (i.e., path loss), an antenna array with high directivity beam gain should be used to overcome the attenuation and absorption. Additionally, steerable and configurable antennas are required to support communications at different user locations [12]. Fortunately, because of small wavelength in the mmWave, the base station (BS) can accommodate large elements in a closely limited space [6]. Consequently, a suitable beamforming technique can be designed to enhance the energy efficiency and transmission range in mmWave networks [13].In mmWave networks, antenna design technologies, for example, multiple-input multiple-output (MIMO) and/or massive MIMO, and spatial division multiple access (SDMA) can be realized [6, 14]. In [15], the authors considered beamforming and SDMA techniques for downlink communications (i.e., IEEE 802.11ac).The existing mmWave communications in the literature focused on point-to-point application scenarios, for instance, wireless personal area network (WPAN) [16], cellular backhaul [17], or wireless local area network (WLAN) [18]. Also, another technique, called beamspace MIMO, has been investigated widely by several authors [19–22]. Very recently, the study [10] proposed the multiuser transmission technique in mmWave communications.Up to date, researches on mmWave communication applications adopted phased-array antennas (PAs). Though the PA can provide directional beam gain, only angular information directions are available for all the ranges (i.e., it has range-independent energy focusing). Note that mmWave propagation channels are usually correlated [23]. For instance, if the channels of two receiving users are highly correlated (i.e., colocated), using the PA, the signal-to-noise ratio (SNR) of the users may be degraded. Consequently, in certain mmWave practical applications, the ability to control users’ information along the angle-range-dependent dimension is extremely useful, especially in mmWave directional beam-steering, suppression of range interference, and user tracking and/or detection. Hence, it is necessary to consider new antenna array beam-steering for mmWave communications.Therefore, we propose the Butler matrix (BM) frequency diverse retrodirective array (BM-FDRA). BM [24, 25] is a Fourier transform technique which consists of M input ports and M output ports with passive four-port power couplers and fixed phase shifters. Note that the BM is another form of beamforming network which can be utilized to excite M array elements to steer the beams towards any of M spatial directions. The BM has been widely applied in communication applications [26, 27]. Another new beam array, namely, FDA proposed in [28], has a small frequency increment across each element to provide angular-range-time-dependent focusing [29]. In contrast, the PA lacks range-time-focusing transmit beam-steering. Actually, the FDA has been widely investigated in the radar fields [30–38]. Recently, because of FDA potentials, it has been utilized in communication applications [27, 39–43]. Thus, using the FDA, we achieve angular-range-time beam dependence. In addition, range-dependent interference or noise suppression is feasible.Moreover, we can improve the beam-steering-focusing ability by employing the retrodirective array (RDA). This is because the RDA has the merit of automatically returning a signal back towards the direction from where it originated. The RDA technology was overviewed in [44]. In the literature, the RDA has been adopted in several applications [45–51]. And this paper has taken the advantage of RDA features to mmWave communication applications for the first time.Therefore, motivated by the BM beam orthogonality feature, FDA angular-range-dependent focusing transmit beamforming, and RDA self-steering arrays (i.e., the prior knowledge on the user locations is not required for the BS), we propose an energy-efficient orthogonal beamforming network for concurrent transmissions towards different users using the Butler matrix (BM) frequency diverse retrodirective array (BM-FDRA) at the BS for mmWave communications. In summary, we present the main novelties as follows:(1)
We propose an energy-efficient directional orthogonal two-dimensional (2D) (i.e., angular-range dimension) beamforming network based on the Butler matrix (BM) frequency diverse retrodirective array (BM-FDRA) in mmWave communications. Note that the prior knowledge on the user locations is not required by the BS. Therefore, we can achieve better energy efficiency at the BS. The beamforming transmission is done concurrently without mutual interferences due to the orthogonality features of the generated beams. Therefore, we can serve multiple users within short range in mmWave communications, as depicted in Figure1.(2)
The secrecy sum rate (SSR) and bit error rate (BER) are analysed for the mmWave communications. Other performance metrics, such as system capacity and energy efficiency, have been examined. We make corresponding comparisons with the PA and other techniques in the literature to validate the achievements of the proposed BM-FDRA scheme in mmWave communications.Figure 1
Schematic illustration of the proposed BM-FDRA in mmWave communications: (a)4×4 BM-FDRA with orthogonal beams for serving four users concurrently; (b) 8×8 BM-FDRA with orthogonal beams for serving eight users concurrently. Note the proposed BM-FDRA attempts to transmit information using the orthogonal beams over the same frequency band simultaneously to different users. This system model can be extended to any M×M.
(a)
(b)


## 2. Brief Introduction of the Frequency Diverse Array (FDA)

### 2.1. FDA Signal Analysis

The FDA beamforming is a nascent array arrangement which has more degrees of freedom than PA beamforming. In the FDA, a small linear/nonlinear frequency incrementΔft is added across each element. Hence, we can formulate the signal radiated from each element as [29](1)smt=wmexp−j2πf0+Δfmtt,where wm denotes the transmit weight and fmt=f0+Δfmt is the radiated frequency, with m=0,1,2,…,M−1. The carrier frequency is denoted as f0, while a nonlinear frequency increment is denoted as Δfmt. In order to generate beamforming towards the desired user location, a propagation time delay constraint expressed as t∈r/c,Td+r/c is required [32], with Td being the pulse duration. Note that when Δfmt=mΔft, a linear frequency increment is obtained. The array factor (AF) of the standard FDA is determined as [29–32](2)AFθ,r,t=∑m=0M−1expj2πmdsinθλ×expj2πΔfmt−rmc⋅t−r−mdsinθc,where c, r, rm≈r−mdsinθ, and d are the light speed, range (i.e., distance), approximated range for the mth element, and element spacing, respectively. More importantly, when Δft=0, (2) can be reduced to the array factor (AF) of the PA determined as(3)AFθ,0,0=∑m=0M−1expj2πmdsinθλ.Using (2) and (3), respectively, we plot both FDA and PA beamforming in Figure 2. Interestingly, we can see in Figure 2(a) that FDA beamforming is very attractive than PA beamforming depicted in Figure 2(b). This is due to the fact that, with the FDA, we achieve decoupling beamforming at a prespecified user position, while in the PA, we achieve a beamforming with single angle information for all the ranges. Hence, FDA beam-steering is more useful in short-range mmWave communications. This is the motivation of this paper.Figure 2
(a) FDA beamforming alongθ,r=41∘,1000m; we adopted the log-frequency increment to support a few hundred meters in mmWave systems. (b) PA beamforming (i.e., only angle-focusing dependence with the same frequency f0 across the element index). Parameters: M = 16, f0=30GHz, and Δf=30kHz.
(a)
(b)


### 2.2. Brief Introduction of the Retrodirective Array (RDA)

Before deriving the proposed BM-FDRA signal formulations in mmWave systems in Section3, we present briefly the RDA principle. The capability of the RDA is that it can retransmit a signal back in the direction(s) along which it was illuminated [48]. This implies that the prior knowledge on the user location(s) is not required by the BS (transmitter). In the RDA, the phase conjugator (PC) is the core element which is used to aid automatic tracking. Herein, we use an example (see Figure 3) to explain briefly the basic principle of the RDA. We consider two users (i.e., user 1 and user 2) located at different positions. These users emit pilot signals, namely, s1t and s2t, at f0 (i.e., incident frequency). At the RDA side (mth element), the detected signal is determined as s1th1t+s2th2t, where h1t and h2t are the channel propagation for user 1 and user 2, respectively. After processing the detected signal via a PC system, we have s1th1t∗+s2th2t∗. Now, reradiating s1th1t∗+s2th2t∗ is weighted by the local signal Ψt at the RDA with the frequency fi. Hence, the received signal at user 1 is described as(4)y1tfi=∑m=0M−1Ψts1th1t∗+s2th2t∗h1t,where ∗ denotes the conjugate operation. When fi=f0, with the assumption of channel reciprocity, (4) can be reformulated as(5)y1tfi=f0=Ψts1∗t∑m=0M−1h1t2+Ψts2∗t∑m=0M−1h2∗th1t.Figure 3
Illustration of the RDA principle.Similarly, the received signal at user 2 can be derived and is omitted here for brevity. Note that, in casefi≠f0, the study [49] pointed out that we can compensate for their frequency differences by directly linking them to make (5) still hold for the proposed scheme.Herein, we highlight the achievements of the proposed BM-FDRA for mmWave networks which is distinct from the existing methods in the literature:(1)
There is no need for user(s) estimation/detection algorithm including CSI because of the adopted RDA in the proposed scheme. Furthermore, there is automatic tracking functionality, i.e., the prior knowledge on the user locations is not required by the BS (transmitter).(2)
Millimeter wave channels are correlated and not independent and identically distributed (i.i.d.) Rayleigh [23]. Hence, it is possible for the user channel to be highly correlated. But in the proposed BM-FDRA, due to the decoupling feature, such correlation is resolved.(3)
The proposed method can serve multiple users concurrently using the beam-steering orthogonality property because of the Butler matrix (BM) beamforming network.


## 2.1. FDA Signal Analysis

The FDA beamforming is a nascent array arrangement which has more degrees of freedom than PA beamforming. In the FDA, a small linear/nonlinear frequency incrementΔft is added across each element. Hence, we can formulate the signal radiated from each element as [29](1)smt=wmexp−j2πf0+Δfmtt,where wm denotes the transmit weight and fmt=f0+Δfmt is the radiated frequency, with m=0,1,2,…,M−1. The carrier frequency is denoted as f0, while a nonlinear frequency increment is denoted as Δfmt. In order to generate beamforming towards the desired user location, a propagation time delay constraint expressed as t∈r/c,Td+r/c is required [32], with Td being the pulse duration. Note that when Δfmt=mΔft, a linear frequency increment is obtained. The array factor (AF) of the standard FDA is determined as [29–32](2)AFθ,r,t=∑m=0M−1expj2πmdsinθλ×expj2πΔfmt−rmc⋅t−r−mdsinθc,where c, r, rm≈r−mdsinθ, and d are the light speed, range (i.e., distance), approximated range for the mth element, and element spacing, respectively. More importantly, when Δft=0, (2) can be reduced to the array factor (AF) of the PA determined as(3)AFθ,0,0=∑m=0M−1expj2πmdsinθλ.Using (2) and (3), respectively, we plot both FDA and PA beamforming in Figure 2. Interestingly, we can see in Figure 2(a) that FDA beamforming is very attractive than PA beamforming depicted in Figure 2(b). This is due to the fact that, with the FDA, we achieve decoupling beamforming at a prespecified user position, while in the PA, we achieve a beamforming with single angle information for all the ranges. Hence, FDA beam-steering is more useful in short-range mmWave communications. This is the motivation of this paper.Figure 2
(a) FDA beamforming alongθ,r=41∘,1000m; we adopted the log-frequency increment to support a few hundred meters in mmWave systems. (b) PA beamforming (i.e., only angle-focusing dependence with the same frequency f0 across the element index). Parameters: M = 16, f0=30GHz, and Δf=30kHz.
(a)
(b)


## 2.2. Brief Introduction of the Retrodirective Array (RDA)

Before deriving the proposed BM-FDRA signal formulations in mmWave systems in Section3, we present briefly the RDA principle. The capability of the RDA is that it can retransmit a signal back in the direction(s) along which it was illuminated [48]. This implies that the prior knowledge on the user location(s) is not required by the BS (transmitter). In the RDA, the phase conjugator (PC) is the core element which is used to aid automatic tracking. Herein, we use an example (see Figure 3) to explain briefly the basic principle of the RDA. We consider two users (i.e., user 1 and user 2) located at different positions. These users emit pilot signals, namely, s1t and s2t, at f0 (i.e., incident frequency). At the RDA side (mth element), the detected signal is determined as s1th1t+s2th2t, where h1t and h2t are the channel propagation for user 1 and user 2, respectively. After processing the detected signal via a PC system, we have s1th1t∗+s2th2t∗. Now, reradiating s1th1t∗+s2th2t∗ is weighted by the local signal Ψt at the RDA with the frequency fi. Hence, the received signal at user 1 is described as(4)y1tfi=∑m=0M−1Ψts1th1t∗+s2th2t∗h1t,where ∗ denotes the conjugate operation. When fi=f0, with the assumption of channel reciprocity, (4) can be reformulated as(5)y1tfi=f0=Ψts1∗t∑m=0M−1h1t2+Ψts2∗t∑m=0M−1h2∗th1t.Figure 3
Illustration of the RDA principle.Similarly, the received signal at user 2 can be derived and is omitted here for brevity. Note that, in casefi≠f0, the study [49] pointed out that we can compensate for their frequency differences by directly linking them to make (5) still hold for the proposed scheme.Herein, we highlight the achievements of the proposed BM-FDRA for mmWave networks which is distinct from the existing methods in the literature:(1)
There is no need for user(s) estimation/detection algorithm including CSI because of the adopted RDA in the proposed scheme. Furthermore, there is automatic tracking functionality, i.e., the prior knowledge on the user locations is not required by the BS (transmitter).(2)
Millimeter wave channels are correlated and not independent and identically distributed (i.i.d.) Rayleigh [23]. Hence, it is possible for the user channel to be highly correlated. But in the proposed BM-FDRA, due to the decoupling feature, such correlation is resolved.(3)
The proposed method can serve multiple users concurrently using the beam-steering orthogonality property because of the Butler matrix (BM) beamforming network.


## 3. Proposed Butler Matrix (BM) Frequency Diverse Retrodirective Array (BM-FDRA) System Model

Figure4 illustrates the proposed BM-FDRA block diagram, while Figure 5 shows the proposed 4×4 BM-FDRA transmitter architecture at the BS. We employ four FDRA array elements connected at the 4×4 Butler matrix (note that it can be extended to any M×M) output with carrier frequencies, namely, 2f0+Δf0t, 2f0+Δf1t, 2f0+Δf2t, and 2f0+Δf3t, respectively. The operation is briefly described as follows: When Port 1 gets excited, the signal path (i.e., A to B to C to D) goes to 2f0+Δf0t and produces 135∘ phase shift. Likewise, we can have 90∘ phase shift between Port 1 and 2f0+Δf1t for the signal path (i.e., A to B to C to E) and so on. For an ideal 4×4 BM, output port 1 and 4 phase differences and output port 2 and 3 phase differences are ±45∘ and ±135∘, respectively [52].Figure 4
Proposed BM-FDRA block system design.Figure 5
Architecture of the proposed4×4 BM-FDRA system.In Figure6, we have shown the antenna configuration of the FDRA connected at the output of the BM. Note that the principle is described in Section 2.2. Importantly, we exploit both the FDA and the RDA to achieve unique properties in mmWave communications. Furthermore, time slots are designed at the BS to facilitate information beamforming along the respective user locations (see Figure 7).Figure 6
Illustration of the FDRA antenna scheme connected at the output of the Butler matrix (BM) system.Figure 7
Proposed BM-FDRA time frame structure at the BS (phase I, i.e.,Tτ for user pilot transmissions to the BM-FDRA transmitter, and phase II, i.e., 1−Tτ BM-FDRA transmitter transmits orthogonal beams to the users).To proceed with the proposed scheme derivations, we make the following standard assumptions in this paper:(1)
The user(s) radiates pilot signaling to the BS.(2)
There is perfect synchronization between the proposed BM-FDRA transmitter at the BS and the users.(3)
Due to the short transmission distance in mmWave communications, we focus on transmitting orthogonal beams to multiple users in range-angle-focusing directions. It should be noted that we ignore the transmission distance effects in this present work.


## 4. 2-Dimensional (Range-Angle-Focusing) Orthogonal Beamforming for mmWave Networks

In this paper, we assume a multiuser communication in mmWave networks. The proposed BM-FDRA transmitter at the BS is equipped with anM-element antenna that transmits orthogonal beams concurrently with information to the respective users. Herein, the users are equipped with a single receive antenna, as illustrated in Figures 5 and 7.The designed time frame structure depicted in Figure7 has two transmission phases. Phase I is allocated time duration Tτ, and phase II is also allocated time duration 1−Tτ. During phase I, the user(s) transmits pilot signals to the BS. This enables downlink channel estimations. In contrast, in phase II, the BS retransmits the signals with modulated communication data using directional orthogonal beams concurrently to the respective users within the short-range-angle dimension in mmWave networks.


### 4.1. BM-FDRA Range-Angle Focusing

During phase I, we assume that the user(s) emits phase of thekth k=0,1,2,…,K−1 pilot signal which is received by the proposed BM-FDRA at the BS with the mth m=0,1,…,M−1 array element at a particular time t. The received pilot signals can be determined as(6)Ψk,mt=2πf0t−rk,mc,where rk,m=r˜−mdsinθink, with r˜ being the range between the kth user and the proposed BM-FDRA first reference element. θink (i.e., kth pilot signal’s incoming angle) originated from the kth user. The proposed BM-FDRA mth array detects the signals which mix with the local oscillator (LO) signal fLOt=2f0+Δfmt, where Δfmt denotes the frequency increment (i.e., Δfm≪f0).In phase II, the retransmitted signals by the proposed BM-FDRA at the BS can be written as(7)ynt;R,θ=∑k=0K−1wm⋅atTmn⋅Bnexpj2πf0+Δfmt−Rk,mc+f0rk,mc,where Bn is the nth Butler matrix input port excitation, wm is the phase weighting, as shown in Figure 6, at is the communication information, and Tmn is defined as [26, 27](8)Tmn=1M⋅exp−j2πMm−M+12n−M+12,m,n=0,1,…,M−1.Note that the noise term was not considered. AndRk,m=R−mdsinθ has the same physical meaning as in (2). The main beam arriving at each user location can be determined in (9), and we assume that at≈at−Rk,m/c is a slow signal.(9)ynkt;R,θ=θink=at−Rk,mcexpj2πf0t−R−r˜c∑k=0K−1∑m=0M−1wmTmn⋅Bnexpj2πΔfmt−Rk,mct−R−r˜−mdsinθc+f0mdsinθ−sinθinkc.It is important to mention that the expressions in (7) and (9), respectively, are based on the nth input port excitation Butler matrix. Hence, the main beam is directed towards Rn,θn directions.Similarly, by replacingn in (7) and (9), respectively, with q, we can also derive expressions for other user locations based on the qth Butler matrix input port excitation with the main beam pointing towards Rq,θq directions as(10)yqkt;R,θ=θink=at−Rk,mcexpj2πf0t−R−r˜c∑k=0K−1∑m=0M−1wmTmq⋅Bqexpj2πΔfmt−Rk,mct−R−r˜−mdsinθc+f0mdsinθ−sinθinkc.It is easy to verify thatynkt;Rq,θq=θink=0 when n≠q. This implies that the yqkt;R,θ=θink main beam is projected along the null radiation direction of ynkt;R,θ=θink. Using (9) and (10), we can achieve orthogonal range-angle beamforming to realize multiuser transmissions in mmWave networks. This implies that the ynkt;R,θ=θink main beam is pointing along user 1 located at a different position, while the yqkt;R,θ=θink main beam is pointing along the null direction towards user 2 concurrently. Note that the issue of mutual interference is alleviated in the proposed BM-FDRA due to the orthogonality property of beamforming.


## 4.1. BM-FDRA Range-Angle Focusing

During phase I, we assume that the user(s) emits phase of thekth k=0,1,2,…,K−1 pilot signal which is received by the proposed BM-FDRA at the BS with the mth m=0,1,…,M−1 array element at a particular time t. The received pilot signals can be determined as(6)Ψk,mt=2πf0t−rk,mc,where rk,m=r˜−mdsinθink, with r˜ being the range between the kth user and the proposed BM-FDRA first reference element. θink (i.e., kth pilot signal’s incoming angle) originated from the kth user. The proposed BM-FDRA mth array detects the signals which mix with the local oscillator (LO) signal fLOt=2f0+Δfmt, where Δfmt denotes the frequency increment (i.e., Δfm≪f0).In phase II, the retransmitted signals by the proposed BM-FDRA at the BS can be written as(7)ynt;R,θ=∑k=0K−1wm⋅atTmn⋅Bnexpj2πf0+Δfmt−Rk,mc+f0rk,mc,where Bn is the nth Butler matrix input port excitation, wm is the phase weighting, as shown in Figure 6, at is the communication information, and Tmn is defined as [26, 27](8)Tmn=1M⋅exp−j2πMm−M+12n−M+12,m,n=0,1,…,M−1.Note that the noise term was not considered. AndRk,m=R−mdsinθ has the same physical meaning as in (2). The main beam arriving at each user location can be determined in (9), and we assume that at≈at−Rk,m/c is a slow signal.(9)ynkt;R,θ=θink=at−Rk,mcexpj2πf0t−R−r˜c∑k=0K−1∑m=0M−1wmTmn⋅Bnexpj2πΔfmt−Rk,mct−R−r˜−mdsinθc+f0mdsinθ−sinθinkc.It is important to mention that the expressions in (7) and (9), respectively, are based on the nth input port excitation Butler matrix. Hence, the main beam is directed towards Rn,θn directions.Similarly, by replacingn in (7) and (9), respectively, with q, we can also derive expressions for other user locations based on the qth Butler matrix input port excitation with the main beam pointing towards Rq,θq directions as(10)yqkt;R,θ=θink=at−Rk,mcexpj2πf0t−R−r˜c∑k=0K−1∑m=0M−1wmTmq⋅Bqexpj2πΔfmt−Rk,mct−R−r˜−mdsinθc+f0mdsinθ−sinθinkc.It is easy to verify thatynkt;Rq,θq=θink=0 when n≠q. This implies that the yqkt;R,θ=θink main beam is projected along the null radiation direction of ynkt;R,θ=θink. Using (9) and (10), we can achieve orthogonal range-angle beamforming to realize multiuser transmissions in mmWave networks. This implies that the ynkt;R,θ=θink main beam is pointing along user 1 located at a different position, while the yqkt;R,θ=θink main beam is pointing along the null direction towards user 2 concurrently. Note that the issue of mutual interference is alleviated in the proposed BM-FDRA due to the orthogonality property of beamforming.


## 5. Theoretical Performance Evaluation

### 5.1. Secrecy Rate Analysis

To evaluate the proposed BM-FDRA scheme secrecy performance, we employ the secrecy sum rate (SSR) [53]. The proposed BM-FDRA SSR can be determined as(11)Cs=∑k=0K−1Ckθk,Rk−maxpCPθep,Rep+.The operatorx+ returns zero if x is negative or else x is returned. And the secrecy rate from the proposed BM-FDRA to the kth desired user can be computed as(12)Ckθdk,Rdk≜Iyθdk,Rdk;ak,θdk,Rdk=log21+γθdk,Rdk,where ak is the receiving useful information. The instantaneous secrecy capacity at the kth user is given as γθdk,Rdk=σdk−2wHaθdk,Rdk2.Similarly, the secrecy rate from the proposed BM-FDRA to thepth eavesdropper ep is determined as(13)Cpθep,Rep≜Iyθep,Rep;ak,θep,Rep=log21+γθep,Rep.The instantaneous secrecy capacity at thepth eavesdropper is γθep,Rep=σep−2wHaθep,Rep2. Note that σdk−2 and σep−2 are the noise powers at desired users and eavesdroppers, respectively.


### 5.2. Bit Error Rate (BER) Analysis

To evaluate the BER of the proposed BM-FDRA scheme, we adopt the quadrature phase-shift keying (QPSK) modulation format. According to the study in [54], we can compute the symbol error probability of QPSK as(14)PQPSK=erfcEs/N02−4−1⋅erfc2Es/N02,where erfc is the complementary error function, determined as erfcx=2/π∫x∞exp−t2dt. Since the proposed BM-FDRA utilizes beamforming information-based transmission, the channel gain will be the square of the absolute array far-field beampattern [55]. Hence, we can write the average symbol error probability as(15)Peθ,R,t=erfcEs/N0AFBM−FDRAθ,R,t22−14erfc2Es/N0AFBM−FDRAθ,R,t22,where  is the expectation operator taken with respect to the number of beams.


### 5.3. Capacity Evaluation

The primary objective of wireless networks is to provide massive connectivity of users with a higher information rate. It is well known that, by improving the signal-to-interference-plus-noise-ratio (SINR), the information rate can be enhanced. The proposed BM-FDRA scheme has the capability to improve the SINR as well as the capacity of the mmWave communication link. The capacity can be expressed as(16)Cy=log21+sinR.Since the proposed BM-FDRA scheme depends on angle, range, and time dimensions, we can derive the SINR in three different ways as follows:(17)SINRBM−FDRA=Φs2wHar,tΦn2wHw,(18)SINRBM−FDRA=Φs2wHaθ=θin,tΦn2wHw,(19)SINRBM−FDRA=Φs2wHaθ=θin,tΦn2wHw.For the comparison purpose, the PA SINR is given as(20)SINRPA=Φs2wHaθ=θinΦn2wHw,where SNR=Φs2/Φn2, with Φs2 being the signal power and Φn2 being the noise power. It is important to mention that the main difference between (17)–(19) and (20) is that the PA method provides the angle-based SINR only, while the proposed BM-FDRA scheme provides the angle-range-time-based SINR. Furthermore, interferences/noise can be suppressed better in the proposed BM-FDRA scheme than in the PA method, thus improving the capacity.


## 5.1. Secrecy Rate Analysis

To evaluate the proposed BM-FDRA scheme secrecy performance, we employ the secrecy sum rate (SSR) [53]. The proposed BM-FDRA SSR can be determined as(11)Cs=∑k=0K−1Ckθk,Rk−maxpCPθep,Rep+.The operatorx+ returns zero if x is negative or else x is returned. And the secrecy rate from the proposed BM-FDRA to the kth desired user can be computed as(12)Ckθdk,Rdk≜Iyθdk,Rdk;ak,θdk,Rdk=log21+γθdk,Rdk,where ak is the receiving useful information. The instantaneous secrecy capacity at the kth user is given as γθdk,Rdk=σdk−2wHaθdk,Rdk2.Similarly, the secrecy rate from the proposed BM-FDRA to thepth eavesdropper ep is determined as(13)Cpθep,Rep≜Iyθep,Rep;ak,θep,Rep=log21+γθep,Rep.The instantaneous secrecy capacity at thepth eavesdropper is γθep,Rep=σep−2wHaθep,Rep2. Note that σdk−2 and σep−2 are the noise powers at desired users and eavesdroppers, respectively.


## 5.2. Bit Error Rate (BER) Analysis

To evaluate the BER of the proposed BM-FDRA scheme, we adopt the quadrature phase-shift keying (QPSK) modulation format. According to the study in [54], we can compute the symbol error probability of QPSK as(14)PQPSK=erfcEs/N02−4−1⋅erfc2Es/N02,where erfc is the complementary error function, determined as erfcx=2/π∫x∞exp−t2dt. Since the proposed BM-FDRA utilizes beamforming information-based transmission, the channel gain will be the square of the absolute array far-field beampattern [55]. Hence, we can write the average symbol error probability as(15)Peθ,R,t=erfcEs/N0AFBM−FDRAθ,R,t22−14erfc2Es/N0AFBM−FDRAθ,R,t22,where  is the expectation operator taken with respect to the number of beams.


## 5.3. Capacity Evaluation

The primary objective of wireless networks is to provide massive connectivity of users with a higher information rate. It is well known that, by improving the signal-to-interference-plus-noise-ratio (SINR), the information rate can be enhanced. The proposed BM-FDRA scheme has the capability to improve the SINR as well as the capacity of the mmWave communication link. The capacity can be expressed as(16)Cy=log21+sinR.Since the proposed BM-FDRA scheme depends on angle, range, and time dimensions, we can derive the SINR in three different ways as follows:(17)SINRBM−FDRA=Φs2wHar,tΦn2wHw,(18)SINRBM−FDRA=Φs2wHaθ=θin,tΦn2wHw,(19)SINRBM−FDRA=Φs2wHaθ=θin,tΦn2wHw.For the comparison purpose, the PA SINR is given as(20)SINRPA=Φs2wHaθ=θinΦn2wHw,where SNR=Φs2/Φn2, with Φs2 being the signal power and Φn2 being the noise power. It is important to mention that the main difference between (17)–(19) and (20) is that the PA method provides the angle-based SINR only, while the proposed BM-FDRA scheme provides the angle-range-time-based SINR. Furthermore, interferences/noise can be suppressed better in the proposed BM-FDRA scheme than in the PA method, thus improving the capacity.


## 6. Simulation and Discussion

In this section, we present several simulations to highlight the potentials of the proposed BM-FDRA scheme in mmWave communication networks.(a)
The proposed4×4 BM-FDRA beamforming network towards k users at distinct locations in mmWave communications: in Figures 8(a) and 8(b), we have shown the proposed BM-FDRA beamforming networks in angle and range profiles, respectively. Here, four users can be considered with four input port excitations, “1,” “2,” “3,” and “4,” respectively. We can see that, in the angle dimension, as depicted in Figure 8(a), we achieve orthogonal beams along four distinct angle directions, namely, 41∘, 76∘, 104∘, and 139∘. Similarly, in Figure 8(b), orthogonal beams along different range values are generated, namely, 6.2km, 8.9km, 11.2km, and 13.9km. Note that based on Figure 8, we can assign users to each directional beam direction.(b)
The proposed8×8 BM-FDRA beamforming network towards k users at distinct locations in mmWave communications: in Figure 9, we can employ 8 users using the proposed BM-FDRA in mmWave networks. Note that Figures 9(a) and 9(b) give a similar trend, as shown in Figure 8. It is important to remark that the proposed BM-FDRA scheme can be extended to any M×M orthogonal beam-steering.(c)
BER beamforming along a particular user: suppose one of the users in the mmWave network is located at41∘,6.2km. Figure 10 demonstrates the BER beamforming along such a location in angle and range dimensions. We observe that Figure 10(a) (i.e., angle dimension) has narrow beamwidth along 41∘ with relatively low sidelobes which can minimize interference with other users. Similarly, in the range dimension along 6.2km (see Figure 10(b)), we notice a similar trend.(d)
The proposed BM-FDRA SSR analysis: Figure11 presents the curves of the SSR against the SNR assuming the desired user and eavesdropper are located in the same direction but distinct ranges. From the figure, the PA SSR curve is zero. This is because the PA is range-independent, hence achieving poor SSR. In contrast, the proposed BM-FDRA scheme with different frequency increment values, namely, Δf=10, Δf=20, and Δf=30, respectively, has better SSR curves than the PA curve. This is due to the fact that the proposed BM-FDRA has the ability to discriminate between two users along angle-range focusing. Additionally, it can be seen that increasing the values of frequency increments has better influence on the SSR curves. Thus, SSR performance can be enhanced significantly.(e)
Energy efficiency analysis: finally, the proposed BM-FDRA energy efficiency is evaluated. According to the study in [56], we can define the energy efficiency Ξ as(21)Ξ=CsPtotal,whereCs is defined in (11) and Ptotal=Pt+MRFPRF+Pbaseband, with Pt denoting the total transmitted power, PRF denoting the power consumption for the radio frequency (RF) chain, MRF denoting the number of RF chains, and Pbaseband denoting the power consumption of the baseband. Herein, we adopt the values from the study [56] as follows: Pt=1W, PRF=300mW, and Pbaseband=200mW. In Figure 12, we show energy efficiency versus SNR. We assume eight users. From the figure, we noticed that the proposed scheme can achieve higher energy efficiency compared to the methods in [10, 22]. More importantly, the proposed scheme further reduced the power consumption since it required a single RF chain [57].(f)
Capacity discussion: Figure13 depicts the cumulative density function (CDF) curves with two distinct SNR values: 5 dB and 20 dB, with different receive antennas Nr and fixed transmit antennas Nt. We notice that as we increase the SNR values, the CDF curves shift to the right side. This implies that when the SINR is improved, the capacity of the system will be enhanced greatly.Figure 8
Illustration of the proposed4×4 BM-FDRA beamforming network along k users in the (a) angle profile and (b) range profile. Parameters: M = 4, f0=30GHz, and Δf=20kHz.
(a)
(b)Figure 9
Illustration of the8×8 beam-steering towards k users in the (a) angle profile and (b) range profile. Parameters: M = 8, f0=30GHz, and Δf=20kHz.
(a)
(b)Figure 10
BER distribution beamforming in the (a) angle profile and (b) range profile. Parameters: user location41∘,6.2km, SNR = 6 dB, f0=30GHz, and Δf=10kHz.
(a)
(b)Figure 11
SSR as a function of SNR with different frequency increment values:Δf=0 (PA), Δf=10, Δf=20, and Δf=30, for the desired user 41∘,6.2km and eavesdropper 41∘,7.5km.Figure 12
Energy efficiency versus SNR. We assume 8 users.Figure 13
Cumulative density function (CDF) curves of capacity for different SNR values: 5 dB and 20 dB.In Table1, we have tabulated the comparisons between the proposed BM-FDRA scheme and the similar representation beamspace schemes investigated in [10, 22].Table 1
Summary of characteristics of the proposed BM-FDRA scheme and methods in [10, 22].
Characteristics
Proposed BM-FDRA
[10]
[22]
Energy efficiency
Very improved
Improved
Good
Automatic self-tracking
Yes
No
No
Range-angle beamforming
Both
Angle only
Angle only
User location
Precise location
Assumed
Assumed
Gain flexibility adjustment
Yes
No
No
Beam orthogonality property
Yes
No
No
Range-dependent interference suppression
Good
No
No


## 7. Conclusion

This paper investigated a different method to achieve multiuser communications in mmWave networks using the Butler matrix (BM) frequency diverse retrodirective array (BM-FDRA). The proposed BM-FDRA can receive and retransmit information at distinct frequencies towards the users. Furthermore, focused angle-range beamforming with the orthogonal property (i.e., no mutual interferences) can be generated by the proposed BM-FDRA to serve multiple users concurrently. The proposed scheme was theoretically evaluated by the following performance metrics, namely, secrecy sum rate, bit error rate (BER), energy efficiency, and system capacity, from the user’s perspective. We verify the achievements of the proposed BM-FDRA over the phased-array scheme and methods in [10, 22], respectively, via simulation results. Herein, we realize that the fundamental limitation of the proposed BM-FDRA scheme is that the number of desired users that can be supported by the system cannot be larger than M×M. One possible solution as a future work is to integrate the nonorthogonal multiple-access (NOMA) concept. Thus, the number of users that can be supported by the proposed BM-FDRA scheme can be larger than M×M.



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*Source: 9408519-2020-02-22.xml*