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

Millimeter-wave (mmWave) networks with the frequency spectrum ranging from 30 GHz (with wavelength 10mm) to 300 GHz (with wavelength 1mm), 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. *is 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 the M × M BM withM elements of the FDRA to create directional beams for concurrent transmission towards different users in range-angle locations. *e 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. *e 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. *e proposed BM-FDRA scheme achievements are highlighted via simulation results.


Introduction
e 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 highfrequency spectrum from 30 GHz (with wavelength 10 mm) to 300 GHz (with wavelength 1 mm) [1][2][3][4][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][7][8][9][10][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].
Up to date, researches on mmWave communication applications adopted phased-array antennas (PAs). ough 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 signalto-noise ratio (SNR) of the users may be degraded. Consequently, in certain mmWave practical applications, the ability to control users' information along the angle-rangedependent 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. erefore, 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. e 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][31][32][33][34][35][36][37][38]. Recently, because of FDA potentials, it has been utilized in communication applications [27,[39][40][41][42][43]. us, using the FDA, we achieve angular-rangetime 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). is is because the RDA has the merit of automatically returning a signal back towards the direction from where it originated. e RDA technology was overviewed in [44]. In the literature, the RDA has been adopted in several applications [45][46][47][48][49][50][51]. And this paper has taken the advantage of RDA features to mmWave communication applications for the first time. erefore, 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. erefore, we can achieve better energy efficiency at the BS. e beamforming transmission is done concurrently without mutual interferences due to the orthogonality features of the generated beams. erefore, we can serve multiple users within short range in mmWave communications, as depicted in Figure 1. (2) e 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.

FDA Signal Analysis.
e FDA beamforming is a nascent array arrangement which has more degrees of freedom than PA beamforming. In the FDA, a small linear/nonlinear frequency increment Δf(t) is added across each element. Hence, we can formulate the signal radiated from each element as [29] where w m denotes the transmit weight and f m (t) � f 0 + Δf m (t) is the radiated frequency, with m � 0, 1, 2, . . . , M − 1. e carrier frequency is denoted as f 0 , while a nonlinear frequency increment is denoted as Δf m (t). In order to generate beamforming towards the desired user location, a propagation time delay constraint expressed as t ∈ [(r/c), T d + (r/c)] is required [32], with T d being the pulse duration. Note that when Δf m (t) � mΔf(t), a linear frequency increment is obtained. e array factor (AF) of the standard FDA is determined as [29][30][31][32] where c, r, r m ≈ r − md sin θ, and d are the light speed, range (i.e., distance), approximated range for the mth element, and element spacing, respectively. More importantly, when Δf(t) � 0, (2) can be reduced to the array factor (AF) of the PA determined as 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). is 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. is is the motivation of this paper.

Brief Introduction of the Retrodirective Array (RDA).
Before deriving the proposed BM-FDRA signal formulations in mmWave systems in Section 3, we present briefly the RDA principle. e capability of the RDA is that it can retransmit a signal back in the direction(s) along which it was illuminated [48]. is 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. ese users emit pilot signals, namely, s 1 (t) and s 2 (t), at f 0 (i.e., incident frequency). At the RDA side (m th element), the detected signal is determined as where h 1 (t) and h 2 (t) are the channel propagation for user 1 and user 2, respectively. After processing the detected signal via a PC system, we have [s * is weighted by the local signal Ψ(t) at the RDA with the frequency f i . Hence, the received signal at user 1 is described as where * denotes the conjugate operation. When f i � f 0 , with the assumption of channel reciprocity, (4) can be reformulated as Similarly, the received signal at user 2 can be derived and is omitted here for brevity. Note that, in case f i ≠ f 0 , 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) ere 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) e proposed method can serve multiple users concurrently using the beam-steering orthogonality property because of the Butler matrix (BM) beamforming network. array elements connected at the 4 × 4 Butler matrix (note that it can be extended to any M × M) output with carrier frequencies, namely, 2f 0 + Δf 0 (t), 2f 0 + Δf 1 (t), 2f 0 + Δf 2 (t), and 2f 0 + Δf 3 (t), respectively. e 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 2f 0 + Δf 0 (t) and produces 135 ∘ phase shift. Likewise, we can have 90 ∘ phase shift between Port 1 and 2f 0 + Δf 1 (t) 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].

Proposed Butler Matrix (BM) Frequency Diverse Retrodirective Array (BM-FDRA) System Model
In Figure 6, 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).
To proceed with the proposed scheme derivations, we make the following standard assumptions in this paper: (1) e user(s) radiates pilot signaling to the BS.
(2) ere 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.

2-Dimensional (Range-Angle-Focusing) Orthogonal Beamforming for mmWave Networks
In this paper, we assume a multiuser communication in mmWave networks. e proposed BM-FDRA transmitter at the BS is equipped with an M-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    is 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.

BM-FDRA Range-Angle Focusing.
During phase I, we assume that the user(s) emits phase of the kth (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.
e received pilot signals can be determined as where r k,m � r − md sin θ k in , with r being the range between the kth user and the proposed BM-FDRA first reference element. θ k in (i.e., kth pilot signal's incoming angle) originated from the kth user. e proposed BM-FDRA mth array detects the signals which mix with the local oscillator (LO) signal f LO (t) � 2f 0 + Δf m (t), where Δf m (t) denotes the frequency increment (i.e., Δf m ≪ f 0 ).
In phase II, the retransmitted signals by the proposed BM-FDRA at the BS can be written as where B n is the nth Butler matrix input port excitation, w m is the phase weighting, as shown in Figure 6, a(t) is the communication information, and T mn is defined as [26,27] Note that the noise term was not considered. And R k,m � R − md sin θ has the same physical meaning as in (2). e main beam arriving at each user location can be determined in (9), and we assume that a(t) ≈ a(t − (R k,m /c)) is a slow signal.
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 (R n , θ n ) directions.
Similarly, by replacing n 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 (R q , θ q ) directions as It is easy to verify that y k n (t; R q , θ q � θ k in ) � 0 when n ≠ q. is implies that the y k q (t; R, θ � θ k in ) main beam is projected along the null radiation direction of y k n (t; R, θ � θ k in ). Using (9) and (10), we can achieve orthogonal range-angle beamforming to realize multiuser transmissions in mmWave networks. is implies that the y k n (t; R, θ � θ k in ) main beam is pointing along user 1 located at a different position, while the y k q (t; R, θ � θ k in ) 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.

Secrecy Rate Analysis.
To evaluate the proposed BM-FDRA scheme secrecy performance, we employ the secrecy sum rate (SSR) [53]. e proposed BM-FDRA SSR can be determined as e operator [x] + 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 where a k is the receiving useful information. e instantaneous secrecy capacity at the kth user is given as c(θ d k , Similarly, the secrecy rate from the proposed BM-FDRA to the pth eavesdropper e p is determined as C p θ e p , R e p ≜ I y θ e p , R e p ; a k , θ e p , R e p � log 2 1 + c θ e p , R e p . (13) e instantaneous secrecy capacity at the pth eavesdropper is c(θ e p , R e p ) � σ − 2 e p |w H a(θ e p , R e p )| 2 . Note that σ − 2 d k and σ − 2 e p are the noise powers at desired users and eavesdroppers, respectively.

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 where erfc is the complementary error function, determined as erfc( 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 where E is the expectation operator taken with respect to the number of beams.

Capacity Evaluation.
e 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. e proposed BM-FDRA scheme has the capability to improve the SINR as well as the capacity of the mmWave communication link. e capacity can be expressed as Since the proposed BM-FDRA scheme depends on angle, range, and time dimensions, we can derive the SINR in three different ways as follows: For the comparison purpose, the PA SINR is given as where SNR � Φ 2 s /Φ 2 n , with Φ 2 s being the signal power and Φ 2 n 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-rangetime-based SINR. Furthermore, interferences/noise can be suppressed better in the proposed BM-FDRA scheme than in the PA method, thus improving the capacity.

Simulation and Discussion
In this section, we present several simulations to highlight the potentials of the proposed BM-FDRA scheme in mmWave communication networks.

Wireless Communications and Mobile Computing
(a) e proposed 4 × 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, 6.2 km, 8.9 km, 11.2 km, and 13.9 km. Note that based on Figure 8, we can assign users to each directional beam direction. (c) BER beamforming along a particular user: suppose one of the users in the mmWave network is located at (41 ∘ , 6.2 km). 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 6.2 km (see Figure 10(b)), we notice a similar trend. (d) e proposed BM-FDRA SSR analysis: Figure 11 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. is 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. is 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 Wireless Communications and Mobile Computing the values of frequency increments has better influence on the SSR curves. us, 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 where C s is defined in (11) and P total � P t + M RF P RF + P baseband , with P t denoting the total transmitted power, P RF denoting the power consumption for the radio frequency (RF) chain, M RF denoting the number of RF chains, and P baseband denoting the power consumption of the baseband. Herein, we adopt the values from the study [56] as follows: P t � 1 W, P RF � 300 mW, and P baseband � 200 mW. 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: Figure 13 depicts the cumulative density function (CDF) curves with two distinct SNR values: 5 dB and 20 dB, with different receive antennas N r and fixed transmit antennas N t . We notice that as we increase the SNR values, the CDF curves shift to the right side. is implies that when the SINR is improved, the capacity of the system will be enhanced greatly.
In Table 1, we have tabulated the comparisons between the proposed BM-FDRA scheme and the similar representation beamspace schemes investigated in [10,22].

Conclusion
is paper investigated a different method to achieve multiuser communications in mmWave networks using the Butler matrix (BM) frequency diverse retrodirective array (BM-FDRA). e 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. e 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. us, the number of users that can be supported by the proposed BM-FDRA scheme can be larger than M × M.

Data Availability
No data were used to support this study.