Revisiting smart antenna array design with multiple interferers using basic adaptive beamforming algorithms: Comparative performance study with testbed results

Smart antennas are becoming popular in the area of cellular wireless communication for capacity enhancement and reduction of the multipath effect and interference. The demand for smart antennas is widely increasing as 5G cellular communication evolves to support the higher data rate and bandwidth. The basic principle of smart antenna design is adaptive beamforming using the best suited digital signal processing algorithms, such as least mean square (LMS), normalized least mean square (NLMS), sample matrix inversion (SMI), and recursive least square (RLS), each having its pros and cons. Among these, the LMS and NLMS are iterative approaches while SMI is a block adaptive method and RLS is a recursive method. The contribution of this article includes easy implementation of four adaptive beamforming algorithms, namely LMS, NLMS, SMI, and RLS. Furthermore, an exhaustive comparative performance analysis is carried out under five interferers and evaluated in terms of beamwidth, null depth, maximum sidelobe level, rate of convergence, error variation for the number of antenna elements, and spacing. Finally, a contrast table is presented to demonstrate the pros and cons of the listed algorithms. Time and dynamic space complexity are studied for RLS and SMI beamforming algorithms. Performance results from a designed reconfigurable testbed model for weight adaptation and smart beamforming are also presented for analysis and a better understanding of the real implementation of the smart algorithms through testbed design. The simulation and testbed results create the ground work for future exploration in the design of smart antenna beamforming.

used in dynamic environments where both the desired signal and the interfering signals arrive from different directions with varying signal levels requiring the design of adaptive antenna arrays. The smart antenna system is also known as an adaptive array antenna, digital antenna array, and recently as time multiple input multiple output systems. 3 Most of the base station antennas in cellular communication are either omnidirectional or sectored type. This leads to power wastage unless the users are not uniformly distributed within the region of base station coverage. 4 Sometimes it causes increased interference to the other users hence degrading the signal to interference plus noise ratio and thus limiting the capacity enhancement. Further, with the forthcoming wave of 5G communication, the demands for multi-beam antennas are increasing to support high bandwidth and data-hungry applications. 5,6 Van Atta in 1959 first introduced adaptive array processing to describe a self-phased array. 7 All elements of the adaptive antenna array are combined optimally so that maximum signal reception is possible in the desired direction and cancel out the signal at the similar frequency coming from different directions. To get the best-suited weight, various adaptive algorithms such as least mean square (LMS), 8,9 normalized least mean square (NLMS), [8][9][10] sample matrix inversion (SMI), 9 and recursive least square (RLS) [11][12][13][14][15] are used earlier to adapt the radio environments. The LMS and NLMS are the steepest descent gradient-based iterative approaches, whereas SMI and RLS are block adaptive and recursive methods respectively. SMI algorithm is suitable for a discontinuous transmission, 9,13 which originates the weights of an array by replacing the array correlation matrix with its estimates. It is more suitable for a dynamic environment as it uses block samples. RLS is an adaptive algorithm that recursively finds the coefficients to minimize the cost function defined by the least square error. It is well known that the rate of convergence of SMI algorithm is much faster than that of LMS and its variant NLMS. 14-16 RLS algorithm is recursive and it is better than LMS, NLMS in terms of complexity. RLS and SMI are the competitive adaptive algorithms for the smart antenna. The recursive process in RLS functions follows a gradual approach to the most suitable weight and utilizes forgetting factor ( ) 17 to progressively adapt the weights, whereas SMI takes the direct approach path to find the best suitable weights. Adaptive antenna array transmits and receives signal simultaneously and hence creates many multipath propagations and multiple angles of arrival (AOA). Therefore, the antenna array needs to determine the true AOA 18 by eliminating the interferers and noise. However, MUSIC 18 (multiple signal classification) and ESPRIT 19,20 (estimation of signal parameters via rotational invariant techniques) algorithms are used to find the AOA.
There are several available works on the performance studies of various adaptive algorithms. In Reference 10, a comparative study is presented between LMS and its variant NLMS for antenna beamforming with null steering toward interference and maximum side lobe level (SLL) with the variation of step size parameter ( ) and antenna elements. Similarly, in Reference 11, a comparative study is performed for LMS, constant modulus, conjugate gradient algorithm, and quasi-Newton algorithm to demonstrate the faster rate of convergence. In References 12-14, performance and different noise level comparisons are done in terms of null depth, SLL, and convergence rate between LMS, NLMS, SMI, and RLS with the variation of the number of array elements and spacing between them in presence of two interferers. Also in Reference 14, a comparison is made between SMI, RLS, LMS/SMI, and hybrid NLMS/RLS with the variation of antenna types and feeding techniques. In Reference 15, mean square error (MSE) 9 and noise level are compared between LMS, conventional RLS, and enhanced RLS algorithms. Simultaneously, the weights of all three algorithms are compared with respect to samples. Comparison between variable step-size adaptive algorithms like variable step-size LMS (VS-LMS), variable step-size sign LMS (VS-SLMS), and variable step-size normalized LMS (VS-NLMS) are made with the variation of antenna elements and spacing between them. 16 The adaptive antenna beamforming occurs using the weight adaptation of the antenna elements. Beamforming at the desired azimuth and null placing to interfere direction is the goal of a smart antenna system. Unlike Reference 12, the number of interferers is increased in the proposed work to observe the impact of the variation of the design parameters on these adaptive algorithms. As the number of mobile users increases day-by-day from the last generations to the current 5G, the usage of smart antenna becomes significant in the presence of more interferers. Finally, knowledge of the simulation process is applied on an experimental testbed for comprehensive analysis.
The new contribution of this paper includes the generations of experimental results for weight convergence of LMS, RLS, and SMI for different step sizes, forgetting factors, and block sizes to validate the simulation results. The novel contribution of this paper lies in the building of an adaptive beamforming system using a hardware testbed in the laboratory with an field programmable gate array (FPGA)-based reconfigurable structure namely wireless open-access research platform (WARP) boards. 21,22 The built-in testbed showcases the adaptive beam shaping results. F I G U R E 1 Functional block diagram of a smart antenna for minimizing the mean square error 9

SMART ANTENNA: CONCEPT AND COMPONENTS
In smart antenna system, the radiation pattern of the antenna is controlled via certain algorithms using digital signal processors. 23 The inputs to the antenna arrays are assumed as the desired signal, interfering signals, and Gaussian noise. Using some adaptive algorithms, the array weights are controlled and subsequently the output error is minimized. 8 The error is calculated by subtracting the output signal from the reference signal which is minimized using adaptive algorithms by controlling the weights. With the suitable design of adaptive algorithms, the signal-to-interference ratio is maximized or minimized according to the different optimization parameters like variance, MSE, 9 interference nulling, and the steering of main beam toward the user. The two most important aspects of the smart antenna are an estimation of direction of arrival (DOA) 18 and the digital beamforming (DBF). 24,25 Figure 1 illustrates the functional diagram of the smart antenna system. There are antenna array units with weight adapter, the DOA unit, and an adaptive algorithm unit.

ESTIMATION OF DOA
The DOA is also known as spectral estimation, bearing estimation, or AOA estimation. 18 In adaptive array processing if one or many transmitters are operating simultaneously, then each source creates many propagation paths and AOA at the receiver. Therefore, the antenna array needs to determine the true AOA by eliminating the interferers and noise signals for greater fidelity. [18][19][20] Figure 1 shows M-element array with "M" potential weights and "N" incident plane waves arriving from "N" directions (where N < M). Each contains adaptive white Gaussian noise (AWGN). The array output is written in the following form: where where W = [W 1 W 2 · · · W M ] T = array weights, u(t) = Z i (t) + n(t) = undesired signal vector, Z D (t) = incident signals vector, Z i (t) = a( i ).I i (t) = interfering signals vector at time 't', a = [a( 1 )a( 2 ) · · · a( N )] = matrix of steering vector a( i ), n(t) = noise vector at each array element. The array correlation matrix can be written as where R ii and R nn are the signal and noise correlation matrix.

AOA estimation by music
In MUSIC algorithm, 9,18 the array correlation matrix is calculated at first assuming uncorrelated noise with equal variances. In this method, the array correlation matrix is given by where R ii = signal correlation matrix, 2 n = eigenvalues of uncorrelated noise, and I = identity matrix. For finding eigenvalues and eigenvectors of R ZZ , it produces "N" eigenvectors associated with the signals and M − N eigenvectors due to the noise. In the case of uncorrelated signals, the smallest eigenvalues are equal to the variance of the noise. It constructs the M × (M − N) dimensional subspace spanned by the noise eigenvectors following the situation. The noise eigenvectors are defined as where E n is orthogonal to the array steering vectors at the AOA. The MUSIC pseudospectrum 9 is defined by

AOA estimation by ESPRIT
The ESPRIT 9,19,20 algorithm is much less computationally intensive and makes the estimation more practical when only a finite number of noisy measurements are required. This algorithm assumes that the signal sources are narrowband and it can be either random or deterministic while the noise is assumed to be random with zero means. It is also assumed that this algorithm is suitable for multiple identical arrays, called doublets. Doublets are produced by the combination of separate arrays. These arrays are displaced translationally but not rotationally. Figure 2 shows a four-element linear array composed of two identical three-element two doublets, where two doublets are translationally displaced by the distance "d." Let the two doublets be represented by array 1 and array 2. The signals induced on each of the arrays are given by where A 1 ( ) = [a 1 ( 1 )a 1 ( 2 ) · · · a 1 ( B )] = Vandermonde matrix of steering vectors for subarray1, B = number of arriving signals, and n 1 (t) = Gaussian noise at subarray1. Similarly, Vandermonde matrix of steering vectors for subarray2, and n 2 (t) = Gaussian noise at subarray2. Here, it is assume that i = | i |e j arg( i ) and i is the angle of arrival. where

DIGITAL BEAMFORMING
DBF 24-26 is a technique for separating the desired signal from interfering signals. With the development of adaptive algorithms, DBF technology has progressed. In the case of DBF, the major advantage is that the phase shifting and array weighting can be performed on the digitized data rather than the implementation in hardware.  Figure 5A shows the DBF process using a digital signal processor. The signal for each antenna element is digitized using an analog to digital converter. DBF receivers use analog RF (radio frequency) translators to shift the RF signal frequency down before the analog-to-digital converters. 24 Figure 5B shows the schematic block diagram of the RF translator. The RF translator consists of a band-pass filter, mixer, local oscillator, and a low-pass filter. The design of the receiver portrays the performance of a DBF array. 25

MINIMUM MSE
From Figure 1, it is seen that one desired signal is arriving at " 0 " angle whereas "N" interfering signals are appearing from angles 1 , 2 , … N . The desired and interfering signals are received by an array of M-elements with "M" potential weights. So the weighted array output for tth time samples can be written by recalling Equation (1) as The best method for optimization of the array weights is found by minimizing the MSE. 9 The signal r(t) is the reference signal which is identical to the desired signal D 0 (t) and uncorrelated with the interfering signal I(t). The error signal (t), as indicated in Figure 1, can be represented as By squaring Equation (11) and taking the expected value, the MSE is given by, where The expression of Equation (12) is a quadratic function of the weight vector W. Since the optimum weights provide the minimum MSE, the extremum is the minimum of this function. Thus, we can write, After simplification, Equation (14) reduces to Wiener solution 9 that is, .,

ADAPTIVE BEAMFORMING
Switched beam antenna and adaptive antenna array are the two types of smart antennas. 8,9,27 According to the literature, the switched-beam system is chosen from one of many predefined patterns to improvise the received signal. Detection of an incoming signal enables the base station to determine the beam which is best aligned in the signal-of-interest direction. It then switches to that beam to communicate with the user. 8 On the other hand, an adaptive array has an infinite number of patterns that are adjusted in the real-time adaption of the radio environment that is, channel condition. The adaptive antenna array is an array of multiple antenna elements that continuously adjusts its pattern with time by collecting feedback from the surrounding environments to keep the array in optimum state. 8,9 If the desired arrival angles continuously change with time, it is necessary to devise such a scheme that can continuously update the array weights. The adaptive beamforming is the best technique for tracking the mobile user continuously in the changing RF environment. In the following subsection the LMS, NLMS, RLS, and SMI will be discussed in flowchart along with the mathematical formulations to understand the background of these algorithms. 23

LMS algorithm
LMS algorithm 10-15 is a gradient-based continuous adaptive approach which we have discussed in the flowchart representation in Figure 6. We know that the gradient method is the best method for determining minima. 9 Here, the performance surface J(W) (cost function) is established by finding the MSE again. By squaring error (t) in Equation (11) and taking the expected value.
The performance surface J(W) is in the shape of an elliptic paraboloid having one minimum. It may employ the gradient method to locate the minimum of the above Equation (17) as where ∇ W is the gradient of the performance surface. The minimum occurs when the gradient of Equation (18) is zero. Thus, the solution for the weights is the optimum as given by We have used an iterative technique called the method of steepest descent to approximate the gradient of the cost function. LMS method helps to find the method of steepest descent in terms of the weights. The steepest descent approximation has obtained as where is a step size parameter and ∇ W is given in Equation (18). If we substitute the instantaneous correlation approximations, we have the LMS solution as where e(t) = r(t) − W H (t)Z(t) = error signal. The convergence of the LMS algorithm in Equation (21) is directly proportional to the step size parameter and * denotes the complex conjugate. For small values of the convergence becomes F I G U R E 6 Flowchart of LMS algorithm very slow whereas for larger values it exhibits very faster convergence. It can be shown that stability is dependent upon the following condition: where max is the largest eigenvalue of the array correlation matrix R ZZ .

NLMS algorithm
The LMS algorithm is the most basic method for calculating weights and the algorithm requires large number of iterations before satisfactory convergence. Also, the stability and convergence time of the LMS algorithm are dependent upon the step-size parameter . To overcome this dependency, the NLMS algorithm is introduced. 14 The NLMS algorithm is used to achieve good stability and faster convergence. The weight update equation for the NLMS algorithm is given as Equation (23) represents the final weight update for the NLMS algorithm where the step-size is divided by the normalized value of the input signal Z(t). In Equation (23), to avoid denominator being zero for no signal condition, a small positive constant ' ' is added to the denominator. Thus Equation (23) can be rewritten as

Sample matrix inversion algorithm
The SMI is a time average estimate algorithm of the array correlation matrix using "T" time samples which are shown in the flowchart in Figure 7. If the random process is averaged over time and space (ergodic) in the correlation, the time average estimate is equal to the actual correlation matrix. In this method, the optimal weights are computed for each input signal block of size "T." Weight adaptations in the SMI algorithm can be done in three different ways. These are block adaptation, block adaptation with overlapping, and block adaptation with memory. In this article, we have used a block adaptation approach. This method is well suited for time-varying signals. Now from Equation (19), we can write, For determining the values of R ZZ and p, the time average is calculated as and Since we use T-length block of data, this method is also called a block adaptive approach. We thus here adapt the weights block-by-block. MATLAB made it easier to calculate the array correlation matrix and the correlation vector by the following procedure: where "t" denotes the block number.
Equation (29) is the final weight update equation for the minimization of error. where

RLS algorithm
The convergence speed of the SMI algorithm depends on the autocorrelation matrix with large eigenvalue which results in slow convergence. This problem is solved using RLS algorithm which minimizes a weighted linear least square cost F I G U R E 7 Flowchart of SMI algorithm function relating to the input signals. [15][16][17]28 However, we can recursively calculate the required correlation matrix and the correlation vector as and where "t" is the block length, R ZZ (t) and p(t) are the correlation estimates ending at time sample "t." Both of the summations in Equations (30) and (31) use rectangular windows, thus they equally consider all previous time samples. Since the signal sources can change or slowly move with time, here it is preferred to work with the recent data samples. This can be done by modifying the two Equations (30) and (31) such that it forgets the previous time samples, which is known as a weighted estimate. Thus and where " " is known as the forgetting factor and its value lies in the range of 0 ≤ ≤ 1.
Thus the next values for the array correlation matrix and the vector correlation matrix can be found by previous values, we can rewrite Equations (32) and (33) as Equations (34) and (35)

COMPARATIVE PERFORMANCE ANALYSIS OF LMS, NLMS, SMI, AND RLS ALGORITHMS IN LIGHT OF ADAPTIVE ARRAY
The performance studies of the above-mentioned algorithms are done by extensive simulations using MATLAB under several interferers (maximum five) in varying parameters for example, null depth, maximum SLL, rate of convergences, etc. In this article, we have studied main beam variation toward the desired user, null steering toward interfering direction along with weight variation and output error for the LMS, NLMS, RLS, and SMI algorithms. Here, initially smart antenna system is designed by taking a uniform linear array (ULA) of 20 elements (M = 20) with inter-elemental spacing (d) equal to half wavelength distance and later we varied the array elements and spacing. The desired signal is taken as a simple cosine signal. We have taken 8000 data samples for simulation purposes. In the example considered here, the desired angle of arrival is 0 • with five interfering signals arriving at −50 • , −30 • , −10 • , 20 • , and 40 • angles. Figure 9 represents the array factor for all four adaptive algorithms with observed deep nulls at −71.1 dB, − 81.85 dB, −97.95 dB, and -112.3 dB, respectively.
It is clear from Figure 9 that the nulls were created in the specific direction of the interferers with good null depth for LMS, NLMS, SMI, and RLS. The maximum value of SLLs is less than −15 dB. Both LMS and NLMS algorithms can iteratively update the weights to force deep nulls toward the interferers and they achieve maximum values toward the desired signal. It has observed that NLMS nulls are as deep as around −81.85 dB at 20 • whereas LMS null is around −71.1 dB. Similarly, the SMI algorithm updates the weights blockwise to force deep nulls toward the interferers and achieves the maximum value toward the desired signal. Moreover, for SMI and RLS algorithms, nulls have been placed in the exact direction of the interferers with further deep nulls of −97.95 dB and − 112.3 dB at −50 • and 20 • , respectively. Thus, both SMI and RLS can produce deeper nulls than LMS and NLMS which is a similar observation as obtained in Reference 12. But LMS and NLMS could not provide better SLLs compared to the SMI and RLS with 20 elements array in the presence of five interferers as opposed to the observation in Reference 12. So with the increased interference, the performance of each algorithm may vary and need to be explored further.  Table 1 summarizes the effect of antenna spacing for 20 elements array beamforming using these four algorithms. The 3 dB beamwidth of LMS and NLMS algorithms are higher than SMI and RLS when antenna spacing is 0.3 although it is almost the same for 0.5 and 0.7. Because of the more interferers and smaller spacing, the energy distribution with LMS and NLMS yields in mutual coupling effect. SMI and RLS are more robust algorithms to mitigate the effect of interference. As the spacing increases, the mutual coupling effect will also decrease. The best null depth occurs when the antenna spacing is 0.5 for all four algorithms that indicate the best choice of element spacing. The magnitude of null depth is the highest for RLS algorithm at 0.5 spacing. The maximum SLL with 20 elements and 0.5 spacing is best in RLS (−13.53 dB) and worst in LMS (−10.41 dB), almost the same for NLMS and SMI (−12.64 dB and − 12.68 dB). On the contrary, LMS algorithm provides the best result in Reference 12. This observation is due to the increased interferer effects in the present work.

Comparison of null depth, beamwidth, and maximum SLL for varying number of antenna elements
For the variation of array elements of the LMS, NLMS, SMI, and RLS algorithms, we have considered fixed element spacing at 0.5 of wavelengths ( ). Figures 13, 14, and 15 illustrate the array patterns of the desired signal and null steering toward interferers with the variation of antenna elements for 15, 20, and 25, respectively. The plots indicate that if the number of antenna elements increases keeping element spacing at 0.5 , the antenna beamwidth becomes narrower. The maximum SLL slightly varies and the maximum null depths remain almost the same validating the observation discussed in Reference 12. However, deeper null depths are obtained in comparison to the Reference 12, with respect to antenna elements. This is due to the complex electromagnetic field interaction between the array and the different interferes. Table 2 presents a summary of the results for varying antenna elements.

Comparison of error for LMS and NLMS algorithms for the variation of antenna spacing
The MSE comparison for LMS and NLMS algorithms is given by varying the array spacing 0.3 -0.7 with 20 elements array, step size parameter = 0.003 and the corresponding results are shown in Figures 16 and 17. In the case of LMS algorithm it is obvious that when the array element spacing varies as 0.3 , 0.5 , and 0.7 , it takes 176, 100, and 60 samples  (iterations) to converge. If the array spacing increases, quicker convergence occurs with an increased initial error. Similarly, in case of NLMS it is observed that when the array spacing is 0.3 , 0.5 , and 0.7 , it takes 150, 90, and 54 samples to converge. If we compare LMS error with NLMS, it is noticed that the LMS error and the required time for convergence is higher than the NLMS algorithm.

Comparison of error for LMS and NLMS algorithms for the variation of antenna elements
Further, the LMS and NLMS error comparison is made by varying array elements for fixed element spacing of 0.5 . The results obtained for both LMS and NLMS algorithms are shown in Figures 18 and 19, respectively. The convergence speed of LMS is improved for an increased number of elements along with error. However, in NLMS, such an effect is not present due to the normalization effect. The convergence speed of LMS and NLMS are almost the same and it is around 55 and 45 iterations respectively for 25 elements.

SMI error for the variation of antenna spacing and elements
The error comparison in SMI algorithm is made by assuming a fixed block size = 1. From Figure 20, it can be observed that if the element spacing increases from 0.3 to 0.7 for 20 elements array, error occurrence reduces. Also, if we compare these with LMS and NLMS, error in SMI is much less and of the order of 10 −10 . This is one of the primary reasons to consider SMI adaptive algorithm for smart antenna beamforming. With an increased number of antenna elements, in Figure 21, SMI error trends to increase but the order remains at 10 −10 .  Table 3 enumerates the comparative performance analysis of the algorithms in terms of maximum null depth, maximum SLL, and error. Null depths are the least gains conjured at the expected positions of interferers. The SLL is the In comparison to all four algorithms, the MSE in SMI is the least of the order of 10 −10 , RLS error is less than LMS and NLMS. However, error in RLS is almost constant at 2.0 × 10 −4 and which is within the error threshold. RLS algorithm has faster convergence among these four algorithms. LMS has the slowest convergence rate. At this point, we may consider the best two algorithms namely SMI and RLS for the least possible MSE and faster convergence rate respectively. Thus for further study, we have calculated the time and space complexity of SMI and RLS only.

COMPLEXITY ANALYSIS OF RLS AND SMI ADAPTIVE ALGORITHMS
The time complexity is accounted to perform the task in MATLAB 2018a platform with 4 core 3.5 GHz processor and 4 GB RAM. The time measurements are taken strictly for this setup and it may vary in other specifications. The space complexity can be viewed with the requirements of data size at run time.

Complexity of RLS process
Let n = number of antenna elements used for the array, m = size of data string for the signal strength of each element, and i = number of iterations. Following the steps of RLS algorithm 29 and pseudocodes as given here, the number of additions and multiplications are calculated, along with the data size. Number of multiplications needed till ith iteration: i(5n 2 + 4n) Number of additions till ith iteration: i(n 2 + 7n + 1) The four equations that are computed stepwise for RLS are:

TA B L E 3 Summary of observations
Step 1: P n = P n−1 − P n−1 x n x T n P n−1 +x T n P n−1 x n Step 2: g(n) = P n x n +x T n P n−1 x n The main four equations for SMI are: Step 1:  Tables 4 and 5, it can be concluded that RLS has a better time complexity but a poorer space complexity than SMI. However, both the complexity heavily depends on the coding of the algorithms, and SMI was optimized for space complexity. Also, for a small number of elements, the SMI algorithm is better in performance in terms of time, space, and MSE in comparison to RLS algorithm.

TESTBED DESIGN FOR SMART ANTENNA: SYSTEM OVERVIEW
A reconfigurable smart antenna system is designed in the laboratory to apply the three basic adaptive beamforming algorithms such as LMS, RLS, and SMI. Finally, we have generated some real-time experimental results for weight adaptation and error. The experimental set-up has the following major components, (a) WARP v3 kit 21,22 , (b) Arduino UNO, (c) Servo motor, (d) Microstrip antenna array, and (e) Semi-anechoic chamber.
Here we have used WARP v3 boards, built by Mango Communications, as the base of building the smart antenna testing system. WARP v3 board is a Virtex-6 FPGA board and suitable to create a mock transmission and receiver node communicating in 2.4 GHz and 5 GHz band following IEEE 802.11 standards. The basic setup diagram to establish the communication system is shown in Figure 24. The availability to use a self-designed antenna in the system as Tx and Rx antenna makes this setup ideal for testing adaptive beamforming algorithms. Figure 25 is the block diagram of the smart antenna system at the receiver side. In order to maximize the signal to noise ratio, the adaptive antenna arrays are armed with signal processors which can automatically adjust the variable antenna weights of a signal processor by a simple adaptive technique. The algorithm to be implemented using any standard digital signal processing tool forms the backbone of such a system. It not only directs the main beam in desired directions but also introduces nulls at interfering directions.

10.1
Hardware setup description Figure 26 is the experimental hardware testbed for smart beamforming in the laboratory environment. From the above Figure 24: 1. Two WARP nodes are connected to the PC via a network switch. 2. A 2 × 2 rectangular planner patch antenna array is used as the receiver for beam formation. Whereas a printed dipole is used as the transmitter. A 2 × 2 patch antenna array is mounted on a servo motor connected to an Arduino UNO. 3. The servo is programmed to rotate from −90 • to +90 • azimuth, while signal strength is measured. The pattern measurement is done by MATLAB in a PC connected to WARP v3 in LAN. 4. The Rx-antenna is placed in a built-in semi-anechoic chamber made of charcoal to reduce reflections.

Novel way of antenna pattern measurement
The WARP v3 board contains the MAX2829/MAX2929 RSSI standard and represents the measured power in terms of received signal strength indicator (RSSI) values. In a certain azimuth angle, the signal strength is measured continuously and later sampled. In Figure 27, the RSSI values are presented and it founds to follow the Gaussian distribution. Thus, the mean RSSI value is taken as a strength indicator. To have more acceptability, the average value between the first " " range is taken, that is, from RSSI − RSSI to RSSI + RSSI range. The RSSI signal (in dBm) is proportional to the signal's "Rx" power. 30

EXPERIMENTAL RESULTS
The

RLS weight adaptation through experiment
Forgetting factor = 0.99; Transmitted signal frequency = 0.5 MHz; Interferer frequency = 5 MHz; and Carrier frequency = 2.4 GHz. A difference can be observed in the experiment with the RLS algorithm. In Figure 29G the error signal is much less in magnitude than LMS ( Figure 28G). This is in the direct contrast from the simulation results (Table 3). The weights perfectly converged near 600 samples ( Figure 29H) and the error was also minimal but due to the algorithmic implementation all the sample iteration was conducted and instability was found in results. Other aspects of the experiment are shown in Figure 29A-F.

Experimental results for SMI
The pivotal role in the SMI algorithm is played by block size, and hence the number of blocks used depends on. SMI: Length of block = 5; Number of blocks = 200; Transmitted signal frequency = 0.5 MHz; Interferer frequency = 5 MHz; and Carrier frequency = 2.4 GHz.
In case of SMI algorithm, as shown in Figure 30A-F the sample number was reduced to 1000, as seen from the simulation result and complexity analysis. Time requires in SMI is influenced by sample numbers and block size. The error signal in Figure 30G Figure 30H. To validate the weight convergence curves of the experimental results with the simulation results, for a 4 element antenna array with 1 MHz signal and interferer of 5 MHz placed at 10 • in the azimuth angle, weight convergence curves using LMS, RLS, and SMI are generated and plotted in Figure 31.
Comparing the weight update curves of the experimental testbed with 4 element antenna array in Figure 28H, 29H, and 30H for LMS, RLS, and SMI respectively with Figure 31, generated by MATLAB simulation in presence of a single interferer placed at 10 • close to main beam (0 • ), close resemblance of all cases are observed. In this experiment, there is an inherent initial transient behavior of the circuit for the estimated signal and introduces time lag to cope up with the desired signal.
In the next section, experimental results obtained by the smart antenna testbed for LMS, RLS, and SMI algorithms with the 2 × 2 planar microstrip array at the receiving end for antenna beamforming through weight adaptation are provided. The detail of the design of the array is out of the scope of this article except for the known normalized measured radiation pattern at hand, measured in an anechoic chamber for the 2 × 2 array at consideration (operating frequency 2.4 GHz). In Figure 32, the radiation patterns of the used microstrip patch array antenna are presented for measured radiation pattern and RSSI-based pattern obtained by the testbed design. The very close similarity of the normalized patterns between the two validates the correctness of the method.   Figure 33 shows the adaptive beamforming using LMS, RLS, and SMI algorithms in presence of interferer using the designed testbed. The actual array pattern is the RSSI-based method shown in the figure which is the desired pattern. The red color represents the antenna pattern modified with the presence of interferer at the position of −80 0 in the azimuth. The difference between the two patterns is denoted as an error. The objective of the smart antenna is to place a null at the interferer position (−80 • ) and the main beam at the desired direction which is −10 • in our experiment. The adaptive algorithm tries to reduce the MSE mitigating the interference effects and generates the desired pattern. In the experiment, we have only used 4 elements antenna array for beamforming (it is the hardware limit). From Figure 33, it is observed that all three algorithms try to cope up with the desired beam pattern placing a deep null at −80 • . But SMI algorithm obtains the maximum null depth at the position of the interferer. Moreover, the beam shaping using SMI is much closer to the desired pattern. Among the three algorithms, SMI has the best beam shaping with 4 element array. The results obtained in this experiment are very promising to explore further in terms of the number of interferers, position (distance and angle) of the interferers, strength of the interferers, and so on. The constraint of the setup is to use only 4 elements array which provide broader beamwidth of the array. But for sharper beamwidth, we may use an array of the array such as an array of Yagi array to compare the beamwidth using different adaptive algorithms. These all are left as our future research ventures.

CONCLUSION
LMS, NLMS, SMI, and RLS are well known adaptive algorithms used for smart antenna beamforming. Still there are some scopes to analyze these algorithms with the presence of multiple interferers in aspects of smart antenna design. Here four adaptive beamforming algorithms are taken into consideration with mathematical analysis and flow-chart representation to assess comprehensively based on the antenna performance parameters, in presence of five interferers. A comparative table was presented by varying the antenna elements spacing and number of array elements. For null depth at −115 dB, RLS algorithm yields far better results compared to the LMS, NLMS, and SMI when antenna spacing is varied. This result is also better than the results presented in Reference 12. Further, LMS and NLMS could not provide better SLLs compared to SMI and RLS with the presence of five interferers as opposed to the observation in Reference 12 with two interferes. The best result is found at d = 0.5 . The convergence rate of LMS, NLMS, and SMI is slower than RLS. NLMS is better than LMS for error and faster convergence rate. Due to the block correlation approach of SMI, it is observed that SMI error is the least (10 −10 ) among the four algorithms. Though RLS error initially is more but the convergence is much faster within only 12 iterations for 20 elements array in comparison of 40 iterations in SMI. From the complexity analysis of RLS and SMI, it is found that RLS is more space complex than SMI and increases with more number of antenna elements. For a small number of antenna elements (ie, 4), SMI is the best in comparison to both time and space complexity. The FPGA-based smart antenna testbed design is the significant contribution of this work and explains clearly the weight adaptation of LMS, RLS, and SMI for interferer frequency, forgetting factor, and block size. Finally, the adaptive beam shaping using this testbed is shown which indicates SMI is the best in beam-shaping for 4 elements array validating the observations in Tables 4 and 5. As a future scope of work, further exploration of smart beamforming will be done using the designed testbed for more number of interferers, the position of interferers in terms of distance, azimuth angle, and beamwidth. To overcome the constraints of 4 elements, the use of an array of antenna arrays may also be used for sharper beam shaping. A sharper beam can initiate point to point communication, almost as an imitation of wired communication. The relatively high difference in strength from peak to null can offer the opportunity in improving the modulation scheme to provide a high data rate and also minimal latency.

PEER REVIEW INFORMATION
Engineering Reports thanks Peter N. Chuku, Karim Djouani, and other anonymous reviewers for their contribution to the peer review of this work.

CONFLICT OF INTEREST
The authors declare no potential conflict of interest.

DATA AVAILABILITY STATEMENT
Research data are not shared.