Design and Analysis of a Multiple and Wide Nulling Collaborative Beamforming Scheme in the Domain of 3-Dimensional Wireless Sensor Networks

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Introduction
Collaborative beamforming (CB) is a useful tool for establishing energy-efcient and reliable communication links between wireless sensor network (WSN) nodes and far-of sinks [1][2][3]. CB in WSNs is vital given that sensor nodes are usually energy-limited and sinks are usually located beyond individual nodes' transmission range. CB serves to overcome the shortcomings associated with multihop transmission in WSNs such as the dependency of transmission quality on individual sensors, high communication overheads, delay, and increased network interference [4]. CB is achieved through appropriate transmit amplitude and phase weighting at a carefully selected set of nodes with an aim of ensuring a constructive combination of individual node radiation energy in the sink's direction. Collaborating nodes more or less form a virtual random antenna array.
CB is often associated with high sidelobes owing to the usual random placement of collaborating nodes [5][6][7]. Te high sidelobes are bound to yield interference at unintended cochannel terminals. Minimizing CB radiation in unintended receiver(s)' direction(s) is of utmost necessity. In centralized antenna arrays, null-steering beamformers have been applied to yield a destructive combination of radiation in the undesired direction(s) while ensuring a constructive combination of radiation in the desired direction(s) [8][9][10][11][12]. Metaheuristic optimization algorithms are essential towards this end. A typical application of a metaheuristic optimization algorithm can be found in [13].
1.1. Related Work. In [8], an adaptive null steering beamformer is designed on the basis of a uniform linear array (ULA). Te bat algorithm (BA) is utilized in the designed beamformer to optimally adjust transmit amplitude at each array element. Te resultant null steering performance has been analyzed against approaches utilizing accelerated particle swarm optimization (APSO) and genetic algorithm (GA). Te BA-driven beamformer is noted to outperform the APSO and GA-driven beamformers in terms of null steering precision, null breadth, and convergence speed. Te essence of deep and broad nulls alongside efective optimizing algorithms is clearly brought to the fore. In [9], a triple-mode circular microstrip patch antenna bearing the capability of forming two nulls within a select hemisphere is presented. Furthermore, the two nulls can be steered independently. Te designed antenna consists of (i) A central circular patch supporting TM11 mode (ii) A TM21 mode shorted annular ring around the central circular patch (iii) A shorted annular ring supporting TM31 mode encircling the other two radiators Te three modes (TM11, TM21, and TM31) are fed using two feed points with an aim of creating right-handed circular polarization. Te resultant radiation pattern is manipulated to yield two independently steerable nulls through apt control of individual feed amplitude and phase. A hybrid particle swarm optimization (PSO) and pattern search algorithm are applied in feed amplitude and phase optimization. A beamforming network consisting of digital variable gain attenuators, digital phase shifters, and low-noise amplifers is utilized in the optimal feed amplitude and phase implementation process. Te designed beamformer bears limited main beam steering. Te research brings to the fore the practical aspects of null steering implementation. In [10], an improved invasive weed optimization (IWO) algorithm is applied in concurrent multiple beamforming and null steering. Te improved IWO algorithm is utilized in optimizing the excitation amplitude at a linear time -modulated antenna array elements to yield optimal beams and nulls. In [11], a coherently radiating periodic structure (CORPS) beamforming network with beam and null steering capability is proposed. Te PSO algorithm is applied in optimizing excitation weights in the CORPS beamforming network to generate an array factor bearing desired sidelobe level, directivity, and null depth. In [12], a combined minimum variance distortionless response (MVDR) and frefy algorithm (FA) approach is utilized in null steering in a linear antenna array. Deep and accurate nulls are obtained. In [14], an array of four patch antennas in conjunction with a set of eight phase shifters is utilized to synthesize steerable nulls. In [15], a distributed beamforming network made up of multiple dual transmitters is investigated in terms of beam and null steering. A groupwise null forming scheme is proposed. Numerical simulations validate the efectiveness of the proposed scheme. Te authors in [16] a set of novel beamforming schemes for synthetic aperture radar are presented. A notable outcome is a trough-like beam pattern with wide nulls. Consequently, interference signals received by synthetic aperture radar can be suppressed efectively. Numerical simulations alongside experimental results validate the efectiveness of the proposed schemes. Null steering is investigated in the planar ring; uniformly distributed and volumetric shell antenna element distributions are shown in reference [17]. In [18], a null steering scheme based on the partitioning of a Mobile ad hoc Network (MANET) node into a set of 2 subarrays is presented. Te authors in [19] present a novel heuristic optimization algorithm christened Fibonacci branch search (FBS). Te algorithm is applied in the design of a low sidelobe and deep nulling adaptive beamformer.
As per the reviews done, research in null steering in conventional (stand-alone) antenna array beamformers is mainly on the basis of uniform array geometry (particularly linear confguration). On the other hand, WSN nodes are usually randomly arranged. In stand-alone antenna array beamforming, there is complete and accurate knowledge of antenna element positioning (geometry) unlike in CB in WSNs.
Te null-steering concept has been extended to CB in WSNs in [20,21]. Application of null-steering is bound to increase Signal to interference and noise ratio (SINR) at the unintended receivers and potentially enhance network security against intercepting terminals.
In [20], a fully distributed nulling steering procedure is proposed in the domain of an Internet of Tings (IoT) network (bearing a decentralized architecture). Herein, nulling steering is applied to enhance secrecy performance. Tis is particularly important when an eavesdropper is located in a high sidelobe region of a CB outcome. It is shown that there exists an optimal degree of nulling that maximizes the secrecy performance. In [21], a node selection-based mechanism (in the sense of yielding a virtual linear array) is utilized in null steering.
In [27], research entailing simultaneous optimization of beamforming, power consumption, and energy harvesting schemes in a WSN is carried out. Te applied beamforming concept entails the use of multiple antennas at base stations, resulting in optimal signal and energy transmission toward intended sinks. Te transmitted signals encompass data and energy that ought to be sent to mobile nodes. Te nodes are expected to split received power into data and energy (energy harvesting). Although the presented research is to a great extent diferent from the null steering research presented in this paper, the authors highlight the signifcance of power saving at WSN nodes (in particular extending node lifetime).
In [6] capacity improvement analysis at unintended receivers (accrued from sidelobe reduction in CB) is analyzed. In the paper, an average reduction of 20 dB in peak sidelobe level and 162 percent capacity improvement is reported in the worst case scenario.

Literature Review Summary.
Observations made in current null steering research (in the context of CB in WSNs) are as per the following listing: (1) Multiple unintended sink(s) in 3-dimension WSN confguration have not been considered (2) Tere is no research on the formation of wide nulls (ideal in scenarios featuring mobile unintended sink(s)) Te contributions brought forward in this paper include (1) Formulation of a wide multiple nulling scheme from the perspective of a 3-dimension WSN using an appropriate metaheuristic optimization algorithm (2) Analysis of capacity performance at unintended receivers upon nulling (3) Nulling performance analysis with changes in CB cluster radius and number of collaborating nodes Utilized performance measures include nulling depth, width, and accuracy.
Practical applications of the outcomes presented in this manuscript would be in (i) Environmental monitoring problems as addressed in [37,38] (ii) Trafc control in cities [39] (iii) Industrial monitoring/control [40] (iv) "Smart" cities [41,42] (v) Health monitoring [43] (vi) "Smart" home applications [44] (vii) "Smart" agriculture applications [45,46] Te rest of the paper is organized as follows: Section 2 presents the methodology. Te sections presented in the methodology include the development of a 3-dimension CB model and the development of a multiple and wide nulling scheme, alongside the general simulation setup. Te obtained results are presented and discussed in Section 3. Te overall concluding remarks are presented in Section 4. Figure 1 illustrates a 3-dimensional random distribution of WSN nodes.

3-Dimension CB Model.
Te considered model encompasses a large variety of practical WSN deployment scenarios wherein sensor nodes are randomly distributed in a 3-dimensional manner. Te considerations made in the model design process are as per the following list: (1) 3-dimension sensor node and sink distribution.
(2) A node is selected from a set of collaborating nodes to act as a cluster head/CB coordinator. Te cluster head also serves as the reference point in mapping out the other collaborating nodes' geometric positions: the nodes are taken as situated at a distance r, azimuth angle ψ, and elevation angle ϑ) with reference to the cluster head. (3) With reference to the cluster head location, the sink is located at an elevated location (A, ϕ, θ). (4) All nodes are synchronized in phase and frequency.
As such, this condition is difcult to meet in practice unless high-precision clocks are utilized in the collaborating nodes.  Te magnetic potential at the far-feld observation point as a consequence of the radiator current density is as per the following equation [47]: where (i) v is representative of the volume integral about the immediate region surrounding the radiator (ii) μ: permeability constant (iii) J is the radiator current density (iv) t: observation time (v) C: wave velocity (vi) d: radiator dimension International Journal of Antennas and Propagation 3 Considering a single-frequency wave, equation (1) takes the form of the following equation: Equation (2) can be written as follows: where k � ω/C � 2π/λ. As per Figure 2, the dimensions PP ′ and PQ of the triangle PQP ′ are approximately equal (considering far distances). PQ � OP − OQ. Te distance R can be expressed as follows: Using the approximation given in equation (4) in the exponential part of (3) and the approximation R≃r in the denominator yields Equation (5) can be written as Te integral factor (equation (7) radiation vector) in equation (6) determines the directional properties of the radiated feld.
where k � kr. Given a three-dimension array of several identical antennas located at positions [R 0 , R 1 , R 2 , . . . R T ], with relative feed current coefcients [a 0 , a 1 , a 2 , . . . a T ], the current density corresponding to the t th antenna is as Te corresponding radiation vector is as Te total radiation vector given T radiators is as follows, the respective array factor being T t�0 a t e jk.R t .
a t e jk.R t F(k).
Given a set of T nodes featuring arbitrary placement/ distribution, the array factor magnitude in the direction (ϕ, θ) may be expressed as where (i) R t : position vector of t th node. R t � r tx a x + r ty a y + r tz a z (ii) r � sin(θ)cos(ϕ)a x + sin(θ) sin(ϕ)a y + cos(θ)a z (iii) • is the dot product operator Upon wavelength normalization, equations (11) can be expressed as where R t � R t /λ.

Proposed Null Steering Scheme.
In the proposed CB scheme, null steering is done concurrently with beam steering. Wide and deep nulls are the intended outcome. Wide nulls are benefcial in the following two ways: (i) Reduction of interference in other distinct CB clusters (collaborating nodes are bound to be spatially distributed) (ii) Reduction of CB frequency in cases of slightly mobile unintended receivers (iii) Let (iv) (ϕ 0 , θ 0 ) be the direction of the intended sink with reference to the cluster head.
Te objectives to be met are as per equations (13)- (16). Te objective function in equation (13) is geared towards beam steering (maximizing radiation in the intended sink direction).
Te objective function in equation (14) ensures minimal radiation power spread in all directions outside of the intended sink direction.
Te objective function in equation (15) is geared towards null steering (minimizing radiation in the unintended sink direction(s)).
Te objective function in equation (16) specifcally caters to wide nulls.
Te overall objective function to be optimized is a weighted (g x ) combination of the objectives given in equations (13)-(16) as per (17) where 4 x�1 g x � 1. Te weighting values (g x ) have been carefully selected to yield the best possible outcome. In particular, the values utilized for g 1 , g 2 , g 3 and g 4 are 03, 0.2, 0.3, and 0.2, respectively. Te values so chosen accord beam steering and nulling comparatively higher weighting in comparison to generalized minimization of radiation in undesired directions and null widening. An exhaustive search entailing various sets of values for g 1 , g 2 , g 3 and g 4 yielded the afore-stated values as optimal in terms of overall CB solution quality.
t n is a weighting factor selected for every optimization iteration in accordance with the value of AF ϕ n ,θ n . Tis allows for a balanced nulling scheme wherein all null values are nearly identical. In particular, for a 3-null case, the utilized values of t n are 0.5, 0.3, and 0.2 for descending values of AF ϕ n ,θ n .
A PSO algorithm variant (Culled-Fuzzy-Adaptive PSO (CFAPSO) algorithm [48]) has been utilized in optimizing (selecting the best possible node weights W) the designed multiobjective function (equation (17)) Te CFAPSO algorithm is described in Section 2.3.

Culled-Fuzzy-Adaptive PSO Algorithm.
PSO algorithm entails a carefully checked movement of a swarm of "particles" (bearing potential solutions to a given problem) in a defned search space [49][50][51]. Equations (18) In the velocity and position update equations above (i) t represents iteration count.
(ii) i denotes a swarm particle.
(iii) x i and v i denote the position and velocity of particle i, respectively. (iv) w denotes inertia weight. w controls the infuence of the immediate previous velocity in the velocity update equation. In the basic PSO algorithm, w is decreased linearly from 0.9 in the frst algorithm iteration to 0.4 in the last algorithm iteration [52]. In the CFAPSO algorithm used in this manuscript, the values of c p and c s are mapped onto the range (2-2.4) and (2.2-2.6), respectively on the basis of iteration count and particle performance index [48]. w is mapped onto the range (0.4-0.9). Te mapping process of c p , c s , and w is done using a fuzzy logic inference system.
In summary, the procedure followed in implementing the CFAPSO Algorithm 1 is as per the following steps:

Simulation Setup.
Te research work laid out in this manuscript has been carried out through appropriate simulations in Matlab software. Te Matlab environment ofers a numeric computing platform suitable for beamforming strategies modeling, optimization, and analysis.
CB analysis entailing varying WSN node count and cluster radius has been carried out. Te utilized WSN arrangements with node count variation are as per Figure 3 and Tables 1 and 2. Te utilized WSN arrangements with cluster radius variation are as per Figure 4 and Table 3. All node distances are normalized with respect to wavelength.
In this research work, ffty independent tests have been used for every optimization procedure. Tis conforms to the requirement that sample sizes greater than 30 are by and large sufcient for a bulk of data distributions for the central limit theorem to hold [54]. Te central limit theorem is crucial in statistical data analysis for two chief reasons [55].
(i) Precise data analysis estimates: sampling distributions of the mean cluster tightly around the population mean with increasing sample size. (ii) Normality assumption: the fact that sampling distributions can approximate a normal (Gaussian) distribution is critical for use of parametric hypothesis tests of the mean.
In this manuscript, the statistical tools used in the results analysis process are as follows: (1) Analysis of variance (ANOVA) test: a test aimed at testing whether or not two or more sample/population means are statistically identical [56] (2) Tukey-Kramer test: a post hoc analysis test aimed at pin-pointing the exact sample/population means that have statistically signifcant diferences [57] Listed below are the general simulation considerations.
(i) 3 null steering directions (ii) A single-beam steering direction (iii) All node distances are wavelength-normalized (iv) Sixty CFAPSO algorithm iterations are utilized for the CB optimization problems addressed (v) A CFAPSO algorithm swarm size of thirty has been adopted (vi) Beamforming outcomes have been presented qualitatively in the form of radiation power pattern plots and quantitatively in terms of null depth values, null width values, nulling accuracy, and radiation power in desired/undesired directions.

Results and Discussion
Herein, the performance of the developed null steering beamformer (as per equation (17)) is weighed against that of a basic null steering beamformer. Te applied basic null steering beamformer does not take into consideration aspects of null width control, null depth fne-tuning, and generalized minimization of radiation in undesired directions as featured in equation (17). Furthermore, nulling performance with changes in the number of collaborating nodes and cluster radius is analyzed. Performance measures utilized include achieved null depth, width, nulling accuracy, and the resultant capacity at unintended receivers in nulling directions.

Performance Analysis against a Basic Null Steering
Beamformer. Te performance of the developed null steering beamformer as per equation (17) is compared against that of the basic null steering beamformer depicted in equation (20). Typical research entailing basic null steering can be found in [8,10,12,21].
where g a � 0.3 and g b � 0.7 are carefully selected constants intended to balance out a beam and null steering, |AF ϕ 0 ,θ 0 (W)| is the magnitude of the array factor in the beam steering direction and 1/N N n�1 |AF ϕ n ,θ n (W)| is the average magnitude of the array factor in the null steering directions.
Te utilized beam steering and null steering directions are as given in Table 4.
Typical radiation power patterns (in the form of mesh plots) obtained upon null steering using the schemes under study are as per Figures 5 and 6.
Typical radiation power patterns (in the form of contour plots) obtained upon null steering using the schemes under study are as per Figures 7 and 8.
A qualitative analysis of the radiation patterns presented in International Journal of Antennas and Propagation (1) Step 1: initialize swarm particles (in a random manner). (2) Step 2: (a) Evaluate the optimization function at all swarm particles. (b) Pick the optimal swarm particle as per the obtained optimization function values. (3) Step 3: if the maximum number of iterations permitted has been exhausted, terminate the algorithm, else proceed to step 4.
Step 4: if the count of iterations is half the specifed maximum value, proceed to step 5, else proceed to step 7.
Step 5: sort and rank swarm particles in accordance with their performance (as per the optimization function). (6) Step 6: cull and randomly reinitialize swarm particles associated with poor performance.
Step 7: update c p , c s , and w values (using a fuzzy logic-based look-up table).     International Journal of Antennas and Propagation Figure 9 comparatively depicts azimuth cut radiation power patterns corresponding to the null steering schemes under study. An outcome associated with conventional beam steering without nulling is also portrayed in the fgure. A qualitative analysis of the azimuth cut radiation patterns indicate the following: (i) Te proposed null steering scheme yields wider nulls (ii) Te proposed null steering scheme yields a slightly narrower main lobe width Table 5 gives the exact null depth values obtained using the proposed and basic null steering mechanisms. Te proposed null steering approach yields deeper and low variance nulls (nearly identical depth values) in comparison to the basic null steering approach.

Performance Analysis with Change in the Number of
Collaborating Nodes. In this section, 3-null placement is considered as per the listing given in Table 6.         Sets of 5, 10, 15, and 20 collaborating nodes are utilized in the null steering process. Figure 10 comparatively illustrates the evolution of the nulling cost function (average outcome of 50 independent runs for the algorithms under study). It can be clearly deciphered that the 20-node confguration outperforms the other node confgurations. Noteworthy, an increase in the number of collaborating nodes in a CB process might be associated with increased phase/time/frequency jitter at the nodes. Tis would inadvertently lead to unpredictable CB performance. It is expected that the performance superiority associated with an increase in collaborating nodes overcomes the performance downgrade associated with phase/ time/frequency jitter at the collaborating nodes.

Beam Pattern Analysis
(1) Radiation Power Pattern. Te azimuth cut of the resultant normalized radiation power pattern is as per Figure 11. Te presented pattern is the average of 50 independent outcomes. It can be observed that an increase in the number of collaborating nodes yields slightly deeper and more accurate nulls. Beam steering performance is roughly identical.
(2) Null Depth Values. Te obtained null depth values are given in Table 7. Te values are the average outcomes of 50 independent runs. Analysis of variance test P-values are 5.0688E − 102, 1.4277E − 132, and 4.5124E − 166 for null 1, 2, and 3, respectively. Going by the low P-values, the null depth values given in Table 7 bear statistically signifcant diferences. Te diferences are captured/summarized in Table 8 following a Tukey-Kramer comparison test. Table 9 presents null depth performance ranking (in accordance with the Tukey-Kramer comparison test results) upon using 20, 15, 10, and 5 nodes. A tie in rank in Table 9 implies statistically equivalent null depth values. Overall, the 20-node nulling procedure yields the best null depth performance. Figure 12 comparatively illustrates the average normalized power observed in the nulling directions against the count of nodes. Tere is an exponential power decrease with an increase in the number of CB nodes.  Table 10. Te values are measures corresponding to the azimuth-cut radiation pattern presented in Figure 11. Tere       Table 11. Te 20 and 15 nodes nulling procedures yield higher nulling precision in comparison to the 10 and 5 nodes nulling.

(5) Power in the Desired and Undesired Directions.
Normalized power values corresponding to the desired (beam steering) direction and all undesired directions are given in Table 12. Te values are the average outcomes of 50 independent runs. Tere is a power performance mix cutting across node counts. For instance, moving from 15 to 10 nodes, there is an unexpected improvement in the power radiated towards the desired direction, but an expected increase in power radiated towards undesired directions. Tis can be attributed to the unpredictable relationship between random node placement and the beam steering direction. A comprehensive performance comparison/trend is given in Table 13. Analysis of variance test P values are 8.4544E − 291 and 0.0000E + 00 for the power in the desired and undesired directions, respectively. Going by the low P values, the power values given in Table 12 bear statistically signifcant diferences. Te diferences are captured/summarized in Table 14 following a Tukey-Kramer comparison test. Table 13 presents the power performance ranking (in accordance with the Tukey-Kramer comparison test results) upon using 20, 15, 10, and 5 nodes. Overall, the 20-node nulling procedure yields the best power performance. Figure 13 gives a comparative view of the average normalized power in the desired and undesired directions against the count of nodes.

Comparative Analysis of Communication Capacity at Unintended Receivers (1) Capacity at Unintended Receivers Positioned in the Nulling Directions.
Herein, communication capacity is evaluated at unintended receivers lying in the direction of the distinct 3 nulling points (as per the listing given in Table 6). Tis is done over a range of SNR values running from 0 to 40 dB.
Given an unintended receiver positioned at (−130 degrees azimuth, 60 degrees elevation), interference values are −29.23, −26.42, −24.71, and −19.61 for 20, 15, 10, and 5 nodes null steering, respectively. Te resultant capacity is as per Figure 14. At a lower range of SNR values (0 to 10 dB), capacity performance is roughly identical for the four cases under comparison. At higher SNR values, 20 nodes of null steering ofer distinctively better capacity performance. Te best capacity improvement realized is 46 percent (at an SNR     International Journal of Antennas and Propagation value of 40 dB); this is as evaluated with the capacity outcomes associated with 5 nodes and 20 nodes null steering. Given an unintended receiver positioned at (10 degrees azimuth, 60 degrees elevation), interference values are −28.92, −29.37, −27.70, and −15.94 for 20, 15, 10, and 5 nodes null steering, respectively. Te resultant capacity is as per Figure 15. At the lower range of SNR values (0 to 7.5 dB), capacity performance is roughly identical for the four cases under comparison. At higher SNR values, 15, 20, and 10 nodes null steering ofer better capacity performance. Te best capacity improvement realized is 86 percent (at an SNR value of 40 dB); this is as evaluated with the capacity outcomes associated with 5 nodes and 15 nodes null steering.
At the lower range of SNR values (0 to 5 dB), capacity performance is roughly identical for the four cases under comparison. Noise is dominant over interference hence the observed outcome. At higher SNR values, a higher node count ofers better capacity performance. Te best capacity improvement realized is 167 percent (at an SNR value of 40 dB); this is as evaluated with the capacity outcomes associated with 5 nodes and 20 nodes null steering. Tis capacity improvement value exceeds that reported in [6] (162 percent).
(2) Capacity at Unintended Receivers Distributed over Approx. All Directions Apart From the Beamsteering Direction. Herein, average communication capacity is evaluated over unintended receivers distributed over all azimuth and elevation directions (less the beam steering direction) at a one-degree precision. Te resultant capacity is as per Figure 17. 20 and 15 nodes null steering ofers better capacity performance. Te best capacity improvement realized is 45 percent (at an SNR value of 40 dB); this is as evaluated with the capacity outcomes associated with 5 nodes and 20 nodes null steering.

Performance Analysis with Change in the Beamforming
Cluster Radius. 3-null placement is considered as per the listing given in Table 15.
Wavelength-normalized CB cluster radius values of 1, 2, 3, and 4 are utilized in the null steering process. Figure 18 comparatively illustrates the evolution of the nulling cost function (average outcome of 50 independent   Table 12).

Beam Pattern Analysis
(1) Radiation Power Pattern. Te azimuth cut of the resultant normalized radiation power pattern is as per Figure 19. Te presented pattern is the average of 50 independent outcomes. An increase in cluster radius is associated with a decrease in null width and main beam width.
(2) Null Depth Values. Null depth values are given in Table 16. Te values are the average outcomes of 50 independent runs. Analysis of variance test P values are 9.8641E − 88, 9.2259E − 40, and 3.6124E − 91 for null 1, 2, and 3, respectively. Going by the low P values, the null depth values given in Table 16 are statistically diferent. Te differences are captured/summarized in Table 17 following a Tukey-Kramer comparison test. Table 18 presents null depth performance ranking (in accordance with the Tukey-Kramer comparison test results) upon using cluster radii 1, 2, 3, and 4. A tie in rank in Table 18 implies statistical equivalence. Overall, the radii 1 nulling procedure yields the best null depth performance. In general, there is no distinct relationship between cluster radius and null depth. Figure 20 comparatively illustrates the average normalized power observed in the nulling directions against cluster radius. Tere is no clear-cut relationship between the power values and cluster radius.  Table 19. Te values are measures corresponding to the azimuth-cut radiation pattern presented in Figure 19. A large cluster radius is associated with narrow nulls.
(4) Nulling Accuracy. Nulling accuracy values are given in Table 20. Increase in cluster radius yields better nulling accuracy.  Table 21. Te values are the average outcomes of 50 independent runs. Analysis of variance test P values are 1.1228E − 197 and 0.0000E + 00 for the power in the desired and undesired directions, respectively. Going by the low P values, the power values given in Table 21 bear statistically signifcant diferences. Te diferences are captured/summarized in Table 22 following a Tukey-Kramer comparison test. Table 23 presents power performance ranking (in accordance with the Tukey-Kramer comparison test results) upon using cluster radii 1, 2, 3, and 4. A tie in rank in Table 23 implies statistical equivalence. Overall, as per Table 23, there is a power performance "mix." Tis can be attributed to the following facts: (i) At a small cluster radius, prominent outcomes are a wide main beam, wide nulls, and few low-leveled sidelobes (ii) At a large cluster radius, prominent outcomes are a narrow main beam, narrow nulls, and a number of average-leveled sidelobes     Figure 21 gives a comparative view of the average normalized power in the desired and undesired directions against cluster radius.

Comparative Analysis of Communication Capacity at Unintended Receivers
(1) Capacity at Unintended Receivers Positioned in the Nulling Directions. Herein, communication capacity is evaluated at unintended receivers lying in the direction of the distinct 3 nulling points (as per the listing given in           Table 15). Tis is done over a range of SNR values running from 0 to 40 dB.
Given an unintended receiver positioned at (−135 degrees azimuth, 35 degrees elevation), average interference values are −32.07, −31.13, −25.02, and −31.83 for 1, 2, 3, and 4 cluster radius confgurations, respectively. Te resultant capacity is as per Figure 22. At the lower range of SNR values (0 to 15 dB), capacity performance is roughly identical for the four cases under comparison. At higher SNR values, the 3-cluster radius confguration ofers better capacity performance. Te best capacity improvement realized is 25 percent (at an SNR value of 40 dB); this is as evaluated with the capacity outcomes associated with cluster radii 1 and cluster radii 3 null steerings.
Given an unintended receiver positioned at (30 degrees azimuth, 35 degrees elevation), average interference values are −32.03, −31.47, −28.26, and −31.44 for 1, 2, 3, and 4 cluster radius confgurations respectively. Te resultant capacity is as per Figure 23. At the lower range of SNR values (0 to 20 dB), capacity performance is roughly identical for the four cases under comparison. At higher SNR values, 1, 2, and 4 cluster radius confgurations ofer roughly identical capacity performance. Te best capacity improvement realized is 11 percent (at an SNR value of 40 dB); this is as evaluated with the capacity outcomes associated with cluster radii 1 and cluster radii 3 null steerings.
Given an unintended receiver positioned at (110 degrees azimuth, 35 degrees elevation), average interference values are −30.48, −28.78, −34.37, and −25.17 for 1, 2, 3, and 4 cluster radius confgurations respectively. Te resultant capacity is as per Figure 24. At the lower range of SNR values (0 to 15 dB), capacity performance is roughly identical for the four cases under comparison. Te best capacity improvement realized is 35 percent (at an SNR value of 40 dB); this is as evaluated with the capacity outcomes associated with cluster radii 3 and cluster radii 4 null steerings.      Te cluster radius 1 confguration ofers slightly better capacity performance. Te best capacity improvement realized is 12 percent (at an SNR value of 40 dB); this is as evaluated with the capacity outcomes associated with cluster radii 1 and cluster radii 4 null steerings.

Conclusion
Tis paper has presented a nulling procedure aimed at reducing interference at unintended receivers upon CB in WSNs. A multiobjective criterion has been developed with the considerations listed as follows: (i) Beam steering.
(ii) Deep and wide nulling. (iii) Minimizing radiation power in all directions outside of the intended sink direction. (iv) Simultaneous optimization of node transmit amplitude and phase to achieve the aforementioned items. In current literature, nulling is performed through phase weighting after a conventional beam steering procedure. (v) 3-dimension WSN node layout.
Te performance of the developed (improved) null steering beamformer has been validated through a comparison with a basic null steering procedure yielding the following outcomes: (i) Lower sidelobes (ii) Lower radiation in all undesired directions (iii) Wider nulls (iv) Slightly narrower main lobe width Te above positive outcomes listed above can be attributed to (i) Introduction of a worthwhile undesired radiation suppression mechanism as per equation (14) (ii) Introduction of a mechanism aimed at null widening as per equation (16) Furthermore, performance analysis has been carried out with the considerations listed as follows: (i) Performance analysis with change in the number of collaborating nodes (ii) Performance analysis with change in beamforming cluster radius Performance measures utilized include null depth, null width, and nulling accuracy. It has been established that an increase in the number of collaborating nodes leads to deeper nulls, a marginal decrease in null width, and increased nulling accuracy. A higher number of collaborating nodes has the advantage of distributing radiation power across more nodes. On the negative side, there is an increase in intra-CB cluster communication. Moreover, an increase in the number of collaborating nodes in a CB process might be associated with increased phase, time, and frequency jitter at the collaborating nodes. Tis would inadvertently lead to unpredictable CB performance. It is expected that the advantages associated with an increase in the number of collaborating nodes overcome performance downgrade associated with phase, time, and frequency jitter at the collaborating nodes. An increase in CB cluster radius leads to slightly deeper nulls at the expense of null width. Increased cluster radius is disadvantageous in terms of an increase in transmission energy usage at collaborating nodes when performing intracluster communication. A node count/ cluster radius balance should be struck to yield an optimal null steering outcome. Deep nulls directly imply reduced interference at unintended receivers. Wide nulls reduce CB frequency in cases of mobile unintended receivers and reduce interference in other distinct CB clusters (collaborating nodes are bound to be spatially distributed). Nulling is associated with a reduction in interference/capacity improvement at unintended receivers. Te best capacity improvement realized is 167 percent (at an SNR value of 40 dB); this is as evaluated with the capacity outcomes associated with 5 nodes and 20 nodes null steering. Tis capacity improvement value exceeds that reported in [6] (162 percent).

Limitations and Future Work
Te research work presented herein assumes that there is no "expiry" or failure of sensor nodes participating in the CB process. Sensor nodes are failure-prone owing to energy exhaustion. A potential area of future research would entail working on CB imperfectness upon failure of a collaborating node. In this research work, it has been assumed that there is no phase/time/frequency jitter at the collaborating nodes. Phase/time/frequency jitter would inadvertently lead to unpredictable CB performance. It is essential to analyze CB performance with phase/time/frequency jitter at the collaborating nodes. As per the observed research outcomes, beam pattern nulls lacked uniformity in null widths. It is