Statistically Thinned Array Antennas for Simultaneous Multi-Beam Applications

Statistically thinned array antennas are usually employed to form single-beam radiation patterns. In this work, the possibility to adopt such type of antennas to obtain multiple-beam patterns is successfully explored. In particular, two schemes are proposed and compared. In the first one, multiple-beam patterns are realized by considering each beam corresponding to a different feeding network. In the second scheme, multiple-beam behavior is achieved by a single feeding network. A key question addressed in this manuscript is given by the analysis of the statistical deviation of the synthesized radiation pattern, as compared to the reference one. To this end, the up-crossing method is employed. In particular, the assumption of symmetric thinned arrays leads to analytical results, but avoids the adoption of the simplified hypothesis which usually give inaccuracy. The proposed approach is verified by a Monte Carlo analysis, and shows very good agreement between empirical data and theoretical predictions.


I. INTRODUCTION
S TATISTICALLY thinned arrays give a type of random array obtained by removing/turning off some elements from the so-called reference filled array, according to a probabilistic law which depends on the amplitude taper of the original reference array [1]- [4]. This type of array is appealing, as it requires a reduced number of elements (with respect to the reference array) to achieve the same resolution, with the peak side-lobe level mainly influenced by the residual elements after thinning operation. Moreover, no amplitude-tapering is needed, so that T/R modules can be used in their optimised configuration [1] [5]. Thinned arrays can be usefully adopted in a variety of applications, including satellite communications, radio-astronomy, groundbased high frequency radars, and interference cancellation by adaptive beam-forming. They can be generally adopted in all applications primarily requiring high resolution and low secondary lobes, rather than high gain [6]- [8]. They could be also exploited in the framework of mm-waves communications [9]- [11].
Thinning operation can be typically realized by the adoption of specific optimisation procedures [12]- [18]. In spite of better performance, these approaches generally entail a high computational cost, which can become cumbersome for large antenna arrays [8].
In this paper, statistically thinned arrays (STA) are considered. They are treated in terms of excitation coefficients typically given by binomial random variables. Other schemes assuming different levels for excitations have been also proposed in the literature [19]. Even if they provide a better approximation for the desired array factor, a more complicated feeding network is however required by this latter strategy.
The array factor of STA is a stochastic process which needs to be characterised by resorting to the probability theory. A statistical characterization is relatively easier in the presence of a high number of antenna elements, when the Central Limit Theorem (CLT) can be applied [3]. This is just the case where the reduction in the number of elements is more relevant. An accurate statistical characterization of a thinned array is generally difficult to obtain. Nonetheless, an a priori estimation of the array pattern (such as in terms of the peak side-lobe level) as a function of the array features and the thinning level, is highly desirable. To satisfy this need, several results have been produced over the years in literature, starting from the papers [1] and [3]. Recently, in [20], more accurate results have been presented for the case of symmetric thinned arrays. In particular, the up-crossing method has been used, but avoiding simplified assumptions which degrade the accuracy.
Statistically thinned arrays are generally adopted for single-beam applications, with possible linear phase excitation if a beam steering feature is required. However, many practical cases exist which impose the presence of simultaneous multiple beams [6]. Therefore, in this contribution, STA is applied to realize multiple-beam patterns. Two schemes are proposed. In the first one, each beam is associated to a different feeding network; hence, simultaneous independent beams can be obtained. The second scheme relies on a single feeding network, so the beams are no longer independent. In both cases, the approach developed in [20] is adopted to estimate the achievable performance in terms of two parameters, namely the array factor variance and the "distance" error between the statistical array factor and the reference one. The former is a local measure of the array factor dispersion around the reference array; the latter provides a "global" metric. In particular, by assuming symmetric STA, the mentioned performance parameters are obtained analytically, even if they account for the non-stationarity of the array factor, often neglected in other studies. Monte Carlo analysis is applied to successfully validate the theoretical predictions. Furthermore, we are dealing with the array factor only and no mutual coupling is assumed between antenna elements [21]. In any case, since thinned arrays are usually obtained from periodic lattices, they still allow a more adequate control of mutual coupling than aperiodic arrays. [4].
The work is organised as follows. Section II contains the fundamental concepts on single-beam statistically thinned arrays. In Section III, the two thinning schemes for multiplebeam array factors are outlined, while in section IV they are tested and validated by numerical analysis. Conclusions and potential future developments are finally reported. In addition, the paper includes an appendix section to support the theoretical derivations.

II. SYMMETRIC STATISTICALLY THINNED ARRAYS
For the sake of argument, we briefly report some basic concepts regarding statistically thinned arrays. Let us consider a linear array of N isotropic radiators arranged along the x axis within the segment [−L/2, L/2], L being the array aperture in terms of wavelength (refer to Fig. 1). N is assumed to be even, and the elements are half-wavelength spaced at x n = −x −n = 0.25 + (n − 1)0.5, with n = [1, 2, ..., N/2] (i.e., there is no element in the position x = 0). Moreover, the amplitude coefficients are chosen so that A −n = A n . The corresponding array factor can be written as where u = cos θ and u 0 = cos θ 0 , with θ and θ 0 being the observation and the steering angles, respectively. Accordingly, the visible space is given by the interval is the so-called reference filled array factor, to be approximated by the thinned array. Herein, we address the case of symmetric thinned arrays, which are obtained by thinning only half the array, and then, for each remaining element, by locating further elements in the position −x n . The corresponding thinned array factor is given by [20] F (u) = 2C where {F n } N/2 n=1 are independent Bernoulli random variables. In particular, P r {F n = 1} = 1 − P r {F n = 0} = F n = p n (P r {·} is a probability measure, while the term F n represents the mean of F n ). Also, 0 ≤ p n = α A n / max n {A n } ≤ 1, with 0 < α ≤ 1 being the thinning factor. Indeed, for α = 1, natural thinning is obtained. C = max n {A n }/α. Note that, since an uniform arrangement has been considered, F (u) is a periodic function.
Since the {F n } N/2 n=1 are random variables, F (u) is a stochastic process whose mean and variance are respectively given as [20] where P (u) = F 2 (u) is the power-pattern of the (symmetric) thinned array, and P (u) gives its mean.
If the number N is sufficiently large, by virtue of the Lyapunov Central Limit Theorem [22], F (u) is Gaussian (for each u), that is F (u) ∼ N F ref (u), σ 2 (u) . Furthermore, since F (u) is periodic, µ(u) and σ(u) are periodic as well [20]. Under these conditions, the cumulative distribution function (cdf ) of the array factor magnitude (and consequently of the power-pattern) is easily found to be expressed as [20] [23] [24] [25]. It can be shown that eq. (5) can be written in a closed-form [23], by exploiting the condition that for positive arguments the Qfunction can be written in a closed-form with very small errors [25].
It must be remarked that the above simple results are achieved due to the assumption of symmetric configurations. For general asymmetric arrays, the distribution of the array factor magnitude can be expressed in terms of a generalised non-central chi-square distribution with two degrees of freedom [26]. In this case, no closed-form can be obtained. Anyway, symmetric and asymmetric thinned arrays return the same mean array factor and the average of N t . Furthermore, the difference in the achievable performance are not so relevant, as shown in [20].

III. MULTI-BEAMS STATISTICALLY THINNED ARRAYS
In this section, two schemes for obtaining a thinned array factor consisting of multiple beams are introduced. Basically, they are obtained by adapting the general statistical thinning approach outlined in the previous section. For convenience, such schemes are addressed in the sequel as scheme 1 and scheme 2.
The starting point is the definition of the multiple-beam reference array factor with B n = M m=1 e −2jπxnum . It is seen that F ref (u) consists of M identical beams which are steered at the directions u m . Since A −n = A n (and real) and B −n = B * n , it is useful to arrange (6) as withÃ n = A n |B n | and −ϕ n = ∠B n , or equivalently as A n [a n cos(2πx n u) + b n sin(2πx n u)] A. SCHEME 1 By this scheme, thinning is achieved as follows It is noted that by this scheme all the M beams share the same random coefficients {F n } N/2 n=1 , which are the same as in (2). This means that the actual excitation coefficients pertaining to the multiple-beam reference array factor have not been employed in defining the binomial random variables and that, simply, all the beams are thinned in the same way. This scheme, however, allows to obtain independent steerable beams, in the sense that each beam can correspond to a different chain of phase shifters and therefore to a different signal [27]. Hence, this scheme can be exploited for simultaneous transmission of multiple signals.
The mean and variance of the above array factor are and with P M1 (u) being the power-pattern related to scheme 1.
. Moreover, the probability distribution of N t is the same as the single-beam case. Therefore, by this thinning scheme, introducing additional beams does not change the distribution of the actual number of antenna elements that remains after the thinning. Finally, as in the classical single-beam case, the array factor "statistically" tends to the reference one when the number of elemental radiators increases. VOLUME 4, 2016 B. SCHEME 2 In this case the multi-beam nature of the array factor is directly considered in thinning procedure. More in detail, the binomial random variablesF n are now set according to the amplitude coefficientsÃ n in (7), that is P r {F n = 1} = F n =p n = αÃ n / max{Ã n } = 1 − P r {F n = 0}. The resulting thinned array factor hence writes as in whichC = max{Ã n }/α. This scheme can be obtained by feeding the antenna elements with a single chain of phase shifters, which provide the N phases {ϕ n }. Therefore, while M × N t phase shifters are needed for scheme 1, here, only N t = 2 N/2 n=1F n phase shifters are required, with N t being still a Gaussian random variable with mean 2 N/2 n=1p n and variance 4 What is more, for scheme 2 the average number of active radiators depends on the number of beams. Also scheme 2 allows to obtain multi-beam array factors without amplitude tapering, though the excitation coefficients are different from the ones pertaining to scheme 1. However, by this scheme, the beams are not independent.
F M2 (u) has again a Gaussian distribution with mean and variance

C. GLOBAL CHARACTERISATION
A rough array factor characterization, and hence a comparison between the two schemes, can be given in terms of the distribution of the array factor magnitude, which, as remarked above, is easy to obtain for the symmetric case. However, the mean and variance of the array factor provides only local information, i.e., for each different value of u. A global metric, instead, should be linked to the distance (in a probabilistic sense) between the actual and the reference array factors. To this end, as in [20], we consider the normalised standardised error, for scheme 1 and i = 2 for scheme 2). In particular, performance is estimated in terms of ϵ(u) magnitude supremum (with u implied) (17) is equivalently recast as Equation (17) gives a measure of the error over the whole visible space, including the beam regions. Indeed, P r {S ≤ ξ} = p% entails that, with a probability of p%, ϵ(u) lies be- can be considered as generalised p-percent level curves [20].
It can be verified that the magnitude of the coefficient of variation [28], |CV (u)| = |σ Mi (u)/µ Mi (u)|, is relatively higher in the region of secondary lobes. Accordingly, it is expected that (18) is mainly contributed by the error in such a region, whereas Finding a closed-form solution for the S-distribution is a very complicated problem. However, for the symmetric thinned arrays under concern, the up-crossing method can be conveniently employed and it has proved to work remarkably well [24].
Let N ξ be the number of times |ϵ(u)| up-crosses (i.e., crosses with positive slope) the level ξ. Accordingly, a first result is P r {S ≤ ξ} = 1 − P r {N ξ ≥ 1} ≥ 1 − N ξ (assuming that |ϵ(u)| is below ξ at u = −1), where the Markov inequality, P r {N ξ ≥ 1} ≤ N ξ has been exploited and N ξ is the mean of the number of up-crossings. Hence, a lower bound for the S-distribution is obtained. However, if N ξ is modeled as a Poisson random variable [30], the Sdistribution as can be analytically estimated as [24] where P r {|ϵ(−1)| ≤ ξ} can be calculated from (5). Here, the final crucial issue is the computation of N ξ . This can be achieved as follows [22] in which f |ϵ||ϵ| ′ (ξ, γ; u) is the joint probability density function of |ϵ(u)| and its first derivative. Since ϵ(u) is a real stochastic process, it follows that the determination of the up-crossings of |ϵ(u)| is equivalent to the simultaneous study of the up-crossings of ϵ(u) and −ϵ(u). Moreover, since ϵ(u) is a Gaussian process [3], then ϵ(u) and its derivative ϵ ′ (u) = dϵ(u)/du are jointly Gaussian [22] (of course, the same holds true for −ϵ(u) and its derivative). Eventually, (20) can be written as (see [20] for details) This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication.
is the standard deviation of dF Mi (u)/du (see Appendix), σ ′ Mi (u) = dσ Mi (u)/du. Note that in (21), we exploited the Bravais-Pearson correlation coefficient between ϵ(u) and its derivative is zero, for each u [20].
It is worth stressing once again the role of symmetric arrays assumption. As shown above, for the case at hand, S-distribution can be accurately determined and is relatively simple to compute since only a one-dimensional integration is required to obtain N ξ . Also, P r {|ϵ(−1)| ≤ ξ} can be calculated in an extremely simple way. The same does not hold true for asymmetric thinned arrays. In fact, in this case, to get tractable expressions, the real and imaginary part of ϵ(u) and their derivatives are considered as being four independent stationary Gaussian processes, with the real and imaginary part of ϵ(u) (resp. the real and imaginary part of ϵ ′ (u)) having the the same variance [31]. These are strong assumptions that generally lead to unreliable results [23].

IV. NUMERICAL ASSESSMENT
In this section, a numerical analysis is presented to check the theoretical findings and compare the proposed statistically thinned array schemes.
To this end, each realisation (sample function [22]) of the stochastic thinned array factor is obtained by employing a sample step in the variable u of 1/(10L), which is 5 times finer than the sampling step required by the bandwidth of the power pattern. In particular, 2000 realisations are employed in the following examples.
Each beam of the reference array factor is obtained by sampling a Taylor n-bar current distribution with n = 5 and side-lobe level equal to −25 dB [21]. Thus, the coefficients {A n } are related to the samples of the corresponding current distribution [4]. Furthermore, as stated above, elemental radiators are half-wavelength spaced.
In order to check the behavior of statistically thinned arrays as the number of beams varies, we consider four reference array factors. The first one consists of a single (that is, M = 1) Taylor beam centered at u = u 1 = 0, for which a (1) This single-beam case can be considered as a kind of touchstone for the other cases. The second case concerns two Taylor beams pointing at u 1 = 0 and u 2 = 0.5, respectively, which corresponds to set VOLUME 4, 2016 This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and The third reference array factor presents three Taylor beams at u 1 = 0, u 2 = 0.5 and u 3 = −0.
We would like to point out that here we just consider linear arrays for the sake of simplicity and for the computational burden purposes. However, the derived theoretical tools and the proposed thinned models can be easily generalised to deal with more general curved arrays. Also, they can be used for planar arrays while considering the cuts of the stochastic array factors, with a similar methodology as done in [1].

A. SCHEME 1
We consider three cases. Case 1 refers to N = 200 and α = 1 (natural thinning), case 2 to N = 200 and α = 5/7 (thinning at 50% percent) and case 3 to N = 280 and α = 5/7 (average number of active elements equal to that of case 1). Although N may seem excessive for linear arrays, it is worth remarking that in the literature linear thinned arrays with elements till 2 × 10 4 [3] have been considered in order to study the properties of statistically thinned arrays.
Results concerning case 1 are shown Fig. 2. In this natural thinning case the average number of active elements is equal to 70% of the maximum number N = 200. Fig. 2a shows the magnitudes (in dB) for the four reference array factors discussed above and the magnitudes of realisations of thinned array factors, all normalised with respect to their supremum. As can be seen, the side-lobes of |F (u)|/max{|F (u)|} increase with the number of beams. This trend was already observed for random aperiodic arrays [24] [29]. However, the main lobes of the actual and reference array factors are very similar. The above results are consistent with the variance behaviours reported in Fig. 2b. Indeed, as the number of beams increases, the variance levels become higher. This entails a greater dispersion around the actual reference array. Fig. 2c shows the comparison between the empirical and the theoretical S-distributions (obtained by exploiting eq.(19)). As can be seen, those curves almost overlap and hence the theoretical estimation works very well. It is interesting also to understand how the performance changes when the average of the actual number of elements, N t , varies having fixed N or when the average of N t is fixed and N increases. The next two examples just allow to shed some light on this question.
For case 2 a more severe thinning is imposed so that the average number of radiators after the thinning is lower than in the previous case. Indeed, now the mean of N t is equal to 100, whereas in the previous case it was 140. Results concerning this case are reported in Fig. 3. As expected, previous considerations still apply. However, now the side-lobe level is in general increased; this is consistent with the variance behaviours that are higher than the corresponding previous cases. This basically confirms that the actual number of elements plays a crucial role in controlling the dispersion of the array factor realisations, [1] [20] [23]- [26]. The point is that such a dispersion can be precisely estimated through the analytical S-distributions (see Fig. 3c).
In the last example (case 3), the maximum number of antenna elements is increased at N = 280, while the thinning is kept at 50%. This way, the mean of N t is the same as case 1. Results are shown in Fig. 4. While the trend is of course the same as the previous cases, it is seen that the performance is in between the one of case 1 and 2 (see, for example, the variance behaviours in 4b). This entails that the achievable performance is not only affected by the average number of elements in the array (after the thinning), as it is often stated in the literature. However, our theoretical estimations of S-distributions works remarkably well and hence gives a general tool to foresee the performance by accounting for all the relevant problem parameters.
As a further validation, scheme 1 was also tested using CST Studio Suite. In particular, Fig. 5 shows the comparison between the CST return and the theoretical directivity, in the azimuth plane, of a linear array of cylindrical halfwavelength dipoles arranged parallel to the z axis of the orthogonal Cartesian system, while the array axis coincides with the x axis. Moreover, N = 200, α = 1, M = 4, the operating frequency is 1 GHz and the spacing between antenna elements is half a wavelength. As can be seen, an excellent matching is observed.

B. SCHEME 2
For the analysis of scheme 2, we consider two cases, N = 200 and N = 280, both under natural thinning conditions. Note that now, the average number of active radiators depends not only on α but also on the number of beams (in general, on the current distribution). Figs. 6 and 7 report the results for such cases. It is seen that, as N increases, the performance improves. As remarked above, this is a general trend that holds true also for scheme 2. However, with respect to scheme 1, by comparing Figs. 2 and 6, it is seen that a worsening occurs. Hence, scheme 1 is better but requires a more complex overall feeding network.

C. DISCUSSION
From previous results, it can be noted that the S-distributions look nearly identical, regardless of the number of beams and the average number of retained elemental radiators. This, of course, does not mean that the distance between the statistically thinned array and the reference one is the same in all the cases. This is because S-distributions refer to standardised processes. Indeed, what they actually measure is the probability that the array factor is globally within a strip (around the reference one) that depends on the stan-dard deviation, σ Mi (u), which in turn depends on the case under consideration. Also, as argued above, since the error is mainly related to the side-lobe regions, the S-distribution gives an estimation of the peak level of secondary lobes.
Statistically thinned arrays are particularly suited for large antenna arrays which are populated by a high number of elements. In these cases, the statistical dispersion around the reference array factor can be made very low [3]. In this regard, we point out that, while the considered linear arrangement is chosen for computational convenience, the 8 VOLUME 4, 2016 This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and theory and the results can be easily adapted to deal with the one-dimensional cuts of a two-dimensional array factor.
Finally, in order to give a general picture of the achievable performance, the results shown above are summarized in Table 1. This table reports the rounded mean value of the number of active elements, N ti , and the normalised standard deviation of the array factor, averaged over u, for scheme 1 and i = 2 for scheme 2). The addressed cases are distinguished by indicating (N, α). Looking at the table, the following conclusions can be drawn: • For a given thinning factor, the variance decreases as the number of active elements increases; • For a given average number of active elements, natural thinning performs better (compare σ M1 for the cases (200, 1) and (280, 5/7); • While for scheme 1 the expected value of the number of active elements is always the same (regardless of the number of beams), this is not the case for scheme 2; • With the same number of beams, N and α, scheme 1 is more efficient than scheme 2; • σ Mi allows to roughly estimate the peak level of the secondary lobes. Concerning the last point, consider, for example, the case in Fig. 2. Here, the highest level of the secondary lobes, for the (200, 1)-two beams sub-case (of scheme 1), is about −15 dB. Hence, the peak side lobe level is actually in between 2.5 × σ M1 = 2.5 × 0.0574 −→ −16.86 dB and 4 × σ M1 = 4 × 0.0574 −→ −12.78 dB (see Table 1 for the value of σ M1 ). Since P r {S ≤ 2.5} ≈ ϵ (with ϵ being a very small positive real number) and P r {S ≤ 4} ≈ 1, 2.5 × 0.0574 and 4 × 0.0574 could be seen as the (statistical) minimum and maximum value of the highest level of the secondary lobes, respectively, if σ Mi can be considered nearly constant for u ∈ [−1, 1]. A more precise characterization can be obtained by considering lower and upper level curves, that is LC(u) = 2.5 × [σ Mi (u)/ max{|F DES M (u)|}] and UC(u) = 4 × [σ Mi (u)/ max{|F DES M (u)|}], that with probability almost equal to 1, contain the peak of secondary lobes for u ∈ [−1, 1].
It is worth mentioning that Table 1 also reports cases with N = 5000, borrowed from [4], which clearly show that performance improves with the number of radiators.

V. CONCLUSION
Statistically thinned arrays have usually been studied for single-beam array factor, considering only a linear phaseshift for beam steering.
Here, we have introduced two statistically thinned array schemes for simultaneous multiple-beam generation. In particular, we have analytically characterized the achievable performance in terms of how the resulting array factor statistically deviates from the reference one. To this end, the array factor variance, which gives local information, and the VOLUME 4, 2016 1: (Rounded) expected value of active elements, N ti , and mean value (with respect to u) of the normalised standard deviation of the array factor, σ Mi , relative to the examples shown above (i = 1 for scheme 1, i = 2 for scheme 2). The vector (200, 1) means N = 200 and α = 1 and the same holds for the other vectors. The acronym n.b. stands for number of beams. supremum of the standardised error magnitude, which gives global information (i.e, over the whole visible space), have been derived and linked to the parameters of the problem, such as number of elements in the reference array, level of thinning, number of beams, etc.
The numerical analysis showed that increasing the number of beams leads to an increase of the distance between the thinned and the reference array factors. Whereas, performance improves as the number of active antenna elements increases. What is more, is that the theoretical findings are in excellent agreement with the outcome of the Monte Carlo numerical analysis. Hence, the obtained results can be actually employed as a tool to set in advance the thinning strategy. In particular, the number of beams can be determined beforehand by checking if it is compatible with the desired performance. .

APPENDIX A
For scheme 1, it is easy to prove that the mean of the derivative of the array factor, F ′ M1 (u) = dF M1 (u)/du, is µ