Notes When Designing Subarray (in AESA Systems)

We will examine, in the following, what, generally, are the criteria for choosing the number and shape of sub-arrays. We will do this by proposing a possible sub-array configuration that seems capable of offering good performance in terms of jammer and clutter cancellation.


INTRODUCTION
Electronic scanning antennas enable powerful clutter and jammer cancellation algorithms to be activated.But this requires that the individual radiating elements are directly accessible, fig. 1.When the numb er of radiating elements is very high (over 100 elements) it could be necessary to opt fo r a sub-array configuration 1 .The sub-array configuration is to be considered sub -optimal with respect to the one in which all N radiating elements are accessible.We will examine, in the fo llowing, what, generally, are the criteria fo r choosing the numb er and shape of the sub-arrays.We do this by proposing a possible sub-array configuration that seems capable of offering good performance in terms of jammer and clutter cancellation.We can say, as a general criterion in the choice of subarrays, that the first things to check are the possible existence of grating lobes and the angular resolution of the subarray.Even the classic monopulse antenna, fig.2c, can be seen as fo rmed by 4 subarrays .
The sub -array configuration that we propose is shown in fig. 3.As can be seen in our case, rather than true sub -arrays, th ese are auxiliary channels, concentrated to the external section of the antenna, we can define it the external ring subarray.The configuration recalls, in effect, that of an SL C, [4], wh ere, in fa ct, the auxiliary channels have the obj ective Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.In fig. 3 the M= 36 external elements of the antenna, fo rmed by N= 13 7 radiating elements, are also used in DBF configuration, i.e. as well as being combined with the other elements to fo rm the 4 monopulse quadrants, they are also brought to the base band, i.e. rendered accessible individually.As we will see, the idea is to use the information about the direction of arrival of the interference collected using only these 36 elements, and th en pass them to the complete antenna (with all 137 elements) through which to carry out the cancellation algorithms.To verify the effe ctiveness of this choice, we will compare the results obtained in the cancellation of interferences, first using all 137 elements and then only the 36.Another possible critical point is that the individual elements are not isotropic but will have their own beam-width which limits the search fo r interference to sectors not exceeding ±45deg.

DESCRIPTION
Interference cancellation algorithms are also called beam fo rming techniques.We can speak, fo r their schematization, about a difference between deterministic and statistical techniques.In the first case known, in some way, the direction of arrival of the interference (s) it is possible to introduce nulls in th ese directions, using an algorithm which essentially consists in the solution of a system of equations, [5], [6].If the directions of arrival of the interferences are not known a priori, we can either estimate th em in some way, fo r example using a DOA algorithm, [7], [8], or we can use a statistical description of these, [6], [8].Therefore a first approach to the interference cancellation problem can be of the one indicated in the fo llowing block diagram:  A second approach, to the problem of interference cancellation, is based on the evaluation of correlation matrix of the signal and interference2 .We, then, proceed by evaluating the correlation matrix of signal, interference and noise.Optimal weights are obtained on the basis of the relation, [6] : (1) where : Rxx = correlation matrix of signal, interference and noise, rxd = correlation vector between received signal and desired signal.
The (1), expressed through the block diagram in fig.S, is obtained from the minimization of the error with respect to a reference signal (with a procedure that recalls that fo r the optimum Wiener filter).

Fig.S -Statistical Approach
Another consideration to make IS that the search fo r interferences, performed in a deterministic or statistical way, takes place using the elements of the subarray in passive mode.In the absence of interference, the presence of the subarray will not be taken into account. IV.

CHECK
Before verifying the effectiveness of the proposed solution, it is important, as mentioned, to verify the angular resolution of our subarray (the identification and cancellation of interferences as well as the depth of nulls will depend on this resolution) .In the specific case we compare the radiation patterns of the complete antenna with that of the sub array only.The N= 13 7 element antenna, shown in fig.3, has a radiation patterns as in fig.6, i.e. it has a beam-width both in azimuth and in elevation equal to about 8.8deg (± 4.4deg) .Based on the Rayleigh limit, [3], when we say that the width of the antenna beam is equal to 8.8deg means that we will be ab le to resolve two obj ects located at an angular distance equal to at least 8.8deg.
The concept is illustrated in fig.7a and 7b which highlight a good separation of the targets as long as these are placed at a distance greater than or equal to the antenna beamwidth.To improve this resolution it is possible to use super resolution algorithms such as Monopulse or Music, [6], or, wh ere possible, increase the size of the antenna.Another approach could be to use several independent beams, capable of overcoming the Rayleigh limit.This can be obtained using a DBF configuration, like the one considered through the our 36 elements3.Now let's see the radiation patterns of our sub-array.As can be deduced from fig. 8 we see that our beam width is equal to 6. 6deg (±3 .3deg) .The side lobe levels are quite high, but the beamwidth is fairly good.
3 The multibeam capability is not used in this article.We verify, in some cases, the interference cancellation obtainable using either the entire antenna, fo rmed by N= 13 7 elements4, or only the M= 36 subarrays.What we want to do, both in the deterministic and in the statistical approach, is to use the information obtained from the 36 elements (respectively direction of arrival and correlation matrix) and then extrapolate th em to the entire antenna, as if this information had been obtained directly fr om the whole antenna.

1) Deterministic Approach
The intrference cancellation, in the deterministic case, can be obtained using the expression, [ 6], [9] : in this V denotes the steering matrix associated with the direction of arrival of the interference and the signal, u is a vector of the type : i.e equal to one only fo r a desired steering direction, and cr 2 1 denotes a noise added to stabilze the convergence.Assume we have identified5 the presence of three signals, a desired one and two interfrences, as fo llows : i1 """ (-25deg , Odeg) , s""" (0, 0) , i2 """ (O deg, +20deg) using the entire antenna we will see something like in fig. 9 4 As if all elements were accessible.5 We note this information is not known a priori, in fa ct fo r the identification of the signals the phase distribution read on the elements of the antenna is used, which can be expressed as the sum of several contributions.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.- •. �-., ,! ' ( � ] -"'.It is therefore possible to obtain a cancellation of interference, at least ;?':3 5dB . To summarize the procedure, the fr rst thing was to identify the signals present on the antenna using the amp litude and phase information on the 36 elements, making up the circular subarray.Starting from this information, (2) was used where the steering matrix is evaluated on all 13 7 elements and consists of a matrix of size [3xl69].

2) Statistical Approach
As before we assume 6 to have two interferences and a desired signal (or desired beam steering), placed at i1= (-25, 0) , d= (0 ,0) and i2=(0, +20).Based on the expression (1), we need to evaluate the correlation matrix of signal, interference and noise.There are several ways to do this.Let us now consider fo ur methods.In a fr rst way it is possible to evaluate the correlation matrix as 1 T T (4) Rxx =-[vsRssvs +�Rii� + Rnn J K vhere : Vs = steering vector of signal s, V; = steering matrix of interferences, Rss = signal correlation matrix, 6 Again it is useful to observe we not know this information a priori, but this can be deduced from the amp litude and phase distribution on the antenna elements (if accessible).
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.If we assume incorrelated interferences it will be possible to write: (5 Rm, =a--I In a similar way to what we saw in the deterministic case, we obtain an estimate of the direction of arrival of the the interferences, by reading the distribution of amplitude and phase on the 36 elements, and we can, starting fr om this information, obtain an expression fo r V relating to all 137 elements and th erefore from ( 1) and ( 4) obtain the coefficients fo r interference cancellation.The procedure is similar, to what we saw in the deterministic case, the difference is that now we will use ( 1) instead of (2).The diagram in fig.12 is obtained, a result very similar to fig.11 c.Let us now consider a second way to compute the correlation matrix, i.e., as th ey say, on the fly [10].Let us consider a certain number, K, of acquisitions, or snapshots, on the 36 elements of our subarray.We get an array of the type: x" (l) x,, x,, (K) It can be shown that in the case of Gaussian data 7, the correlation fu nction can be evaluated as : K k Once we have obtained the correlation matrix we can reuse (1 ), but this time we will have only 3 6 coefficients, not 13 7 (corresponding to the entire antenna).What we will do is calculate the weights M' to steer the antenna beam as if there was no interference, using all the elements (1 3 7 in our example) and then add th em to tho se obtained fo r interference cancellation (obtained using only the 36 subarray elements).To verify the results of this procedure, let's start by seeing what would be obtained fr om the SMI method if all 13 7 elements were accessible.Fig. 13a considers an example in which 128 snapshots have been acquired fo r a signal s(t) and an interference i(t) respectively positioned angularly at (0,0) and 05.0).In fig.13a we see the two targets, in the 13b we can see the cancellation of the interference.Finally, if, as already said, we add up the 36 coefficients obtained for the cancellation of interference with those that, starting from all the elements, we can evaluate to steer the antenna beam in a certain desired direction, we obtain, the quite satisfactory, result shown in fig.l5.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.A fmt ber consideration to make is that if the jammer enters tbe main beam (as when the sun enters directly into the eyes when driving) the introduction of a null risks transforming the sum diagram into a delta.It is necessary to adaptively cancel one or more jammers in the main beam by placing an adaptive spatial null in the direction of the escort jammer(s) wbile trying to maintain acceptable pattern gain elsewbere in the main lobe, [7].Another technique could be to reduce the width of the main beam by adjusting the taper of tbe elements.Finally, as said, one more approach could be to use several independent beams, capable of overcoming tbe Rayleigb limit.This can be obtained, in fa ct, using a DBF configuration like the one considered through tbe our 36 elements.
A tbird way to calculate the conelation matrix consists simply in assigning it, on the basis of reasonable assumptions about the degree of conelation of tbe interferences.The main beam steering (direction of anival of desired signal) is contained in the vector rxd and the conelation matrix, of the Toeplitz type, can be estimated as: We also note tbat in this third way, fo r the calculation of the conelation matrix, all tbe elements of the anay were directly involved, not only those of the subanay.Furthermore, it sbould be added that the latter technique produces shallower nulls, but is quite independent of the direction of ani val of the interference ( s )8.Finally, a fu rther way to obtain the coefficients w fo r interference cancellation is based on a recmsive approach, [6].In this way the inversion of the conelation matrix is avoided, see expression (1), and the optimal weights, fo r interference cancellation, are obtained in a iteratively way.In this Least Mean Square (LMS) technique we obtain the optimal weigbts based on the relationships Tbe relations (1 1) and (12) are repeated recmsively, starting from the initial condition of zero weights w.The choice of ,u 8 Somehow the latter approach is equivalent to requiring a Taylor distribution in the sidelobes, i.e. low values in order to obtain an acceptable and almost tmiform cancellation.147 determine s the convergence of tbe algorithm, and it can be sbown tbat a criterion for convergence requires that it be: where Amax indicates the maximum eigenvalue of the correlation matrix Rxx.

VI. CONCLUSIONS
A configmation bas been proposed in which the sub anays are not made up of groups of elements, but are in fa ct coincident with single radiating elements, placed at the edge of the main antenna.These, while contributing to the formation of the sum and delta beams, are also connected to fo rm a DBF structtu•e through which it is possible to estimate the origin of the interferences.What must be higbligbted is that the estimation of the direction of ani val is made using tbe reduced numb er of elements (positioned on the edge of the main antenna), but the cancellation of int erference is implemented using all the elements of tbe main antenna .In the deterministic approach, the elements of tbe subanay are used to estimate the direction of anival of tbe interference, on the basis of which a system of equations is then solved using all the elements of the antenna.In tbe statistical approach we initially calculate the coefficients to steer the beam in a cettain desired direction using all antenna elements.Then we find the 36 coefficients, for deleting the interferences, and fi nally the two sets of coefficients are added together.VII.
Fig.l -Digital Beam Forming Architecture

Fig. 8 -
Fig.8 -SubArray Radiation Pattern , and like in fig.lO when using the subarray.While the signals in fig.9 are clearly visible, instead, in fig.lO, the somewhat elevated sidelobes can mask the real targets, which could lead to introducing nulls wh ere there would be no need.

)Fig. 9 -i
Fig.9 -Signal plus interference (s) seen by all 137 elements (as if all elements were accessible)