# Equivalence of some schemes for constructing adaptive spatial filtering in phased antenna arrays

### Abstract

#### For citation:

Kiryakmasov A.K. Equivalence of some schemes for constructing adaptive spatial filtering in phased antenna arrays. *Journal of «Almaz – Antey» Air and Space Defence Corporation*. 2019;(1):13-17.
https://doi.org/10.38013/2542-0542-2019-1-13-17

## Introduction

Наиболее известным вариантом схемы адаптивной пространственной фильтрации (АПФ) является однородная схема в пространстве элементов (рис. 1), в которой в тракте каждого элемента имеется адаптивный регулируемый весовой коэффициент w_{i} и выходной сигнал всей решетки у является весовой суммой входных сигналов [1]:

у = w^{H} x, (1)

where w - vector of weighting factors, w = (w_{i}, ..., w_{N}), defined by the relationship (not considering normalisation):

w = R^{-1}s; (2)

x - N-dimensional (by the number of array elements) vector of signals from element outputs,

x = … (x_{1}, x2, ..., x_{N})^{T}

Here, correlation matrix (CM) of the input signals;

s - reference vector, consisting of ones;

()^{Н} - reference vector, consisting of ones

- sign of statistical averaging.

If the number of array elements N is considerably high, the amount of computations by (2), proportional to N^{3}, turns out to be too large, so a problem arises requiring to decrease the computation amount by reducing dimensionality of the task.

**Fig. 1.** Homogeneous scheme of ASF in element space

Possible options for reducing task dimensionality can be switching to the scheme with a dedicated main channel or to ASF in the beamspace. In either case, two options for constructing

respective ASF schemes are possible. The objective of this paper is to prove equivalence of the two options in each one of these cases.

## ASF in the scheme with a dedicated main channel

In the scheme with a dedicated main channel, the adaptive weighting factors are included in paths L of the array elements (L<N), which can be called auxiliary channels. In this case, two construction options are possible: in the first case, in an N-element antenna array, N L− elements belong to the main channel (Fig. 2, а), and in the second case, all N array elements belong to the main channel (Fig. 2, b), with L of them remaining adaptive.

**Fig. 2.** Two options for constructing ASF scheme with the dedicated main channel:

а – N–L elements in the main channel;

b – N elements in the main channel

In the scheme with a dedicated main channel, the weighting factor is defined by the following relationship [2]:

w_{ L} = R_{L}^{-1 }a

where R_{L} - CM of the signals from outputs L of the adaptive elements,

a – correlation vector (CV),

у_{0} - signal from the output of the main channel;

( )* - sign of complex conjugation;

x_{L} - L-dimensional vector of signals from the outputs of the adaptive channels.

Then the resultant weighting vector for the entire array in the first option of ASF scheme construction (see Fig. 2, a) looks as follows:

w_{1} = (w_{0}, -w_{L1}) (4)

and in the second option (see Fig. 2, b):

w_{2} = (w_{0}, s_{L} - w_{L2}) (5)

where w_{0} - fixed weighting vector for N-L non-adaptive elements;

s_{L} - L -element reference vector, consisting of ones.

Let us show that w_{1} = w _{2}. Apparently, CM R_{L} and its opposite RL^{-1} will be similar for both options, and in case of influence of a single interference source, they appear as:

under the influence of M sources:

where σ_{0}^{2} - power of intrinsic noise in array element;

σ_{i}^{2} - power of the i-th interference source in array element;

- signal vector from adaptive elements, corresponding to the i-th interference source;

I – unity matrix.

The la equatstion in formula (7) represents a recurrent algorithm for finding an inverse CM.

For the two scheme options, the CV will be different, due to correlation or non-correlation of intrinsic noise in the main and auxiliary channels. For the case of influence of a single interference source, the CV for the first and second options is equal to, respectively:

under the influence of M sources:

where- signal vector from non-adaptive elements, corresponding to the i-th interference source;

s_{0} - (N-L)-element reference vector, consisting of ones.

Substituting formulas (6), (8), (9) in equation (3), we obtain for the first option

for the second option

in the first case, vectors and s_{0} are (N–L)-element, and in the second case, N-element ones. Transforming the expression in parentheses and reducing similar terms, relationship (13) can be represented as

with x_{01}^{-H} and S_{0} in expression (14) being already (N–L)-element vectors.

Substituting formulas (12) and (14) in equations (4) and (5), respectively, we obtain similar expressions for w_{1} and w_{2}.

For the case of influence of M sources, we shall prove equality of vectors w_{1} and w_{2} by the method of mathematical induction. For the number of interferences M–1, the weighting vector in the first and second options of scheme construction will be equal to, respectively

Let us proceed from assumption that w_{1} (M -1) = = w_{2} (M -1). Then

Now let us rewrite expression (15) with consideration of formula (3), as

or, multiplying both parts by R_{L} (M -1), as

It remains to be shown that in case of M interferences, relationship (16) is likewise fair, i. e.

It remains to be shown that in case of M interferences, relationship (16) is likewise fair, i. e.

Substituting formulas (18)–(20) in (17), we have

In the last expression, in the left-hand part, vectors and S_{0} are ( ) N L− -element, and in the right-hand part, N -element ones, which in fact proves the correctness of relationship (17).

## Beamspace

Under ASF in the beamspace, transition from element space to the beamspace is provided by means of transformation matrix B, whose columns are orthonormalised vectors of amplitude-and-phase distribution across the antenna aperture for formation of respective beams. In other words, L-element vector of signals from the outputs of beams x_{л}, CM R_{л}, and reference vector s_{л} in the beamspace are formed as follows:

where x – N-dimensional vector of signals from element outputs;

R – CM of the input signals;

s – reference vector in the signal space.

Due to orthogonality of the auxiliary beams, reference vector in the beamspace will take the following view (for definiteness, the element relating to the signal beam is placed in the first position):

s_{л} =(√N, ...,0,0)^{T} (21)

The formed beams correspond to L processing channels, i.e. their number is less than the number of elements N in the array (L <N). Normally, one of the channels is referred to as a signal channel, and the rest L–1 channels – compensation, or auxiliary ones. Such scheme is also referred to as a multi-channel autocompensator [3][4]. In the beamspace too, two options of ASF scheme construction are possible: a homogeneous scheme (all beams in Fig. 3, а are adaptive) and a scheme with a dedicated main channel, where the main (non-adaptive) is considered to be a single selected beam (beam 1 in Fig. 3, b).

**Fig. 3.** Two options for constructing ASF scheme in the beamspace:

а – homоgeneous scheme;

b – scheme with the dedicated main channel

An optimal solution in a homogeneous scheme in the beamspace looks as follows [5]:

In a scheme with the dedicated main channel, similarly to the ASF in the element space, the weighting factor in the beamspace can be written as

where R_{L} - correlation matrix of signals from L adaptive beams;

a_{L} - correlation vector, ;

у_{0} - signal from the output of the main beam;

x_{L} – vector of signals from the outputs of L adaptive beams.

The total (composite) weighting vector in this case will be equal to

w_{л} = (1, w_{L}) (24)

Let us show the equivalence of expressions (22) and (24). In the beamspace, in case of the scheme with the dedicated main channel, the number of adaptive weighting factors is equal to L–1, i. e. one unit less than the total number of beams L. Hence, matrix R_{л}^{-1} can be found by the method of bordering matrix R_{L}^{-1} [6]. Let us write matrix R_{л} as

Then the inverse matrix appears as

Substituting formula (25) in (22) and considering expressions (21) and (23), we have

which matches (24) to the accuracy of normalisation.

## Conclusion

Hence, the equivalence of two options for constructing an ASF scheme in the beamspace and the equivalence of two options of ASF scheme with a dedicated main channel is proved. It determines a possibility to undertake study for one of the options only, namely, for that one where it is simpler and more convenient in a given situation.

### About the Author

**A. K. Kiryakmasov**Russian Federation

### Review

#### For citation:

Kiryakmasov A.K. Equivalence of some schemes for constructing adaptive spatial filtering in phased antenna arrays. *Journal of «Almaz – Antey» Air and Space Defence Corporation*. 2019;(1):13-17.
https://doi.org/10.38013/2542-0542-2019-1-13-17