Minimax frequency invariant beamforming - Electronics ... - IEEE Xplore

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Introduction: The frequency invariant beamformer (FIB) is a beam- former of which the response is approximately invariant for a designed frequency range.
then the optimal vector of the complex beamformer coefficients is defined as one that minimises the L1 norm of e:

I.D. Dotlic´

wopt : minfmax jejg ¼ minfmax jXw  rd jg w

Introduction: The frequency invariant beamformer (FIB) is a beamformer of which the response is approximately invariant for a designed frequency range. The technique mostly used for the design of an FIB is FIR or IIR filter coefficient optimisation that employs some analytical relations between frequency responses of filters located on different antenna array elements of the FIB, with utilisation of a differentiation filter at the beamformer output and with or without utilisation of multirate techniques [1, 2]. Another approach is to define some desired beamformer pattern over angle and frequency, and optimise the vector of the beamformer coefficients, in order to minimise the Lk norm of difference between the desired and realised pattern [3]. Attractiveness of the L1 norm minimisation, or the minimax approach, is because it is able to shape the array pattern sidelobes’ envelope in a desired way while minimising its level at the same time. In the field of narrowband beamforming, the L1 pattern synthesis problem has been solved in different ways, but this has not been achieved in the field of wideband beamforming. In our previous work [4] we applied the minimax algorithm, originally developed for the compressive FIR filter design in pulse compression radars [5], to the problem of L1 norm narrowband antenna array beamforming. Since the wideband beamformer with FIR filters can be viewed as a structure that is a combination of the antenna array and the FIR filters, we have decided to try to extend the application of our minimax algorithm to the field of wideband beamforming.

Method formulation: The physical structure that is being addressed here is a wideband beamformer with N antenna array elements and an M complex coefficient FIR filter on each array element. For this purpose, no constraints are being put on the antenna array geometry, i.e. the array elements’ positions and patterns are arbitrary. Our problem is to find the vector of the complex beamformer coefficients w ¼ ½w1;0 ; w1;1 ; . . . ; w1;M 1 ; w2;0 ; w2;1 ; . . . ; T

w2;M 1 ; . . . ; wN ;0 ; wN ;1 ; . . . ; wN ;M 1 

ð1Þ

which, for the set of discrete optimisation points over angle and frequency, makes the L1 norm of the difference between the realised beamformer pattern and the desired one minimal. In view of the general array theory [3], the beamformer response for the CW signal from direction y with a frequency f and amplitude equal to one, r(y, f ) can be represented as rðy; f Þ ¼ dðy; f ÞT w

ð2Þ

where d(y, f ) is the so-called steering vector [3]. For the set of discrete optimisation points {(yi, fi)}, 1  i  P, (2) can be arranged in the linear algebraic form r ¼ Xw

ð3Þ

r ¼ ½rðy1 ; f1 Þ; rðy2 ; f2 Þ; . . . ; rðyp ; fp ÞT

ð4Þ

X ¼ ½dðy1 ; f1 Þ; dðy2 ; f2 Þ; . . . ; dðyp ; fp ÞT

ð5Þ

where

If we define the vector of the desired array responses rd for the same set of optimisation points rd ¼ ½rd ðy1 ; f1 Þ; rd ðy2 ; f2 Þ; . . . ; rd ðyp ; fp ÞT

wopt : minfmax jdiagðgÞejg

ð9Þ

w

Here, g is a real and positive difference weighting vector. Since the problem defined by (9) can be easily reduced to (8), the minimax algorithm may be used in this case as well. Minimax FIB: The mathematical formulation (8), (9), along with application of the minimax algorithm gives the designer various possibilities in pattern synthesis. By changing the number and positions of the optimisation points along with the corresponding rd and g elements’ values, different effects may be acquired. Here we will limit our discussion to the case of the FIB. In this case, we propose that optimisation should be performed for the set of discrete frequencies, uniformly distributed between mimimum and maximum frequency of optimisation, with corresponding elements of the rd and the g vectors that are frequency invariant, i.e. with the same value for the points that are placed in the same direction, but for different frequencies of optimisation. For each frequency of optimisation we propose that the optimisation points should be placed over angular dimension as Fig. 1 shows, similar to the case of narrowband beamforming [4], i.e. the main lobe shape is not constrained, but left to form freely, since in the main lobe area optimisation points are placed only in the directions of the main lobe maximum and its borders. The values of the corresponding g vector elements are very high for these main lobe area points, which gives very low corresponding e element magnitudes at the end of the optimisation. In this way the main beam width is precisely controlled and the frequency selectivity of the main beam maximum is minimised. In the area of sidelobes, optimisation points are distributed uniformly with the g vector elements level that represents inversion of the desired sidelobes’ envelope shape. Since the minimax solution has a tendency to have multiple elements of the jdiag(g)ej vector with the values practically equal to the jdiag(g)e)j, corresponding array pattern points define the sidelobes’ envelope. 1.0

100 80 60 40

optimisation points along with rd elements' values

g vector elements' level (in area of sidelobes inverse to desired sidelobes' envelope shape)

0.8 0.6 0.4

main lobe area 0.2

20 0 -90

0 -60

-30 0 30 angle of arrival, deg

60

90

Fig. 1 Optimisation points distribution along angular axis with corresponding rd and g vectors elements’ values

Numerical examples: For the purpose of illustrating our method we selected linear unequally spaced array geometry optimised for broadband purposes [6] similar to the geometries that have been used in [1, 2]. The selected array has a total of 16 omnidirectional elements and is optimised for frequency range of 0.65fref to fref. Its geometry is shown in Fig. 2.

Fig. 2 Array geometry for examples

ð6Þ

and define the vector of the difference between r and rd as e ¼ r  rd

The minimax algorithm [4, 5] was designed to solve the problem formulated as (8). Additional possibilities in the beamforming may be acquired by solving the weighted minimax problem

elements of g level, dB

A novel method for the minimax pattern synthesis of FIR filter based wideband beamformers is presented. Emphasis is placed on frequency invariant beamforming, although intentional frequency variant patterns may be synthesised as well. The method is based on the algorithm originally developed for the compressive FIR filter design in pulse compression radars.

ð8Þ

w

elements of rd value

Minimax frequency invariant beamforming

ð7Þ

The sample rate is set to be fref. FIR filters on each of the array elements have seven coefficients, which have been empirically found to be sufficient for synthesis of the array pattern in the frequency

ELECTRONICS LETTERS 16th September 2004 Vol. 40 No. 19

range of 0.65fref to fref, i.e. a further increase of the filter length does not enhance the array pattern performance significantly. Optimisation has been performed in 20 frequencies, uniformly distributed between 0.65fref and fref with the 100 optimisation points per frequency. For both examples we set the main lobe width to 30 . In the first example we set the main lobe maximum at 20 and desired the sidelobes’ pattern to be flat. As Fig. 3 indicates, the realised pattern has a main lobe maximum level and width constant over frequency. In addition, the pattern sidelobes’ envelope level is approximately constant over both angle and frequency.

Conclusions: A novel method for minimax synthesis of the wideband beamformer patterns based on the minimax algorithm originally developed for the compressive FIR filter design for pulse compression radars is presented. Although the emphasis is placed on the FIB, the method may be utilised for any problem that involves an FIR filter based wideband beamformer pattern synthesis. The method is also applicable to classic narrowband beamformer pattern synthesis over angle and frequency, or angle alone, which are special cases of the wideband beamformer pattern synthesis problem. In the numerical examples presented, our method showed good results in shaping the sidelobes’ envelope in a desired way and suppressing it in the minimax sense while keeping the main lobe width and direction equal to user-predefined values over the frequency range of optimisation. # IEE 2004 Electronics Letters online no: 20045472 doi: 10.1049/el:20045472

19 May 2004

I.D. Dotlic´ (IMTEL—Institute of Microwave Techniques and Electronics, Bul. Mihajla Pupina 165-B, 11070 Belgrade, Serbia and Montenegro) References 1 2

Fig. 3 Array pattern over angle and frequency in first example

3 4

In the second example, to further illustrate the ability of the minimax method to shape sidelobes, we selected the desired sidelobes’ shape to be 20 dB lower between 40 and 40 than otherwise, and because of that, the level of the g vector elements corresponding to the sidelobes’ area optimisation points which reside between 40 and 40 are set to be 20 dB higher than for other sidelobes’ area optimisation points, as shown on Fig. 1. Here, the main lobe maximum is set to be at 0 . As Fig. 4 indicates, similar to the previous example, the main lobe width and direction are precisely controlled and the sidelobes’ envelope approximates the desired one.

5

6

Ward, D.B., et al.: ‘FIR filter design for frequency invariant beamformers’, IEEE Signal Process. Lett., 1998, 3, (3), pp. 69–71 Forcellini, S., and Kohno, R.: ‘Frequency invariant beamformer using a single set of IIR filter coefficients and multirate techniques’. Proc. IEEE 6th Int. Symp. on Spread-Spectrum Techniques and Applications, New Jersey, USA, September 2000, pp. 575–579 Van Even, B.D., and Buckley, K.M.: ‘Beamforming: a versatile approach to spatial filtering’, IEEE ASSP Mag., 1988, 5, (2), pp. 4–24 Dotlic´, I.D., and Zejak, A.J.: ‘Arbitrary antenna array pattern synthesis using minimax algorithm’, Electron. Lett., 2001, 37, (4), pp. 206–208 Zrnic´, B.M., et al.: ‘Range sidelobe suppression for pulse compression radars utilizing modified RLS algorithm’. Proc IEEE ISSSTA’98, IEEE 5th Int. Symp. on Spread Spectrum Techniques and Applications, Sun City, South Africa, September 1998, pp. 1008–1011 Doles, J.H., and Benedict, F.D.: ‘Broad-band array design using the asymptotic theory of unequally spaced arrays’, IEEE Trans. Antennas Propag., 1983, 36, (1), pp. 27–33

Fig. 4 Array pattern over angle and frequency in second example

ELECTRONICS LETTERS 16th September 2004 Vol. 40 No. 19