tribution." l E E E Trcrtis. Acou.sf.. Sprc,ch. Signal PmcxJ.ssir1g. vol. 36. pp. 417-420. 1988.

[ I31 B. Boashash, "Nonstationary signals and the Wigner-Ville distribu- tion. '' in Proc. IASTED I n / . Symp. Signal Processing Applic~utioiis (Brisbane, Australia), 1987, pp. 261-268.

[ 141 A. Papoulis. Probability, Roridoin Varicihles aizd Srochastic Pro- cesses. 2nd ed. New York: McGraw-Hill, 1984.

Prescribed Sidelobes for the Constant Beamwidth Array

R. J . WEBSTER A N D T. N. LANG

Abstract-A linear array having constant beamwidth over an oc- tave range can be constructed from a special arrangement of element positions and filters which alter the effective aperture with frequency. However, sidelobe level control is complicated by the appearance of an anomalous lobe when the array is steered away from broadside.

INTRODUCTION The beamwidth of a linear array is inversely proportional to the

product of frequency and aperture. A fixed aperture array is there- fore frequency dependent because its beamwidth increases with de- creasing frequency. However, it is possible to combine the outputs of two separate arrays, one having twice the aperture of the other, ' so as to achieve constant beamwidth over an octave range. This results in a net improvement for both broad-band and narrow-band processing within the constant beamwidth octave.

This technique has been described by a number of authors [ 11- [7]. Each array is beam formed in the usual way; that is, the ele- ments are delayed, weighted, and summed to produce a beam pat- tern with a prescribed main beam direction and peak sidelobe level. The two beam-formed outputs are then passed through comple- mentary filters and summed to yield the final output of the com- posite array. The filters, designed to satisfy the constant beamwidth constraint, act by creating a virtual aperture which varies with fre- quency between the two physical apertures. In other words, the filters implement a kind of frequency dependent aperture weighting function, shortening the aperture as the frequency increases.

It is reasonable to ask whether there is any conflict between the virtual aperture weighting, which is intended for constant beam- width, and the physical aperture weighting which is intended for sidelobe control. D o the sidelobe levels prescribed by the physical aperture function remain intact under constant beamwidth process- ing?

The answer is that for the most part they do, but with a notable exception. Depending on the frequency and steering direction, the first grating lobe of the large aperature (low-frequency) array may intrude into the visible region of the pattern function. It is not a grating lobe in the usual sense because it is not a simple replica of the main lobe. As we show, its height varies with frequency but not significantly (or barely) with the physical aperture function. Thus the prescribed peak sidelobe level can be inadvertently de- feated. Since none of the cited references mentions this grating lobe problem at all, it seems worthwhile to examine it in more detail.

Manuscript received August 16, 1988; revised June 15, 1989. The authors are with Computing Devices Company, Ottawa, ON, Can-

IEEE Log Number 8934015. ' I n practice, the two arrays can be merged; see Fig. 1.

ada, K I G 3M9.

IEEE TRANSACTIONS ON ACOUSTICS. SPEECH. A N D SIGNAL PROCESSIUG VOL 3X. NO 4. APRIL IYYU 721

0096-35 18/90/0400-0727$01 .OO 0 1990 IEEE

i Fig. 1. Constant beamwidth array for N = 5 showing steering delays IZT

and physical aperture weights w,.

T H E CONSTANT BEAMWIDTH TECHNIQUE A schematic for constant beamwidth processing is shown in Fig.

1 (the example shows two merged 5-element arrays), and its op- eration can be summarized as follows. B ( $) denotes the pattern function of the high-frequency array for general N (odd) elements and physical aperture weighting function M?,,, and is defined by

(N- l ) / 2

( 1 ) B ( $ ) = WO + 2 z: w,, cos (n$) , I = I

where in the usual notation

27rd ,

$ = (sin 0 - sin 0,)

d is the interelement spacing, X is the wavelength, and 0 and 0, are the incidence and steering angles, respectively, measured from broadside.

From the design wavelength AD = 2d we define and use for con- venience a frequency ratio v = hD/X; thus $ = 7rv( sin 0 - sin 0 , ) with v = 1 at the design frequency. On this scale the pattern func- tion of the low-frequency array is B ( 2 $ ) , with v = 1 / 2 at its de- sign frequency. Obviously, B ( $ ) at v = l and B ( 2 $ ) at v = 1 / 2 are the same function. The beam pattern of the composite array is the sum of the patterns of the individual arrays weighted by the filter function H ( v ) and its complement, thus

The beamwidth we wish to make constant in the composite array is the angular half-power half-width Bo which applies, for any given steering angle OJ, to each separate array at its design frequency. It is therefore defined by the condition B ( $ , ) / B ( O ) = 1 /& where (j0 = *(sin 0, - sin O T ) . Then, with Bo and 0, constant, the con- dition is made to apply at arbitrary frequencies in the composite array through the constraint

with $' = nv(sin Bo - sin e , ) , leading to

(4 )

Note in (5) that H ( 1 ) = 1 and H ( f ) = 0 as required in (3) to make A ( $ ) = B ( $ ) at v = 1 a n d A ( $ ) = B ( 2 $ ) at v = 1 / 2 . In this way, constant angular beamwidth over the octave 5 v 5 1 is achieved. It is worth emphasizing that the filter function is the same for all steering directions.

The value of Go, and the form of H ( v ) , will depend on the phys- ical weighting function w,,. In the discussion which follows we will

__ T

.

728

5

6

7

x 9

1 0

I E E t rRANSACTlONS ON ACOUSTICS. SPEECH, A N D SIGNAL PROCESSING. VOL 3 8 . NO 4, APRIL IYYO

0 0

0 381

0.618

0 7 8 1

0 902

1 0

use the power-cosine family of weightings defined by

as being representative of the many options available [8]. The peak sidelobe is - 13 d B at a = 0 (rectangular), -23 dB at a = 1, and -32 dB at a = 2 (Hanning). Some numerical values of $o and H ( v ) are given in Table I for the case N = 13. Clearly, H ( v ) is a weak function of a and, as will be seen later, of N .

T H E ANOMALOUS SIDELOBE Constant beamwidth beam patterns illustrating the grating lobe

problem are shown in Fig. 2 , again for N = 13. Each individual graph plots 1 A ( $ ) / A (0) 1 ' in decibels on the vertical axis against sin 8 - sin 8,, on the horizontal axis. On this scale the patterns are independent of the steering angle. For a beam steered to O,, the physical visible region extends from sin 8, - 1 to sin 19, + 1. Thus, broadside corresponds to the interval - 1 to + 1 ; end fire, - 2 to 0, o r 0 to + 2 . The three columns from left to right correspond to a = 0, 1 , and 2 in order. The six rows from top to bottom correspond to v = I , 0.9, . . . , 0.5 in order. For any a , the patterns at v = 1 and v = 1 / 2 must be identical under constant beamwidth since these are the patterns of the individual high- and low-frequency arrays at their respective design frequencies. In between, the 3-dB angular beamwidth remains constant but the remainder of the pat- tern is unconstrained except by the physical aperture function. The level of the sidelobe immediately adjacent to the main lobe, which is normally the peak sidelobe as determined by a , is seen to be maintained, or bettered, in all of the plots. But in the region 1 5

1 sin O - sin $,I 5 2 an anomalous sidelobe appears, moving from I s i n O - s i n O , / = l a t v = l , t o I s i n O - s i n O , I = 2 a t u = 1 / 2 , systematically increasing in height. At v = 1 / 2 it is recognizable as the regular first grating lobe. In wave number this is always located at $ = T , and in angle at I sin 8 - sin 8, I = l / v , so its height will be given, from (3), by

( 7 )

Some numerical values are given in Table I. Since H ( v ) is rela- tively insensitive to a , it follows that the height of the anomalous sidelobe will not be responsive to the physical aperture weighting and must be accounted as a fixed feature of the constant beamwidth beam pattern.

It might appear in Fig. 2 that the special case of broadside steer- ing is immune. Although the peak of the anomalous sidelobe is always outside the visible region, its shoulder may intersect the edge of the visible region at a height that exceeds the design peak sidelobe level and so again, in a technical sense, the prescribed level is defeated. Obviously this effect becomes more pronounced as a increases since the prescribed level becomes smaller, while the anomalous lobe does not, and the width of the main lobe, and so of the anomalous lobe, becomes larger. In wave number, the broadside edge is at $ = a v while the anomalous lobe is centered at $ = T . As v decreases from one, the anomalous lobe is increas- ing in height and moving away from the visible region at the same time. Thus, the level a t the band edge goes through an extremum. An example is plotted in Fig. 3 for (Y = 2 and N = 13. It is a transitory effect in frequency and may be inconsequential to the overall performance for broadside steering. The anomalous side- lobe is essentially a problem for the steered array. The way it dom- inates the end-fire visible region is made dramatically clear in a three-dimensional display. Figs. 4 and 5 compare the normal and constant beamwidth beam patterns for a = 2 , and again N = 13.

THE FILTER FUNCTION H ( v ) We have seen from (7) that the height of the anomalous sidelobe

is principally influenced by the filter function H( v). This function, in all the examples we have studied numerically, seems remarkably

TABLE I

SIDELOBE LEVEL ( N = 13) NUMERICAL V A L U E S OF THE F I L T E R F U N C ~ I O N A N D T H E ANOMALOUS

a = O a = 1 n:? 1 ' I (v"=0214635) 1 ('?"=(1,26Y(x)6) 1 (Yg-O323?XS) " r - llf"l

-22.28 ~37.93

insensitive to both N and a. We can show how this comes about by using simple approximations. For example, for large N and a = 0. we have

B ( $ ) = SI cos (. . ?) dx - I

and

(9)

L

The solution of sin 9 / q = 1 /" is 9 = 1.39156, and since N $ o / 2 + 9 and N $ ' / 2 -+ u9 as N + 03, then

sin 9 sin 2vq

9 2v9

v9 2v9

lim H ( v ) = N - W sin v q sin 2 ~ 9 '

For a 2 0 we have [9]

and

Now Go is a function of a and there is a corresponding N $ o / 2 -+

9a as N -+ 03. Table I1 lists values of 9e determined from the so- lution of

1 - _ r ( I + - + - ; :) r ( I + - - - ; :)-"

L

along with the corresponding peak sidelobe levels which the reader may find useful to have at hand. The angular beamwidth for finite N can be approximated from NT (sin Bo - sin 8,) = 9a. The nu- merical values show that as a increases, l + a / 2 >> 9 , / ~ . With this assumption, and using Stirling's approximation for large a in

IEEE TRANSACTIONS ON ACOUSTICS. SPEECH. AND SIGNAL PROCESSING. VOL. 38. NO 4. APRIL 1990 729

ummu-) 1.1

0

J .I0 I -I0 1:: /-a -30

4 4

N-I3 Ilr 0 I 9

1 6 .I0 I I-\ I

N-13 Ilr 0 I 8

N-11 C 0 - 7 N.13 -1.0 I 7

I\

N-11 0 N 6 I

RI

N-I1 C I D l r . 6 / \ I

umw-i 1.1

0

I

I:: 4

u m w ~ r ~ in

Fig. 2 . Beam patterns illustrating the anomalous sidelobe in the end-fire visible region I 5 I sin 8 - sin 8, I 5 2 .

730 IEEE TRANSACTIONS ON ACOUSTICS. SPEECH, A N D SIGNAL PROCESSING. VOL. 38. NO 3. APRIL 1990

20

m U _J w > w

30 m

12 3 (0

0 90 a 95 1 00

RATIONAL FREQUENCY v

Fig. 3. The anomalous sidelobe level at the broadside visible region edge ( a = 2 ) .

, sin H sin ( 1 ,

Fig. 4. Normal frequency-azimuth beam pattern ( a = 2 )

\ = I I

1 I I

sin 0 -sin 0

Fig. 5 . Constant beamwidth frequency-azimuth beam pattern ( a = 2) .

TABLE I1

LEVEL, N$,/2 --t q, NUMERICAL VALUES OF THE ASYMPTOTIC BEAMWIDTH AND PEAK SIDELOBE

1 0 . A

O m 0

RATIONAL FREQUENCY v

Fig. 6. Constant beamwidth filter function relatively insensitive to both a and N.

(13), we find that 4, cancels in the ratio and

4u’ - 1 lim H ( u ) = ~

N - m 3 2 ’

a--

Fig. 6 plots H ( v ) for cy = 0 and N = 3, along with the limiting version (10) annotated N = 03, cy = 0, and version (14) annotated N = m , cy = W . The differences are small and for practical pur- poses, H ( U ) can be treated as a function of U alone.

CONCLUSION

When a normal aperture shading function is applied to the sep- arate physical apertures of a constant beamwidth array, the pre- scribed peak sidelobe level is not necessarily realized because of the intrusion of an anomalous sidelobe, originating as the first low- frequency array grating lobe, into the visible region of the array. Moreover, the level of this sidelobe is virtually unaffected by the physical aperture function. This is an important consideration for the use of the constant beamwidth technique. The remedy is not apparent, but would likely require an elaboration of the conven- tional constant beamwidth array geometry as well as the inclusion of more array elements.

REFERENCES

[ I ] J . C. Morris and E. Hands, “Constant beamwidth arrays for wide fre- quency bands,” Acusfica, vol. 1 1 , pp. 341-347, 1961.

[2] R. P. Smith, “Constant beamwidth receiving arrays for broad-band sonar systems,” Acusfica, vol. 23, pp. 21-26, 1970.

[3] R. J. Holden and E. L. Hixson, “An octave bandwidth constant beam- width underwater receiving array,” Acoustics Res. Lab., Univ. of Texas at Austin, tech. memo 34, May 1972.

141 F. Pirz, “Design of a wide-band, constant beamwidth, array micro- phone for use in the near field,” Bell Sysf . Tech. J . , vol. 58, no. 8, pp. 1839-1850, Oct. 1979.

[ 5 ] J. Lardies and J.-P. Guilhot, “An optimal broad-band constant beam- width end-fire line array.” Acousf. Leu. , vol. 10, no. 5 , pp. 70-73, 1986.

161 J . Lardies and J.-P. Guilhot, “Realization of a broad-band constant beamwidth end-fire line array.” Acousf. Lef t . , vol. 10, no. 8 , pp. 122- 127, 1987.

[7] J . Lardies and J.-P. Guilhot, “Octave bandwidth constant beamwidth acoustical end-fire line array,” Rev. Sr i . Insfrum., vol. 58 . no. 9 , pp. 1727-1731, Sept. 1987.

[8] F. J . Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE, vol. 66, pp. 51-83, Jan. 1978.

191 R. J . Webster. “A generalized Hamming window,” IEEE Trans. Acoust.. Speech. Signul Processing. vol. ASSP-26. pp. 269-270, June 1978.