Layout-optimized Cylindrical Sonar Arrays
Jan Egil Kirkebø, Andreas Austeng and Sverre Holm
Department of Informatics, University of Oslo
P.O. Box 1080, Blindern, N-0316 Oslo, Norway
Corresponding author: janki@ifi.uio.no
Abstract— The cost of sonars scales to the number of active
elements. Therefore, it is favorable to reduce the number of
elements without loss in imaging quality. This is a combinatorial
problem, but of such a large dimension, even for small arrays,
that an exhaustive search is futile.
In this work, we have found layout-optimized sparsed cylindrical arrays, i.e. arrays with binary weights. In underwater
applications, with demands for omni-directional imaging, cylindrical arrays have shown to have beneficial qualities, and are
commonly used in fishery.
We have found that to optimize the layout of the cylindrical
array it is sufficient to optimize a small sub-array. With a
reasonable number of elements in the elevation direction, finding
the optimal array is then possible through an exhaustive search.
The cylindrical array was then sparsed so that each line of
elements in the elevation direction were chosen to correspond
to the sub-array array.
I. I NTRODUCTION
Cylindrical sonar arrays are often used in fishery industry to
image the sea in all directions around a boat. The sonar used
is a 3D matrix array consisting of a large number of elements
arranged in a pattern covering the cylinder surface.
In transmit mode, all transducer elements are excited to form
an umbrella-shaped beampattern illuminating all 360◦ at the
same time. In reception mode, single line beampatterns are
constructed using some or all elements pointing in the imaging
direction. For both transmit and receive mode, the imaging
performance will scale with the number of elements. Besides
the total number of elements, the layout of the 3D matrix
transducer array is another limitation to the possible performance of a sonar. At the same time the cost and complexity
of the system scales with the number of elements/channels in
the transducer array. Therefore, it is desirable to choose the
number of elements and position of these elements to attain
a low cost and low complexity system with optimal image
quality.
Array optimization has a long history in radar literature
dating back almost 50 years. One-dimensional (1D) and twodimensional (2D) array optimization has been applied for
sonar applications more recently [1]–[3]. The complexity of
the problem to solve scales with the number of elements in
the array and, therefore, array optimization of a 3D array is
a difficult and challenging task which in the general case can
be hard to solve. In this paper an approach is presented which
simplifies the optimization problem so that in most cases brute
force can be applied to find the optimal solution. To ensure the
omni-directional property of the arrays, this is done by making
the presented designs periodic in the azimuth direction in
blocks of two lines in the elevation direction. These blocks are
repeated 16 times in the azimuth direction, to form a cylinder.
Sparse array layouts where the maximal sidelobe level can be
traded against the mainlobe width is presented. Compared to
solutions having all elements arranged in a belt, solutions with
both lower sidelobes and narrower beamwidth are presented.
To calculate the beampatterns of all the permutations a method
is presented which reduces the computational cost significantly
by a differential update.
The article is organized as follows. In Section II sparse
arrays and the model for the calculation of beampatterns of
arrays with directive elements are introduced. This is followed
by results in Section III, and the conclusion in Section IV.
II. M ETHOD
A. Sparse array optimization
In spite of a long history, most methods of array optimization reported during the last century show similarities.
Common for the different approaches are the assumption of
a regular underlying grid of the arrays, each element having
continuous wave (CW) excitation and each element being an
omni-directional point element.
The number of permutations for an array with M elements
and K active elements are
M!
M
=
.
(1)
K
K!(M − K)!
For the array presented in Section III, with M = 320
elements sparsed to K = 160 elements, this corresponds to
approximately 1095 permutations. An exhaustive search of all
the permutations, for all but very low values of M and K, is
futile. Therefore, another approach is necessary to reduce the
search space.
In fishery it is desireable to have omni-directional imaging properties. Using a cylindrical array, the same level
of performance in all directions can be obtained by having
periodicity in the layout in the azimuth direction of the array.
The block of elements which are periocically repeated forms
a 1D array. Analysis of the beamforming performance in
each direction can then be reduced to studying the periodic
extension of the 1D array wrapped around a cylinder.
B. Beam Pattern
The far-field continuous wave (CW) beampattern of an array
with M omni-directional elements focused in infinity is given
of sparsed arrays, for which the number of differing elements
between consecutive arrays is small. Define the weight vector
w = [w1 , . . . , wM ], and
 o

A1 (θ, φ)ej2πsθ, φ · x1 /λ


..
v(θ, φ) = 
.
.
o
j2πsθ, φ · xM /λ
A (θ, φ)e
wavefront
z
φ: azimuth
θ: elevation
k
sφ,θ
φ
M
y
The beampattern can then be written as
W (θ, φ) = w(θ, φ)v(θ, φ).
θ
Sample the beampattern in the directions (θn , φn ), n =
1, . . . , N , and let sn denote the corresponding unit direction
vector. By introducing the extended matrix
v = v(φ1 , θ1 ), . . . , v(φn , θn )
(4)
xm
x
Transducer array
Fig. 1.
the beampattern matrix for all sampled angles (θn , φn ) can
be written as
W = wv.
(5)
Coordinate system for 2D arrays.
by [4]:
M
X
W (k) =
wm ejk · xm ,
m=1
where xm are the element locations and wm are the corresponding weights. k is the wavenumber vector, which has
the amplitude |k| = 2π/λ, where λ is the wavelength. The
element directivity function (i.e. smoothing function) for the
mth element is [4]:
Z ∞
am (x)ejk · x dx,
(2)
Am (k) =
An important observation is that row m in (4) corresponds
to the contribution of element m to the total beampattern in
(5). Thus, to calculate the response with element m added or
removed, one only needs to remove or add the row with the
values
T
 o
Am (φ1 , θ1 )ej2πs1 · xm /λ


..
(6)


.
o
j2πsN · xm /λ
A (φ , θ )e
m
N
N
where Aom (k) is the element directivity function shifted to
the origin. This can be shown through “the space-shifting
property” of the spatial Fourier transform, which corresponds
to the time-shifting property of the Fourier transform in timefrequency-space.
Writing the unit direction vector as
to the matrix v in (4) and do the corresponding operation on
the weight vector w. By choosing the permutations carefully
the number of changes in elements between each iteration will
in almost all cases be limited to one or two. This limits the
computational cost of updating v in (5) to one or two times
2M complex additions, respectively.
To evaluate the different layout candidates, three measures
of imaging quality performance have been used. The main
focus has been on the peak sidelobe level, which indicates the
signal-to-noise ratio of the array. The other two measures are
beamwidth and integrated sidelobe ratio (ISLR). The beamwidth defines the lateral resolution, while the ISLR relates to
the contrast of the imaging system [6]. It is given as the ratio
of the energy in the sidelobes to the energy in the mainlobe,
where in this work the mainlobe-sidelobe transition is given
at the −6 dB level.
sθ,φ = (cos θ sin φ, sin θ sin φ, cos φ)
III. R ESULTS
in rectangular coordinates (Figure 1), the wavenumber vector
becomes k = 2πsθ,φ /λ.
For a circular element with radius d it is
2πd
J1 (kxy d),
A(kxy ) =
kxy
Simulations were done on an array with 10 elements in
elevation on a staggered layout, and 32 elements in the azimuth
direction (M = 320). This is depicted in Figure 2, where
elevation is along the x-axis and azimuth is oriented parallel
to the surface of the cylinder at x = 0. Each element was
circular, with a diameter of 0.008 m. The center frequency was
f0 = 120 kHz, and the speed of sound was c = 1500 m/s. The
diameter of the cylinder was 0.075 m, and the height 0.84 m.
The array was sparsed by up to a factor of 50 % (K = 160)
and all elements had unity weighting.
−∞
where am (x) is the aperture function over the element. The
CW beampattern for directive elements then becomes
W (k) =
M
X
wm Aom (k)ejk · xm ,
(3)
m=1
where kxy represents radius in the (kx , ky ) plane. Jm (x) is
the m-order Bessel function of the first kind.
In [5] a method is presented which can be used to reduce the
computational time for calculating the beampattern of a series
Array = SonarCyl10x32, No. els. = 320
Array = SonarCyl10x32, No. els. = 320
0
40
−5
30
−10
20
Worst−case cut [dB]
−15
y [mm]
10
0
−20
−25
−30
−10
−35
−20
−40
−45
−30
−50
−40
−50
−40
Fig. 2.
−30
−20
−10
0
x [mm]
10
20
30
40
0
0.1
0.2
0.3
0.4
0.5
2
2 1/2
(u + v )
0.6
0.7
0.8
0.9
1
50
Cylindrical array with 10 × 32 elements in a staggered grid.
Fig. 4. Worst-case cut of beampattern for cylindrical array with 10 × 32
elements in a staggered grid.
Array = SonarCyl10x32Belt, No. els. = 160
Array = SonarCyl10x32, No. els. = 320
0
0
−1
−0.8
−5
−0.6
−10
−0.4
−15
−0.2
−20
0
−25
0.2
−30
0.4
−35
0.6
−40
0.8
−45
−5
−10
Worst−case cut [dB]
−15
−20
−25
−30
−35
−40
−45
−50
1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
−50
Fig. 3. Beampattern of cylindrical array with 10×32 elements in a staggered
grid.
The angles were simulated from θ ∈ [−90◦ , 90◦ ] (elevation)
and φ ∈ [−90◦ , 90◦ ] (azimuth). The axes of the beampattern
are u = cos θ sin φ and v = sin θ sin φ. The beamwidth is
given by its width at −6 dB. The beampattern of the full array
in Figure 2 is shown in Figure 3. The beamwidth in the uvdomain is 0.172, corresponding to 9.9◦ . The worst-case cut of
the beampattern is defined as
W C(φ) = max W (θ, φ),
θ>θ0
where θ0 corresponds to the angle of the first zero. The worstcase cut of the array in Figure 2 is shown in Figure 4. The
maximum sidelobe level is −10.5 dB.
An obvious way to reduce the number of elements in a
0
0.1
0.2
0.3
0.4
0.5
2
2 1/2
(u + v )
0.6
0.7
0.8
0.9
1
Fig. 5. Worst-case cut of beampattern of cylindrical array with 5 × 32
elements in a staggered grid.
cylindrical array is to remove the outer elements in elevation,
so that the new array consists of the central belt of the original
array. A 50 % reduction of elements in the array in Figure 2
in the x-direction gives the worst-case cut in Figure 5. The
beamwidth is 0.326, corresponding to 18.8◦, and the maximum
sidelobe level is −10.7 dB. This comforms with the predicted
effect from theory, i.e. an increased beamwidth because of
reduced aperture.
To reduce the number of permutations to be investigated
for the sparse arrays, given by (1) with M = 320 and K =
160, it is fruitful to consider the array as the product of two
“linear” arrays; a 1 × 16 cylindrical array in azimuth, and the
10 × 2 staggered layout array in elevation shown in Figure 6.
10
5
y [mm]
Since the array should have omni-directional properties, all
the arrays in elevation should have the same sparsing. A 50 %
reduction of elements in the elevation sub-array, i.e. M = 20
and K = 10, leaves 184 756 permutations to investigate.
0
−5
Array = SonarHexCirc10x02, No. els. = 20
−10
−50
10
−40
−30
−20
−10
0
x [mm]
10
20
30
40
50
−40
−30
−20
−10
0
x [mm]
10
20
30
40
50
y [mm]
5
0
−5
−10
10
−30
Fig. 6.
−20
−10
0
x [mm]
10
20
30
40
10 × 2 array in a staggered layout.
5
y [mm]
−40
0
−5
−10
−50
To make the search space even more narrow the elevation
arrays can be assumed to be symmetrical, giving M = 10
and K = 5 and thus 252 permutations. Calculating the
beampattern for all the configurations is easy to do on today’s
computer systems.
For the case of a cylindrical array periodic in the azimuth
direction, further reductions in the computational burden can
be made in the method culminating in (6) by observing that
the addition or removal of an element in the staggered layout
array in Figure 6 gives an equivalent operation on all the
corresponding elements in the cylindrical periodic extension
of this array. The simplification is the done by adding all
these rows, of the form given in (6), into one single row
which contains the necessary information of all the elements.
Hence, the beampattern of the cylindrical sparse array can be
represented by a matrix w which contains the same number
of rows as the staggered layout array.
As the example with the belt-array showed earlier, it is
important to take the beamwidth into consideration when
optimizing with respect to the maximum sidelobe level. In
order to find the optimal array layouts, the full beampattern
of all 252 symmetrical designs were calculated and evaluated.
The Opt. 1 array on the top was chosen as the best without
restrictions on the beamwidth. With a restriction on the beamwidth to be less than 0.17, the Opt. 2 had the best solution
with respect to the maximum sidelobe level. Figure 7 shows
the layout of the two solutions. The vital statistics for these
arrays, along with the previously mentioned, are summed up
in Table I.
The Opt. 1 array on the top was chosen as the best without
restrictions on the beamwidth. With a maximum sidelobe level
of −10.5 and a beamwidth of 0.178, corresponding to 10.2◦,
it outperforms the belt-array with respect to both beamwidth
and sidelobe levels. The Opt. 2 array was the best array with a
beamwidth of less than that of the full array, with a maximum
sidelobe level of −8.7 dB
The beampatterns of the Opt. 1 and Opt. 2 arrays are shown
in Figures 8 and 9, respectively. Though there is a tradeoff between a narrow beamwidth and low sidelobe levels, it
is not that significant for the example presented here. This
is because of the small amount of elements in the azimuth
Fig. 7.
Layouts of optimized arrays for different beamwidths.
Array
Full
Belt
Opt. 1
Opt. 2
# els.
320
160
160
160
−6 dB BW
0.172
0.326
0.178
0.169
Max. SL [dB]
-10.5
-10.7
-10.5
-8.7
TABLE I
P ROPERTIES OF THE ARRAYS .
direction, which dictates a minimum beamwidth of about 0.17,
independent on the number and configuration of elements in
the elevation direction. With more elements in the azimuth
direction, and a greater diameter of the cylinder, this would
give more freedom in the choosing the trade-off outlined
above.
When choosing the trade-off outlined in the paragraph
above, there is another factor which should be taken into
consideration. In Figure 8, which corresponds to the optimized
array with the widest beamwidth and lowest sidelobe level,
the energy levels are more uniform and at a higher level.
Thus, the array will receive more energy from the different
directions than e.g. the array corresponding to the beampattern
in Figure 9. This relates to the ISLR, which was mentioned in
the introduction. ISLR decribes an arrays abillity to image low
contrast media, which for instance would be the case when the
algae concentration is large and the algae would act as false
targets reflecting energy from all directions. For the Opt. 1
the ISLR is 19.6 dB, and for the Opt. 2 array it is 20.7 dB,
confirming that the latter has better contrast.
From Table I we see that it is well worth considering the
use of sparse arrays in cylindrical sonar arrays. Compared
to the belt, the sparse arrays give freedom to have improved
maximum sidelobe levels compared to both the full and the
belt arrays, or a beamwidth comparable to the full array.
IV. C ONCLUSION
A method of finding well-performing cylindrical arrays has
been presented. Through simulations a sparse 10 × 32 cyl-
Array = SonarCyl10x32Opt1, No. els. = 160
0
−1
−0.8
−5
−0.6
−10
−0.4
−15
−0.2
−20
0
−25
0.2
−30
0.4
−35
0.6
−40
0.8
−45
1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
−50
Fig. 8. Beampattern of optimized 10 × 32 sparsed with half of the elements,
with no restriction on beamwidth.
ACKNOWLEDGMENTS
We would like to thank Trym Eggen at Simrad AS, Horten,
Norway for fruitful discussions.
Array = SonarCyl10x32Opt2, No. els. = 160
0
−1
−0.8
−5
−0.6
−10
−0.4
−15
−0.2
−20
0
−25
0.2
−30
0.4
−35
0.6
−40
0.8
−45
1
−1
indrical array has been shown to have comparable performance
to the corresponding full array with respect to both beamwidth and maximum sidelobe levels, even with a reduction
of elements of 50 %. Compared to the trivial reduction of
elements of keeping just a central belt of elements there is a
50 % reduction in beamwidth for the same maximum sidelobe
level. The suggested periodic array along with the differential
updating scheme for calculating the beampattern, has made
possible a full search of the possible solutions. The approach
has shown viable of finding solutions with good performance
properties.
Works such as [3], [7] have shown that there are significant
gains in allowing non-symmetry and weighting all elements.
For future work it would be of interest to look at the potential
gains in allowing asymmetric configurations in elevation, and
possibly adding weights to the elements.
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
−50
Fig. 9. Beampattern of optimized 10 × 32 sparsed with half of the elements
with beamwidth restricted to less than 0.17.
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