1. Introduction
Based on the capabilities of 360° coverage from using multiple beams, an omnidirectional
beam, or a narrow beam steering over 360°, cylindrical arrays have been used in many
applications of radar, sonar, and navigation systems [1,2]. By removing the elements from a fully cylindrical array, the sparse cylindrical
sonar array (SCSA) can keep the aperture and image quality almost unchanged, compared
with a fully cylindrical array. Therefore, SCSAs are utilized in many products of
sonar, such as UMS 4110 and KINGKLIP Mk2 of Thales group, to mitigate the complexity
of the hardware and its operation [2,4,5].
To ensure speed of observation in the azimuth plane, sonars usually produce an umbrella-shaped
beam pattern or a sector beam pattern in transmit mode, and produce single beams that
simultaneously steer over 360° in receiver mode [2]. The sector beam patterns combined with the movement of the sonar can be used in
some mine-hunting sonars, for example, the SA 9510S and SA 9520, the TSM 2022 MkIII,
and the MG-89DSP [6]. The sector beam pattern is also used for cylindrical arrays (or a part of the cylindrical
array) in transmit mode when it does not need to observe 360° coverage (avoiding a
blind direction or focusing on a particular sector) as done with the Kongsberg Simrad-Mesotech
SM 2000 [7]. Although sector beam patterns can be applied in many sonars using an SCSA, only
a small amount of the published academic literature has focused on describing the
mathematical expression for the sector beam pattern of the SCSA in depth.}
By constructing the mathematical expression of the beam pattern and analyzing the
simulation results, this study proposes a solution for generating the sector beam
pattern (using a beam to observe in the fan sector) in an SCSA. With the explicit
expression and simulation software, the parameters of the fan beam, such as the width
or position, can be determined according to the number and position of the active
elements. The effectiveness of the proposed solution is evaluated in MATLAB by comparing
the directivity from the sector beam pattern with that from the omnidirectional pattern
of a full SCSA. Moreover, the parameters of the sector beam pattern obtained from
the proposed solution are also compared with those derived from the conventional solution.
This paper is organized as follows. Section 2 explains the configuration of the SCSA.
Section 3 constructs the expression illustrating the sector beam pattern. Section
4 presents simulation results for the sector beam pattern. Finally, Section 5 concludes
the paper.
2. Configuration of the SCSA
The SCSA configuration is shown in Fig. 1, with N columns and M elements in each column. M is also the number of double rows
(each double row includes two staggered circles). The radius of the circles on the
SCSA is R, and the distance between two adjacent elements in each column is 2h. Therefore,
with N columns featuring a 360° viewing angle, the angle between two consecutive columns
is$\Delta \theta =2\pi /N$, and the total number of elements in the SCSA is N${\times}$M.
With the choice of the first element at A$_{1}$(R,0,0), the s$^{th}$ element in the
n$^{th}$ column (1 ${\leq}$ n ${\leq}$ N) and the m$^{th}$ double row (1 ${\leq}$
m ${\leq}$ M) have a position satisfying s= M(n-1)+m. The array’s elements are numbered
from top to bottom and from left to right (viewed clockwise from the Oz axis). Therefore,
the coordinates of the s$^{th}$ element in the SCSA are determined as follows:
Eqs. (3a) and (3b) can be merged to become
where $\left\lceil t\right\rceil $ and $\left\lfloor t\right\rfloor $ denote the round
functions toward integers of arbitrary real number t: $\left\lceil t\right\rceil =\min
\left\{\mathrm{p}\in \mathrm{Z},\mathrm{p}\geq t\right\}$ or $\left\lceil t\right\rceil
=ceil(t),$ and $\left\lfloor t\right\rfloor =m\mathrm{ax}\left\{\mathrm{r}\in \mathrm{Z},\mathrm{r}\leq
t\right\}$ or $\left\lfloor t\right\rfloor =floor(t)$.
Fig. 1. Configuration of an SCSA with N=16 and M=2.
3. Illustrating the Sector Beam Pattern
Consider point P in the far-field in which the direction is represented by directive
unit vector $\overset{\rightarrow }{u}=$ $(\cos \theta \cos \varphi ,\,\,\,\sin \theta
\cos \varphi ,\,\,\,\sin \varphi )$, with azimuth direction $\theta $ and elevation
direction $\varphi $. The path-length difference between elements A$_{{s}}$ and A$_{{1}}$
directed to P is $\Delta l\,$, calculated as [1,5]:
Substituting directive unit vector $\bar{u}$ and the coordinates of the s$^{th}$ element
in (1-2, 4) into Eq. (5), the expression of the path-length difference is given by
Choosing the first element, A$_{1}$, as the phase reference, (${\psi}$$_{1}$= 0),
the difference between the s$^{th}$ element, A$_{\mathrm{s}}$, and element A$_{1}$
in the direction of vector $\bar{u}$ is
where $k=\frac{2\pi }{\lambda }$ is the wave number, and$\lambda $is the wavelength.
The beam pattern of the array with isotropic elements, which is called the array factor
(AF), is determined as follows:
where $\psi _{s}$ is the phase distribution of elements in the SCSA. To steer the
main beam to any desired direction, all elements in the array must be excited in phase
based on that direction. For example, if it is necessary to steer the main beam to
point P, then the phase distribution of elements in the SCSA has to satisfy $\psi
_{s}=-\Delta \psi _{s}$. When steering to angle $\varphi _{0}$ in the elevation plane,
phase distribution $\psi _{s}$ is determined as seen in Eqs. (9)-(12). In the array, a$_{s}$ is the amplitude distribution of the elements, which can be
separated into the product of distributions in column a$_{m}$ and on double row a$_{n}$.
To reduce the side-lobe level in receiver mode, the amplitude distribution, a$_{n}$,
can be chosen [7].
When taking into account the beam pattern of each element, the element patterns of
the s$^{\mathrm{th}}$ element for the azimuth and the elevation are suitably denoted
cos($\theta_{es}$) and cos($\varphi_{es}$), respectively [1]. When the transducer is attached to the cylindrical bulb, the emission intensity
in the tail direction is very small in comparison with the head direction. Therefore,
the beam pattern of each element in the SCSA in the pattern can be considered in the
azimuth plane and elevation plane using the forms cos($\theta_{es}$) and cos($\varphi_{es}$)
with back lobes equaling 0. With the formulas of the coordinates as (1-2, 4), the
element patterns of the s$^{\mathrm{th}}$ element in the azimuth plane and the elevation
plane are expressed as:
With consideration of the element pattern, the beam pattern of the SCSA is determined
as
To produce the omnidirectional pattern in the azimuth plane (the horizontal plane)
and to steer to angle $\varphi _{0}$ in the elevation plane (the vertical plane),
the amplitude distribution for the elements on each circle is uniform, and the phase
distribution for all elements in the SCSA is calculated according to directive unit
vector $\overset{\rightarrow }{u_{0}}=(0,0,\sin \varphi _{0})$ as follows:
Eq. (12) shows that to generate an omnidirectional pattern in the azimuth plane, the elements
on a circle need to be excited in phase. Based on this characteristic, the sector
bean pattern can be generated by choosing the number of columns, Q (Q≤N), in the SCSA
and by exciting the elements in phase on each circle. The sector width and the sector
position are dependent on the number and position of the active columns, respectively.
With the proposed solution, the expression of the sector beam pattern is depicted
as follows:
where a$_{q}$ is the amplitude distribution for the active columns (including Q columns)
in the SCSA, and q is the ordinal number of the active columns (1 ${\leq}$ q ${\leq}$
Q).
Based on expression (13) and simulation software, it is possible to generate the sector beam pattern for the
SCSA by choosing the number of active columns on the cylinder. With the mathematically
explicit formula, the fluctuations in the intensity of the transmitted beam in the
azimuth plane are mitigated thanks to a suitable window function (a$_{q}$). To change
the fan beam’s position, the starting position of the active columns can be shifted
from 1 to N. With each shift, the sector beam pattern moves to the angle $\Delta \theta
=2\pi /N$.
4. Simulation of the Sector Beam Pattern
To highlight the effectiveness of the proposed solution, this study considers an SCSA
configuration that is the same as the SX90 fishery sonar [9]. In the configuration, the radius of the SCSA is R = 0.185 m with 32 columns (N =
32). The number of elements in a vertical column is 8 (M = 8), and the distance between
the centers of two vertical successive elements is 2h = 4.15 cm. Therefore, the total
number of elements in the SCSA is 256, and the angle between two consecutive columns
is $\Delta \theta =2\pi /32=\pi /16$. With the center frequency of this sonar between
20 kHz and 30 kHz, the simulations were carried out at a carrier frequency of 25 kHz
and by assuming that the sound velocity in seawater is 1500 m/s.
In the elevation plane, the SCSA becomes a uniform linear array antenna (ULAA) when
considering circles as uniformly spaced elements [8]. There are many research papers concentrating on the production of beam patterns
with requirements for half-power beam width (HPBW) and the side-lobe level (SLL) for
the ULAA [10-12]. Therefore, this paper only focuses on generating the sector beam pattern for the
SCSA in the azimuth plane, and assumes it is necessary to generate the sector beam
pattern with widths of 180°, 120°, and 60° in the azimuth plane.
Based on phase distribution formula (12) and the formula of the beam pattern from (13), the SCSA can generate sector beam patterns with different widths by choosing the
number of columns from the simulations. By observing the simulation results when varying
the number and the position of columns, it is necessary to select 18 elements to generate
a fan beam with an approximate width of 180°. Fig. 2 depicts in MATLAB the sector beam pattern with an approximate width of 180° at an
elevation angle of 0° with the number of columns selected, Q, being 18 (i.e., from
1 to 18).
Fig. 2(a) illustrates the three-dimensional (3D) beam pattern of the SCSA generated by 18 columns
in the array when steering at the elevation angle of 0°. With the requirement for
generating a wide beam, the gain of the fan beam (approximately 180°) is determined
by comparing it with the gain of the omnidirectional pattern (360°). To compare beam
patterns, the omnidirectional pattern generated from the full SCSA is plotted with
a solid curve and the sector beam pattern is plotted with a dashed curve in Fig. 2(b), showing the two self-normalized beam patterns. Fig. 2(c) illustrates relatively normalized beam patterns according to the peak of the omnidirectional
pattern.
From Fig. 2(b), the HPBW of the sector beam pattern is 178.0$^{\circ}$ (approximately 180°). The
SLL of the beam pattern in the azimuth plane is -18.95 dB. With active sonars, it
is possible to reduce the SLL by applying window functions for elements in the array
when generating narrow beams in receiver mode [6,8]. As a result, the total SLL is the sum of two SLLs when receiving and transmitting
in decibels. The ripple (fluctuation) on the main beam is 2.62 dB (less than 3 dB).}
The gain of the sector beam pattern compared with the omnidirectional pattern in Fig. 2(c) is 1.22 dB. Since the number of columns (or elements) for generating a sector beam
pattern with a width of 180$^{\circ}$ is only more than half that for generating the
omnidirectional pattern, the gain of the sector beam is not much larger compared to
the gain of the omnidirectional beam. To obtain a large gain, it is possible to increase
the number of elements in the array and generate narrow beam patterns in the rotational
directional transmission mode or receive mode of sonars.}
By using several elements in the SCSA, the energy required is less than that needed
from all the elements when emitting towards a particular sector. To change the fan
angle position, the position of the sector with the active columns in the SCSA can
be shifted thanks to the excitation of the columns. With 32 columns, the generated
fan-beam position total is 32, which corresponds to the initial position and 31 displacements
of active elements.
To produce the sector beam pattern with a width of 120$^{\circ}$, it is also possible
to observe the simulation results obtained from formulas (12) and (13) when varying the number and position of columns. By observing the simulation results
for generating a sector beam pattern with a width of 120° in MATLAB, the number of
active columns selected is 13. The active columns are shifted to change the fan angle
position. Fig. 3 depicts the beam pattern with a beam width of 120° at an elevation of 5° when shifting
the fan angle to the 10th position (corresponding to columns 10 to 22).}
The 3D beam pattern of an SCSA with the above choice of columns is shown in Fig. 3(a). Fig. 3(b) depicts two self-normalized beam patterns with a beam width of 120$^{\circ}$ and
the omnidirectional pattern. From Fig. 3(b), the fan beam width is 122.2° (approximately 120°). The SLL of the beam pattern with
a beam width of 120° in the azimuth plane is -19.01 dB. The ripple on the main beam
with the sector at approximately 120° is 2.44 dB (also less than 3 dB). The gain of
the sector beam increases compared with the omnidirectional beam in Fig. 3(c) with an approximate value of 1.24 dB. With the number of selected columns at approximately
one-third, compared with the omnidirectional beam, the gain of the fan beam with a
width of 120° is also only slightly larger than for the omnidirectional beam.}
Similarly, Fig. 4(a) expresses the sector beam pattern with about a 60° width at an elevation of 10° when
shifting the fan angle to the 20$^{\mathrm{th}}$ position thanks to the choice of
eight active columns (from the 20$^{\mathrm{th}}$ column to the 27$^{\mathrm{th}}$
column). With a uniform amplitude distribution, the depression in the intensity of
the main beam at the middle position is more than 3 dB compared with the peak of the
main beam. To reduce the ripple on the main beam, some amplitude distributions having
a peak at the middle position can be used for the active columns.
Based on investigation of the amplitude distributions having a peak at the middle
position, such as the Dolph-Chebyshev window, the Taylor window, the Gaussian window,
the Hanning window, and the Kaiser window, this paper determined the amplitude distribution
for decreasing the fluctuation in the main beam by less than 3 dB, ensuring a beam
width of nearly 60$^{\circ}$, and increasing directivity gain. With the above requirements,
the chosen distribution is the Kaiser window with the coefficient ${\beta}$=1.1.
Fig. 4(b) shows the sector beam pattern with the Kaiser window (${\beta}$=1.1). From this figure,
the beam width is 59.8° (approximately 60°), and the ripple on the main beam is 2.69
dB (less than 3 dB). The SLL of the beam pattern when using the Kaiser window (${\beta}$=1.1)
is -17.68 dB. With relatively normalized beam patterns as seen in Fig. 4(c), the directivity gain from the sector beam increases by more than 1.00 dB compared
with the omnidirectional beam. Similar to the previous two cases, with only one-quarter
of the total columns generating the omnidirectional beam pattern, the gain also slightly
increased, compared to the omnidirectional beam. The 3D beam pattern of the SCSA generated
from the selection of eight active columns with the Kaiser window is shown in Fig. 4(d).
To evaluate the effectiveness of the proposed solution, this paper compares the parameters
of the beam patterns from the proposed solution and from the conventional solution
[13]. The beam patterns of uniform circular arrays (or cylindrical arrays) in [13] were synthesized by transforming the uniform circular array into a virtual uniform
linear array. According to this solution, a Dolph-Chebyshev window can be chosen with
a side-lobe attenuation of -17 dB to generate a fan beam of 60°. The window from the
proposed solution is the Kaiser window with ${\beta}$=1.1. With the two windows, two
self-normalized beam patterns are shown in Fig. 5(a), and the relatively normalized beam patterns based on the peak of the omnidirectional
pattern are illustrated in Fig. 5(b).
The parameters of the beam patterns are listed in Table~1. From Table 1, the Dolph-Chebyshev window produces a fan beam with HPBW of 56.0°. Owing to the
design for optimizing narrow beams, a fan beam of 60° cannot be generated by the Dolph-Chebyshev
window so that both SLL and ripple do not increase. According to Table 1, the window from the proposed solution can produce a beam pattern having an SLL of
-17.68 dB and a ripple of 2.69 dB. The SLL and ripple are smaller than from the conventional
solution [13] with values of -16.35~dB and 2.98 dB, respectively. The gain of the array from the
proposed solution is slightly larger, compared with the conventional solution [13]. Despite generating a wider beam, the proposed solution produces a beam pattern with
a larger gain and smaller ripple than the conventional solution [13].
The simulation results show that the proposed solution can provide a basis for selecting
the fan angle and position of elements in the cylindrical (or circular) array when
observing the target in a particular sector to optimize the hardware. The proposed
solution can be used not only for sonar systems but also for radar or navigation systems
in emission mode with the sector beam. In receive mode, the targets can be observed
simultaneously by a multi-beam array.
To generate narrow beam patterns in receive mode or transmit mode in both the azimuth
plane and the elevation plane, select several columns and the total circles in the
SCSA, and feed power to the columns so they are in phase in the desired direction.
The phase distribution for narrow beam generation is determined from Eqs. (7) and (8) in Section 3. With this phase distribution, it is possible to determine the amplitude
distribution reducing the SLL to less than the required value and generating the narrowest
HPBW, as described in [8].
Fig. 2. Sector beam pattern with an approximate width of 180°: (a) a 3D beam pattern; (b) the directivity of beam patterns; (c) relatively normalized beam patterns.
Fig. 3. Sector beam pattern with an approximate width of 120°: (a) a 3D beam pattern; (b) the directivity of beam patterns; (c) relatively normalized beam patterns.
Fig. 4. Sector beam pattern with an approximate width of 60°: (a) beam patterns with uniform distribution; (b) beam patterns with a Kaiser window for the active columns; (c) relatively normalized beam patterns; (d) a 3D beam pattern with a Kaiser window.
Fig. 5. Sector beam pattern generated from the two solutions: (a) beam patterns from the solutions; (b) relatively normalized beam patterns.
Table 1. The parameters of the beam patterns.
|
Parameters
|
HPBW
|
SLL
|
Ripple
|
Gain
|
|
Conventional [13]
|
56.0°
|
-16.35 dB
|
2.98 dB
|
0.83 dB
|
|
Proposed
|
59.8°
|
-17.68 dB
|
2.69 dB
|
1.00 dB
|
5. Conclusion
The paper proposed a solution that provides the sector beam pattern for an SCSA by
analyzing the mathematical expression of the beam pattern as well as simulation results.
With the proposed solution, the width and the position of the sector beam were determined
according to the number and position of active columns in the SCSA. The parameters
of the sector beam from the proposed solution were evaluated according to the omnidirectional
pattern in the azimuth plane. The validity and the effectiveness of the proposed solution
were demonstrated with simulation results in MATLAB.