PROCEEDINGS of the23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Method to determine far-field beampattern of long array fromsubarray beampattern measurements
Donghwan JUNG1; Jeasoo KIM2; Hongju GU3; Sungyong MOON4
1,23

Korea Maritime and Ocean University, Korea

Daewoo Shipbuilding & Marine Engineering Co., Ltd, Korea
4SonarTech Co., Ltd, Korea

ABSTRACT
Beampattern measurement is essential to verify the performance of an array sonar. However, common
problems in beampattern measurement of arrays are the constraints to achieve the far-field condition andreach plane waves due to aspects such as limited measurement space. We aimed to measure an array
beampattern in limited space and evaluate the performance through the –3 dB beam width. Hence, wedevised a method to measure the beampattern of a discrete line array in limited space based on the subarray
method. Specifically, a discrete line array whose measurement space does not satisfy the far-field condition isdivided into several subarrays that satisfy this condition, and the beampattern of the line array is thendetermined from subarray measurements. The proposed method is verified by simulations, and we performed
a measurement experiment on a line array with 256 elements and a design frequency of 455 kHz. The
proposed method accurately determines the beampattern of the line array through a simple approachcompared to beampattern conversion obtained from measurement data from the near up to the far-field.
Keywords: Beampattern, Near-field

1. INTRODUCTION
Beampattern measurement is essential to verify the performance of array sonar. However, suchmeasurement for long arrays generally needs a huge testing field or a complex setup (1,2). Especiallyfor long arrays operating at high frequencies, it is difficult to measure beampatterns in acoustic tanks.
We intended to measure the beampattern in a limited space for performance evaluation through the–3 dB beam width of the array. Near-field scanning can be applied to a very large (or long) array, forwhich the far-field distance becomes prohibitive regarding the achievable experimental test range ordimension of the acoustic tank. This way, effective performance measurements of a large array can beachieved. However, as near-field scanning is very time-consuming (3), we propose a method based on
subarrays to measure the beampattern of a long line array in a limited space. We verified the proposed
method through numerical simulation and then experimentally measured the beampattern of an array.

2. METHODS
2.1 Beampattern
A beampattern represents the intensity variation of a beam as function of its direction and distancefrom its source. Beamforming is a technique used to send (receive) signals over (from) a specificdirection. It can be achieved through spatial filtering by using a sensor array or through signal
processing. The beampattern of a discrete array can be determined by summing the products of thephase differences and the delay functions of all of the array elements by considering the constructive
and destructive interference of signals at a specific angle (4). When the same omnidirectional elements
1

wjdehdghkss@kmou.ac.kr
jskim@kmou.ac.kr
3 guhhjj@dsme.co.kr
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are uniformly arranged over a plane, as shown in Figure 1, and assuming no mutual coupling among
elements, the beampattern is given by𝑀

𝐵(𝜃, 𝜙) = ∑𝑁∑𝑚=1𝐿∑
𝑛=1𝑔(𝑥, 𝑦, 𝑧) ∙ 𝑒𝑥𝑝[(𝑚 − 1)𝜓𝑥 + (𝑛 − 1)𝜓𝑦 + (𝑙 − 1)𝜓𝑧 ]

(1)

𝑙=1

where angles 𝜃 and 𝜙 are as described in Figure 1, m, n, and l correspond to values along axes M, N,and L in Figure 1, respectively, 𝑔(𝑥, 𝑦, 𝑧) is the aperture function, and (𝜓𝑥 , 𝜓𝑥 , 𝜓𝑥 ) are expressed as𝜓𝑥 = 𝐾𝑑𝑥 cos 𝜃 cos 𝜙𝜓𝑦 = 𝐾𝑑𝑦 sin 𝜃 cos 𝜙(2)𝜓𝑧 = 𝐾𝑑𝑧 sin 𝜃with 𝐾 being the wavenumber of the detected signal frequency and 𝑑 the interval between elements(4–6). Beampattern B is defined for plane waves of sound because they satisfy the far-field conditions.

Figure 1 – Geometry of uniform rectangular array, where each element is a red diamond
2.2 Proposed Measurement Method
For a discrete line array whose measurement space does not satisfy the far-field condition, theproposed method for obtaining the beampattern from near-field measurements divides the array into
several subarrays, each satisfying the far-field condition. Then, it is possible to derive the beampatternof the array by assuming each subarray to be one element contributing a beampattern. The aperturefunction of the line array consisting of subarrays is given by
𝑁𝑔𝑛 (𝑥 − 𝑥𝑛 )𝑔(𝑥) = ∑(3)𝑛=1

where 𝑥𝑛 is the position of the subarray and 𝑔𝑛 is the aperture function of subarray n. If theaperture functions for all the subarrays are the same and for a sampling function as in Equation 4, theaperture function of the array can be written in the convolution form shown in Equation 5, where ∗denotes convolution.
𝑁𝐷(𝑥) = ∑𝛿(𝑥 − 𝑥𝑛 )(4)

𝑛=1(5)
𝑔(𝑥) = 𝑔𝑛 (𝑥) ∗ 𝐷(𝑥)As convolution in spatial coordinates is multiplied by the angular area of the beampattern, the total
beampattern can be obtained as(5)𝐵(𝜃, 𝜙) = 𝐵𝑛 (𝜃, 𝜙) × 𝐵𝐷 (𝜃, 𝜙)which corresponds to the beampattern of the subarray multiplied by the beampattern of the sampling
function.

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3. SIMULATION AND EXPERIMENTS
3.1 Numerical simulation
To verify the effectiveness of the proposed method, we first performed a simulation on a line arraywith 256 omnidirectional antenna elements at a design frequency of 455 kHz. The array was divided
into 16 subarrays, each having 16 elements. The beampattern for the far-field condition and that using
the phase difference of each element at the near-field (r = 2.44 m) are shown in Figure 2. Although theideal far-field beampattern corresponds to a sinc function, the beampattern using the phase difference
for the near-field in Figure 2 substantially differs from the far-field beampattern.

Figure 2 – Beampattern simulation for far-field and near-field (r = 2.44 m)
The beampattern of each subarray and that of the sampling function to obtain the total beampatternusing the proposed method for the far-field are shown in Figure 3. The beampattern obtained from theresults of Figure 3 through the proposed method are shown in Figure 4. The calculated and simulated(Figure 2) beampatterns suitably agree, confirming the correctness of the proposed method.

Figure 3 – Beampattern of subarray (top) and sampling function (bottom) in far-field

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Figure 4 – Far-field beampattern verifying the proposed method
The beampattern obtained from the near-field (r = 2.44 m) phase difference using the proposed
method is shown in Figure 5 and confirms that the far-field beampattern can be obtained by applying
the proposed method to near-field measurements.

Figure 5 – Beampattern for near-field obtained by proposed method and beampattern for far and near field
3.2 Experiments
To obtain a real beampattern and further verify the proposed method, we performed near-fieldmeasurement experiments on an array as that from the simulation placed within a 4 × 4 × 5 m acoustic
tank. For near-field measurements, the distance between the transmitter and receiver was 2.66 m, andthe measurement angle ranged from –50 to 50° at intervals of 1°. In addition, we considered ameasurement interval from –10 to 10° for the main lobe with resolution of 0.1° and a sampling
frequency of 10 MHz. We obtained the beampattern from near-field measurements to determine theperformance through the –3 dB beam width of the array, and hence we limited the range ofmeasurement angles.
Figure 6 shows the examples of beampatterns obtained using data measured in the near-field forvarious subarrays. Although all the elements that comprise the array are nearly omnidirectional, thisfeature is not ideal, and there are measurement errors that cause a small differences in beampatterns
among subarrays. To obtain the total beampattern of the array through the proposed method, weassumed that the beampatterns of all subarrays are equal. Therefore, in this experiment, the average
beampattern from the measured subarrays shown in Figure 7(a) was used as representative
beampattern, and the sampling beampattern shown in Figure 7(b) was used to apply the proposedmethod.

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Figure 6 – Examples of subarray beampatterns obtained from near-field measurements

Figure 7 – Subarray beampattern to obtain total beampattern using the proposed method. (a) Subarray
beampattern obtained from near-field measurements and (b) sampling beampattern
The main lobe of the array beampattern obtained from the proposed method with the results in
Figure 7 is shown in Figure 8 (blue line). For comparison, the figure also shows the beampatternobtained from simulation (red dashed line) for an array of 256 omnidirectional elements. The5420

proposed method can suitably retrieve an array beampattern from near-field measurements and beemployed to evaluate the performance of the array given the –3 dB beam width.

Figure 8 – Beampatterns obtained from simulation and proposed method

4. CONCLUSIONS
We aimed to measure the beampattern of an array to evaluate its performance over the –3 dB beamwidth. However, the usually limited available space for beampattern measurements impedes satisfying
the far-field condition of the array. We overcome this problem using a method to obtain the
beampattern of the arrays through near-field measurements when the far-field condition cannot beachieved due to limited space. Through a simulation, we verified the effectiveness and accuracy of theproposed method. In addition, we experimentally verified the proposed method by obtaining thebeampattern of an array through near-field measurements. When measuring beampatterns of arrays,
the far-field condition is usually not satisfied, and far-field measurements present various problems
related to the transmission of sound waves. Therefore, the near-field beampattern measurements
enabled by the proposed method can be used in a variety of practical cases.

ACKNOWLEDGEMENTS
This research was part of the project titled “Development of Ocean Acoustic Echo Sounders andHydro-Physical Properties Monitoring Systems” funded by the Ministry of Oceans and Fisheries,
Korea (Grant No. 20130056).

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