An ADS-Based Sparse Optimization Method for Sonar Imaging Sensor Arrays

applied sciences

Article An ADS-Based Sparse Optimization Method for Sonar Imaging Sensor Arrays

Jiancheng Liu 1,2,*, Feng Shi 1,2, Yecheng Sun 1,2 and Peng Li 1,2,3

1 Collaborative Innovation Center of Atmospheric Environment and Equipment Technology in Jiangsu, Nanjing 210044, China; [email protected] (F.S.); [email protected] (Y.S.); [email protected] (P.L.) 2 Jiangsu Key Laboratory of Meteorological Observation and Information Processing, Nanjing University of Information Science and Technology, Nanjing 210044, China 3 Binjiang College, Nanjing University of Information Science and Technology, Wuxi 214105, China * Correspondence: [email protected]

Received: 26 February 2020; Accepted: 28 April 2020; Published: 2 May 2020

Abstract: The Mills Cross sonar sensor array, achieved by the virtual element technology, is one way to build a low-complexity and low-cost imaging system while not decreasing the imaging quality. This type of sensor array is widely investigated and applied in sensor imaging. However, the Mills Cross array still holds some redundancy in sensor spatial sampling, and it means that this sensor array may be further thinned. For this reason, the Almost Diﬀerent Sets (ADS) method is proposed to further thin the Mills Cross array. First, the original Mills Cross array is divided into several transversal linear arrays and one longitudinal linear array. Secondly, the Peak Side Lobe Level (PSLL) of each virtual linear array is estimated in advance. After the ADS parameters are matched according to the thinned ratio of the expectant array, all linear arrays are thinned in order. In the end, the element locations in the thinned linear array are used to determine which elements are kept or discarded from the original Mills array. Simulations demonstrate that the ADS method can be used to thin the Mills array and to further decrease the complexity of the imaging system while retaining beam performance.

Keywords: underwater; sonar imaging; sensor array; ADS; sparse optimization

1. Introduction Sonar imaging is widely used for underwater detection applications, such as military defense, engineering maintenance, accident rescue, topographic survey, and so on [1]. A variety of array sensor systems play an important role in sonar imaging. The conventional underwater array imaging systems generally include two main forms: front-view or side-scan sonar imaging. Both imaging approaches employ a one-dimensional (1D) linear array sensor to obtain two-dimensional (2D) imaging, and then further to get three-dimensional (3D) imaging [2]. Recently, a sonar imaging approach with a 2D array sensor has been in use, which adopts a subarray system and implements a multibeam technique. However, the imaging quality based on the subarray system is related to the number of subarrays and array elements. The higher the imaging quality, the more subarrays and array elements are required. This results in a very complex imaging system and increased manufacturing cost. One of the ways to solve this contradiction is to use sparse array sensor and synthetic aperture imaging technology. The ﬁrst task is to design a sparse layout of array sensor elements while maintaining beam performance and imaging quality [3]. The sparse layout optimization of the sonar array sensor resembles that of a radar array antenna. Therefore, sparse methods for the sonar array sensor can be developed from the sparse radar array antenna [4]. These methods are mainly divided into two categories: random and determined. For the random methods, the distribution characteristics of beam response can be utilized in sparse

Appl. Sci. 2020, 10, 3176; doi:10.3390/app10093176 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 3176 2 of 16 optimization. Yan Tao employed an improved genetic algorithm (GA) to obtain 2D sparse planar arrays, in which he established an optimization model with the goal of minimizing the peak side lobe level (PSLL). This method not only provides a specific sparse ratio of the array elements, but also maintains the array performance [5]. Rodríguez used the simulated annealing algorithm (SAA) to synchronously optimize the locations and the weighting coefficients of linear array elements when the side lobe of the beam pattern meets the relevant requirements [6]. The Convex Optimization (CO) method proposed by D’Urso can globally optimize the positions and weighting coefficients of the array elements to obtain the maximal thinned array, and effectively reduce the computational complexity and PSLL values [7]. For a rectangular array with unequal amplitude excitation, Yuan Zhihao employed the improved Particle Swarm Optimization (PSO) algorithm to perform sparse optimization of the array, in which the inertia weighting strategy is utilized for the global optimal particle perturbation and hopping. The improved algorithm greatly improves the convergence speed and accuracy [8]. The random array optimization algorithms are implemented repeatedly in the iterative search strategy to achieve one global optimal solution under constrained conditions. However, the common disadvantage of the random methods is the huge computational workload and the lack of real-time capacity in practical applications. Liu Xuesong proposed a low-complexity cross-type array, in which the multifrequency (MF) emission and parallel subarray (PS) beam-forming algorithms are used. The obtained sparse array can greatly bring down the hardware cost and complexity of the imaging system [9]. This scenario uses the Cyclic Difference Set (CDS) to perform sparse optimization on Multiple Inputs and Multiple Outputs (MIMO) arrays. The CDS method can obtain the prior estimates of PSLL and the position of array elements. However, this method cannot provide a solution to those arrays with an arbitrary aperture size due to the binary property of the autocorrelation function of the CDS [10]. Dong Jian adopted the Almost Difference Set (ADS) to perform sparse optimization of MIMO uniform linear arrays. The ADS method can determine the position of each array element of the transmit array and receive array, and perform a prior estimation on the side-lobe level of the beam pattern. Therefore, the ADS method greatly enhances the service efficiency of the array element and can be popularized for large arrays [11]. Because deterministic algorithms do not require iteration or fewer iteration times, the solution speed is faster and the convergence is better. Therefore, the ADS method may have great potential in the sparse optimization of the array sensor. Luo Yi-yang constructed a nonequidistant sparse antenna array on the Latin square array structure by using the cyclic difference set as the unit. This method can ensure high sparsity with low lateral radiation [12]. Matteo Carlin analyzed the robustness of the analytical refinement algorithm based on the Almost Difference Set when there are defective elements in the linear array, and derived the PSLL statistics of the damage layout from the values of different aperture sizes [13]. The ADS method possesses many advantages, such as obtaining a prior estimation on the side-lobe level of the array beam power pattern, enhancing the using efficiency of array element, fitting the large size array, and avoiding the binary property of the autocorrelation function. Therefore, the ADS method is suggested to further thin the Mills Cross array sensor, which is based on virtual array element technology in this paper. The purpose is to further reduce the number of array elements, the manufacturing cost, and the system complexity when using the cross array sensor while maintaining beam pattern and imaging quality.

2. Almost Diﬀerence Set and Sparse Optimization of Mills Cross Array

2.1. Almost Diﬀerence Sets Let D be a K subset of an Abelian group G of order N, and the diﬀerences between any two nonzero elements in D will form a set F. If t nonzero elements in G appear in each of Λ times, and the remaining (N-t-1) nonzero elements appear (Λ + 1) times each, then D is said to be (N, K, Λ, t)-ADS [14,15]. Appl. Sci. 2020, 10, x FOR PEER REVIEW 3 of 16

2. Almost Difference Set and Sparse Optimization of Mills Cross Array

2.1. Almost Difference Sets Appl. Sci. 2020, 10, 3176 3 of 16 Let D be a K subset of an Abelian group G of order N, and the differences between any two nonzero elements in D will form a set F. If t nonzero elements in G appear in each of Λ times, and the remainingIf the set (ND-t-1)is ( N,nonzero K, Λ, t elements)-ADS, and appear there ( isΛ a + binary 1) times sequence each, then of length D is saidN that to be satisﬁes (N, K, DΛ,, wheret)-ADS the[14,15].n-th element is represented as: ( If the set D is (N, K, Λ, t)-ADS, and there is a1, binaryn D sequence of length N that satisfies D, where an = ∈ (1) the n-th element is represented as: 0, other Then, the characteristic binary sequence (that corresponding1, nD∈ to D) S = s , n Z is expressed as: =  n an  { ∈ } (1) ( 0, other 1, modN(n) D sn = ∈ =∈{ } (2) Then, the characteristic binary sequence (that0, othercorresponding to D) SsnZn, is expressed as: where, modN(n) means taking the modulus of N, and there exists a periodic autocorrelation function ( ) 1, mod ()nD∈ Cs z about Sn: =  N sn N 1 (2) X−0, other Cs(z) = SnSn+z, z Z (3) ∈ where, modN (n ) means taking the modulusn=0 of N, and there exists a periodic autocorrelation () functionThe autocorrelation Czs about functionSn : is characterized as “three-valued”:  N −1 = Cz ()K,= SS z 0 , z∈ s  nnz+ (3) Cs(z) =  Λ + 1,n=0 z L , K Λ+1 (4)  ∈ ≥  Λ, other The autocorrelation function is characterized as ‘‘three-valued’’:

In the period z [0, N 1], L being a set ofKN, 1 z=0t elements (i.e., L = {lp Z; p = 1, ..., N 1 t }). ∈ − =Λ+∈ − − ≥Λ ∈ − − For illustrative purposes, let us considerCzLKs (z) the examples 1, of ADSs , reported +1 in Table1 of the literature [ 16(4)]  together with the corresponding binary sequences Λ， and other autocorrelation functions. In the interest of saving space, however, only two ADS sequences D1 and D2 are given. They are {5, 6, 9} and {0, 1, 3, 13, In the period zN∈−[0, 1] , L being a setADS of Nt−−1 elements (i.e., L = { l ∈ ; 16, 17}, respectively. The autocorrelation functions Cs (z) of D1 and D2 are shown in Figure1. p pNt=−−1,..., 1 }). For illustrative purposes, let us consider the examples of ADSs reported in Table 1 of the literatureTable [16] 1. All together Peak Side with Lobe the Levels corre (PSLLspondings) for the binary arrays thinned sequences by ADS. and autocorrelation functions. In the interest of saving space, however, only two ADSopt sequences D1 and Dopt2 are given. K Sparse Ratio E{Φmin} PSL (dB) PSL (dB) N min maxADS () They are {5, 6, 9} and {0, 1, 3, 13, 16, 17}, respectively. The autocorrelation functions Czs of D1 20 50% 2.65 13.5 7.57 and D2 are shown in Figure 1. − −

Figure 1. Autocorrelation function CADSADS(z()) of D. Figure 1. Autocorrelation function Czs s of D.

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ADS Through the comparison and analysis, we can see that the autocorrelation function Cs (z) of a (N, K, Λ, t)-ADS is close to that of the (if any) corresponding (N, K, Λ)-CDS. ( K z = 0 CCDS(z) = (5) s Λ other

In addition, the ADSs share several other properties with the CDSs; for example, both ADS and ADS cannot be deﬁned for every value of N, K, Λ and t. However, it should be noted that the autocorrelation function of CDS is “binarity”. As a result, the condition limitations of ADS are fewer than those of CDS. Therefore, ADS is able to generate more sets and arrays, and thus it is easier to obtain the array form achieving the designed beam pattern [16].

K(K+1) = tΛ(N 1 t)(Λ + 1) (6) − − Another characteristic of ADS is cyclic shift, assuming that D(σ) is a sequence obtained by cyclic shifting σ of the original ADS sequence. These shifted ADS sequences are called the isomorphic ADS sequences and have the same parameters as the original ADS sequences (N, K, Λ, t). Since the maximum number of cyclic shifts σ is the length N of the original ADS sequence, N diﬀerent isomorphic ADS sequences can be obtained by cyclic shift, which are expressed as: (σ) = (σ) = ( + ) = D dk modN dk σ , k 1, ..., K (7)

Here, σ Z. ∈ 2.2. ADS-Based Arrays According to the investigations from Oliveri [17], ADS can be used to thin the linear array sensors. The basic idea is to arrange N array elements into a one-dimensional linear array and express them according to the ADS sequence, and then the power function and PSLL estimation are obtained by the autocorrelation function of the linear array and its “ternary value” characteristics. Through the power function and the isomorphism ADS sequences, the discarded or reserved elements in the original linear array are determined. This will ensure that the thinned array sensor achieves or approximates the required beam performance. Suppose that the weighting coeﬃcient w(n) is used to indicate whether the array element exists, here w(n) 0, 1 , n = 1, ..., N. When w(n) is equal to 1 and 0, respectively, the reservation or desertion ∈ { } of the array element are indicated. The array factor can be expressed as:

N 1 X− S(u) = w(n) exp(i2πnγu) (8) n=0 where γ is the ratio of the array element spacing to the signal wavelength, u = sin θ, and θ represents the angle between the signal and the normal direction. In addition, the weighting coeﬃcients of the ADS array by can be obtained according to Equations (1) and (2): ( 1, n D w(n) = ∈ (9) 0, other

2 From Equation (8), when the sampling interval u = 1/Nγ, the beam power pattern S(u) at u = k/Nγ is expressed as: 2 S(k/Nγ) = Ξ(k), k = 0, 1, ..., N 1 (10) − Appl. Sci. 2020, 10, 3176 5 of 16

According to the ADS autocorrelation function, the beam power pattern S(u) at u = k/Nγ, that is, the sampling of S(u), is the Fourier transform of the autocorrelation function Cs(z) [17]:

N 1 X− Ξ(k) =, Cs(z) exp(i2πzk/N) (11) z=0

It is further expressed as: Ξ(k) = K Λ + NΛδ(k) + ψ(k) (12) − For Equation (12), δ(k) is the discrete impulse function [δ(k) = 1 if k = 0, and δ(k) = 0 otherwise], PN 1 t r r Ψ(k) , IDFT ψ(z) where ψ(z) , − − δ(z z ) and z , r = 1, ..., N 1 t are the indexes at which p=1 − − − Cs(z) = Λ + 1 (an analogous relationship holds true in the linear case [17]). According to Equation (4), the ADS sequence exhibits a three-level autocorrelation function. The PSLL of the beam power direction satisﬁes the inequation [18]:

opt opt opt opt PSLL PSLL PSLLopt D PSLL PSLL (13) MIN ≤ DW ≤ { } ≤ UP ≤ MAX n o n o opt = (σ) opt = min opt = where PSLL minσ [0,N 1] PSLL D , PSLLDW max PSLL∞, PSLL , PSLLUP ∈ − n mino opt opt n mino n mino E Φ PSLL∞, PSLL = PSLL∞ and PSLL = E Φ PSLL∞ , being E Φ 0.8488 + N MIN MINn o MAX N MAX N ≈ min = min ( ) opt 1.128 log10 N [18] and PSLL E ΦN minn PP ,n /PP ,0. It should be pointed out that PSLLDW opt ∞ ∞ and PSLLUP are determined when the ADS sequence is available since they require the knowledge of opt opt the coefficients PP ,n. On the contrary, PSLL and PSLL can be always computed a priori from ∞ MIN MAX Equations (14) and (15). p opt K Λ 1 + t(N t)/(N 1) PSLL = − − − − (14) MIN (N 1)Λ + k 1 + N t − − − p opt n o K Λ 1 + t(N t) PSLL = E Φmin − − − (15) MAX N (N 1)Λ + k 1 + N t − − − According to the relevant theory and research of ADS, it is extremely difficult to deduce a more opt opt accurate PSLLDW and PSLLDW as the limits of PSLL estimation completely from mathematics, while it opt opt is relatively easy to use PSLLDW and PSLLUP, and it can meet the requirements of PSLL prior estimation of the beam power pattern [18]. For illustrative purposes, let us use the cases in the literature [16]. (0) When using the (42, 21, 10, 31)-ADS sequence, the ADS-based MIMO transmitting array is D21 = (0, 1, 2, 4, 7, 8, 11, 12, 13, 14, 16, 17, 20, 22, 25, 27, 35, 36, 37, 38, 39) and 41 other isomorphic sequences are obtained by applying cyclic shifting to the original ADS. Then, calculate the PSLLs of different ADS-based virtual arrays to obtain the optimal PSLL.

3. Mills Cross Array Optimization Using ADS Method The Mills Cross array sensor is a kind of regular array, which is obtained by virtual array element technology. It has been widely investigated in radar and sonar imaging [19]. Figure2 shows a Mills Cross array with nine transmit and nine receive elements, which form a transmit and receive array, respectively, arranged on the X and Y axes of the Cartesian coordinate system. According to the equivalent phase principle, the Mills Cross array can determine a 9 9 virtual array sensor under the × condition of far ﬁeld, as shown in Figure3. Therefore, this will allow to greatly decrease the complexity of the imaging system compared with the common real array. However, there is still some degree of redundancy in the Mills Cross array sensor when the number of array elements reaches a certain scale. Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 16 Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 16

degree of redundancy in the Mills Cross array sensor when the number of array elements reaches a degree of redundancy in the Mills Cross array sensor when the number of array elements reaches a Appl.certain Sci. scale.2020, 10 , 3176 6 of 16 certain scale.

z P P

y θ

x x

Figure 2. Mills Cross array with nine transmit and nine receive elements. Figure 2. Mills Cross array with nine tran transmitsmit and nine receive elements.

Figure 3. Virtual array form Mills Cross array in FigureFigure2 2.. Figure 3. Virtual array form Mills Cross array in Figure 2.

Because thethe Mills Mills Cross Cross array array sensor sensor is one is two-dimensionalone two-dimensional planar planar array, therearray, will there be awill problem be a Because the Mills Cross array sensor is one two-dimensional planar array, there will be a ofproblem large computational of large computational complexity complexity if the existing if the ADS existing optimization ADS optimization strategies strategies are directly are used directly [20]. problem of large computational complexity if the existing ADS optimization strategies are directly Forused this [2120 reason,]. For this this paper reason, proposes this apaper diﬀerent proposes optimization a different strategy optimization to avoid this strategy problem. to Firstly,avoid this the used [2120]. For this reason, this paper proposes a different optimization strategy to avoid this two-dimensionalproblem. Firstly, virtualthe two-dimensional array sensor from virtual the Mills array Cross sensor array from should the beMills divided Cross into array several should virtual be problem. Firstly, the two-dimensional virtual array sensor from the Mills Cross array should be lineardivided arrays into several with equal virtual length. linear As arrays shown with in Figureequal 3length., the virtual As shown circle in represents Figure 3, the virtual locations circle of divided into several virtual linear arrays with equal length. As shown in Figure 3, the virtual circle virtualrepresents array the elements, locations and of eachvirtual dashed array frame elements, is one and virtual each linear dashed array frame of the is sameone virtual length. linear According array represents the locations of virtual array elements, and each dashed frame is one virtual linear array toof arraythe same theory length. [21], According each virtual to lineararray arraytheory can [2221 be], regarded each virtual as a linear “virtual array large can array be regarded element”. as In a of the same length. According to array theory [2221], each virtual linear array can be regarded as a this“virtual way, large in the array direction element”. perpendicular In this way, to in the the virtual direction linear perpendicular array, a linear to array the virtual composed linear of array, virtual a “virtual large array element”. In this way, in the direction perpendicular to the virtual linear array, a linearlinear arrayarray cancomposed be formed. of virtual According linear to array the expected can be formed. beam pattern According and theto the estimated expectedPSLL beam, the pattern linear linear array composed of virtual linear array can be formed. According to the expected beam pattern arrayand the can estimated be thinned PSLL by,the the ADSlinear method array can tochoose be thinned a virtual by the linear ADS array, method and to then choose some a virtual virtual linear array and the estimated PSLL, the linear array can be thinned by the ADS method to choose a virtual linear elementsarray, and are then gained some in thevirtual same array way. elements As such, are the sparsegained optimizationin the same ofway. two-dimensional As such, the virtualsparse array, and then some virtual array elements are gained in the same way. As such, the sparse arrayoptimization sensors canof two-dimensional be realized from thevirtual azimuth array X andsensors Y. Finally, can be the rea positionlized from of these the resultantazimuth sensorsX and Y. is optimization of two-dimensional virtual array sensors can be realized from the azimuth X and Y. usedFinally, for the reverse position calculation of these to resultant reserve and sensors discard is used the array for reverse elements calculation in the original to reserve Mills and Cross discard array. Finally, the position of these resultant sensors is used for reverse calculation to reserve and discard Asthe aarray result, elements a two-dimensional in the original array Mills with Cross a more array. sparse As a result, and nonuniform a two-dimensional “cross” shapearray with is obtained. a more the array elements in the original Mills Cross array. As a result, a two-dimensional array with a more Thesparse optimization and nonuniform implementation “cross” shape is described is obtained. in Figure The 4optimization. implementation is described in sparse and nonuniform “cross” shape is obtained. The optimization implementation is described in Figure 4. Figure 4.

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FigureFigure 4. 4. AlmostAlmost Different Diﬀerent Set Setss (ADS) optimization procedure for a Mills Cross array.array.

Because the beam performance of the array is greatestgreatest when the sensors are spaced at half the wavelength of the transmitting transmitting sound sound wave wave [2221 [21],], the the virtual virtual array array elementselements fromfrom thethe MillsMills Cross array are also on thethe gridgrid atat halfhalf wavelength.wavelength. Seco Secondly,ndly, after the ﬁnitefinite length linear array from the virtual array elements is is thinned thinned by by the the ADS ADS method method,, the the sensor sensor arrangement arrangement ofof the the sparse sparse array array is isrepresented represented by bya binary a binary sequence, sequence, that that is, 1 is, is 1re ispresented represented as array as array element element retention retention and 0 andas array 0 as arrayelement element abandonment. abandonment. The obtained The obtained K arrayK array elements elements are are sparsely sparsely distributed distributed at at NN half-wave positions. Moreover, the advantages of the ADS methodmethod are shown by the low redundancy of fewer array elements.elements. InIn thisthis paper, paper, 40 40 transmitting transmitting and and 40 receiving40 receiving elements elements were were used used in the in original the original Mills CrossMills Cross array. array. Because the thinned ratio in ADS research is usually 50%, that is, K is 20, 20, the the parameters parameters ( (N,K,Λ,,tt) in the ADSADS librarylibrary areare onlyonly (40,(40, 20,20, 9,9, 10),10), andand thethe correspondingcorresponding originaloriginal sequencesequence is:is: ()0 (0) =() D20 = (1,3,4,5,7,8,11,14,16,17,18,19,22,24,25,26,27,31,32,36) D20 1, 3, 4, 5, 7, 8, 11, 14, 16, 17, 18, 19, 22, 24, 25, 26, 27, 31, 32, 36 The PSLL range of the corresponding array expressed by this sequence is shown in Table 1. The PSLL range of the corresponding array expressed by this sequence is shown in Table1. After the MillsTable Cross 1. All array Peak withSide Lobe 40 transmitting Levels (PSLLs) elements for the arrays and 40thinned receiving by ADS. elements is further thinned by the (N, K, Λ, t)-ADS method, the element locations in the thinned array are determined, as min E{Φ } opt opt K Sparse Ratio N PSLmin (dB) PSLmax (dB)

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Appl. Sci. 2020, 10, 317620 50% 2.65 −13.5 −7.57 8 of 16 After the Mills Cross array with 40 transmitting elements and 40 receiving elements is further thinned by the (N, K, Λ, t)-ADS method, the element locations in the thinned array are determined, shown in Figure5a. The two-dimensional virtual array formed from the thinned array is shown in as shown in Figure 5a. The two-dimensional virtual array formed from the thinned array is shown in Figure5b. The vacant position on the grid in Figure5 is a discarded array element. Figure 5b. The vacant position on the grid in Figure 5 is a discarded array element.

(a)

(b)

FigureFigure 5. 5.Thinned Thinned array array from from the the Mills Mills Cross Cross array array and and the the corresponding corresponding virtual virtual array. array. (a ()a ADS) ADS thinnedthinned array array form form the the Mills Mills Cross Cross array; array; (b ()b Virtual) Virtual Array Array formed formed from from subﬁgure subfigure (a ).(a).

4.4. Analysis Analysis of of ADS ADS Thinned Thinned Array Array

4.1. Eﬀect of Cycle Index 4.1. Effect of Cycle Index According to the PSLL estimation inequality of the beam pattern in the ADS method, the PSLL According to the PSLL estimation inequality of the beam pattern in the ADS method, the PSLL boundsbounds of of transmitting transmitting and and receiving receiving arrays arrays can can be be obtained obtained for for the the original original Mills Mills Cross Cross array. array. Based Based onon the the equivalent equivalent relation relation of of product product theorem, theorem, the the beam beam pattern pattern of of virtual virtual array array can can be be expressed expressed as: as:

Pvirtual(θ,θϕϕ) = Pt( θϕθ, ϕ) ⋅ Pr( θϕθ, ϕ) (16) PPPvirtual(, )=(, t )· r (, ) (16) where Pvirtual(θ, ϕ), Pt(θ, ϕ), and Pr(θ, ϕ) represent the normalized power pattern of the virtual array, the transmitting array, and the receiving array, respectively.(θ, ϕ) indicates the angle of the beam. For

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Appl. Sci. 2020, 10, 3176 θ ϕ θ ϕ θ ϕ 9 of 16 where Pvirtual (, ), Pt (, ), and Pr (, ) represent the normalized power pattern of the virtual array, the transmitting array, and the receiving array, respectively. (,θ ϕ ) indicates the angle theof normalizedthe beam. For beam the power normalized pattern beam of the power transmitting pattern or of receiving the transmitting array, the or maximum receivingPSLL array,is the 1 (mainmaximum lobe region PSLL oris 1 grid (main lobe), lobe which region satisﬁes or grid the lobe), virtual which array satisfies power the pattern: virtual array power pattern:

minminmin minmax maxmax max PSLLPSLLPSLL≤≤≤ PSLLPSLL PSLLPSLL PSLL (17)(17) r r≤ virtual virtual≤ virtual virtual≤ r r

BecauseBecause the the obtained obtained virtual virtual array array also also satisfies satisfies the prior the priorPSLL PSLLprediction prediction of the ADSof the linear ADS array, linear thearray, prediction the predictionPSLL inequality PSLL inequality of the transmitting of the transmitting and receiving and arrayreceiving can bearray used. can Figure be used.6 is Figure a power 6 is cross-sectionala power cross-sectional view of an view obtained of an arrayobtained beam array pattern beam in pattern the x direction in the x direction when θ = when0, ϕ =θ 0=and0 , ϕ the= 0 numberand the of number cycles is of 0, cycles 15, and is 0, 28, 15, respectively and 28, respective (in orderly to (in make order the to curvemake clear,the curve only clear, the beam only patterns the beam ofpatterns three isomorphic of three isomorphic sequences sequences are given, becauseare given, the because different the array different element array layout element have layout different have powerdifferent patterns, power including patterns, the including optimal one).the optimal It can be one). seen fromIt can Figure be seen6 that from the Figure beam performance 6 that the beam of = theperformance array sensor of is the diff arrayerent sensor with di isff erentdifferent cycles. with When differentσ 28cycles., the PSLLWhenof σ the=28 array, the power PSLL of pattern the array is thepower lowest. pattern Therefore, is the inlowest. order Therefore, to obtain ain sparse order array to obtain with a the sparse optimal arrayPSLL with, the the number optimal of PSLL cycles, the needsnumber to be of correct. cycles Inneeds Figure to6 ,bePSLL correct.-max andIn FigurePSLL-min 6, PSLL are the-max estimated and PSLL maximum-min are and the minimum estimated PSLLmaximum, respectively. and minimum PSLL, respectively.

FigureFigure 6. 6. PowerPower pattern pattern when when KK == 20,20 φ, φ==0,0 and, andϕ =ϕ =0;0σ; =σ0, 15, = 0, and 15, 28,and respectively. 28, respectively.

4.2. Azimuth Angle Performance 4.2. Azimuth Angle Performance In order to evaluate the beam performance of the obtained array in different directions, the 3D In order to evaluate the beam performance of the obtained array in different directions, the 3D power patterns were simulated in different directions. In this paper, the beam power diagrams are power patterns were simulated in different directions. In this paper, the beam power diagrams are given in Figure7 when θ and ϕ are 0 , 30 , 45 , 60 , and 75 , respectively. The main lobe width is about given in Figure 7 when θ and ϕ◦ are◦ 0°, 30°,◦ 45°,◦ 60°, ◦and 75°, respectively. The main lobe width is 2.6 in the case of 3 dB, and increases slightly as the azimuth angle increases. The slight change of the ◦ − mainabout lobe 2.6° width in the is case probably of −3 dB, a result and increases of the size slightly of the as array the azimuth aperture angle being increases. small when The compared slight change to theof detectionthe main range, lobe onlywidth about is probably 30 cm under a result a carrier of the frequency size of ofthe 100 array KHz. aperture In addition, being the small amplitude when compared to the detection range, only about 30 cm under a carrier frequency of 100 KHz. In addition, of the main lobe remains consistent when the azimuth angle is less than 45◦ and drops significantly the amplitude of the main lobe remains consistent when the azimuth angle is less than 45° and drops when the azimuth angle exceeds 45◦. Nevertheless, the PSLL also remains relatively steady from 11.83significantly to 11.85 when when the the azimuth azimuth angle angle changes. exceeds It 45°. can alsoNevertheless, be seen from the thesePSLLsimulation also remains results relatively that thesteady width from and 11.83 amplitude to 11.85 of when the main the lobeazimuth and angle the PSLL changes.change It can slightly also withinbe seen the from scope these of simulation azimuth angularresults variation.that the width and amplitude of the main lobe and the PSLL change slightly within the scope of azimuth angular variation.

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(a) (b)

(e) (f)

(g) (h)

Figure 7. Cont.

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(i) (j)

θ ϕ Figure 7. 3D and 2D power(i) patterns of the obtained sparse array after ADS.(j) (a) and are 0°; θ ϕ θ ϕ θ ϕ θ ϕ (b) and are 0°; (c) and are 30°; (d) and are 30°; θ(e) andϕ are FigureFigure 7.7. 3D3D andand 2D2D powerpower patternspatterns ofof thethe obtainedobtained sparse array after ADS. ( a) θ and andϕ are 0 ◦are;(b )0°;θ 45°; (f) θ and ϕ are 45°; (g) θ and ϕ are 60°; (h) θ and ϕ are 60°; (i) θ and ϕ and ϕθare 0 ;(ϕc) θ and ϕ are 30θ;(d) θ andϕ ϕ are 30 ;(e) θ andθ ϕ areϕ 45 ;(f) θ and ϕ areθ 45 ;(gϕ) θ and (b) and◦ θ are 0°;ϕ (c) ◦ and are 30°;◦ (d) and ◦are 30°; (e) and◦ are ϕareare 75°; 60◦;( (hj)) θ and andϕ are60 are◦;( i75°.) θ and ϕ are 75◦;(j) θ and ϕ are 75◦. 45°; (f) θ and ϕ are 45°; (g) θ and ϕ are 60°; (h) θ and ϕ are 60°; (i) θ and ϕ θ ϕ InareIn order order75°; ( toj )to further further and identify identify are 75°. the the performance performance of of the the array array thinned thinned by by ADS, ADS, we we compare compare the the beam beam pattern of the target array with those of the original Mills Cross array with 40 40 elements and the pattern of the target array with those of the original Mills Cross array with 40× × 40 elements and the sparsesparseIn array orderarray by toby the further the CDS CDS identify method. method. the When Whenperformance using using the the CDSof theCDS method, array method, thinned the parametersthe byparameters ADS, (weN, K,compare(N,Λ )K, are Λ) set theare as beam set (40, as 21,pattern(40, 1) 21, to ofmake1) theto make thetarget element the array element countwith countthose of the ofof thinned thethe originthinned arrayal Millsarray approximately Crossapproximately array the with same the 40 same as× 40 that elementsas after that ADS.after and ADS. The the PSLLsparseThe valuesPSLL array values of by the the of beam CDSthe patternsbeam method. patterns at When various at usingvarious steering the steerin CDS angles gmethod, angles are given arethe in givenparameters Table in2. Table Meanwhile, (N, 2. K, Meanwhile, Λ) theare beam set theas pattern(40,beam 21, pattern graphs1) to make graphs of the the Mills ofelement the Cross Mills count arrayCross of andthe array thinned the and thinned the array thinned array approximately byarray CDS by are CDS the also sameare presented also as thatpresented after here. ADS. Thehere. beamTheThe PSLL beam patterns values patterns are of given theare beamgiven in the patterns in case the of case θatand variousof ϕθ atand 45steerin◦ ϕand atg are angles45° shown and are are ingiven shown Figures in Tablein8 andFigures 2.9 .Meanwhile, 8 and 9. the beam pattern graphs of the Mills Cross array and the thinned array by CDS are also presented here. Table 2. PSLL values of the beam patternsθ forϕ three arrays at various steering angles. The beam patternsTable 2.are PSLL given values in the of the case beam of patterns and for threeat 45° arrays and are at various shown steering in Figures angles. 8 and 9.

SteeringSteering Angle Angle 0◦ 0° 30◦ 30° 45°45 ◦ 60° 60◦ 75° 75◦ Array Table 2.Array PSLL values of the beam patterns for three arrays at various steering angles. Original MillsOriginalSteering Mills Angle13.27 −13.27 13.27−13.27 −13.2613.26 −13.22 13.22−13.25 13.25 − 0° − 30° 45°− 60° − 75° − Array Thinned by CDSThinned by CDS12.33 −12.33 12.35−12.35 −12.3212.32 −12.12 12.12−12.07 12.07 Original Mills− −13.27 − −13.27 −−13.26− −13.22 − −13.25 − Thinned by ADSThinned by ADS11.85 −11.85 11.84−11.84 11.8511.85 −11.88 11.88−11.83 11.83 Thinned by CDS− −12.33 − −12.35 −12.32− −12.12 − −12.07 − Thinned by ADS −11.85 −11.84 −11.85 −11.88 −11.83 0

-5 X: 0.5725 0-10 Y: -13.26

-5-15 X: 0.5725 -10-20 Y: -13.26

-15-25

-20-30 Power pattern/dB Power

-25-35

-30-40 Power pattern/dB Power -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 sin(θ) -35

-40 (a) -1 -0.8 -0.6 -0.4 -0.2 (b0 ) 0.2 0.4 0.6 0.8 1 sin(θ) FigureFigure 8. 8. Beam power pattern of the original Mills Cross array.a (a) 3D pattern;b (b) 2D section pattern. Beam power pattern(a) of the original Mills Cross array. ( ) 3D pattern;(b) ( ) 2D section pattern.

FigureThese 8. results Beam powerindicate pattern that ofthe the original original Mills Mills Crossarray array.has the (a) best 3D pattern; beam performance(b) 2D section comparedpattern. to the other thinned arrays with roughly equal beam width. The major difference is the PSLL values of the beamThese patterns.results indicate This difference that the originalis due to Mills a larger array thinned has the ratio best in beam CDS performance and ADS sequence compared results to thein theother spatial thinned undersampling arrays with roughlyof the array. equal Comparison beam width. of The the major two arrays difference thinned is the by PSLL CDS valuesand ADS of thedemonstrates beam patterns. this point.This difference In our investigation, is due to a la thrgere thinned thinned ratios ratio of in the CDS CDS and and ADS ADS sequence methods results are 21 in the spatial undersampling of the array. Comparison of the two arrays thinned by CDS and ADS demonstrates this point. In our investigation, the thinned ratios of the CDS and ADS methods are 21

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and 20, respectively, and the element counts of the two virtual arrays are 441 and 400, respectively. Therefore, the PSLL values of the beam patterns following CDS method are slightly lower than those using ADS. In addition, ADS is able to generate more sequence sets and arrays compared to CDS, and thus it is easier to obtain an estimated array. Unfortunately, we cannot deduce other sets due to our weaker mathematical ability, but we will contact Oliver’s research group and invite them to guide Appl.us in Sci. creating2020, 10, 3176 a new set. At that time, we will provide the further investigation results. Here,12 of 16the results are just to demonstrate that the ADS method can be applied to sparse the sensor array.

(a) (b)

FigureFigure 9. 9.Beam Beam power power pattern pattern of of the the obtained obtained array array after after employing employing the the CDS CDS method. method. ( a()a) 3D 3D pattern; pattern; (b()b 2D) 2D section section pattern. pattern.

4.3. TheseAzimuth results Resolution indicate that the original Mills array has the best beam performance compared to the other thinned arrays with roughly equal beam width. The major difference is the PSLL values of In this paper, the MVDR direction spectrum estimation method is used to evaluate the azimuth the beam patterns. This difference is due to a larger thinned ratio in CDS and ADS sequence results resolution of the thinned array after applying the ADS method. The azimuth resolution in the radar in the spatial undersampling of the array. Comparison of the two arrays thinned by CDS and ADS field is defined as the minimum azimuth angle between two objects that can be distinguished by demonstrates this point. In our investigation, the thinned ratios of the CDS and ADS methods are 21 radar in the same radar detection direction. Using MVDR direction spectrum estimation, we and 20, respectively, and the element counts of the two virtual arrays are 441 and 400, respectively. determine the azimuth resolution based on the azimuth angle when the minimum value of the Therefore, the PSLL values of the beam patterns following CDS method are slightly lower than those direction spectrum of the two sources is increased by 3 dB, that is, the lowest depression between the using ADS. In addition, ADS is able to generate more sequence sets and arrays compared to CDS, and direction spectra of two sources is raised by 6 dB. It is important to note that the azimuth resolution thus it is easier to obtain an estimated array. Unfortunately, we cannot deduce other sets due to our has a direct relationship with the width of the main lobe of the beam pattern. Figure 10 displays the weaker mathematical ability, but we will contact Oliver’s research group and invite them to guide us MVDR direction spectra estimations of different arrays with the steering angle at ϕ = 0° and θ = in creating a new set. At that time, we will provide the further investigation results. Here, the results ° are0 , just where to demonstrate the subfigures that a theandADS d respectively method can correspond be applied to to the sparse sparse the arrays sensor after array. application of the ADS and CDS methods to the original Mills Cross array with 40 × 40 and 20 × 20 elements, 4.3.respectively. Azimuth Resolution Here, we indicate that the ADS and CDS methods still correspondingly use the parameters above. According to Figure 10, the azimuth resolutions of the various arrays in subfigures In this paper, the MVDR direction spectrum estimation method is used to evaluate the azimuth a and b are roughly 5.2°, 5.5°, 5°, and 10.2°. To fully describe the azimuth performance of these four resolution of the thinned array after applying the ADS method. The azimuth resolution in the radar arrays, Table 3 gives the average values and standard deviations of the azimuth resolutions of the field is defined as the minimum azimuth angle between two objects that can be distinguished by radar arrays at the different steering angles. Here, the acoustic wavelength λ is set as 100 KHz. in the same radar detection direction. Using MVDR direction spectrum estimation, we determine the azimuth resolution based on the azimuth angle when the minimum value of the direction spectrum of the two sources is increased by 3 dB, that is, the lowest depression between the direction spectra of two sources is raised by 6 dB. It is important to note that the azimuth resolution has a direct relationship with the width of the main lobe of the beam pattern. Figure 10 displays the MVDR direction spectra estimations of different arrays with the steering angle at ϕ = 0◦ and θ = 0◦, where the subfigures a and d respectively correspond to the sparse arrays after application of the ADS and CDS methods to the original Mills Cross array with 40 40 and 20 20 elements, respectively. Here, we indicate that the × × ADS and CDS methods still correspondingly use the parameters above. According to Figure 10, the azimuth resolutions of the various arrays in subfigures a and b are roughly 5.2◦, 5.5◦, 5◦, and 10.2◦. To fully describe the azimuth performance of these four arrays, Table3 gives the average values and standard deviations of the azimuth resolutions of the arrays at the different steering angles. Here, the acoustic wavelength λ is set as 100 KHz.

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Table 3. Average values and standard deviations of azimuth resolutions of the arrays.

Items Transmit Receive Aperture Azimuth Azimuth Array Elements Elements Size Average Deviations Thinned by ADS 20 20 19.5 λ 5.25° 0.31°

Thinned by CDS 21 21 18.5 λ 5.62° 0.28°

Mills Cross 40 40 19.5 λ 5.00° 0.16°

Mills Cross 20 20 9.5 λ 10.21° 0.12°

From these results, one can observe that the azimuth resolution of the thinned array using ADS is superior to that using CDS, although the CDS method can produce a better PSLL of the thinned array. The reason for this disadvantage of the CDS method is that the autocorrelation function of the CDS method has the ‘binarity’ property. Besides, this work also implies that the same azimuth resolution can be achieved by the ADS method using half the numbers of the original Mills array. WithAppl. Sci. these2020 results,, 10, 3176 there is reason to believe that the ADS method can produce one sparse array13 with of 16 unchanged beam performance when the right ADS sequence is created.

(a) (b)

Figure 10. Azimuth resolution of four didifferentﬀerent arrays. ( a)) Sparse array after ADS implementationimplementation (40,20,9,10); ( (bb)) Sparse Sparse array array after CDS implementation (40,21,1); ( (aa)) Sparse Sparse array array after ADS implementation (40,20,9,10); (d) Mills Cross array with 2020 × 20 elements. × 4.4. FrequencyTable Performance 3. Average values and standard deviations of azimuth resolutions of the arrays. It is essentialItems to investigateTransmit the frequencyReceive performanceAperture of the sparseAzimuth array obtainedAzimuth via the ADS Array method, so that this sparse optimizationElements methodElements can be appliedSize to futureAverage practical projects.Deviations Therefore, Thinned by ADS 20 20 19.5λ 5.25◦ 0.31◦ Thinned by CDS 21 21 18.5λ 5.62◦ 0.28◦ Mills Cross 40 40 19.5λ 5.00◦ 0.16◦ Mills Cross 20 20 9.5λ 10.21◦ 0.12◦

From these results, one can observe that the azimuth resolution of the thinned array using ADS is superior to that using CDS, although the CDS method can produce a better PSLL of the thinned array. The reason for this disadvantage of the CDS method is that the autocorrelation function of the CDS method has the ‘binarity’ property. Besides, this work also implies that the same azimuth resolution can be achieved by the ADS method using half the numbers of the original Mills array. With these results, there is reason to believe that the ADS method can produce one sparse array with unchanged beam performance when the right ADS sequence is created.

4.4. Frequency Performance It is essential to investigate the frequency performance of the sparse array obtained via the ADS method, so that this sparse optimization method can be applied to future practical projects. Therefore, Appl. Sci. 2020, 10, 3176 14 of 16 Appl. Sci. 2020, 10, x FOR PEER REVIEW 14 of 16 thethe aboveabove obtainedobtained sparse sparse array array using using the the (40,20,9,10)-ADS (40,20,9,10)-ADS sequence sequence structure structure is used is used to produce to produce some somefrequency frequency examinations, examinations, and the and results the areresults compared are compared with those with of thethose original of the Mills original Cross Mills array Cross with 40 40 elements. The relevant parameters are set as follows: center frequency (design frequency) array× with 40 × 40 elements. The relevant parameters are set as follows: center frequency (design f0 = 300 kHz, signal frequency range fmin = 20 kHz, fmax = 400 kHz, frequency step 40 kHz. Main frequency) f 0 = 300 kHz, signal frequency range f min = 20 kHz, f max = 400 kHz, frequency step lobe width (MLW) and PSLL are adopted as performance indices. The experimental results are shown 40 kHz. Main lobe width (MLW) and PSLL are adopted as performance indices. The experimental in Figure 11. The size of array aperture is speciﬁed as (k 1) λ/2 in the simulation experiments. results are shown in Figure 11. The size of array aperture is specified− · as (1)/2k −⋅λ in the simulation Otherwise, the grating lobes must appear when the wave frequency increases to a certain extent if experiments.using the ﬁxed Otherwise, size of array the aperture. grating lobes must appear when the wave frequency increases to a certain extent if using the fixed size of array aperture.

(a) (b)

Figure 11. 11. ReceivingReceiving performances performances of of two two arra arraysys at at different different frequencies. frequencies. ( (aa)) PSLLPSLL changeschanges at at different different frequencies;frequencies; ( b) main lobe width (MLW) changeschanges atat didifferentfferent frequencies.frequencies.

As shown in Figure 11,11, the PSLL values of of the the two arrays arrays are are different diﬀerent in the given frequency range, except except for for 60 kHz below, while their MLW havehave consistentconsistent values.values. The PSLL deviations are always 1.5 dB dB at at the the different diﬀerent frequencies. frequencies. The The devi deviationsations are are caused caused by by the the aforementioned aforementioned overlarge overlarge thinnedthinned ratio ratio resulting resulting from the spatial undersamplin undersamplingg of the array sensor. This illustrates that the trendstrends of PSLL and MLW of the sparse array after the ADS method are in line with the original Mills Cross array in the given frequency range.

5. Conclusions Conclusions This paper proposes the ADS method to further thin Mills Cross arrays while keeping the beam pattern unchanged, in in order order to to reduce reduce the the comple complexityxity and and cost cost of of array array sonar sonar imaging imaging systems. systems. Simulations ofof thethe thinned thinned arrays arrays were were conducted conducted and and compared compared with with those those of the of original the original Mills CrossMills Crossarray. Thearray. results The demonstrateresults demonstrate that the PSLLthat thevalues PSLL of thevalues beam of patternthe beam are pattern affected are by theaffected thinned by ratio the thinnedand the cycleratio and number. the cycle The number. beam patterns The beam do not patterns decrease do not as long decrease as the as ADS long parameters as the ADSare parameters correctly areassigned. correctly In addition,assigned. the In MLWaddition, and PSLLthe MLWvalues and remain PSLL almost values unchanged remain almost with azimuth unchanged angle with and azimuthfrequency angle variation. and frequency The azimuth variation. resolution The azimuth is maintained resolution when is using maintained the given when ADS using sequence, the given but ADSit is possible sequence, that but the it azimuth is possible resolution that the is influencedazimuth resolution by the thinned is influenced ratio. However, by the thinned it is true ratio. that However,the azimuth it is resolution true that andthe azimu the MLWth resolution are affected and by the the MLW aperture are sizeaffect ofed the by array the aperture sensor. Fromsize of these the arrayexperimental sensor. results,From these it is concludedexperimental that results, the ADS it methodis concluded can further that the decrease ADS method the redundancy can further of a decreaseMills Cross the array redundancy while maintaining of a Mills the beamCross pattern. array Awhile mathematically maintaining deduced the beam ADS sequencepattern. forA mathematicallyMills Cross array deduced sensor sparse ADS sequen optimizationce for willMills be Cross provided array in sensor our future sparse work. optimization will be provided in our future work. Author Contributions: Conceptualization, J.L.; Methodology, F.S.; Software, F.S.; Validation, Y.S.; Formal Analysis, AuthorF.S.; Writing-Original Contributions: Draft Conceptualization, Preparation, P.L.; Writing-ReviewJ.L..; Methodology, & Editing, F.S.; Software, J.L.; Project F.S.; Administration, Validation, Y.S..; P.L.; FundingFormal Acquisition, J.L. All authors have read and agreed to the published version of the manuscript. Analysis, F.S.; Writing-Original Draft Preparation, P.L.; Writing-Review & Editing, J.L.; Project Administration, P. L..; Funding Acquisition, J.L.

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Funding: Project supported by the science and technology development fund project of Wuxi (Grant No. N20191008), the social development project of the key development plan in Jiangsu Province (Grant No. BE2015692), National natural science foundation of China (Grant No. 41075115), and the second-phased construction projects of superior subjects in Jiangsu universities. Conﬂicts of Interest: The authors declare no conﬂict of interest. These funders provide the supports for the preliminary study of this investigation, including document retrieval, hardware system build and technical advisory. Certainly, they provide funds for the publication of this article.

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