# An ADS-Based Sparse Optimization Method for Sonar Imaging Sensor Arrays
## 2 Almost Difference Set and Sparse Optimization of Mills Cross Array
### ADS-Based Arrays  
According to the investigations from [PERSON] [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)\  
According to the ADS autocorrelation function, the beam power pattern \(S(u)\) at \(u=k/N\gamma\), that is, the sampling of \(S(u)\), is the Fourier transform of the autocorrelation function \(C_{s}(z)\)[17]:  
\[\Xi(k)=\triangleq\sum_{z=0}^{N-1}C_{s}(z)\exp(i2\pi zk/N) \tag{11}\]  
It is further expressed as:  
\[\Xi(k)=K-\Lambda+N\Lambda\delta(k)+\psi(k) \tag{12}\]  
For Equation (12), \(\delta(k)\) is the discrete impulse function [\(\delta(k)=1\) if \(k=0\), and \(\delta(k)=0\) otherwise], \(\Psi(k)\triangleq IDFT[\psi(z)]\) where \(\psi(z)\triangleq\sum_{p=1}^{N-1-t}\delta(z-z^{r})\) and \(z^{r}\), \(r=1,...,N-1-t\) are the indexes at which \(C_{s}(z)=\Lambda+1\) (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]:  
\[PSLL^{opt}_{MIN}\leq PSLL^{opt}_{DW}\leq PSLL^{opt}\{D\}\leq PSLL^{opt}_{UP} \leq PSLL^{opt}_{MAX} \tag{13}\]  
where \(PSLL^{opt}=\min_{\omega\in[0,N-1]}\{PSLL\big{(}D^{(\omega)}\big{)}\}\), \(PSLL^{opt}_{DW}=\max\{PSLL^{\infty},PSLL^{\min}\}\), \(PSLL^{opt}_{UP}=E\Big{\{}\Phi_{N}^{\min}\Big{\}}PSLL^{opt}_{MIN}=PSLL^{\infty}_ {MIN}\) and \(PSLL^{opt}_{MAX}=E\Big{\{}\Phi_{N}^{\min}\Big{\}}PSLL^{\infty}_{MAX}\), being \(E\Big{\{}\Phi_{N}^{\min}\Big{\}}\approx 0.8488+1.128\log_{10}N\)[18] and \(PSLL^{\min}=E\Big{\{}\Phi_{N}^{\min}\Big{\}}\min_{n}(PP_{\infty,n})/PP_{ \infty,0}\). It should be pointed out that \(PSLL^{opt}_{DW}\) and \(PSLL^{opt}_{UP}\) are determined when the ADS sequence is available since they require the knowledge of the coefficients \(PP_{\infty,n}\). On the contrary, \(PSLL^{opt}_{MIN}\) and \(PSLL^{opt}_{MAX}\) can be always computed a priori from Equations (14) and (15).  
\[PSLL^{opt}_{MIN}=\frac{K-\Lambda-1+\sqrt{t(N-t)/(N-1)}}{(N-1)\Lambda+k-1+N-t} \tag{14}\]  
\[PSLL^{opt}_{MAX}=E\Big{\{}\Phi_{N}^{\min}\Big{\}}\frac{K-\Lambda-1+\sqrt{t(N- t)}}{(N-1)\Lambda+k-1+N-t} \tag{15}\]  
According to the relevant theory and research of ADS, it is extremely difficult to deduce a more accurate \(PSLL^{opt}_{DW}\) and \(PSLL^{opt}_{DW}\) as the limits of \(PSLL\) estimation completely from mathematics, while it is relatively easy to use \(PSLL^{opt}_{DW}\) and \(PSLL^{opt}_{UP}\), 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]. When using the (42, 21, 10, 31)-ADS sequence, the ADS-based MIMO transmitting array is \(D_{21}^{(0)}=\) (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 \(PSLL\)s of different ADS-based virtual arrays to obtain the optimal \(PSLL\).