Other parameters are the same as in Fig. 2. where ω is the mode frequency, T is the tunnel matrix element defined by the overlapping of electric field intensity of neighbour modes.
This fact originates from the modification of the dipole matrix elements defined in Equations14, 15, and 20 when the Coulomb interaction is considered.
where H s (a) is the nonlinear operator that sets all but the largest (in magnitude) s elements of a to zero, μ t is a step size, and W t is an m×m diagonal matrix with each element defined as mathsf{W}_{t}(i,i =frac{gamma^{2}}{gamma^{2}+ y_{i}-{phi^{T}_{i}} mathbf{x}^{(t)})^{2}},ldotsldots,m, where ϕ i denotes the column vector obtained by transposing the i-th row of Φ.
The first subarray contains sensor 1 to M s, while the second subarray contains sensor M − M s + 1 to M. Obviously, the array manifold matrices associated with these two subarrays are related by A 1 Φ = A 2, where Φ is a K × K diagonal matrix with its diagonal element defined by ϕ k = e j 2 π d s sin θ k / λ.
where A 1 again denotes the array manifold for subarray 1, Ψ 1 is a K × K diagonal matrix with its diagonal element defined by ψ k = e j 2 π Δ sin θ k / λ, and Ψ n = Ψ 1 n, n = 1, …, N − 1. Denote the set of L beam pointing angles as θ ¯ l, l = 1, …, L. The conventional beamforming weight matrix for each subarray is then W b = a b ( θ ¯ 1 ) a b ( θ ¯ 2 ) ⋯ a b ( θ ¯ L ) (11).
H d)and H(c)are N × N matrices with ij th element defined as H i j ( d ) = H ( d ) i j, and H i j ( c ) = H ( c ) i j.
We can then define the matrix elements of ({mathbf {D}}): begin{aligned} {mathbf {D}}_{ij} = mu _{ij}left| vec {d}_{ij}right| end{aligned}where (vec {d}_{ij} = vec {a}_i - vec {a}_j) and (mu _{ij}) is the reduced mass of (a_i) and (a_j).
We defined the matrix elements at the ith row and the jth column, A ij, in the similarity matrix, A, as The second term is the dissimilarity matrix with the numerator denoting the Euclidean distance between centroids i and j.
Matrix A is then transformed into a new matrix B with elements defined as b ij = a ij /h j (i = 1,…, n; j = 1,…, m) and displayed graphically in Fig. 3c.
We then define the next 16 matrix elements as (12) where (13) according to the order given above.
