Nonlinear Blind Identification with Three-Dimensional Tensor Analysis

Algorithm 1

Different steps of the new blind identification algorithm-based cumulant tensor analysis.

Initialize ℎ₁ and ℎ₂ as random variables (estimates ℎ₁⁽⁰⁾;ℎ₂⁽⁰⁾).

For 𝑛≥1,

 (i) build Hankel matrices using (4.5), for 𝑝=1;2,

              𝐻₁^(𝑛)=ℋ(ℎ₁^(𝑛)),𝐻₂^(𝑛)=ℋ(ℎ₂^(𝑛)),

 (ii) compute matrices estimate 𝐴 and 𝐵 as

𝐴^(𝑛)=8𝛾³₂{(𝐻₂⋄𝐻₁⋄𝐻₂)+(𝐻₂⋄𝐻₂⋄𝐻₁)+(𝐻₁⋄𝐻₂⋄𝐻₂)}^(𝑛),

𝐵^(𝑛)={8𝛾³₂((𝐻₂⋄𝐻₁⋄𝐻₁)+(𝐻₁⋄𝐻₁⋄𝐻₂)+(𝐻₁⋄𝐻₂⋄𝐻₁))+48𝛾⁴₂((𝐻₂⋄𝐻₂⋄𝐻₂))}^(𝑛),

 (iii) minimize cost functions (6.1) and (6.2) so that

              ℎ₁^(𝑛+1)=𝐴^(𝑛)#(vec(𝐶_𝑈)−𝐵^(𝑛)ℎ₂^(𝑛)),

              ℎ₂^(𝑛+1)=𝐵^(𝑛)#(vec(𝐶_𝑈)−𝐴^(𝑛)ℎ₁^(𝑛)),

 (iv) reiterate until parametric error convergence

              ‖vec((ℎ₁^(𝑛+1)ℎ₂^(𝑛+1))−(ℎ₁^(𝑛)ℎ₂^(𝑛)))‖‖vec(ℎ₁^(𝑛+1)ℎ₂^(𝑛+1))‖≤𝜀.