Nonlinear Digital Filters

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REVIEWS

MATHEMATICAL PRELIMINARY

Eigen decomposition

Suppose an n × n matrix A has n linear independent eigenvectors, denoted as ξ_i for I = 1, 2,…, n and the corresponding eigenvalues are λ_i. Denote T ≡ [ξ₁, ξ₂,…, ξ_n] and D ≡ diag(λ₁, λ₂,…, λ_n). Then A is diagonalizable and A = TDT⁻¹.

By employing the eigen decomposition, it can facilitate the evaluation of a power of a matrix. For example, if A has n linear independent eigenvectors, then A^k = TD^kT⁻¹. If ∃ {1, 2,…, n} such that |λ_j| > 1, then some of the elements in lim_{k → + ∞} A^k would be unbounded. Hence, the BIBO stability condition of a discrete time linear system becomes all eigenvalues confined inside the unit circle.

Inverse of a map

A map F: X → Y is said to be injective if ∃