Spectral factorization of a spectral density with arbitrary delays

Abstract

An impulse response f (with support on the nonnegative real numbers) is said to be in A_ if, on í t ≥ 0,f(t) = f _a (t) + f _sa (t) where its regular functional part f _a is such that $\int_0^\infty {|f\left( t \right)|} \;exp \left( { - \sigma t} \right)dt < \infty$ and its singular atomic part ${f_{sa}}: = \sum\nolimits_{i = 0}^\infty {f\delta \left( {. - {t_i}} \right),}$ where t _o = 0,t _i > 0 for i ≥ 1 and f _i ∈ ℂ for i ≥ 0 with $\sum\nolimits_{{i = 0}}^{\infty } {|f\left( t \right)|\exp \left( { - \sigma {{t}_{i}}} \right) < \infty } ,$ for some σ < 0. Â_ denotes the algebra of distributed parameter system proper-stable transfer functions, i.e. Laplace transforms of elements in A_.