Extracting outer function part from Hardy space function

Abstract

Any analytic signal f a (eit ) can be written as a product of its minimum-phase signal part (the outer function part) and its all-phase signal part (the inner function part). Due to importance of such decomposition, Kumarasan and Rao (1999), implementing the idea of the Szegő limit theorem (see below), propose an algorithm to obtain approximations of the minimum-phase signal of a polynomial analytic signal $${f_a}\left( {{e^{it}}} \right) = {e^{i{N_0}t}}\sum\limits_{k = 0}^M {{a_k}{e^{ikt}}} ,$$ (0.1) where a 0 ≠ 0, a M ≠ 0. Their method involves minimizing the energy $$E\left( {{f_a},{h_1},{h_2}, \ldots ,{h_H}} \right) = \frac{1}{{2\pi }}\int_0^{2\pi } {{{\left| {1 + \sum\limits_{k = 1}^H {{h_k}{e^{ikt}}} } \right|}^2}{{\left| {{f_a}\left( {{e^{it}}} \right)} \right|}^2}dt} $$ (0.2) with the undetermined complex numbers h k ’s by the least mean square error method. In the limiting procedure H → ∞, one obtains approximate solutions of the minimum-phase signal. What is achieved in the present paper is two-fold. On one hand, we rigorously prove that, if f a (eit ) is a polynomial analytic signal as given in (0.1), then for any integer H ≥ M, and with |f a (eit )|2 in the integrand part of (0.2) being replaced with 1/|f a (eit )|2, the exact solution of the minimum-phase signal of f a (eit ) can be extracted out. On the other hand, we show that the Fourier system e ikt used in the above process may be replaced with the Takenaka-Malmquist (TM) system, \({r_k}\left( {{e^{it}}} \right): = \frac{{\sqrt {1 - {{\left| {{\alpha _k}} \right|}^2}} {e^{it}}}}{{1 - \overline {{\alpha _k}} {e^{it}}}}\Pi _{i = 1}^{k - 1}\frac{{{e^{it}} - {\alpha _j}}}{{1 - \overline {{\alpha _j}} {e^{it}}}},k = 1,2, \ldots ,{r_0}\left( {{e^{it}}} \right) = 1\), i.e., the least mean square error method based on the TM system can also be used to extract out approximate solutions of minimum-phase signals for any functions f a in the Hardy space. The advantage of the TM system method is that the parameters α 1,...,α n ,... determining the system can be adaptively selected in order to increase computational efficiency. In particular, adopting the n-best rational (Blaschke form) approximation selection for the n-tuple {α 1,...,α n }; n ≥ N, where N is the degree of the given rational analytic signal, the minimum-phase part of a rational analytic signal can be accurately and efficiently extracted out.