Variable step-size pseudo affine projection algorithm for censored regression

Abstract

The censored observations of adaptive signal processing have widely occurred in plenty of utility applications. Using traditional adaptive algorithms to recognize systems will confront convergence reduced under these circumstances. To address the above problem, the least mean square algorithm for censored regression (CR-LMS) has been proposed. However, the CR-LMS algorithm will converge slowly under colored inputs. In this paper, a pseudo affine projection algorithm based on censored regression (CR-PAP) is present to process colored input signals. Moreover, the variable step-size strategy is used to enhance the convergence performance. Computer simulations verify the better convergence of the proposed algorithm over the CR-LMS algorithm in system identification scenarios.

Appendix

The conditional density function of the cumulative distribution function of the random variable $x$, which satisfies the formula in Table

Table 2 Probability density function $f(x)$ and cumulative distribution function $F(x)$ of three different background noises

2, can be written as follows:

$$ \begin{gathered} p(x|x > a) = \frac{{dP(V < x|x > a)}}{dx} \hfill \\ \, = \frac{d}{dx}\frac{P(V < x,x > - a)}{{P(x > a)}} \hfill \\ \, = \frac{d}{dx}\frac{{\int_{a}^{x} {f(u)du} }}{P(x > a)} = \frac{f(x)}{{1 - F(a)}} \hfill \\ \end{gathered} $$

where $f(x)$ and $F(x)$ are given in Table 2. According to the probability theoretical calculated the conditional expectation, the $E[x|x > a]$ can be expressed as follows:

$$ \begin{gathered} E[x|x > a] = \int_{a}^{\infty } {xp(x|x > a)} dx = \int_{a}^{\infty } {\frac{xf(x)}{{1 - F(a)}}} dx \hfill \\ \, = - \frac{1}{1 - F(a)}f(x)|_{a}^{\infty } = \frac{f(a)}{{1 - F(a)}} \hfill \\ \, = \frac{f(a)}{{F( - a)}} = R(a) \hfill \\ \end{gathered} $$