Optimization and Estimation of Complex-Valued Signals: Theory and applications in filtering and blind source separation

Complex-valued Bayesian parameter estimation via Markov chain Monte Carlo

“…According to (2), the conditional pdf p(x|y) can be interpreted as the joint distribution of the real and the imaginary part of x conditioned on y, i.e.…”

“…On the other hand, a variety of practical systems, such as those in communication, electromagnetic, oceanography and optics, are more preferable to be described by complex-valued parametric models [1,2,14,19,30]. Traditionally, to estimate the complexvalued parameters, we usually formulate the complex-valued signals as bivariate and real-valued signals, and then perform the MCMC sampling directly in real parameter space.…”

“…For example, it is more simple and efficient to present widely-linear or nonlinear transformations. Besides, it also brings economies and insights into the underlying physics that would not be achieved by real-valued representation [1,2,30]. So, it is more preferable to perform the MCMC sampling in complex space as to the complex-valued parameter estimation.…”

Filtering and tracking with trinion-valued adaptive algorithms

“…The calculation of trinion-valued gradient is important for adaptive algorithm derivation. In the complex domain, the gradient is based on the assumption that a function of variable z is a function of z and its conjugate (Brandwood, 1983;Adalı and Schreier, 2014). A similar prerequisite in the quaternion domain is that a function of variable q is a function of q and its three involutions (Jiang et al, 2014).…”

“…We will see that for the most general case, a pure quaternion model will not work, while trinion algebra provides a convenient and compact solution. For the first contribution, the augmented second-order statistics are also considered (Adalı and Schreier, 2014). This paper is organised as follows.…”