Complex-valued Signal Processing

Complex-valued signals arise frequently in applications as diverse as communications, radar, and biomedicine, as most practical modulation formats are of complex type and applications such as radar and magnetic resonance imaging lead to data that are inherently complex valued. Processing in the complex domain presents a number of challenges, and as a result, the vast majority of algorithms developed for the complex domain have taken shortcuts limiting their usefulness. Our group works on developing algorithms for complex-valued signal processing such that the full potential of complex-valued signal processing can be realized and many simplifying assumptions can be eliminated.

MRI acquisition yields complex-valued data...
MRI acquisition yields complex-valued data
(Picture redrawn and modified by Vince Calhoun based on the original in I. Khateed's dissertation)

Active Projects:

    Collaborative Research: Complex-Valued Signal Processing and its Application to Analysis of Brain Imaging Data
    Funded by NSF-CCF (Award no: 0635129)
    We establish a framework for complex-valued signal processing such that all computations can be carried out in the complex domain eliminating the need for many simplifying assumptions, such as the circularity of signal, both in the derivation and the analysis of the algorithms. We demonstrate the application of the framework for deriving a new class of efficient algorithms for performing complex ICA, and in particular, for studying brain function using the medical imaging data in its native, complex form.

    Collaborative Research: SEI: Independent Component Analysis of Complex-Valued Brain Imaging Data
    Funded by NSF-IIS (Award no: 0612076)
    We develop a class of complex ICA algorithms, in particular for analysis of biomedical imaging data and demonstrate the power of joint data analysis as well as performing the analysis on the complete set of data, i.e., by utilizing both the magnitude and the phase information. We focus upon three image types, functional magnetic resonance imaging (fMRI), structural MRI (sMRI) and diffusion tensor imaging (DTI). These three imaging data provide complementary information about brain connectivity, and all can benefit from the incorporation of a complex-valued data processing approach.

Key references:

  • T. Adali, H. Li, M. Novey, and J.-F. Cardoso, "Complex ICA using nonlinear functions," IEEE Trans. Signal Processing, in press.
    • We introduce a framework based on Wirtinger calculus for nonlinear complex-valued signal processing such that all computations can be directly carried out in the complex domain. The two main approaches for performing independent component analysis, maximum likelihood and maximization of non-Gaussianity--which are intimately related to each other--are studied using this framework. The main update rules for the two approaches are derived, their properties and density matching strategies are discussed along with numerical examples to highlight their relationships.
  • T. Kim and T. Adali, "Approximation by fully-complex multilayer perceptrons," Neural Computation, vol. 15, no. 7, pp. 1641-1666, July 2003.
    • We extend the result on the approximation ability of the multilayer perceptron to the complex domain by classifying nonlinear functions based on their singularities. The approximation theorems for the first two classes of elementary transcendental functions are very general and resemble the universal approximation theorem for the real-valued feedforward multilayer perceptron that was shown almost concurrently by multiple authors in 1989 (Cybenko, 1989; Hornik and Stinchecombe, and White, 1989; Funahashi, 1989). The third approximation theorem for the complex multilayer perceptron is unique and related to the power series approximation that can represent any complex number arbitrarily closely in the deleted neighborhood of a singularity.

Project team:

Recent publications:

  • J.-F. Cardoso and T. Adali, “ The maximum likelihood approach to complex ICA,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing (ICASSP), Toulouse, France, May 2006.
    • We derive the form of the best non-linear functions for performing independent component analysis by maximum likelihood estimation and discuss several special cases for the score function as well as adaptive scores.
  • M. Novey and T. Adali, “Stability analysis of complex-valued nonlinearities for maximization of nongaussianity,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing (ICASSP), Toulouse, France, May 2006, (best student paper in MLSP category).
    • Complex maximization of nongaussianity is an approach we introduced recently and have shown to provide reliable separation of both circular and noncircular sources. In this paper, we present conditions for the local stability of the algorithm and use these conditions of stability to quantify convergence performance for a number of complex nonlinear functions.

Resources: