Complex-valued Datasets Research Articles

Generalized Inversion of Electrochemical Impedance Data for Battery Systems

Electrochemical Impedance Spectroscopy (EIS) is a key technique for taking a snapshot of an electrochemical system near equilibrium conditions. Due to the richness of the complex-valued dataset and its simple experimental operation, it is currently being investigated as a test to diagnosis battery health and degradation details. Nevertheless, challenges with use of EIS is in interpreting the measured impedance. Current methods for analysis involving either fitting simplified equivalent circuit models or inverting distribution of relaxation times with a singular basis function. Because of these methods' simplifications, their results sometime fail to capture both the complex and heterogeneous underlying physics.Here, we present a novel algorithm for the inversion of EIS data for battery systems, which generalizes the inversion problem to any given linear equivalent circuit model, allowing for the analysis of complicated electrochemical systems. Importantly, this algorithm provides two clear benefits. First, it allows for the analysis of both bounded and unbounded processes, something quite common to battery systems and is vital in understanding non-obvious relationships in degraded batteries. Second, the distribution of independent processes, which can correspond to different physical phenomena, are also inverted by this algorithm. These distributions contain vital features in the identification and prediction of battery state-of-health. Overall, this tool will be indispensable in the analysis of degradation extent over battery aging, and provide an algorithmic framework for interpreting EIS measurements of complicated electrochemical systems.

High-resolution human brain functional χ mapping reveals focal and bidirectional BOLD responses

A blood oxygenation-level dependent (BOLD) functional magnetic resonance imaging (fMRI) study produces a four-dimensional (4D) complex-valued data set. Conventional magnitude-based brain functional mapping is an indirect measure of brain activity due to a dataflow in a cascade of MRI transformations (e.g., a dipole convolution). The MRI transformations also impose a MRI parameter dependence to BOLD fMRI data (e.g. the dependence on B0 and TE). By solving an inverse MRI problem, we may reconstruct a brain magnetic susceptibility source distribution (denoted by χ) from a brain MR image, thereby achieving a more direct representation of the vascular origin than the MRI data representation. For a complex-valued BOLD fMRI dataset, we can extract net BOLD phase responses (δP) through a complex division, and then reconstruct BOLD susceptibility responses (δχ) therefrom. In this paper, we compare the functional maps depicted in different dataspaces (magnitude image, δP image, δχ source) based on numeric characterizations of activation blobs by pattern analysis. Through an experimental demonstration with high-field (7 T) and high-resolution (0.5 mm in-plane) finger-tapping experiment, we show that the δχ-depicted functional map reveals task-evoked bidirectional BOLD δχ responses at the motor cortex region. The positive and negative activations are spatially separated and balanced, indicating conservation of concurrent excitations and inhibitions. We also observe that the source-depicted activation spots are more compact than the image-depicted spots, which is attributed to the dipole effect removal by inverse MRI.

Complex-valued neural networks for nonlinear complex principal component analysis

Principal component analysis (PCA) has been generalized to complex principal component analysis (CPCA), which has been widely applied to complex-valued data, two-dimensional vector fields, and complexified real data through the Hilbert transform. Nonlinear PCA (NLPCA) can also be performed using auto-associative feed-forward neural network (NN) models, which allows the extraction of nonlinear features in the data set. This paper introduces a nonlinear complex PCA (NLCPCA) method, which allows nonlinear feature extraction and dimension reduction in complex-valued data sets. The NLCPCA uses the architecture of the NLPCA network, but with complex variables (including complex weight and bias parameters). The application of NLCPCA on test problems confirms its ability to extract nonlinear features missed by the CPCA. For similar number of model parameters, the NLCPCA captures more variance of a data set than the alternative real approach (i.e. replacing each complex variable by two real variables and applying NLPCA). The NLCPCA is also used to perform nonlinear Hilbert PCA (NLHPCA) on complexified real data. The NLHPCA applied to the tropical Pacific sea surface temperatures extracts the El Niño-Southern Oscillation signal better than the linear Hilbert PCA.

Maximum likelihood estimation of signal amplitude and noise variance from MR data.

In MRI, the raw data, which are acquired in spatial frequency space, are intrinsically complex valued and corrupted by Gaussian-distributed noise. After applying an inverse Fourier transform, the data remain complex valued and Gaussian distributed. If the signal amplitude is to be estimated, one has two options. It can be estimated directly from the complex valued data set, or one can first perform a magnitude operation on this data set, which changes the distribution of the data from Gaussian to Rician, and estimate the signal amplitude from the obtained magnitude image. Similarly, the noise variance can be estimated from both the complex and magnitude data sets. This article addresses the question whether it is better to use complex valued data or magnitude data for the estimation of these parameters using the maximum likelihood method. As a performance criterion, the mean-squared error (MSE) is used.

Maximum likelihood estimation of signal amplitude and noise variance from complex valued data

Complex valued data may be obtained by means of acquisition systems such as magnetic resonance imaging (MRI) or digital communication systems. If the signal amplitude is to be estimated from these (inevitably noise corrupted) complex data, one has two options. Either the signal amplitude is directly estimated from the complex valued data set, or, the complex data is first transformed into a magnitude data set after which the signal amplitude is estimated. Similarly, the noise variance can be estimated from both data sets.This paper addresses the question whether it is better to use complex valued data or magnitude data for the estimation of these parameters using the maximum likelihood method. As a performance criterion, the mean-squared error (MSE) is used.