A new approach for determining the coefficients of a complex-valued autoregressive (CAR) and complex-valued autoregressive moving average (CARMA) model coefficients using complex-valued neural network (CVNN) technique is discussed in this paper. The CAR and complex-valued moving average (CMA) coefficients which constitute a CARMA model are computed simultaneously from the adaptive weights and coefficients of the linear activation functions in a two-layered CVNN. The performance of the proposed technique has been evaluated using simulated complex-valued data (CVD) with three different types of activation functions. The results show that the proposed method can accurately determine the model coefficients provided that the network is properly trained. Furthermore, application of the developed CVNN-based technique for MRI K-space reconstruction results in images with improve resolution.

Parametric modeling technique has been applied to almost all fields of endeavor, these include but not limited to the the field of biomedical signal processing [

In recent times, the introduction of complex-valued neural networks (CVNNs) has widened the scope and applications of artificial neural network (ANN) [

This evolving paradigm has gained much attention not only because there are situations where CVNNs are inevitably required or greatly effective than its counterpart, the real-valued neural network (RVNN), but because of its usefulness which is enshrined in the fundamental theorem of Algebra [

If the coefficients in (

Despite the success of parametric models especially AR and ARMA models in various areas of applications [

The accuracy of the resulting model parameters highly depends on the methods used in determining the unknown model parameters. The use of inaccurate parameters often leads to introduction of artifacts, erroneous peaks and valleys, outrageous predicted values, system instability, and the list continues [

Several methods have been suggested for ARMA model coefficients determination, these can be broadly divided into two groups, namely optimal and suboptimal techniques [

In overcoming the computational problems associated with the optimal technique, a suboptimal method which takes advantage of the existing linear relationship between the estimated autocorrelation matrix and the AR coefficients has been suggested. One of the most successful suboptimal-based autocorrelation approaches is the modified Yule Walker (MYW) method. In this approach, the AR coefficients are firstly computed, which is then followed by the determination of MA coefficients [

The second problem associated with parametric modeling approach is the appropriate method of model order estimation. Because the optimal model order is not known a priori, the traditional approaches have always been to evaluate various model orders based on some error measure criteria. Several model order determination techniques have been suggested in literature among which are final prediction error (FPE), Akaike information criterion (AIC), Minimum description length (MDL), and Hannan and Quinn (HNQ) [

An optimal technique for determining ARMA model coefficients using RVNN approach has been reported in [

The remaining part of this paper is organized as follows. Development of CVNN-based parametric model for CAR and CARMA is contained in Section

ANN (i.e., RVNN and CVNN) is a global search technique that emulates the biological neurons of the human body. It is a general mathematical computing paradigm that models the operations of biological neural systems with unique characteristic of having massively parallel distributed structures and high capability of learning and generalization [

A typical ANN structure shown in Figure

Artificial neural network structure.

The combination of the large sets of connection weights and nonlinear activation functions makes ANN an ideal tool in estimating, classification, and predicting some of the linear and nonlinear systems [

CVNN also shares similar characteristics and properties with RVNN, the only difference is the nature of data being processed. Similar to RVNN, CVNN training can also be broadly divided into two classes namely supervised and unsupervised learning. In a supervised learning, the network is trained by presentation of sets of input and output data (target data) to the system. During this phase, the weights are successively adjusted based on a set of inputs and desired output presented to the network. The error from learning is back propagated for weight adjustment and the most popular method of error optimization is based on the minimization of the output mean squared error (MSE) [

The major challenge mitigating against the use of complex activation in CVNN is the issue of boundedness and differentiability nature of the intended activation function. In overcoming these problems, two major approaches have been reported, namely, fully complex (FC) and split complex (SC) activation function [

The FC approach uses an activation function that can satisfy the conflicting requirements of boundedness and differentiability of a complex function [

Furthermore, in overcoming the unbounded problem associated with complex activation function, Georgiou and Koutsougeras in [

The second approach that avoids the unboundedness in CVNN activation functions and which has been proved to be a special case of the FC is the use of SC approach. In SC approach, two real-valued activation function (RVAF) are used to process the in-phase and quadrature components of the input signal [

The general form of a CARMA model shown in Figure

Schematic diagram of CARMA model.

Decomposing and rearranging (

A two-layer CVNN for estimating CARMA model coefficients using split complex-valued weight and adaptive activation functions has been proposed here. The basic assumption guiding this proposal is that either the real part of the output

CVNN-based CARMA model network diagram.

Similarly, comparing (

Furthermore, neglecting value of

The PSD associated with the rational ARMA or CARMA model transfer function is given as

Similarly, the PSD of a AR/CAR system is given as

The performance analysis of the proposed modeling approach on CVD using CVNN-based CAR and CARMA model is investigated in this section.

Consider a CVD,

The PSD plot obtained is shown in Figure

PSD plot of complex sinusoid in noise using CVNN-based CAR model.

Consider a second-order CAR of the form

The estimated CVNN-based CAR model coefficients are shown in Tables

Comparison of results for mixed CAR model.

Mixed CAR signal | |||
---|---|---|---|

LHRP | LHIP | LS | |

Effect of activation function on mixed CAR model.

Mixed CAR signal | |||
---|---|---|---|

TANH | CLF | LAF | |

PSD: Mixed CVNN-based CAR model.

Consider a CARMA model described by the difference equation

Comparison of results for CVNN-based CARMA model.

Coefficients | Actual value | CARMA | ||

LHRP | LHIP | LS | ||

Effect of activation function on CARMA model in noise.

Coefficients | Actual value | CARMA | ||

TANH | CLF | LAF | ||

MRI is used to produce images of the internal section of the human body [

TERA Modeling Technique.

Detailed information regarding TERA and its variants for MRI reconstruction is as contained in [

split each row or column of the MRI K-space data

each series is modeled as the output of an IIR filter by estimating the transfer function from the generated finite data set.

The ARMA model can be regarded as a cascade of MA and AR filter.

The unit impulse sequence

The component

estimate the AR and MA coefficients of the Hermitian and anti-Hermitian series.

obtain the IDFT of the original image using

Suppose the CVNN-based CAR model is represented as

Figure

K-space data extrapolation for a typical K-space row: (a) Negative Real Axis, (b) Positive Real Axis, (c) Negative Imaginary Axis, (d) Positive Real Axis, (e) Negative Absolute Value, (f) Positive Absolute Value.

MRI reconstructed images using (a) FFT-based technique, (b) CVNN-based CAR model, (c) CVNN-based TERA model technique.

A new method of obtaining CARMA, CAR, and CLPC coefficients from a CVNN with split adaptive linear activation function for a CVD data has been developed in this paper. The results obtained from evaluation of LHRP and LHIP shows that any of the two techniques is appropriate for determination of model coefficients for a properly trained network. Similarly, it was observed that the use of TANH function and CLAF gives better result compared with the result obtained using LAF. Images with improved resolution when compared with the FFT technique has be obtained by the use of the proposed technique though with a far time of completion when compared with FFT technique. There is an ongoing work to reduce the algorithm computation time so as to be comparable with that of FFT technique. Other areas of application of this work aside from MRI reconstruction include nonlinear signal modeling and prediction, shape modeling and identification, crack modeling and prediction for automated building system and seismic signal processing.

The authors would like to express their appreciation to Professor K. Chon for his support and encouragement. Great appreciation goes to the reviewers of this paper for their constructive reviews and suggestions for improving this paper. This work is supported by Malaysia E-Science Grant 01-01-08-SF0083.