Abdussamad Umar Jibia

An Appraisal of Gardner Transform-Based Methods of Transient Multiexponential Signal Analysis

There are several problems in applied science in which experimental observations can be accurately represented by a sum of exponential decay functions in which the amplitudes, decay rates and number of components have different physical interpretations and need to be estimated. A parameter estimation technique of multicomponent exponential functions that has undergone many modifications is the Gardner transform in which a nonlinear transformation is used to convert the data signal into a convolution model containing the parameters of interest. Modifications of this early technique include modification of the original transform or deconvolution procedure and additional processing of the deconvolved data to obtain better estimates of the desired parameters. This paper presents an appraisal of Gardner transform and its variants. It discusses major modifications and their implications to the overall results of analysis.

Cramer-Rao Lower Bound for Parameter Estimation of Multiexponential Signals

The Cramer Rao Lower Bound on the mean square error of unbiased estimators is widely used as a measure of accuracy of parameter estimates obtained from a given data. In this paper, derivation of the Cramer-Rao Bound on real decay rates of multiexponential signals buried in white Gaussian noise is presented. It is then used to compare the efficiencies of some of the techniques used in the analysis of such signals. Specifically, two eigendecomposition-based techniques as well as SVD-ARMA (Singular Value Decomposition Autoregressive Moving Average) method are tested and evaluated. The two eigenvector methods were found to outperform SVD-ARMA with minimum norm being the most reliable at very low SNRs (Signal to Noise Ratios). Index Terms—Cramer Rao bound, multiexponential, MUSIC, minimum norm, SVD I. INTRODUCTION The Cramer-Rao Lower Bound (CRLB) for the mean square error of unbiased estimators has been widely accepted as a measure of the efficiency of parameter estimates of a given data. The CRLB gives valuable insight into the performance of estimators and is useful for experimental design since it allows the optimization of the sample positions or, when some parameters are to be estimated with a specific precision, minimum number of acquisition averages can be predicted. In particular, it has been used as a measure of efficiency in parametric estimation of overlapping peaks (2), autoregressive (AR) modulated harmonics (8), Damped Sinusoidal process (7), Single damped exponential (6) and in Surface plasmon experiments in Biophysics (5), etc. Generally, computation of the CRLB requires the determina- tion of Fisher Information Matrix (FIM) whose size equals the number of real-valued parameters to be estimated. The major problem faced by researchers on multicomponent signals is the fact that these signals do not form an orthogonal base. Various measures have been devised to solve this problem. In our approach, the original signal is converted into a discrete convolution model whose input is a train of weighted delta functions containing the signal parameters to be determined. Fourier transformation and deconvolution then generate data consisting of a sum of complex exponentials in noise. The data is analyzed using spectral estimation techniques ((12),(4) and (11)). The procedure used by researchers to compare different algorithms is to apply them to one or several simulated sets of input data to compare the accuracy of the solution, the sensitivity to noise in the input data, and the time required to compute the solutions. However, results from different researches revealed that this procedure which works well for well-posed problems return very poor results when applied to the ill-posed problem of exponential analysis (9). In this paper, Cramer Rao Lower Bound is derived and used for the first time to test a number of techniques for the analysis of damped exponentials. Applications include, but are not limited to, fluorescence decay analysis, deep-level transient spectroscopy and reaction kinetics. In section II of this paper, the signal preparation procedure is presented. It involves the conversion of the original signal into a convolution model, followed by deconvolution that generates a sum of complex exponentials with the same number of components as the original signal in noise. The CRLB is computed in section III. The computed CRLB is used to evaluate three methods using a test signal and the results are presented and discussed in section IV.

Parameter Estimation of Transient Multiexponential Signals Using SVD-ARMA and Multiparameter Deconvolution Techniques

Much has been reported about the analysis of transient multiexponentials data. In a previous paper, for example, this analysis was done using autoregressive moving average model which was applied to the deconvolved data arising from the application of Gardner transform followed by optimal compensation deconvolution to the original signal. Optimal compensation deconvolution uses a single parameter noise-reduction parameter. In this paper, a deconvolution parameter incorporating multiple noise-reduction parameters is used instead. Simulations and experimental results show that the proposed combination, despite its limitations supersedes several existing methods.

Effect of multiple deconvolution parameters on the resolvability of decay rates of multiexponential signals

Noise reduction in deconvolution process has been a challenge to researchers in the field of signal processing. The problem is ill-posed and various algorithms have been developed to reduce noise enhancement. The effect of using multiple noise-compensating parameters in the deconvolution of multiexponential signals is considered in this paper. Three parameters are simultaneously adjusted to obtain optimal reduction in noise. It is shown that this approach performs better than a single parameter approach.

Effect of sampling on the parameter estimates of multicomponent transients

The need to estimate the parameters of transient multiexponential signals frequently arises in different areas of applied science. A classical technique that has been frequently used with different modifications is the Gardner transform. Gardner transform is used to convert the original data signal into a convolution model. Converting this model into a discrete type for further analysis depends on the selection of correct sampling conditions. Previously, a relationship between the sampling frequency and the weighting factor in the modified Gardner transform was derived. In this paper, the effect of this relationship on the accuracy of parameter estimates is investigated.

Analysis of multicomponent transient signals using MUSIC superresolution technique

The problem of estimating the parameters of transient signals consisting of real decay constants has for long been a subject of study by many researchers. Such signals arise in many problems in Science and Engineering like nuclear magnetic resonance for medical diagnosis, deep-level transient spectroscopy, fluorescence decay analysis, etc. Many techniques have been suggested by researchers to analyse these signals but they often produce mixed results. A new method of analysis using modified MUSIC (multiple signal classification) subspace algorithm is successfully applied to the analysis of this signal. A noisy multiexponential signal is subjected to a preprocessing procedure consisting of Gardenerspsila transformation and inverse filtering. Modified MUSIC algorithm is then applied to the deconvolved data. The parameters of focus in this paper are the number of components and decay constants. It is shown that with this technique parameter estimates do not significantly change with signal to noise ratio. The superiority of this algorithm over conventional MUSIC algorithm is also shown.

Matlab-Based Algorithm for Real Time Analysis of Multiexponential Transient Signals

Multiexponential transient signals are particularly important due to their occurrences in many natural phenomena and human applications. For instance, it is important in the study of nuclear magnetic resonance (NMR) in medical diagnosis (Cohn-Sfetcu et al., 1975)), relaxation kinetics of cooperative conformational changes in biopolymers (Provencher, 1976), solving system identification problems in control and communication engineering (Prost and Guotte, 1982), fluorescence decay of proteins (Karrakchou et al., 1992), fluorescence decay analysis (Lakowicz, 1999). Several research work have been reported on the analysis of multicomponent transient signals following the pioneer work of Prony in 1795 (Prony, 1975) and Gardner et al. in 1959 (Gardner, 1979). Detailed review of several techniques for multicomponent transient signals’ analysis was recently reported in (Jibia, 2010).