**2. Techniques of multiexponential transient signal analysis**

424 MATLAB – A Fundamental Tool for Scientific Computing and Engineering Applications – Volume 1

requires efficient tools for its development and implementation in real-time.

important that these parameters are accurately estimated.

equation (1), it is not sufficient that equation (1) approximates data accurately; it is also

There are many problems associated with the analysis of transient signals of the form given in equation (1) due to the nonorthogonal nature of the exponential function. These problems include incorrect detection of the peaks, poor resolution of the estimated decay and inaccurate results for contaminated or closely-related decay rate data as reported in (Salami et.al., 1985). These problems become increasingly difficult when the level of noise is high. Although Gardner transform eliminated the nonorthogonality problem, it introduced error ripples due to short data record and nonstationarity of the preprocessed data. Apart from these problems, analysis of multiexponential signal is computationally intensive and

To overcome these problems, modification of Gardner transform has been proposed recently with Multiple Signal Classification (MUSIC) superposition modeling technique (Jibia et al., 2008); with minimum norm modeling technique (Jibia and Salami, 2007), with homomorphic deconvolution, with eigenvalues decomposition techniques, and the Singular Value Decomposition (SVD) based-Autoregressive Moving Average (ARMA) modeling techniques (Salami and Sidek, 2000; Jibia, 2009). As reported in (Jibia, 2009), performance comparison of these four modeling techniques has been investigated. Though, all the four techniques were able to provide satisfactory performances at medium and high signal-to-noise ratio (SNR),

the SVD-ARMA was reported to have the highest resolution, especially at low SNR.

analysis using MATLAB software package is examined in this chapter.

real-time analysis of multiexponential transient signal.

recommendation for future study.

Hence, the development of SVD-ARMA based algorithm for multiexponential signal

MATLAB provides computational efficient platform for the analysis and simulation of complex models and algorithms. In addition, with the aid of inbuilt embedded MATLAB Simulink block, it offers a tool for the integration of developed algorithm/model in an embedded application with little programming efforts as compared to the use of other programming languages (Mathworks, 2008). This functionality is explored in integrating the developed MATLAB-based algorithm into National Instrument (NI) Labview embedded programming tool. Hence, an integrated MATLAB-Labview software interface is proposed for real-time deployment of the algorithm. To this end, the analytical strength of MATLAB together with simplicity and user-friendly benefits of the National Instrument (NI), Labview design platforms are explored in developing an efficient, user-friendly algorithm for the

The rest of the chapter is organized as follows. Section 2 provides brief review of techniques for multiexponential signal analysis. The MATLAB algorithm development for the signal analysis is presented in section 3. The development of an integrated MATLAB-Labview realtime software interface is then examined in section 4. Section 5 presents sample real-time data collection together with results and analysis. The chapter is concluded in section 6 with Several techniques have been reported for the analysis of transient multiexponential signal. They are classified as: (i) time domain or frequency domain, and (ii) parametric or nonparametric techniques. The main objective of these techniques in analyzing the multiexponentially decaying signals is to estimate the signal parameters as accurately as possible and to get better display of the signal spectra. Time-domain techniques constitute the oldest methods of multiexponential signal analysis prior to the advent of Gardner transformation technique (Gardner et al, 1959; Salami, 1985). Gardner transformation is one of the most important methods of the transient signal analysis based on spectral analysis. Generally, spectral analysis involves transformation of a time-domain signal to a frequency domain so that certain features of the signals that characterized them are easily discerned such as its decay constant, *<sup>k</sup>* and amplitude, *Ak*. In other words, it is the process of obtaining the frequency content (spectrum) of a signal (Proakis and Manolakis 1996). The spectral analysis approach is further categorized into nonparametric and parametric techniques. Nonparametric technique is a frequency-domain technique that obtains the signal spectra directly from the deconvolved data, while the parametric technique obtains the signal spectra indirectly by determining a finite set of parameters that defines a closed form mathematical model for the deconvolved data. Therefore the techniques of multiexponential transient signal analysis are sub-divided into time-domain, nonparametric frequency domain and parametric frequency domain techniques as shown in Figure 1 with their associated methods.

Among the earliest time-domain technique is the peeling technique. However, this technique produces poor results when *S*( ) contains more than two components. Other time-domain techniques such as Prony's method and its variants produce better performance than the peeling technique, however they are very sensitive to noise (Smyth, 2002; Salami, 1995). The nonlinear least squares technique (Smyth, 2002) is computationally inefficient and the solution sometimes fails to converge, which means the estimate of the signal parameters cannot be accurately obtained.

The nonparametric techniques of spectral analysis are introduced to overcome some limitations of the time-domain techniques. The Gardner transformation technique produces error ripples which obscure the real peaks of the spectrum due to the cutoff points. This technique is good in analyzing signal with high SNR. The fast Fourier transform (FFT) technique, which is an improvement over the original Gardner transformation produces improved resolution. However, the problem of error ripples still exists. The extension of this technique involving the use of digital signal processing and Gaussian filtering is sensitive to noise and its data range has to be limited to get accurate estimates of the signal parameters. Whilst the modified FFT technique is better than the previous methods, it often fails to estimate *<sup>k</sup>* correctly especially when the peaks are closely related. The differential technique (Swingler, 1977) provides some improvements over the existing techniques such as better resolution display but it is not suitable for analyzing noisy signal. Furthermore, modifying the FFT technique by incorporating integration procedure (Balcou,1981) does not produce better results as compared to the previous digital technique and the modified FFT technique. Moreover, this technique is still affected by error ripples.

**Figure 1.** Overview of Techniques for Multiexponential signal analysis

Parametric techniques such as Autoregressive (AR), Moving Average (MA) and ARMA models are introduced to the analysis of multiexponential signals to alleviate the drawbacks of the nonparametric techniques. The AR modeling technique requires less computation than the ARMA modeling technique as its model parameters are relatively easy to estimate. However, it is sensitive to noise due to the assumed all-pole model. On the other hand, MA parameters are difficult to estimate and the resultant spectral estimates have poor resolution. Although, the ARMA modeling technique is much better in estimating noisy signal than the AR modeling technique, it requires a lot of computation. A detailed review of these techniques can be obtained in (Jibia, 2009; Salam and Sidek, 2000).
