Wavelet Derivative: Application in Multicomponent Analysis of Electrochemical Signals

The wavelet derivative was developed for resolving overlapped signals in electrochemical analysis. It has been shown that the wavelet derivative could be used as an effective tool in resolution enhancement of the electrochemical signals. Since signal‐to‐noise ratio (SNR) does not degrade significantly at high order derivatives, the wavelet derivative is much better than the conventional numerical derivative in resolving practical signals with low SNR, particularly in the case when the overlapped degree of complex is high and high order derivatives are required for the further resolution enhancement. In this article, applications of the wavelet derivative in the voltammetric determination of small mounts of substances in the presence of large mounts of potentially interfering species are presented, and the advantages of this technique are discussed as well.     

在线小波变换用于伏安分析仪的研制

将在线小波变换应用于伏安信号的在线处理,研制了具有小波变换功能的伏安分析仪。该系统利用在线小波变换对采集到所数据进行实时处理,具有控制灵活,分辨率高,数据处理方便等特点。通过对低浓度Ｃｙ＾２＋和Ｆｅ＾３＋混合样品的阶梯斜坡扫描伏安珠滤噪及定性,定量分析,结果表明可获得满意的分析结果。     

A novel method to calculate the approximate derivative photoacoustic spectrum using continuous wavelet transform

A novel method based on continuous wavelet transform (CWT) using Haar wavelet function for approximate derivative calculation of analytical signals is proposed and successfully used in processing the photoacoustic signal. An approximate nth derivative of an analytical signal can be obtained by applying n times of the wavelet transform to the signal. The results obtained from four other different methods – the conventional numerical differentiation, the Fourier transform method, the Savitzky-Golay method, and the discrete wavelet transform (DWT) method – were compared with the proposed CWT method; it was demonstrated that all the results are almost the same for signals without noise, but the proposed CWT method is superior to the former four methods for noisy signals. The approximate first and second derivative of the photoacoustic spectrum of Pr(Gly)₃Cl₃· 3 H₂O and PrCl₃· 6 H₂O were obtained using the proposed CWT method; the results are satisfactory. Received: 21 December 1999 / Revised: 28 February 2000 / Accepted: 7 March 2000     

Resolution of global signals using ratio differential pulse polarograms: Determination of p-nitroaniline and p-nitrotoluene in their mixture

Ratio differential pulse polarograms obtained by dividing the multianalyte and single analyte signals are proposed as a tool for resolution of global signals and quantification of the analytes from a qualitatively known mixture by differential pulse polarography (DPP) and related electroanalytical techniques. The influences of shape and position of the resolving function (DP polarograms of individual analyte) on the efficiency of resolution are discussed on simulated and experimental results. The method is applied for the determination of p-nitroaniline (NA) and p-nitrotoluene (NT) from their mixture in N,N′-dimethylformamide solutions with 0.1 M tetrabutylammonium iodide as supporting electrolyte, using an external calibration diagram and internal standard addition methods. NA and NT give one-electron DP polarographic peaks with 93 mV of peak separation and, therefore, show significant overlapping which depends on the concentration ratio of NA and NT in the mixture. The method is especially suitable for quantification of one analyte in the presence of a large excess of another analyte, because by division the component in excess is removed and the pseudo-ratio DPP of the minor component is clearly revealed in a way which is not possible by deconvolution using polynomial division or deconvolution by Fourier transforms.     

Minimisation of instrumental noise in the acquisition of FT-NIR spectra of bread wheat using experimental design and signal processing techniques

Spectral resolution (R) and number of repeated scans (S) have a significant effect on the S/N ratio of Fourier transform-near infrared (FT-NIR) spectra, but the optimal values of these two parameters have to be determined empirically for a specific problem, considering separately both the nature of the analysed matrix and the specific instrumental setup. To achieve this aim, the instrumental noise of replicated FT-NIR spectra of wheat samples was modelled as a function of R and S by means of the Doehlert design. The noise amounts in correspondence to different experimental conditions were estimated by analysing the variance signals derived from replicate measurements with two different signal processing tools, Savitzky–Golay (SG) filtering and fast wavelet transform (FWT), in order to separate the “pure” instrumental noise from other variability sources, which are essentially connected to sample inhomogeneity. Results confirmed that R and S values leading to minimum instrumental noise can vary considerably depending on the type of analysed food matrix and on the different instrumental setups, and helped in the selection of the optimal measuring conditions for the subsequent acquisition of a wide spectral dataset.     

Study on spline wavelet self‐convolution in filtering and resolving overlapped voltammetric signals simultaneously

The normal distribution function is used as resolving factor and spline wavelet self‐convolution is used to resolve the overlapped peaks containing noise. The resolution of several kinds of overlapped peaks (SNR from 5 to 50) simulated by computer has been discussed in details. It is known that the overlapped peaks with noise can be separated (SNR > 50) directly and the noise is removed at the same time in time domain. Base‐line separation can be achieved. The relative errors of peak area and position are less than 5.0%. The satisfactory results are also obtained in resolving voltammetric overlapped signals by this method.     

A novel method to calculate the approximate derivative photoacoustic spectrum using continuous wavelet transform

A novel method based on continuous wavelet transform (CWT) using Haar wavelet function for approximate derivative calculation of analytical signals is proposed and successfully used in processing the photoacoustic signal. An approximate nth derivative of an analytical signal can be obtained by applying n times of the wavelet transform to the signal. The results obtained from four other different methods--the conventional numerical differentiation, the Fourier transform method, the Savitzky-Golay method, and the discrete wavelet transform (DWT) method--were compared with the proposed CWT method; it was demonstrated that all the results are almost the same for signals without noise, but the proposed CWT method is superior to the former four methods for noisy signals. The approximate first and second derivative of the photoacoustic spectrum of Pr(Gly)3Cl3.3H2O and PrCl3.6H2O were obtained using the proposed CWT method; the results are satisfactory.     

阶跃伏安法中干扰信号的消除

以地阶跃伏安法中干扰信号的消除提出了一种新方法，即将条小波和ＲＬＴ滤波技术相结合既可以滤去随机噪声，又可以滤去充充电电流。方法效果较好。     

Wavelet transform as a new approach to the enhancement of signal-to-noise ratio in anodic stripping voltammetry

De-noising signals is a frequent aim achieved by signal processing in analytical chemistry. The purpose is to enable the detection of trace concentrations of analytes. The limit of detection is defined as the lowest amount of analyte that still causes signals greater than the background noise. Appropriate de-noising decreases only the noise and maintains the measurement signal, so that signal-to-noise ratios are enhanced. One adequate mean of signal processing for this purpose is wavelet transform, which still is not a common tool in analytical chemistry. In this paper, the ability of de-noising by wavelet transform is shown for measurements in anodic stripping voltammetry using a hanging mercury drop electrode. The calculation of limits of detection and signal-to-noise ratios on the basis of peak-to-peak noise is exercised to quantify the performance of de-noising. Furthermore, signal shape with regard of easing the application of base lines is discussed. Different wavelet functions are used, and the results are compared also to Fourier transform. Coiflet2 was found out to reduce noise by the factor of 330 and is proposed as the adequate wavelet function for voltammetric and similar signals.     

Wavelet transform-based Fourier deconvolution for resolving oscillographic signals

Fourier self-deconvolution is an effective means of resolving overlapped bands, but this method requires a mathematical model to yield deconvolution and it is quite sensitive to noises in unresolved bands. Wavelet transform is a technique for noise reduction and deterministic feature capturing because its time-frequency localization or scale is not the same in the entire time-frequency domain. In this work, wavelet transform-based Fourier deconvolution was proposed, in which a discrete approximation (such as A(2)) obtained from performing wavelet transform on the original data was substituted for the original data to be deconvolved and another discrete appropriate approximation (such as A(5)) was used as a lineshape function to yield deconvolution. Again, instead of the apodization function, the B-spline wavelet was used to smooth the deconvolved data to enhance the signal-to-noise ratio. As a consequence, this method does not suffer as badly as Fourier self-deconvolution from noises in the original data. Thus, resolution enhancement can be increased significantly, especially for signals with higher noise level. Furthermore, this method does not require a mathematical model to yield deconvolution; it is very convenient to deconvolve electrochemical signals.     

小波包分析用于重叠分析化学信号的处理

对小波包分析的算法进行了改进,并将此算法成功地应用于多组分重叠色谱信号的解析.结果表明,本文提出的算法解决了MRSD算法的不足,更适合处理分析化学信号,用于重叠信号的解析时不需重构(逆变换),简化了数据处理步骤,加快了数据解析速度,具有较强的解析能力.对于重叠色谱信号的解析,小波包分析比小波分析具有更强的解析能力.     

Resolution of overlapping signals in spectrometry using a wavelet packet transform and an Elman recurrent neural network

A novel method named a wavelet packet transform based Elman recurrent neural network (WPTERNN) was proposed for the simultaneous UV–visible spectrometric determination of Cu(II), Cd(II) and Zn(II). This method combined wavelet packet denoising with an Elman recurrent neural network. A wavelet packet transform was applied to perform data compression, to extract relevant information, and to eliminate noise and collinearity. An Elman recurrent network was applied for nonlinear multivariate calibration. In this case, using trials, the kind of wavelet function, the decomposition level, and the number of hidden nodes for the WPTERNN method were selected as Daubechies 14, 3, and 8, respectively. A program (PWPTERNN) was designed that could perform the simultaneous determination of Cu(II), Cd(II) and Zn(II). The relative standard errors of prediction (RSEP) obtained for all components using WPTERNN, a Elman recurrent neural network (ERNN), partial least squares (PLS), principal component regression (PCR), Fourier transform based PCR (FTPCR), and multivariate linear regression (MLR) were compared. Experimental results demonstrated that the WPTERRN method was successful even where there was severe overlap of spectra. The results obtained from an additional test case also demonstrated that the WPTERNN method performed very well. Figure The part of WP coefficients obtained by wavelet packet transforms     

Comparative Analysis of Mathematical Methods for Noise Filtering in Stripping Voltammetry

Some mathematical methods for the elimination of noise in stripping voltammetric curves are compared. On the basis of the obtained data, the wavelet transform method is proposed for noise filtering.     

小波滤噪用于示波计时电位信号处理的研究

根据信号（Ｓ）和噪声（Ｎ）在小波变换下表现出的截然不同的性质,提出了一种新的滤噪方法,研究了噪音在不同细节的性质、滤噪阈值的确定及平滑与滤噪的特点。对不同Ｓ／Ｎ的示波计时电位信号进行了实验结果表明;小波滤噪在大大提高Ｓ／Ｎ的同时,信号强度基本不损失,优于传统的滤噪方法。     

二维小波变换与子窗口因子分析法结合用于含噪声HPLC-DAD数据的解析

将二维小波变换与子窗口因子分析法相结合,用于模拟的高噪声HPLC-DAD数据的解析.首先应用二维小波变换滤除噪声,然后采用子窗口因子分析法解析重叠峰.结果表明,信噪比为10的高噪声基本被滤除掉.     

A Background and noise elimination method for quantitative calibration of near infrared spectra

A new hybrid algorithm is proposed to eliminate the varying background and noise simultaneously for multivariate calibration of near infrared (NIR) spectral signals. The method is based on the use of multi-resolution, which is one of the main advantages provided by wavelet transform. The signals are firstly split into different frequency components, which keep the same data points of the original signals. In conjunction with a modified uninformative variable elimination (mUVE) criterion, the new method can be used to remove the low-frequency varying background and the high-frequency noise simultaneously. The method is successfully applied to simulated spectral data set and experimental NIR spectral data, resulting in more parsimonious multivariate models with higher precision. In addition, the proposed strategy can be applied to other spectral signals as well.     

连续样条小波变换用于分解重叠峰的研究

以B-样条小波为分析小波，提出了用于分析化学重叠信号解析的新方法——连续样条小波变换，结果表明：连续样条小波变换应用于分析化学信号的处理，能使峰变窄同时还能提高信号的信噪比，是一种新型有效的重叠信号解析方法，能从含噪声重叠信号中直接得到重叠峰的峰数目及其相应的峰位置。     

Dedicated mother wavelet in the determination of antimony in the presence of copper

In this paper, a new signal-processing procedure is applied to the optimization of voltammetric determination of antimony in the presence of copper and the parallel determination of these two elements. The proposed numerical algorithm for the separation of the overlapping peaks utilizes the continuous wavelet transform and the inverse continuous wavelet transform. As the base function, the specially defined dedicated mother wavelet is used. In its construction the ideal, simulated voltammetric peak is intensively exploited. This approach, corresponding to the wavelet theory, gives satisfactory signals separation, even in the cases when they constitute one peak. The possibility of Sb(III) determination is presented in the case of different distances between copper and antimony peaks and also in 10-, 20- and 50-fold excesses of Cu. The parallel determination of Cu(II) and Sb(III) is possible even in the case of a 10-fold excess of copper. The quality of the obtained results fulfills the requirements of validation.     

An On-Line Wavelet Transform for De-Noising of High Performance Liquid Chromatograms

ABSTRACT

A novel algorithm of wavelet transform, called the on-line wavelet transform, which can be used in on-line processing of signals of an instrumental analysis, is proposed. And application of the on-line wavelet transform in denoising of high performance liquid chromatograms was investigated. Results showed that the noise in chromatograms was cleanly removed by the method, peak position after the on-line de-noising dose not change, and the linearity of calibration curves of concentration versus peak area remained and even was improved. Quantitative determination of three mixed samples were investigated with five standard samples; the recoveries were between 94.0-105.0%.     

Multidetermination of thiamine HCl and pyridoxine HCl in their mixture using continuous daubechies and biorthogonal wavelet analysis

A new graphical method based on the one-dimensional wavelet transform (WT) was proposed and tested on mixture of thiamine hydrochloride (THI) and pyridoxine hydrochloride (PYR) in the presence of strongly overlapping signals. We selected from the data of the UV-VIS absorption spectra a signal consisting of 1150 points corresponding to the concentration range 8-32 mg ml⁻¹ for each vitamin and we subjected it to Daubechies8 (DAUB8) and Biorthogonal6.8 (BIOR6.8) wavelet transforms. Since the peaks of the transformed signals were bigger than original ones a zero crossing method was applied to obtain the calibration graphs. In addition, the validity of Beer-Lambert law was assumed for the transformed signals. An appropriate scale setting was choosing to obtain an alternative calibration for each method. Matlab 6.5 software was used for one-dimensional wavelet analysis and the basic concepts about wavelet method were given. The obtained results were successfully compared among each other as well as with those obtained by other literature methods. The method developed in this paper is rapid, easy to apply, not expensive and it is suitable for analyzing of the overlapping signals of compounds in their mixtures without any chemical pre-treatment.