Could you please look through some fragments of my paper
The present paper is devoted to harmonic wavelets and their application to the processing and detailed analysis of different oscillating signals including multiharmonic signals and transitional processes. Harmonic wavelets are defined as a specific class of functions and a fast algorithm for determining wavelet coefficients is introduced, which is advantageous for working with big datasets. Harmonic wavelets are intended for change-point detection in complex signals, which is also illustrated in the paper. Finally, we describe several approaches to signal denoising on the basis of harmonic wavelets with the help of special thresholding. At the end an example of a contaminated signal is demonstrated, where unwanted noise is successfully removed and the effectiveness of the developed algorithm is thus confirmed.
In the given paper we will consider a specific class of wavelets known as harmonic wavelets. They are mainly directed at processing oscillating signals in the presence of singularities and multiple change points. The reason why these wavelets are suitable for such signals is that their shape and that of the signals mentioned (oscillating signals) are very much alike. In many cases the most effective basis is the one that resembles most investigated signals, for example, multiharmonic signals or transitional processes. Signal decomposition in such a basis will have a minimum number of components among other bases.
We will mostly focus our attention on multiharmonic signals with different numbers of harmonics. Furthermore, we will introduce new approaches to a ubiquitous practical task – signal denoising, which will be dealt with by means of thresholding. In brief, the contents of the paper can be represented in the following way:
1) Introduction of harmonic wavelets and a two-stage procedure for the calculation of wavelet coefficients;
2) Demonstration of multiharmonic signal analysis in the presence of noise (with zero mean and some standard deviation) and multiharmonic signal together with a transitional process;
3) Illustration of signal denoising on the basis of harmonic wavelets with the use of thresholding.
Consider a signal basis that includes wavelets whose spectra have a rectangular shape in a particular frequency band. For instance, for the zero level there is the following expression for the power spectral density (PSD) of a wavelet:
Next it is necessary to determine the corresponding basis function in the time domain by means of an Inverse Fourier Transform (IFT):
The real and imaginary parts of the numerator of expression (2) make a linear combination of harmonic functions. Since here we use a complex form of a wavelet according to (1), it becomes possible to obtain two real expressions, which is similar to what is done in the classical harmonic (Fourier) analysis where the function is used to produce two real ones: and respectively by using the Euler formula. It is obvious that the real part of (2) is even, whereas the imaginary one is odd, which is also analogous to the complex exponent in the Fourier analysis. Now consider a general analytical expression of a wavelet at j-th level with the translation index j in the frequency and time domains:
1) Harmonic wavelets have compact support in the frequency domain and therefore they are very helpful for the localization of frequency singularities in signals.
2) Fast algorithms for wavelet coefficient calculation and signal reconstruction exist and they are based on the Fourier transform.
The main drawback of harmonic wavelets is their insufficiently good localization properties in the time domain in comparison with other types of wavelets. Since the spectra of harmonic wavelets are rectangular they decay as [IMG]file:///C:/DOCUME%7E1/Dima/LOCALS%7E1/Temp/msohtml1/01/clip_image002.gif[/IMG] in the time domain, which might prove to be not enough for extracting short-time singularities in a signal.
Consider the effectiveness of harmonic wavelets for the tasks of processing two main types of signals:
1) Combination of multiharmonic and transitional processes;
2) Additive mixture of multiharmonic signals and Gaussian noise with zero mean value and variance .
First, we will illustrate the example with multiharmonic and transitional processes, which is shown in Fig. 1 .
Fig.1 makes it clear that the entire signal contains two different processes alternating with one another: a monoharmonic process (samples ranging from 1 to 1024 and from 2048 to 3072) and a transitional process. The task of discovering change points in a complex signal is doubtless of great importance in practice and the harmonic wavelet transform is a suitable means of doing this. Below in Fig.2 there is the harmonic wavelet transform for the signal in question.
The jargon makes it difficult to read, but everything looks good. I don't know how often "respectively" is used to mean separately, if that's what you mean here. It might be clearer to use "separately".