## 3 Harmonic Wavelets  
Wavelet transform uses wavelets as the basis functions. An arbitrary function can be obtained from one function (\"mother\" wavelet) by using translations and dilations in the time domain. The wavelet transform is commonly used for analyzing non-stationary (seismic, biological, hydroacoustic etc.) signals, usually together with various spectral analysis algorithms [16; 17]. Consider the basis of harmonic wavelets whose spectra are rectangular in the given frequency band [15; 16]. Harmonic wavelets are usually represented in the frequency domain. Wavelet-function (mother wavelet) can be written as:  
\[\Psi(\omega)=\left\{ \begin{array}{c}\frac{1}{2\pi},\;2\pi\leq\omega<4\pi\\ 0,\;\omega<2\pi,\omega\geq 4\pi\end{array} \right.\;\;\Leftrightarrow\;\psi(x)= \int\limits_{-\infty}^{\infty}\Psi(\omega)e^{i\omega x}d\omega= \frac{e^{i4\pi x}-e^{i2\pi x}}{i2\pi x} \tag{1}\]  
There are some techniques that allow us to decompose input signals using different basic functions: wavelets, sine waves, damped sine waves, polynomials, etc. These functions form the atom dictionary (basis functions) and each function is localized in the time and frequency domains. Often the dictionary of atoms is full (all types of functions are used) and redundant (the functions are not mutually independent). One of the main problems in these techniques is the selection of basic functions and dictionary optimization to acheive optimal decomposition levels [17]. Decomposition levels for wavelets can be defined as:  
\[ \begin{array}{l}\Psi_{jk}(\omega)=\left\{\begin{array}{c}\frac{1}{2\pi}2^{-j}e ^{-\frac{\mathrm{i}\omega}{2^{j}}},\ 2\pi 2^{j}\leq\omega<4\pi 2^{j}\\ 0,\ \omega<2\pi 2^{j},\omega\geq 4\pi 2^{j}\end{array}\right.\\ \psi_{jk}(x)=\psi(2^{j}x-k)=\int\limits_{-\infty}^{\infty}\Psi_{jk}(\omega)e^{ i\omega x}d\omega=\frac{e^{i\pi(2^{j}x-k)}-e^{i2\pi(2^{j}x-k)}}{i2\pi(2^{j}x-k)}\end{array}  \tag{2}\]  
where \(j\) is decomposition level and \(k\) is dilation. Very often, wavelets are basis functions because of their useful properties [14] and the potential to process signals in the time-frequency domain. The Fourier transform of a scaling function can be written as:  
\[\Phi(\omega)=\left\{ \begin{array}{c}\frac{1}{2\pi},\ 0\leq\omega<2\pi\\ 0,\ \omega<0,\omega\geq 2\pi\end{array} \right.\ \Leftrightarrow\ \phi(x)=\int\limits_{-\infty}^{\infty}\Phi(\omega)e^{i\omega x}d\omega= \frac{e^{i2\pi x}-1}{i2\pi x} \tag{3}\]  
We can formulate the following properties of harmonic wavelets, which relate them with other classes of wavelets:  
* Harmonic wavelets have compact support in the frequency domain, which can be used for localizing signal features. * There are fast algorithms based on the fast Fourier transform (FFT) for computing wavelet coefficients and reconstructing signals in the time domain. The drawback of harmonic wavelets is their weak localization properties in the time domain in comparison with other types of wavelets. The spectrum in the form of a rectangular wave leads to decay in the time domain as \(1\!/\!x\), which is not sufficient for extracting short-term singularities in a signal in the time domain. Wavelet Transform in the Basis of Harmonic Wavelets  
Detailed coefficients \(a_{jk},\overline{a}_{jk}\) and approximation coefficients \(a_{\phi k},\overline{a}_{\phi k}\):  
\[ \begin{array}{l}a_{jk}=2^{j}\int\limits_{-\infty}^{\infty}f(x)\overline{\psi }(2^{j}x-k)dx\quad\overline{a}_{jk}=2^{j}\int\limits_{-\infty}^{\infty}f(x) \psi(2^{j}x-k)dx\\ a_{\phi k}=2^{j}\int\limits_{-\infty}^{\infty}f(x)\overline{\phi}(x-k)dx\quad \quad\overline{a}_{\phi k}=2^{j}\int\limits_{-\infty}^{\infty}f(x)\phi(x-k)dx \end{array}  \tag{4}\]  
where \(j\) is the decomposition level; \(k\) is the dilation. If \(f(x)\) is a real-valued function, then: \(\overline{a}_{jk}=\overline{a}_{jk},\overline{a}_{\phi k}=\overline{a}_{\phi k}\).