Signal denoising using wavelet-based methods  (Page 3/9)

Characteristic of wavelet expansions

The properties and hence advantages of a familiy of wavelets depend upon the mother wavelet features. However, a common set of features are shared by the most useful of them citep( ).

• A wavelet expansion is formed by a two-dimensional expansion of a signal. It should be noticed that the dimension of the signal itself is not determinant in the wavelet representation.
• A wavelet expansion provides a dual time-frequency localization of the input signal. This implies that most of the energy of the signal will be captured by a few coefficients.
• The computational complexity of the discrete wavelet transform is at most $O\left(nlog\right(n\left)\right)$ i.e. as bad as for the discrete Fourier transform (DFT) when calculated using the Fast Fourier Transform (FFT). For some particulartypes of wavelets, the complexity can be as low as $O\left(n\right)$ .
• The basis functions in a wavelet expansion are generated from the mother wavelet by scaling and translation operations. The indexing in two dimensions is achieved using this expression:
  $${\psi }_{j,k}\left(t\right)=2^{j/2}\psi (2^{j}t-k)j,k\in \mathbb{Z}$$
• Most wavelets basis functions satisfy multiresolution conditions. This property guarantees that if a set of signals can be represented by basis functions generated from a translation $\psi (t-k)$ of the mother wavelet, then a larger set of functions, including the original, can be represented by a new set ofbasis functions $\psi (2t-k)$ . This feature is used in the algorithm of the fast wavelet transform , FWT the equivalent of the FFT algorithm for wavelets decomposition.
• The lower resolution coefficients can be calculated from the higher resolution coefficients using a filter bank algorithm. This property contributes to the efficiency of the calculation of the DWT.

Denoising with wavelets

If $y\left(t\right)$ is an empirically recorded signal with and underlying description, $g\left(t\right)$ , a model for the noise addition process transforming g(t) into y(t) is described by Equation .

where ${ϵ}_i$ are independent normal random variables $N(0,1)$ and $\sigma$ represents the intensity of the noise in $y\left(t\right)$ . Using this model, it follows that the objective of noise removal is, given a finite set of $y_i$ values, reconstruct the original signal $g$ without assuming a particular structure for the signal.

The usual approach to noise removal models noise as a high frequency signal added to an original signal. Fourier transform could be used to track this high frequency, ultimately removing it by adequate filtering. This noise removalstrategy is conceptually clear and efficient since depends only on calculating the DFT of a given signal. However, there are some issues that must be taken into account. The most prominent of such issues ocurrs when the original signal hasimportant information associated to the same frequency as the noise. When a frequency domain representation of the signal is obtained, filtering out this frequency will induce noticeable loss of information of the target signal.

In cases as the one described, the wavelets approach is more appropiated due to the fact that the signal will be studied using a “dual” frequency-time representation, which allows separating noise frequencies from valuable signalfrequencies. Under this approach, noise will be represented as a consistent high frequency signal in the entire time scope and so its identification will be easier than using Fourier analysis.