The original and transformed signal are both shown in Fig. 2.1. we have chosen to join the given points by straight line segments to get a good visualization of the signals.
I perform a more drastic compression. this time we choose the threshold equal to 9. the computations are given in Table 2.3, and the graphs are plotted in Fig. 2.2. the peaks in the original signal have been flattened. I note that the signal now is represented by only four non-zero entries.
The transformation performed by calculating means and differences of a signal is an example of the discrete wavelet transform. It can be undone by simply reversing the steps performed. we have also seen that the transformed signal may reveal features not easily seen or detected in the original signal.
For a wavelet, a window function is defined whose time radius is large at low frequencies; So that it is suitable for low frequency decay, and also its time radius is small at high frequencies; To be suitable for high frequency content analysis.
In wavelets, we have two types of functions: the wavelet function and the scale function
The wavelet function is more suitable for analyzing medium and high frequencies or low scales due to its mid-pass filtering property. We use the scale function to analyze wavelets at high scales or very low frequencies.
Dear Hassan Nasser
I want to know what is the relation between scale function (or wavelet function) and 8 samples that have been decomposed.
Dear Siavash:
As you know: Wavelets are mathematical functions that cut up data into different frequency com- ponents, and then study each component with a resolution matched to its scale. They have ad- vantages over traditional Fourier methods in analyzing physical situations where the signal contains discontinuities and sharp spikes.
-- Wavelet transform has become a powerful tool for signal processing. It offers time-frequency analysis to decompose the signal in terms of a family of wavelets or a set of basic functions, which have a fixed shape but can be shifted and dilated in time.
-- On each step of the wavelet transform, the signal is separated into two components, vi and wi. Elements of high-frequency components are wavelet coefficients. The lowest scale i0 corresponds to the case when a low-frequency component consists of the only element.
-- The outputs of this transform A and D are the reconstruction wavelet coefficients: A: The approximation output, which is the low frequency content of the input signal component. D: The multidimensional output, which gives the details, or the high frequency components, of the input signal at various levels (up to level 6).
-- wavelet transform of a function belonging to ℒ2{ℝ}, the space of the square integrable functions, is its decomposition in a base formed by expansions, compressions, and translations of a single mother function ψ(t), called a wavelet.
-- Theoretically, the maximum de- composition level (M) can be calculated as: M = log2 (N), where N is the series length. When conducting a wavelet-based ANN model, it needs to determine the most suitable decomposition level from 1 to M. ... their mean values, respectively; n is the series length.
There's a type called: Dyadic wavelet transforms which are scale samples of wavelet transforms following a geometric sequence of ratio 2. Time is not sampled. This transform uses dyadic wavelets. It is implemented by perfect reconstruction filter banks.
&& - Indeed i think you could benefit from this valuable book about your topic:
"Wavelet Theory and Application in Communication and Signal Processing"
By Nizar Al Bassam, Vidhyalavanya Ramachandran and Sumesh Eratt Parameswaran.
Published: January 21st 202.
Chapter.3 & chapter.4
https://www.intechopen.com/chapters/74766.
I hope it will be helpful...
With my best regards ...