Represented by Wavelet Transformation

First, let's talk about the Cartesian coordinate system.
The three-dimensional Cartesian coordinate system is a well-known X, Y, and zcoordinate system. In this coordinate system, the vector is decomposed by the projection length of the vector to the Three Coordinate Systems multiplied by the unit vector.
A deep understanding of the three components can bring more physical meaning. For example, a combination of any two components can be seen as a vector projection to a plane. The meaning of this combination is the approximation of the vector on the projection plane (approximate), and the remaining independent components are the details of the vector after projection (detail ). The significance of this coordinate system is actually the decomposition of Vectors under a group of orthogonal basis. This base is the three direction vectors of the coordinate system.

Then we want to project the vector to another coordinate system to analyze the features of the vector.
In this case, we need to find different orthogonal bases. We need to use orthogonal methods to find different bases. For example, Schmidt orthogonal method. The formula for determining the component of this orthogonal method is physical and does not need to be memorized. The result of each orthogonal base is calculated based on the projection of the previous Group of base components, that is, approximate + detail.

By extending the Vector Analysis Method to signal processing, we can quickly understand Fourier transformation. Fourier transform is an orthogonal basis signal decomposition based on sin and cos. The signal is projected into this space to analyze the signal features. After the projection, the signal features are the signal's frequency domain features. The Fourier Transform Formula can also be memorized from the physical projection features of approximate + detail.

Fourier transform can process the frequency characteristics of signals. However, after analysis, we can find that some signals are very intuitive using Fourier transform, while some signals are not adaptive to Fourier transform. Therefore, we need to introduce two concepts: stable and non-stable signals.

The difference is that the frequency is characteristic of the stable signal and the non-stable signal.
Stable signal: The frequency component does not change with time. Each frequency component exists at any time.
Non-stationary signal (non-stationary signal) is characterized by changing frequency components during different periods. For example, bird's name

Fourier transform is suitable for processing stable signals because the frequency components do not change with time. However, Fourier variation does not give the time-domain features of the frequency component, for example, the difference between the 3 m-> 30 m-> 300 m Signal Fourier spectrum and the-> 30 m-> 3 m Signal Fourier spectrum is very small.

To solve this problem, some people think that it is not good to segment the time domain signal. If we use a window function to divide the signal into different segments for Fourier change, don't you get the frequency information for different time periods? So STFT was invented, that is, short-term Fourier transformation, also called Window Fourier transformation.

Short-term Fourier transformation can solve some problems. However, for Signal Analysis with different frequency overlays, if the size of the selected window becomes a problem, the window is too small to analyze the low frequency component, if the window counter is large, the high-frequency components cannot be analyzed, that is, the resolution problem occurs.

Finally, the concept of wavelet transform is proposed, so that the window can be transformed over time to obtain the same signal frequency and time domain information. Different wavelet functions (mother wavelet and sub-wavelet) can be used to obtain different analysis results. In fact, different wavelet functions are different orthogonal bases.