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🔥 Content Introduction
In the vast field of signal processing, accurately decomposing signals and clearly presenting their inherent characteristics is crucial. Symplectic Geometry Modal Decomposition (SGMD), as an innovative method that integrates symplectic geometry theory with signal processing technology, stands out due to its unique mathematical structure and efficient decomposition capabilities. Visualizing the signal components after SGMD decomposition provides an intuitive and powerful tool for us to deeply understand the essence of signals and uncover potential information.
1. SGMD Symplectic Geometry Modal Decomposition: Technical Core Analysis
1.1 Basics of Symplectic Geometry Theory
Symplectic geometry is a special type of differential geometry that uses symplectic forms as its core concept, describing the spatial characteristics of spaces with specific geometric structures. The introduction of symplectic geometry theory into the field of signal processing is due to its ability to accurately characterize the geometric features and dynamic behaviors of signals in phase space. For instance, when dealing with nonlinear and non-stationary signals, symplectic geometry can reveal the inherent laws of signals from a geometric perspective, providing new ideas and methods for signal decomposition.
1.2 Decomposition Principle of SGMD
SGMD constructs a variational model based on symplectic geometry theory, and by solving this model, the original signal is adaptively decomposed into multiple intrinsic mode functions (IMFs) with different frequency characteristics and physical meanings. Compared to traditional signal decomposition methods, such as Empirical Mode Decomposition (EMD), SGMD overcomes issues like mode mixing and endpoint effects. In constructing the variational model, SGMD utilizes the structural preservation characteristics of symplectic geometry to ensure energy conservation and geometric invariance during the decomposition process, thereby more accurately extracting the essential features of the signal. During the iterative solving process, parameters are continuously adjusted to ensure that the decomposed modal components reflect the different frequency components of the original signal to the greatest extent.
2. Methods for Visualizing Signal Decomposition Components
2.1 Time Domain Visualization
Time domain visualization is the most basic and intuitive visualization method. Displaying each intrinsic mode function obtained from SGMD decomposition in the time domain allows for a clear observation of how each component changes over time. By observing the waveform shapes, amplitude sizes, and trends of each component, one can preliminarily judge the time distribution of different frequency components in the signal. For example, in the decomposition of mechanical vibration signals, time domain visualization can quickly identify periodic vibration components and sudden impact components, providing important evidence for fault diagnosis. Additionally, color coding, transparency adjustments, and other methods can be used to simultaneously display multiple components, facilitating comparative analysis of their interrelationships.
2.2 Frequency Domain Visualization
Using tools like Fourier Transform, the components after SGMD decomposition can be converted to the frequency domain for visualization. The frequency domain graph uses frequency as the horizontal axis and amplitude or energy as the vertical axis, intuitively presenting the distribution and intensity of various frequency components in the signal. In frequency domain visualization, one can clearly see the frequency range corresponding to each intrinsic mode function and the contribution of different frequency components to the original signal. In power system signal analysis, frequency domain visualization can accurately detect harmonic components and their frequencies, aiding in the assessment of power quality and diagnosing power equipment faults. Advanced visualization forms such as waterfall plots and three-dimensional frequency spectrum plots can also be combined to display the frequency domain changes of the signal at different time points, providing a more comprehensive analysis of the signal’s dynamic characteristics.
2.3 Time-Frequency Domain Visualization
Time-frequency domain visualization methods, such as Short-Time Fourier Transform (STFT) and Wavelet Transform, can simultaneously display the changes of signals in both time and frequency dimensions. Visualizing the components after SGMD decomposition in the time-frequency domain can yield time-frequency distribution maps, where color or grayscale represents the energy intensity of the signal at different time and frequency points. This visualization method is particularly important for analyzing non-stationary signals, as it can capture the changes in signal frequency over time, showcasing the local time-frequency characteristics of the signal. In speech signal processing, time-frequency domain visualization can clearly present the pitch variations and formant information of speech, aiding in applications such as speech recognition and synthesis. Commonly used time-frequency domain visualization tools include Matplotlib, Seaborn, and the PyWavelets library in Python, which can easily implement various time-frequency domain visualization effects through corresponding code.
⛳️ Running Results
📣 Partial Code
🔗 References
🎈 Some theoretical references are from online literature; please contact the author for removal if there is any infringement.
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