1. Definition of Winger-Ville distribution (WVD)
The self-Wigner distribution of a signal x(t) is defined as the Fourier transform of its instantaneous correlation function with respect to the lag :
Winger proposed and used quantum mechanics in 1932. In 1948, Ville was applied to signal analysis, and discussed the mathematical foundation, the unified representation of time-frequency distribution, the definition and properties of WVD, etc. The paper and results were called "the mother of all time-frequency distributions".
The WVD of the two time domain signals after convolution is equal to the convolution of the respective WVD on the time axis
The multiplied WVD of the two signals is equal to the convolution of the respective WVD on the frequency axis. When we add a window to truncate an infinitely long time-domain signal, it only affects its frequency-domain resolution, not its time-domain resolution.
The WVD after the addition of the two signals is not equal to the addition of the respective WVD, but also the mutual WVD of the two signals. These mutual WVDs are interferences to the added signal WVDs, and in the time-frequency distribution, they are called cross-term interferences.
Disadvantages of WVD
1. WVD has the existence of cross items, so that the distribution of the sum of the two signals is no longer the sum of the respective distributions of the two signals.
2. Since WVD is the distribution of signal energy over time and frequency, theoretically speaking, it should always be positive. But since
it is
a Fourier transform, it is guaranteed to be real-valued, but not necessarily guaranteed to be non-negative.
3. WVD of commonly used signals
4. Realization of discrete WVD
Like DFT, WVD must be implemented on the computer: (1) windowing and intercepting x(t) (2) discretization of t and Ω
The meaning of WVD
1. Understand the time -frequency representation as the time-frequency energy distribution (or "instantaneous power spectrum"). The quantized time-frequency representation combines the concepts of
instantaneous power
and spectral energy density .
However, due to Heisenberg's uncertainty principle, the concept of time-frequency energy density at each point on the time-frequency plane cannot be established. Therefore, the quadratic form of time-frequency representation is an intuitive and reasonable assumption, because energy itself is a quadratic signal representation.
2. The linear characteristic of the quadratic signal of the time-frequency distribution is destroyed, that is, the time-frequency distribution of the sum of the two signals is not the sum of their respective time-frequency distributions, that is, a cross term (coherent term) is generated. The quadratic formation of WVD (coherent formation of signals) has relatively good time-frequency resolution (or time-frequency aggregation), and the cross term is relatively small.
5. Performance evaluation of time-frequency distribution
Most applications of time-frequency signal analysis are related to multi-component extraction of non-stationary signals. It is generally desirable for time-frequency signal analysis to have the following capabilities.
1. Able to determine the number of signal components existing in the signal;
2. Able to identify signal components and cross terms;
3. Ability to distinguish signal components that are very close to each other on the time-frequency plane;
4. It can estimate the instantaneous frequency of each component of the signal.
Performance Evaluation of Time-Frequency Analysis Methods: Time-Frequency Clustering and Cross-Terms
The time-frequency distribution of the chirp signal is as follows:
Reference video: