1. Background
Fourier transform is only suitable for stationary signals whose statistical properties do not change with time, but the statistical properties of actual signals are often time-varying, and such signals are collectively called non-stationary signals.
Since the statistical properties of non-stationary signals change with time, it is necessary to understand their local statistical properties for the analysis of non-stationary signals.
Fourier transform is a global transformation of the signal, so for non-stationary signals, Fourier transform is no longer an effective analysis tool.
On the other hand, both the time domain description and the frequency domain description of the signal can only describe part of the characteristics of the signal. In order to accurately describe the local characteristics of the signal, it is often necessary to use the two-dimensional joint representation of the time domain and frequency domain of the signal.
The joint time-frequency domain analysis of non-stationary signals is called time-frequency analysis of signals .
In some applications, it is required to use quadratic time-frequency representation to describe the energy density distribution of the signal, and the time-frequency representation of the energy density distribution in this stricter sense is called the time-frequency distribution of the signal .
Second, the limitations of the Fourier transform
The width of a rectangular window is inversely proportional to the width of its spectral main lobe. Since the rectangular window plays a role in signal truncation in signal processing, if the signal is shortened in the time domain, that is, it maintains a high resolution in the time domain, then the resolution in the frequency domain is widened due to the main lobe of It is bound to fall. Reflects the inherent contradiction in the time domain and frequency domain resolution in the Fourier transform.
Fourier transform cannot be used for local analysis of signals
A famous example is the δ(t) function introduced by Dirac. The point pulse in time has a uniform spectrum with positive and negative infinite extension in the frequency domain. Therefore, the signal x(t) and the spectrum X(Ω) describe each other as a whole and cannot reflect the characteristics of each in a local area, so they cannot be used for local analysis of the signal.
The necessity of time-frequency analysis
It can be seen from the above two examples that the Fourier transform cannot reflect the characteristics of the signal frequency changing with time. For non-stationary signals whose frequency changes with time, the Fourier transform can only give a total average effect. In order to analyze and process non-stationary signals, it is necessary to use the two-dimensional joint representation of the time domain and frequency domain of the signal, that is, frequency analysis .
Joint Time-Frequency Distribution of Signals
The basic purpose of time-frequency analysis is to construct a time-frequency joint distribution that can reflect the time-varying characteristics of the signal, which can describe the time-frequency joint characteristics of the signal. Specifically, for a given signal x(t), it is hoped to find a two-dimensional function , which should have the following basic properties:
is the joint distribution function of time f and frequency;
It can reflect the characteristics of the energy of the signal x(t) changing with time t and frequency;
It not only has good time resolution, but also has good frequency resolution.
3. The basic concept of resolution
"Resolution" includes both the time domain and the frequency domain of the signal. It refers to the minimum interval in the time domain or frequency domain that can distinguish the signal (also known as the minimum resolution cell)
The quality of resolution depends on the characteristics of the signal, and on the algorithm used
For signals with transients in the time domain, we hope that the resolution in the time domain is better (the observation interval in the time domain should be as short as possible) to ensure that the moment when the transient signal occurs and its transient form can be observed.
4. Signal duration and bandwidth
5. Uncertainty principle
Significance of the theorem——For a given signal, the product of its duration and bandwidth is a constant.
When the time width of the signal decreases, its bandwidth will increase accordingly. When the time width decreases to infinity, the bandwidth will become infinite, such as the pulse signal in the time domain; vice versa, such as the sinusoidal signal in the time domain.
That is to say, the duration and bandwidth of the signal cannot be infinitely small at the same time.
The application of the theorem: the constraint relationship between time resolution and frequency resolution; seeking the best time-frequency resolution
If the duration of the signal x(t) is limited, it is called tight support, and its duration interval (range) is t1<t<t2, which is called the support range, and the frequency is the same.
6. Instantaneous frequency
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