The physical meaning of negative frequency in the spectrum (I.)

On the physical meaning of negative frequency in spectrum

                                                     --Reading analysis

* Statement: This article is a large number of references to Chen Hui deep, Fang Haiyan of the relevant paper content
* This article is the best I've ever read to explain the negative frequency of the spectrum, just to pay tribute to the two authors in this blog post.

Absrtact: This paper discusses the physical meaning of the spectrum obtained by Fourier transform of the signal, which focuses on the negative frequency component. In many of the teaching materials of signals and systems, negative frequency components are considered to be of no physical significance. This paper proves that the negative frequency component not only has the definite physical meaning, but also has the important engineering application value in many examples. In this paper, MATLAB program is used to illustrate how to obtain Fourier inverse transform by using the geometric method, and the lumped spectrum is synthesized into the signal of the last domain, and the meaning of negative frequency component can be clearly seen. 1 negative frequency and complex signal

The original definition of frequency is the number of occurrences per second that can be used to measure the frequency of mechanical motion, electrical signals, and even the repetition of any event, which of course does not have a "negative" concept. The concept of angular frequency ω is generated when the frequency is used to describe the circular motion (i.e. into a two-dimensional signal plane). Starting from the mechanical rotational motion, the Ω=DΘDT is defined as the angular velocity. For periodic motions, angular velocity is also the angular frequency. Usually counterclockwise is positive, so the rotation of the positive frequency is counterclockwise rotation angular velocity, the negative frequency is the clockwise rotation angular velocity. The positive and negative signs are very naturally formed, and there is no physical problem.
The unit vector of electricity (voltage or current) revolves around the origin and can be expressed in u=ejθ=ejωt+θ0, which is clear in the circuit. The physical meaning of Theta's positive and negative representation is never controversial, and its derivative ω=dθ/dt the physical meaning of self-evident. Take positive take negative does not affect the definition, why take negative will lose the physical meaning?
* * * * * * * * * * * * * * * * * * * * * * * * * * * Look at the picture carefully to see the meaning of it * * * * * * * * * *
In the course of signals and systems to simplify the problem, it is easy for beginners to master the concept, the scope of the study is limited to the real signal f (t), that is, in the voltage rotation vector U=ejωt=sin (ωt) +jcos (ωt), only a projection of sin (ωt) or COS on the real plane or the imaginary plane. (ωt), the study of the characteristics of complex signal ejωt and only the real signal of sin (ωt) or cos (ωt) is two different levels. The former (complex signal) is a reflection of the overall characteristics of the signal in space, as shown in Figure 1. The latter (real signal) only studies the characteristics of the signal projecting on a plane (a plane consisting of x-t or y-t). This is bound to lose some important information, so that X=sin (ωt) and sin (−ωt) in the z-t plane of the waveform does not have any difference, this is one of the direct reasons for the negative frequency of the significance of questions. Obviously, in the plane of x-t or y-t, it is impossible to see the rotation. Neither Theta nor Omega can be seen. These two rotation parameters can only be seen on the X-y plane.Note that the above-mentioned X-plane corresponds to the real part, the Y-plane corresponds to the imaginary part, that is, this article is standing in the plural expression has the corresponding physical meaning of the premise of the discussion, then I gave J an effective theoretical support, circuit theory in the j,-j,-1, can be seen as a rotational factor. Complex F times J, equivalent to F in the complex plane counterclockwise rotation 90 degrees, complex f times-j, equivalent to clockwise rotation 90 degrees, F times or divided by-1, equivalent to f clockwise or counterclockwise rotation 180 degrees
Figure 1 Complex Signal ejωt
2 complex signal and real signal spectrum
Similarly, the result of Fourier transform using ejωt, sin (ωt) or cos (ωt) as a nucleus is also a comprehensive signal, one-sided signal. When Fourier transform the real signal, if the exponential ejωt is used as the nucleus, the bilateral spectrum will be obtained. Taking the cosine signal of angular frequency n as an example, it has a frequency characteristic that is at ±n two, amplitude is 0.5, and phase angle is zero. Its relations can be represented in Figure 2. Two vectors with a length of 0.5: rotating at the same speed as the real axis, their synthetic vector is the zero signal along the direction of the imaginary axis in the cosine vector. It can be seen that there must be a vector of negative frequencies in order to constitute a purely real signal. So the Euler formula cos (ωt) =0.5 (ejωt+e−jωt) has its definite geometrical meaning (i.e. physical meaning). In the literature [1] The animation is given, and the geometric interpretation of the positive and negative digital frequencies is given.
Fig. 2 The real signal is synthesized by the positive and negative frequency complex vectors

4) Doppler frequency
Doppler frequency is also an example of a negative frequency, if the source of the signal is moving to us, then the Doppler frequency is the positive frequency, if the source of the signal is away from us, then the Doppler frequency is the negative frequency, where the positive and negative frequencies are clear physical meaning. Doppler frequency is a differential frequency, it is represented as the envelope frequency of the synthesized signal, so it still conforms to the above principle, in the real signal domain can only find the size of the Doppler frequency, but it does not detect its positive or negative. To get the negative frequency, it must be considered from the complex signal domain. Visible, do not understand this point, can not find the Doppler speed of the principle block diagram.
In the final analysis, the corner and the frequency of the positive and negative, must be in the X-y plane or two-dimensional signal can be observed. It is an epistemological error, not a scientific method, because the method of observation is wrong and the meaning is not seen, thus denying its existence. This is what the "elephant" story says, the man who feels like a leg denies that he has a nose, and that the fault is in his method of verification. He always wanted to find the elephant's nose (negative frequency) in the elephant's leg (real signal field), and of course he never found it. The correct way is to change the angle, touch the other parts (complex signal domain) in order to get the full