The catch is that concepts in electrical engineering are sometimes pretty difficult to visualize. For example, how do you visualize propagating waves interacting with an antenna surface? What about negative frequencies? Maybe you intuitively think about frequency as the rate of repetition.
What could it possibly mean to have a negative repetition? This article will discuss some concepts about the frequency spectrum, negative frequencies, and complex signals. Remembering that physically, sinusoids are waves, the sign of the frequency represents the direction of wave propagation. Simply put, negative frequencies represent forward traveling waves, while positive frequencies represent backward traveling waves. This sign relation is by convention. To calculate the spectra theoretically, Fourier methods are usually used.
The Fourier transform yields a spectrum which contains positive and negative frequencies. The exponential form of the Fourier series also yields a spectrum which contains positive and negative frequencies. It's the same X rpm if it's spinning clockwise or anti-clockwise. You've invented negative rpms! Negative frequency is no different from the above simple example. A simple mathematical explanation of how the negative frequency pops up can be seen from the Fourier transforms of pure tone sinusoids.
For a pure sinusoid real , we have from Euler's relation:. However, it has only one frequency and a real sinusoid actually has two.
For sinusoidal repetition only positive frequencies makes sense. The physical interpretation is clear. For complex exponential repetition both positive and negative frequencies makes sense. It may be possible to attach a physical interpretation to negative frequency. That physical interpretation of negative frequency has to do with direction of repetition.
The definition of frequency as provided on wiki is: "Frequency is the number of occurrences of a repeating event per unit time". If sticking to this definition negative frequency does not make sense and therefore has no physical interpretation.
However, this definition of frequency is not thorough for complex exponential repetition which can also have direction. Negative frequencies are used all the time when doing signal or system analysis. The sinusoidal repetition is normally of interest and the complex exponential repetition is often used to obtain the sinusoidal repetition indirectly. That the two are related can be easily seen by considering the Fourier representation written using complex exponentials e.
So instead of considering a positive 'sinusoidal frequency axis', a negative and positive 'complex exponential frequency axis' is considered. On the 'complex exponential frequency axis', for real signals, it is well known that the negative frequency part is redundant and only the positive 'complex exponential frequency axis' is considered.
In making this step implicitly we know that the frequency axis represents complex exponential repetition and not sinusoidal repetition. The complex exponential repetition is a circular rotation in the complex plane. In order to create a sinusoidal repetition it takes two complex exponential repetitions, one repetition clock-wise and one repetition counter clock-wise.
If a physical device is constructed that produces a sinusoidal repetition inspired by how the sinusoidal repetition is created in the complex plane, that is, by two physically rotating devices that rotates in opposite directions, one of the rotating devices can be said to have a negative frequency and thereby the negative frequency has a physical interpretation.
In many common applications negative frequencies have no direct physical meaning at all. Consider a case where there is an input and an output voltage in some electrical circuit with resistors, capacitors, and inductors.
There is simply a real input voltage with one frequency and there is a single output voltage with the same frequency but different amplitude and phase. The ONLY reason why you would consider complex signals, complex Fourier Transforms and phasor math at this point is mathematically convenience.
You could do it just as well with entirely real math, it would just be a lot harder. The Fourier Transform uses a complex exponential as its basis function and applied to a single real-valued sine wave happens to produces a two valued results which is interpreted as positive and negative frequency. There are other transforms like the Discrete Cosine Transform which would not produce any negative frequencies at all.
You should study the Fourier transform or series to understand the negative frequency. Indeed Fourier showed that we can show all of waves using some sinusoids.
Each sinusoid can be shown with two peaks at the frequency of this wave one in positive side and one in negative. So the theoretical reason is clear.
But for the physical reason, I always see that people say negative frequency has just mathematical meaning. But I guess a physical interpretation that I'm not pretty sure; When you study the circular motion as the principal of discussions about the waves, the direction of speed of the movement on the half-circle is inverse of the another half. This can be the reason why we have two peaks in both sides of the frequency domain for each sine wave.
What is the meaning of negative distance? One possibility is that it's for continuity, so you don't have to flip planet Earth upside down every time you walk across the equator, and want to plot your position North with a continuous 1st derivative.
Same with frequency, when one might do such things as FM modulation with a modulation wider than the carrier frequency. How would you plot that? An easy way of thinking about the problem is to imaging a standing wave. Here comes the answer on why you have two frequency components in the FFT. FFT is basically a sum convolution of many of such oppositely traveling waves that represent your function in time domain.
Used to be to get the right answer for power you had to double the answer. But if you integrate from minus infinity to plus infinity you get the right answer without the arbitrary double. So they said there must be negative frequecies. But no one has ever really found them. They are therefore imaginary or at least from a physical point of view unexplained.
After reading the rich multitude of good and diverse opinions and interpretations and letting the issue simmer in my head for sometime, I believe I have a physical interpretation of the phenomenon of negative frequencies. And I believe the key interpretation here is that fourier is blind to time. Search only containers. Search titles only. Search Advanced search…. New posts. Search forums. Log in. Install the app.
Contact us. Close Menu. Welcome to EDAboard. To participate you need to register. Registration is free.
0コメント