Why negative frequencies

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. Electrical engineers define wave propagation as the following:. The most notable case is for physics. Physicists define wave propagation in exactly the opposite sense, with positive frequencies propagating forward.

The following two spectrums show the location of a forward-traveling and backward-traveling wave, respectively. So what happens when plot the spectrum of a cosine? The direction of propagation is much more obvious when looking at the complex form of sinusoids. So what does it all mean, and how do I relate all this to real signals traveling down wires?

Whether or not you should care about positive and negative frequencies depends on how you got your time-domain data and your application. If your samples are just real numbers, then you can ignore half the spectrum and double just one side. If your samples are complex I and Q , you need both sides of the spectrum since the spectrum is asymmetric.

For applications such as simple real-valued filters, you can just use cosines and look at a single-sided spectrum. In that case you care about both sides of the spectrum. 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. Expaning on this further:. While the overarching insights of the authors saying this is not lost, this statement is nontheless inconsistent with the definition of temporal frequency, so first we must define our terms very carefully.

For example:. Now all the sudden we are in the business of measuring number of rotations per unit time, a vector quantity that can have direction , VS just the number of repititions of some physical oscillation. Thus when we are asking about the physical interpretation of negative frequencies, we are also implicitly asking about how the scalar and very real measures of number of oscillations per unit time of some physical phenomenon like waves on a beach, sinusoidal AC current over a wire, map to this angular-frequency that now all the sudden happens to have direction, either clockwise or counterclockwise.

From here, to arrive at a physical interpretation of negative frequencies two facts need to be heeded. Thats great, but so what? Well, the complex tones are rotating in directions opposite to each other.

See also Sebastian's comment. But what is the significance of the 'directions' here that give our angular frequencies their vector status? What physical quantity is being reflected in the direction of rotation? The answer is time. Time is going backwards. Keeping this in mind and taking a quick diversion to recall that temporal frequency is the first derivative of phase with respect to time, simply the change of phase over time , everything begins to fall into place:.

My first realization was that fourier is time-agnostic. That is, if you think about it, there is nothing in fourier analysis or the transform itself that can tell you what the 'direction' of time is. Now, imagine a physically oscillating system ie a real sinusoid from say, a current over a wire that is oscillating at some scalar temporal-frequency, f.

Imagine 'looking' down this wave, in the forwards direction of time as it progresses. Now imagine calculating its difference in phase at every point in time you progress further. This will give you your scalar temporal frequency, and your frquency is positive. So far so good. But wait a minute - if fourier is blind to time, then why should it only consider your wave in the 'forward' time direction?

There is nothing special about that direction in time. Thus by symmetry, the other direction of time must also be considered. Thus now imagine 'looking' up at the same wave, ie, backwards in time , and also performing the same delta-phase calculation. What Fourier is really saying, is that this signal has energy if played forward in time at frequency bin f, but ALSO has energy if played backwards in time albeit at frequency bin -f. In a sense it MUST say this because fourier has no way of 'knowing' what the 'true' direction of time is!

So how does fourier capture this? Well, in order to show the direction of time, a rotation of some sort must be employed such that a clockwise roation dealinates 'looking' at the signal in the forward arrow of time, and a counterclockwise roation dealinates 'looking' at the signal as if time was going backwards.

The scalar temporal frequency we are all familiar with should now be equal to the scaled absolute value of our vector angular frequency. But how can a point signifying the displacement of a sinusoid wave arrive at its starting point after one cycle yet simultaneously rotate around a circle and maintain a manifestation of the temporal frequency it signifies? Only if the major axes of that circle are composed of measuring displacement of this point relative to the original sinusoid, and a sinusoid off by 90 degrees.

This is exactly how fourier gets his sine and cosine bases the you project against every time you perform a DFT! And finally, how do we keep those axes seperate? The 'j' guarantees that the magnitude on each axis is always independant of the magnitude on the other, since real and imaginary numbers cannot be added to yield a new number in either domain.

But this is just a side note. The fourier transform is time-agnostic. It cannot tell the direction of time. This is at the heart of negative frequencies. As our universe has shown before , it is precisely because Fourier does not know the direction of time, that both sides of the DFT must be symmetric, and why the existence of negative frequencies are necessary and in fact very real indeed.

Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams?

Learn more. What is the physical significance of negative frequencies? Ask Question. Asked 10 years, 1 month ago. Active 3 years, 4 months ago. Viewed 84k times. Improve this question. Spacey Spacey 9, 8 8 gold badges 38 38 silver badges 77 77 bronze badges. I have provided an answer to my own question and would like to share it with that group too. I dont seem to have access to that area I am a chemist.

I deal with molecules. The negatives frequencies indicate the instability in the molecules or, in other words, saddle points on the potential energy surface.

A stable molecule should have no imaginary frequencies, a transition state should have one 1st order saddle point. Why not stable molecule should have negative frequencies imaginary frequencies after all it is the complementary to the real frequency. An imaginary frequency turns an oscillating, bounded complex exponential into an exponentially increasing or decreasing ordinary exponential. A negative frequency, as the answers below indicate, refers to the "handedness" of the oscillation.

They are still bounded functions, so I imagine it would still be "stable". Add a comment.