Negative Frequency - Best Explanation

If you have a physical tone at some frequency and it is DFT'd, you get a result in both the positive and
negative
frequencies - why and how does this occur? What does it mean?
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:
There has been a lot of talk about the 'direction' of the frequency, and thus how it can be +ve or -ve. 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:
 Distance is a scalar (can only ever be +ve), while displacement is a vector. (ie, has direction, can be
+ve or -ve to illustrate heading).
 Speed is a scalar (can only be +ve), while velocity is a vector. (ie, again, has direction, and can be
+ve or -ve).
Thus by the same tokens,
 Temporal Frequency is a scalar, (can only be +ve)! Frequency is defined as number of cycles per
unit time. If this is the accepted definition, we cannot simply claim that it is going in 'a different
direction'. Its a scalar after-all. Instead, we must define a new term - the vector equivalent of
frequency. Perhaps 'angular frequency' would be the right terminology here, and indeed, that is
precisely what a digital frequency measures.
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. The
first one is that as Fourier pointed out, an oscillatory real tone with scalar temporal frequency, f, can be
constructed by adding two oscillatory complex tones, with vector angular frequencies, +w and -w together.
cos(ω0t)=eȷω0t+e−ȷω0t2
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.
In the first complex tone, time is travelling in the +ve direction, and in the second complex tone, time is
travelling in the -ve direction. 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:
The physical interpretation of negative frequencies is as follows:
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. Since time is going backwards now, and your
frequency is change-of-phase/(negative time), your frequency will now be negative!
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).
Thus in summary:
The fourier transform is time-agnostic. It cannot tell the direction of time. This is at the heart of negative
frequencies. Since frequency = phase-change/time, anytime you take the DFT of a signal, fourier is saying
that if time was going forwards, your energy is located on the +ve frequency axis, but if your time was going
backwards, your energy is located on the -ve frequency axis.
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.

Negative Frequency - Best Explanation

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If you have a physical tone at some frequency and it is DFT'd, you get a result in both the positive and

Thus by the same tokens,

The physical interpretation of negative frequencies is as follows:

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