This has been one of the holes in my cheddar cheese block of understanding DSP, so what is the physical interpretation of having a negative frequency?
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?
Edit: Oct 18th 2011. I have provided my own answer, but expanded the question to include the roots of why negative frequencies MUST exist.
Negative frequency doesn't make much sense for sinusoids, but the Fourier transform doesn't break up a signal into sinusoids, it breaks it up into complex exponentials (also called "complex sinusoids" or "cisoids"):
$$F(\omega) = \int_{-\infty}^{\infty} f(t) \color{Red}{e^{- j\omega t}}\,dt$$
These are actually spirals, spinning around in the complex plane:
Spirals can be either left-handed or right-handed (rotating clockwise or counterclockwise), which is where the concept of negative frequency comes from. You can also think of it as the phase angle going forward or backward in time.
In the case of real signals, there are always two equal-amplitude complex exponentials, rotating in opposite directions, so that their real parts combine and imaginary parts cancel out, leaving only a real sinusoid as the result. This is why the spectrum of a sine wave always has 2 spikes, one positive frequency and one negative. Depending on the phase of the two spirals, they could cancel out, leaving a purely real sine wave, or a real cosine wave, or a purely imaginary sine wave, etc.
The negative and positive frequency components are both necessary to produce the real signal, but if you already know that it's a real signal, the other side of the spectrum doesn't provide any extra information, so it's often hand-waved and ignored. For the general case of complex signals, you need to know both sides of the frequency spectrum.
Let's say you had a spinning wheel. How would you describe how fast it is spinning? You'd probably say it's spinning at X revolutions per minute (rpm). Now how do you convey in what direction it's spinning with this number? It's the same X rpm if it's spinning clockwise or anti-clockwise. So you scratch your head and say oh well, here's a smart idea: I'll use the convention of +X to indicate that it's spinning clockwise and -X for anti-clockwise. Voila! 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.
Consider the Fourier transform pair of a complex sinusoid: $e^{\jmath \omega_0 t}\longleftrightarrow \delta(\omega+\omega_0)$ (ignoring constant multiplier terms). For a pure sinusoid (real), we have from Euler's relation:
$$\cos(\omega_0 t)=\frac{e^{\jmath \omega_0t}+e^{-\jmath \omega_0 t}}{2}$$
and hence, its Fourier transform pair (again, ignoring constant multipliers):
$$\cos(\omega_0 t)\longleftrightarrow \delta(\omega+\omega_0) + \delta(\omega-\omega_0)$$
You can see that it has two frequencies: a positive one at $\omega_0$ and a negative one at $-\omega_0$ by definition! The complex sinusoid of $ae^{\jmath \omega_0 t}$ is widely used because it is incredibly useful in simplifying our mathematical calculations. However, it has only one frequency and a real sinusoid actually has two.
Currently, my viewpoint (it is subject to change) is the following
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 fundamental reason for this being the Euler formula $$e^{j\omega n} = \cos( \omega n) + j\, \sin(\omega n)$$ and the fact that complex exponentials are eigenfunctions of LTI systems.
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.g. $$ x[n]= \frac{1}{2\pi} \int_{-\pi}^{\pi}\!\!\!d\omega \;\;\;\;X(e^{j\omega}) e^{j\omega n} $$
However, this is equivalent to
$$ x[n] = \frac{1}{2\pi} \int_{0}^{\pi}\!\!d\omega \;\;[a(\omega) \cos(\omega n) + b(\omega) \sin(\omega n)] = \frac{1}{2\pi} \int_{0}^{\pi}\!\!d\omega \;\; \alpha(\omega) \sin(\omega n + \phi(\omega))] $$
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.
There are different types of time/frequency transforms. 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. Again, it’s a matter of mathematical convenience; the Fourier Transform is often the quickest and most efficient way to solve a specific problem.
even though everything important has been said I wanted to add some code and more visual keys on why the following formulas require positve and negative frequencies to make clear that negative frequencies are important for canceling out the counterparts in the inverse dft.
So lets see for
$\sin z = \frac{1}{2\mathrm{i}} \left(\mathrm{e}^{\mathrm{i}z} - \mathrm{e}^{-\mathrm{i}z} \right)$
$\cos z = \frac{1}{2} \left(\mathrm{e}^{\mathrm{i}z} + \mathrm{e}^{-\mathrm{i}z} \right)$
lets first produce a complex sinusoids and plot it:
# sine wave parameters
freq = 1; # frequency in Hz
ampl = 1; # amplitude in a.u.
phas = 0; # phase in radians
# generate the sine wave
pos_csw = ampl * np.exp( 1j* (2*np.pi * freq * time + phas) );
neg_csw = ampl * np.exp( 1j* (2*np.pi * (- freq ) * time + phas) );
# plot the results
plt.plot(time,np.real(pos_csw),label='real')
plt.plot(time,np.imag(pos_csw),label='imag')
plt.xlabel('Time (sec.)'), plt.ylabel('Amplitude')
plt.title('Complex positive freq sine wave projections')
plt.legend()
plt.show()
# plot the results
plt.plot(time,np.real(neg_csw),label='real')
plt.plot(time,np.imag(neg_csw),label='imag')
plt.xlabel('Time (sec.)'), plt.ylabel('Amplitude')
plt.title('Complex negative freq sine wave projections')
plt.legend()
plt.show()
# now show in 3D
fig = plt.figure()
ax = fig.add_subplot(projection='3d')
ax.plot(time,np.real(pos_csw),np.imag(pos_csw))
ax.plot(time,np.real(neg_csw),np.imag(neg_csw))
ax.set_xlabel('Time (s)'), ax.set_ylabel('Real part'), ax.set_zlabel('Imag part')
ax.set_title('Complex sine wave in all its 3D glory')
plt.show()
now for computing cosine = pos_csw + neg_csw results in
while computing sine = pos_csw - neg_csw results in:
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.
In the real World, actual event frequency is fundamentally unsigned
Frequency is measured in Hz, which is defined as the inverse of the second, and is used to count how often an event repeats. Time is assumed to flow only towards the future, thus for a periodic event, period $\small T$ is fundamentally an unsigned quantity, and frequency $\small f=1/T$ is therefore fundamentally unsigned.
Negative frequency sinusoid: Mixing with frequency of the complex exponential
Talking about a sinusoidal waveform with a negative frequency is inaccurate. We are mixing the sinusoid waveform with its usual mathematical breakdown into a handy sum of complex exponentials: $\small \cos(\omega t) = \frac {1} {2} e^{j\omega t} + \frac {1} {2} e^{-j\omega t}$ of same and opposite frequencies, $\small \omega$ and $\small -\omega$.
Indeed one of the complex exponentials has a negative frequency, still the term of this mathematical breakdown cannot be transposed into the physical world. In DSP, this complex exponential exists only as a sequence of numbers. To reach the real world, and represent an actual phenomenon, it must be converted into a physical signal (a waveform), e.g. using a DAC. A DAC just takes either the real or the imaginary part of the complex sequence, both are regular sinusoids where the notion of negative frequency is meaningless (see picture below).
Negative frequencies are similar to the square root of -1, they are only handy mathematical concepts, but they have no physical counterparts.
Negative sign is often a convention, not a physical property
Using negative frequencies is not different than using other negative quantities.
E.g. we can measure a room occupancy by looking at people entering and leaving the room, counting the former positively and the latter negatively. The number of people in the room is the sum. It would be meaningless to find a difference between people, based on "their sign", because this sign is a methodological one, artificially defined.
Negative frequencies are only that. An observer may declare a wheel turns at 10 Hz or at -10 Hz. The unsigned value 10 represents something real, the measured angular velocity, but the sign is only a convention from the observer. If the observer looks from the other side of the wheel, the sign might change while the wheel still rotates the same way. The convention may be different for another observer. On a car, do the wheels turn at a positive or a negative rate?
Negative frequency: Blame Euler
From Euler's formula we know sinusoids, sines and cosines, can be split into two complex carriers of opposite frequencies, multiplied by conjugate phasors. The carriers are functions of time. (More about phasor and carrier).
$\small A\cos(\omega t +\phi) = Ae^{j \phi} \times \frac {1} {2} e^{j\omega t} + Ae^{-j \phi} \times \frac {1} {2} e^{-j\omega t}$
The carriers can be represented as helices, turning at the angular velocity defined by their frequency. If the frequency sign are different, the helices turn in opposite directions:
From the pictures, we see each helix can be split into two sinusoids, which are nothing else than the helix real and imaginary parts. The phasors only determines the amplitude and the starting angle (phase at time t=0).
To match Euler's formula, the frequencies and the start angels must be exactly opposite and the amplitude must be identical. In that case:
The carriers turn at the same velocity, the vertical sinusoids are opposite and cancel each other when added.
On the other hand, the horizontal sinusoids are in phase, and when added result in a sinusoid with amplitude doubled.
This breakdown of cosine function into conjugate phasors/carriers is massively used in Fourier transform used to convert a signal into its spectral components. FT being a basic operation in signal processing, negative frequencies are always present.
However as mentioned, we must be careful: The possibility of this mathematical breakdown of a sinusoid doesn't mean it has a physical equivalence, these helices don't exist physically. The only real thing in the signal is the starting point: A sinusoidal variation with its unsigned frequency.
Negative frequencies in a spectrum
The principle of the discrete Fourier transform (DFT) is to find the phasors associated with a series of carriers. Note the DFT doesn't compute the sample values for the carriers, it just assumes the carrier frequencies are symmetrical about zero (conjugates), are in the same number than the samples in the input, are spaced by the inverse of the sampling rate and have all a unit amplitude. The DFT rather computes the phasors which, in this context, are called the spectral coefficients. Each phasor is the complex exponential corresponding to the amplitude and phase at time zero of a carrier.
For an input which is real-valued (e.g. samples from a physical sensor), conjugate carriers have conjugate phasors. So these two spectral components nicely sum up into a cosine of twice the amplitude and the same (unsigned) frequency.
But we do not need this property to explain why negative frequencies are not found into the real world. After all we can build mathematical signals with negative frequencies from scratch, where the conjugate carriers don't have conjugate coefficients, or we can mathematically shift a symmetrical spectrum entirely to the negative side by a simple multiplication with a mathematical sinusoid of negative frequency.
The very reason these signals cannot reach the real world is a rotating phasor has no actual existence, in DSP it's only numbers, in analog SP it exists as two separate sinusoidal sources. This fictive phasor must be converted from its digital or dual form into a single analog waveform. So the engineer must find a way to convert the virtual helix into a regular sinusoid at some point, prior to the conversion. This regular sinusoid puts a halt to the mathematical negative sign, which never reaches the real world.
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. The standing wave (in time domain) can be represented as a sum of two oppositely moving traveling waves (in frequency domain with positive and negative k vector, or +w and -w which is equivalent). 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.
The significance is s p i n
(The first part of this answer assumes familiarity with complex "spin"; supplementary explanations are referenced at bottom, and should be consulted before reading the main body if one's unfamiliar.)
Spectral asymmetry is spin asymmetry. Negatives dominating positives guarantees net-clockwise traversal about the axis of evolution (which is also the axis of revolution), and vice versa.
One can have all the ingredients, but the missing piece can be the right perspective. The key is to recognize that spin is a separate degree of freedom.
The Fourier transform collapses the time axis and only speaks of frequencies. Its ability to directly represent variation, in that behavior is read off directly from coefficients faithfully, does not extend to frequencies or amplitudes that change over time. More generally, we have time-frequency, whose coefficients directly represent frequency and amplitude over time - in best case exactly, in worst case bounded by Heisenberg's uncertainty. Within it, positives are negatives are both more distinct and more natural:
We have what can be described as pulses - bursts of oscillations - and a usual spectrogram, except also showing negatives, and this time the intensities aren't mirrored as for real-valued signals. Each half of the spectrogram responds to unique variation.
Indeed, lack of mirroring guarantees the signal isn't real-valued. This may be a surprise - per conventional wisdom of STFT modulus discarding phase, one may figure we can mirror magnitude but have phase be not Hermitian-symmetric. But we can't! To do so would mean we have counter-clockwise and clockwise traversal over same time instants, which by definition cancel. The only alternatives are complete sign reversal, which cancels to zero, or aligned rotation, which doubles up either on positives or negatives. Indeed, STFT isn't FT, and its modulus is strongly invertible (within global phase shift) - if such mirroring were possible, much more info would be lost.
More yet, CWT says positives and negatives are infinitely apart. In an important sense, CWT is the one with fixed resolution, not STFT - and follows ratio-based (logarithmic) decomposition, more natural for many structures. If we try to cross from negatives to positives, with linear frequency growth, we see:
The traversal stops, momentarily. This is required, since we're crossing the DC bin, which is of course the constant bin with zero derivative - and it further shows the fundamental distinction: no amount of speeding up or slowing down, or changing of intensity, results in changed direction of rotation.
The morale is simple: spin encodes irreducible variation that is inexpressible in terms of frequency - rate of oscillation - or amplitude - intensity of oscillation. As a more familiar analogy, it's like using a microphone that only records volume, without frequency: no matter how we do it, volume is just volume - an entire degree of freedom is missing.
So what's it all mean? What of the "real world"?
The real world:
Source: Vsauce
The world orbits the sun, the sun orbits Sagittarius A* - in turn, our Earth traverses a helix about the sun's orbit. If we take January 1 to be 0 degrees with respect to the sun and start recording, then by December 31, the Earth will traverse 358.77 degrees counterclockwise about the sun with a mean radius of 149,600,000 km, and 7,233,408,000 km with respect to Saggitarius A*. The Discrete Fourier Transform of this path peaks at $+3.16 \times 10^{-8}\ \text{Hz}$, with amplitude of around 149,600,000 km, and no negative frequencies.
If aliens hand us a graph of the complete Fourier transform of Earth's orbit (and cite relativity for how they saw future paths), and the graph shows a peak at a negative frequency, we'll ask if they have room aboard.
If we want to compare slinkies, or springs,
Source: sparktec, e-Bay
If the spring on left is meant to be inserted from the wide end, then its Fourier transform peaks negative (clockwise), else positive. If we want coiling with even spacing, we can check if there's multiple peaks close in amplitude, as with springs on right. If we want windings to have the same radius, or compare spacings between windings, we can look at their spectrogram, which is a locally weighted unfolding of the Fourier transform along time, that'll reveal the instantaneous radius and rate of winding over the length of the spring.
If we want to measure the performance of an all-terrain vehicle (ATV),
Source: carbuzz.com
the 2D Fourier transform (rather, 3D spectrogram) of a recording of a point on the wheel over time, and over distance the ATV travels on muddy roads, will reveal the efficiency with which the wheels' rotations result in actual motion as opposed to fighting the mud (distance vs time amplitude), and we can compare performance for different wheel speeds (frequency), and when the car is going forward vs backwards (negative vs positive frequency).
The list goes on. Now, some may object. Yet, the "metaphysical status" of complex numbers bears no difference. The positive and negative frequencies of the Fourier transform meaningfully and usefully describe physical phenomena, and each is distinct and necessary for a complete description.
Fine. Actually, in your ears:
Source: Britannica
The Joint Time-Frequency Scattering transform shares bioplausibility with STRF (Spectro-Temporal Receptive Fields), where close correspondence with auditory responses of animals and humans is shown by neurophysiological and behavioral evidence (V. Lostanlen, et al, D. Klein, et al, F. Theunissen, et al). Methods include comparing direct measurements of auditory cortex responses with the model's coefficients to a range of stimuli, and correlating perceptual recognition of human listeners of musical sounds with nearest neighbor search directly on coefficients. T. Chi, et al estimate $Q_2 \approx 2$ (one of key transform parameters) in humans.
It's a fundamentally complex transform. It applies complex 2D wavelets upon the modulus of complex wavelet transform to discriminate rises and falls in frequency. The input is standard real-valued audio, but its projection to complex space is necessary to obtain the discriminatory, invariance, and stability properties of the transform that are responsible for the notable success of its features fed straight to a linear classifier on audio tasks. The linear classifier is the final stage of nearly all modern machine learning methods; success of fed features indicates that the relevant variability is distributed in such a way in the transformed space that distinct classes are grouped left and right (for example) and can be picked out by naked eye - or in short, strong explanatory power.
Simply put, there's strong evidence for the ear doing equivalently complex-valued processing. As for where exactly it happens, physically, I'm no cochlea expert at all.
What's required, physically, for distinct use of negative frequencies and complex numbers, is three degrees of freedom - with two of said degrees evolving in terms of the third, while forming a pair for which "angle" is a valid parameter. This is suggestive of rotation, and while valid rotational interpretation directly follows, something physically rotating is not required - the rotation can instead take place in the resulting projection, for example electric and magnetic components of an EM wave. Of course, physical circular behavior (not necessarily over time) is ideal.
Is negative real? Can you hold negative two apples?
"I can have debt, that's negative money" - no, that's what the banker says to remove your positive money. Indeed, that's what "negative" means - a quantity such that, when combined with a positive of equal size, cancels.
An imaginary is a quantity such that, when squared and combined with a real squared, cancels.
Imaginaries "exist" as much as reals do. They don't, or they do. The name is what I consider to be the greatest misnomer in all of mathematics. And what I did above isn't semantic tricks - it's precisely how many mathematical concepts are defined - in terms of properties. The unit impulse is a distribution such that, when integrated over its entire domain, is unity. And I consider it to be a lot more "unreal" than imaginaries, yet it forms the basis of real-world signal processing.
It depends.
If the data is provided in complex form, odds are, it was fed through stages of processing to end up that way, and said processing determines the interpretation. I/Q data, for example, can be used to efficiently transmit two independent signals, or one and describe its amplitude and phase - positive and negative frequencies have no meaning here.
Worse yet, "positives" can lose their original meaning entirely - the Fourier transform is fundamentally a numeric manipulation, it need not come with any "meaning" attached. This can happen with a frequency shift to "baseband" (f=0), for example, that's done for engineering purposes. Then the interpretation could be efficient compression, sparse encoding, or other.
Either data must be measured as complex as in above examples, or the processing of real-valued data must involve operations that distinguish between positive and negative frequencies, as with JTFS.
There are some great and enlightening answers here, but I feel like one simple and important perspective is missing. Negative frequencies can be understood in the same way as any other negative quantity (of apples for example): not as absolutes, but as relative positions below some arbitrary reference value.
For example, the Fourier transform of a window function (see many illustrations on Wikipedia) reveals how a particular method of slicing a short duration out of a signal will "leak" energy present at each frequency in that signal into specific frequencies symmetrically distributed higher and lower than the original frequency.
Hann window and its Fourier transform, public domain image from Olli_Niemitalo
Multiplying your time-domain signal by the left-hand window (i.e. chopping out a section with this envelope) is equivalent to pasting an amplitude-scaled copy of the right-hand image at each distinct frequency present in the frequency domain (convolution). This Fourier transform extends above and below zero because it represents effects at frequencies situated above and below each of those present in the original signal.
Similarly, amplitude modulation (as used in AM radio) creates sidebands symmetrically above and below the carrier frequency (again, see good illustrations on Wikipedia).
It is physically measurable that when you broadcast AM (multiplying a carrier signal by an audio signal in the time domain) it consumes twice the bandwidth of the original audio, and it does so at frequencies situated symmetrically above and below the carrier, at distances independent of the carrier itself.
Spectrogram of 558 kHz carrier wave with symmetric audio sidebands, public domain image from Mysid
If the same sound were broadcast on any other carrier frequency (AM radio channel) the spectrogram would look the same, with even the horizontal scale unchanged. It would consist of two sidebands with content offset above and below the carrier frequency by amounts equal to the frequencies present in the audio. Positive and negative frequencies seem like a perfectly reasonable way to represent this.
Combining the windowing and AM radio examples, abruptly switching an AM signal on and off creates something called spectral splatter. This plot shows a symmetrical distribution which I would expect to be similar to the Fourier transform of the boxcar/rectangular window function, above and below the intended transmission band.
Spectrogram of radio signal with switching noise, image CC BY-SA from Hp.Baumeler
To make this more concretely perceptible: if you quickly toggle an audio signal on and off (even physically, with a switch) you can measure and even discern with your ears a click of broad-spectrum "switch noise" at frequencies both higher and lower than the audio itself. The spectrum should essentially follow the Fourier transform of the boxcar or rectangular window function. Try silencing many small bits of an audio file containing a pure high-frequency tone. When you play it back, you should hear noises lower than the pitch you originally synthesized. That's negative (relative) frequency.
Switch noise on 10 kHz sine wave (generated in Audacity)
Now, to tie this relative idea of negative frequency together with other ideas of negative frequency discussed in other answers. What happens when the negative lobe of the Fourier transform is interpreted relative to a low enough reference frequency that the result actually falls below zero Hz?
This should happen for example if a carrier frequency is lower than that of its modulator. If you generate a 2 kHz tone and then modulate it with another tone sweeping up to 2.5 kHz and back down, this is what you get:
Sweeping modulator exceeding carrier frequency. Audio signal generated with Reaktor and plotted with Audacity.
The Fourier transform of the modulating signal consists of two peaks equally above and below zero, yielding two tones equally above and below the carrier tone. They spread farther apart (around zero) as the modulating frequency increases. When the modulator exceeds 2 kHz, the Fourier transform still has two peaks equally spaced around zero. But the lower sideband of the resulting audio tone folds and the tone reflects off of zero!
This is not a theoretical result. If you listen to this modulated signal, you will hear two tones moving apart in opposite directions, then when the modulator exceeds the carrier frequency you will hear one tone reverse direction and both will be rising.
You can imagine what will happen as you shift the carrier down to zero. The entire lower sideband will fold up until it perfectly overlaps the upper sideband yielding double the power.
Sweeping carrier down to zero, folding lower sideband onto upper. Generated with Nyquist code below.
define function modulator ()
return 0.5 * hzosc(4000) + 0.5
define function carrier_sweep ()
return 10000 * (0.5 * hzosc(0.2) + 0.5)
define function carrier ()
return hzosc(10000 - carrier_sweep())
return modulator() * carrier()
Our ideas of waves and frequencies are very general, and can describe many different kinds of motion. As discussed in other answers, the positive and negative frequencies can be interpreted as waves with clockwise and counterclockwise spin. In light or radio this corresponds to polarization. However, only waves oscillating in directions perpendicular to their direction of travel can exhibit this property. Transverse waves (including pressure waves like sound) that oscillate along their axis of propagation exhibit no spin. They can be modeled like other waves, but the negative frequency components with opposite spin must balance the positive ones because their axis of oscillation doesn't allow spin to physically manifest.
Essentially, as the lower tone in my example reaches zero, it slows down to zero rotational velocity and then speeds back up in the opposite direction (as nicely illustrated in a previous answer) but because this is audio (a longitudinal wave), those two spins correspond to identical physical effects.
So, my understanding is that the negative and positive parts of your Fourier transform are simply the clockwise and counterclockwise circularly polarized components of your original signal, where your original signal is unpolarized (i.e. an equal mix both spins). When those negative values are summed with some reference frequency having a higher absolute value, they behave just like "normal" negatives, yielding a positive result with a lower absolute value. But when they're interpreted down at zero, the negative and positive sidebands again yield clockwise and counterclockwise circularly polarised components. These are perceived in combination as unpolarized waves, either because they're mixed in equal quantities, or because the wave is oscillating along its axis of propagation so simply cannot exhibit spin.
If a wave is vibrating in a direction that does allow it to possess spin, you can create signals whose Fourier transforms have unbalanced positive and negative sides. This distinction between negative and positive frequencies is definitely physically significant and even highly useful: wifi and even analog TV rely on it, as they use quadrature amplitude modulation which intentionally employs an asymmetric spectrum. In TV, if I understand correctly an asymmetry between positive and negative frequencies was used to independently add and remove red and blue from the main, higher-resolution black-and-white image. And (though less practically useful) I believe in the case of light or radio these two components can in fact exert torque on physical objects in opposite directions.
As a thought experiment for someone more versed in the math and physics than me: If you had a circularly polarized radio carrier and amplitude-modulated it with frequencies exceeding that of the carrier, would the part of the lower sideband that folds around zero have opposite circular polarity to the upper sideband? That would neatly tie together most of the answers here.
This has turned out to be quite the hot topic.
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,
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.
$$ \cos(\omega_0 t)=\frac{e^{\jmath \omega_0t}+e^{-\jmath \omega_0 t}}{2} $$
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.
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.
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