Negative frequencies are just a mathematical construct to allow us to analyse real signals using a complex number framework, which is used when looking at double-sided spectra. A complex number can only be made real if you add to it its conjugate, e.g. (a+bj) + (a-bj) = 2a. Hence a real cosinusoid can be represented using complex exponentials via the sum of e^(jωt) and its complex conjugate e^(-jωt). Hence this is where the negative frequency comes from. Of course, ωt represents the phase angle in radians varying over time, so -ωt means the phase angle of the complex number is decreasing (clockwise rotation) with respect to time.

If I remember well, this was already discussed in RG several weeks ago.

But your wording is correct: It is just a "concept" - nothing else.

The fictitious extension of the term "frequency" to negative values has some advantages (e.g. mathematical description of the spectral content of a signal), but this does not mean that negative frequencies physically "exist" (their existence cannot be verfied by measurements).

It is easier to write e^jwT than the combination of sines and cosines. Well, as w= 2*pi*f and T= 1/Fsam then you have e^(j*2*pi*f)/Fsam, where f can go from 0 to Fsam (so your angular argument goes from 0 to 2*pi). In discrete systems positive and negative frequencies are needed to write the sine and cosine ( cos(theta)=(e^theta+e^-theta)/2 and sin(theta)=(e^theta-e^-theta)/2j ). So, instead of dealing with frequencies from 0 to Fsam one deals with frequencies -Fsam/2 to +Fsam/2, which physically represent frequencies 0 to Fsam/2, of course. Is this enough?

Negative frequency has a perfectly valid physical description. Imagine a wheel spinning clockwise (as opposed to counterclockwise).

That is, you can have either "forward" cycles per second or "reverse" cycles per second, at a given rate.

https://en.wikipedia.org/wiki/Negative_frequency

@Dr. Bryce Ready,

Sorry, but I don`t agree.

There is a clear definition of the term "frequency": Number of events per time unit.

(For my opinion, originally defined for sinusoidal events only, in contrast to "repetiton rate").

And the time is counted as positive, does it not?

Is there anybody who has verified negative frequency by measurements?

But, of course, we are free to define a new variable - based on the spinning wheel - containing the direction as an additional information. But this should not be called "frequency".

Renoh, u r very much correct frequency by its definition can never be negative. Now, frequency has 2 different definitions. One is that, you mentioned. The other one is that obtained from Fourier series or Fourier Transform. In FT you consider frequency interval of -infinity ton+infinity for ease of integration. These gives the frequency spectra in the negative domain. This is an image and should be ignored for all practical applications.

Dear Mala Mitra

The spectrum for negative values of f is only a mirror image of the spectrum for positive frequencies if the signal in time domain is real.

Yes - and the amplitude of both -negative and positive lines - are only half if compared with the on-sided (positive) spectrum.

@Lutz von Wangenheim

There's nothing wrong with your definition of frequency, but it's a definition. It has its uses, and it has its drawbacks. Much like numbers, or vectors, or any other mathematical construct.

There is similarly nothing wrong with the definition I'm talking about. My definition is perfectly physically verifyable: if I turn my clock one way at a certain rate, versus the other way at the same rate, I can observe positive and negative frequency, and tell the difference between them (presuming I'm sampling fast enough to avoid degenerate cases like having only a single sample per revolution). This is physically meaningful, and it usefully applies to any number of practical systems.

Frankly, what I think we're running into here is a difference in the way the term is used across disciplines/applications, and thus this back and forth is probably valuable to the OP and anybody else looking at this question: there are certainly applications where negative frequency makes no physical sense. My point is that there are applications where the concept of negative frequency has been in wide use for many years, and is perfectly valid.

Quote 1: "There's nothing wrong with your definition of frequency."

Quote 2: "There is similarly nothing wrong with the definition I'm talking about."

For my opinion, a definition never can be "wrong". It can be helpful - or not.

On the other hand, we cannot have two different definitions for the same quantity.

Thefefore my suggestion to use another word for the "number of events with respect to their directions".

Quote 3: "My point is that there are applications where the concept of negative frequency has been in wide use for many years, and is perfectly valid."

Did I argue against the "concept" of negative frequencies and it`s use? No, in contrary. I have mentioned the advantages conncted with the introduction of neg. frequencies.

But: I think, that it is a concept - a mathematical construct.

More than that, I think now it is necessary to be very exact (pedantic):

A "frequency" never can "exist" in a physical sense.

In reality, we have a periodic event (voltage, current, or something else) which does "exist" and which can be described by some characteristic properties: Amplitude, shape and - FREQUENCY. This is the DEFINITION of the term under discussion (cycles per second).

Thus, the frequency is a pure number without any direction or any other attribute (sign).

@Lutz von Wangenheim

First, I should say thank you for this ongoing discussion. As I mentioned before, i think that this philosophical discussion adds value to this thread, and deepens the answer to the OPs original question.

I certainly did not mean to imply that any definition can be wrong: as you correctly point out, it's all about utility.

I do take issue with this statement:

Quote: "On the other hand, there cannot exist two different definitions for the same quantity."

That is (alas!) not so! Unfortunately, it's not even the case in mathematical treatises ("abuse of notation" anyone?) Just like any other term in human language, "frequency" can in fact have many definitions. You emphasize the "definition" of the term frequency, but my point is that your choice of that particular definition is arbitrary.

It seems to me (correct me if I'm wrong) that you are giving your opinion that the community should coin a different term for what I'll call "2-d frequency" to distinguish it from regular frequency. I happen to disagree with that opinion, but beyond that, the fact is that the community does not do this currently (hence the OP's question), and there are many people who use the term frequency in the 2-d sense I have discussed. Whether you or I or anyone thinks this is a good idea, the questioner should at least be made aware of this fact.

Beyond that, my bigger point (and I think the real source of this disagreement) is this:

Quote: "Is there anybody who has verified negative frequency by measurements? "

Perhaps I am missing some subtlety, but it seems to me like the physical verification is trivial: I could, for instance, take a series of clocks, some spinning clockwise and some counterclockwise, and have a human (or a computer with an attached camera for that matter) verify that there is a difference, and categorize clockwise and counterclockwise spins with 100% accuracy.

Now if I'm looking at a sine wave, I agree that no "positiveness" or "negativeness" of its frequency can be physically verified (though if I had the corresponding cosine wave it could be). The fact that one meaning of "frequency" relies on a two dimensional construct and thus needs two dimensional data to verify it does not make it unverifyable. If I'm missing some subtlety here that you could clarify, I think the OP and I would both be grateful.

My point is that there are multiple ways to talk about "frequency": one is simply a matter of cycles per second - a one dimensional construct. The other requires a two dimensional construct, and thus two dimensional data to observe it.

As you originally and correctly pointed out, the 2-d definition is very useful. This is even true in 1-d situations, in which situations I agree that it is just a construct with no provable bearing on reality (I cannot with any justification say that *every* sine wave is the projection of some phasor somewhere). My only objection was this statement in your first post: (Quote) "this does not mean that negative frequencies physically "exist" (their existence cannot be verfied by measurements)."

Yes, concept wise we can define negative frequency 2-D definition) but frequency is uses an absolute value. The definition implies that how many times it completes a cycle in one second; so where the positive or negative is involved here.

Quote B. Ready: "That is (alas!) not so! Unfortunately, it's not even the case in mathematical treatises ("abuse of notation" anyone?) Just like any other term in human language, "frequency" can in fact have many definitions. You emphasize the "definition" of the term frequency, but my point is that your choice of that particular definition is arbitrary."

Yes - you are right. I must confess that - as somebody with an engineering education in the field of electronics - my view is somewhat restricted. That means: By using the term „frequency“ always and immediately I think of a sinusoidal voltage or current or electro-magnetic wave.

However, the situation may change taking into account mathematical or even philosophical viewpoints.

Quote B. Ready: "It seems to me (correct me if I'm wrong) that you are giving your opinion that the community should coin a different term for what I'll call "2-d frequency" to distinguish it from regular frequency. I happen to disagree with that opinion, but beyond that, the fact is that the community does not do this currently (hence the OP's question), and there are many people who use the term frequency in the 2-d sense I have discussed. Whether you or I or anyone thinks this is a good idea, the questioner should at least be made aware of this fact."

Also agreed. However, by using the 2-d term „negative frequency“ I hope that the community at the same time is aware (as I do) that this is a „mathematical construct“ only.

Quote B. Ready: "Perhaps I am missing some subtlety, but it seems to me like the physical verification is trivial: I could, for instance, take a series of clocks, some spinning clockwise and some counterclockwise, and have a human (or a computer with an attached camera for that matter) verify that there is a difference.."

Is this really a „physical verification“ for the existence of negative frequencies? Of course, we can observe two different rotations (clock- and counterclock-wise). And if I only count the events per second (without regard to the direction) I arrive at a pure number without sign. This is in full agreement with the classical definition for „frequency“, is it not?

For my opinion, in your example we simply observe two clocks spinning in different directions. That`s all - and that`s sufficient to describe the properties of this configuration.

Quote B. Ready: "My point is that there are multiple ways to talk about "frequency": one is simply a matter of cycles per second - a one dimensional construct. The other requires a two dimensional construct, and thus two dimensional data to observe it. "

Yes - full agreement, in particular with the expression „two dimensional construct“. My interpretation: It is a „construct“ which makes sense, but cannot be observed as a property of a real existing physical process.

There is another nice example - rather close related to our discussion: The introduction of a complex frequency s=sigma+jw.

For example, calculating the pole distribution for an ideal oscillating system we arrive at a conjugate-complex pair which gives the solution in the time domain (frequency of the oscillating output signal). However, only the positive part can be verified by measurements.

I must confess that I am not sure if the original question has been answered sufficiently. Thus, I like to summarize the main points (from my engineering view):

* A periodic signal of any shape can be described using the FOURIER series which contains sinus and cosinus expressions.

A much more simplified method to write the FOURIER series is based on the well-known EULER form for sinusoidal signals:

exp(jwt)=cos(wt)+j*sin(wt).

* This allows to write:

sin(wt)=Qp/2j - Qn/2j and cos(wt)=Qp/2j + Qn/2j

with Qp=exp(jwt) and Qn=exp(-jwt).

This method to write the FOURIER series in complex form was the background to speak about positive and negative frequencies. However, to circumvent these terms, which might cause problems to understand the meaning of negative frequencies, we can avoid the word "frequency" in this context.

Instead, we can speak about positive and negative PHASORS (Qp and Qn, respectively). Of course, both phasors rotate in different directions - however, both with the same frequency (which still is a pure number).

As a consequence, the two-sided phasor spectrum which describes the harmonic content of a periodic signal has positive and negative portions - each with an amplitude that is 50% of the one-sided spectrum (based on conventional sinusoidal expressions). I hope this partly answers the question of this thread.

Ya that makes some sense Mr.Lutz von Wangenheim. The concept of phasors must be applied here right? The two phasors rotating in two opposite directions?

Yes - that´s what I have tried to explain in the above contribution.

The concept of negative frequencies is intimately related to complex numbers. In the "real" world, you don't need negative frequencies (because, as mentioned before, a real signal always consists of equal parts at positive and negative frequencies, for example, the sine is a difference of two exponentials with a positive and a negative frequency with the same absolute value). Therefore, the question whether a negative frequency is measurable is immediately answered negatively by the fact that we cannot measure imaginary parts. Therefore, you can think of both, imaginary parts and negative frequencies, as mathematical construct, if you wish.

Interestingly enough, a signal that has no contributions at negative frequencies is not real but analytic in the time domain, i.e., for each sine in the real part, there is a cosine in the imaginary part (the imaginary part is the Hilbert transform of the real part).

Negative frequencies are just a mathematical construct to allow us to analyse real signals using a complex number framework, which is used when looking at double-sided spectra. A complex number can only be made real if you add to it its conjugate, e.g. (a+bj) + (a-bj) = 2a. Hence a real cosinusoid can be represented using complex exponentials via the sum of e^(jωt) and its complex conjugate e^(-jωt). Hence this is where the negative frequency comes from. Of course, ωt represents the phase angle in radians varying over time, so -ωt means the phase angle of the complex number is decreasing (clockwise rotation) with respect to time.

Dealing with rotors, the question can be seemingly enlightened by considering Noether´s theorem. Simply stated, it asserts that every continuous symmetry of the dynamical behaviour of a system implies a conservation law for that system. Frequency actually translates the dynamical equation of the rotor, meaning that its change of sign can be considered if associated to some adequate conservation law. By considering rotation around the z-axis, rotation sense changes upon reflection on the x-y plane, distinguishing clockwise from anticlockwise rotation. One conveniently assign different signs to both senses, and correspondingly, to the frequency. The mechanical energy of the rotor does not depend on the rotation sense, what gives some support to the physical concept of negative frequency.

Quote C. Queiroz: "One conveniently assign different signs to both senses, and correspondingly, to the frequency."

To me, this is not logical (..."correspondingly").

There is a definition for PHASORS Qp and Qn (positive and negative)

Qp=exp(jwt) and Qn=exp(-jwt)

rotating in different directions.

Here, the negative sign carries the direction information.

If we would introduce, in addition, a negative value for w in the expression for Qn, we have again Qp=Qn.

Dear Lutz von Wangenheim:

Please note that I was referring to circular motion frequency; not to signal frequency. As a rotor can be the source of a signal, motion frequency can match signal frequency. However, they are not conceptually equivalent.

For the rotation of a rotor in 3-D we have a fixed axis, which is the axis of rotation. The plane of rotation is orthogonal to this axis. If the rotation takes place around z-axis, the plane of rotation is the xy-plane, implying that everything in that plane it kept there.

A phasor represents sinusoidal signal in the complex plane. For a real periodic signal to be represented, every positive frequency complex sinusoid considered should be summed with a negative frequency sinusoid of equal amplitude, so that the imaginary parts cancel each other, rendering just the real signal. We can, perhaps, better understand this by saying that counterclockwise circular motion of a phasor should be accompanied by equal and opposite clockwise circular motion.

Yes - agreed.

However, I think the question was related to the "classical" frequency of a signal (in Hz). But I think, everything has been answered now. Thank you.

Just modest contribution to this discussion. There is nothing called negative frequencies in real life. They just meant to be a mathematical expansion of Fourier transform in the two sided frequency spectrum. Their contribution in reality is concerned with their energy content only, which simply interpreted as doubling the amplitudes of positive frequency spectrum side. Their bandwidth also to be considered in case of frequency shift or modulation in pass band systems .

The concept of negative and positive frequency can be as simple as a wheel rotating one way or the other way. A signed value of frequency can indicate both the rate and direction of rotation. The rate is expressed in units such as revolutions (aka cycles) per second (hertz) or radian/second (where 1 cycle corresponds to 2π radians)

For sinusoids:

Similar to the rotating vector (Fig 1), the complex-valued function:

exhibits different properties for positive and negative values of the parameter ω. When ω > 0, R(t) appears to lead I(t) by ¼ cycle (= π/2 radians). When ω < 0, the roles are reversed. Fig 2 depicts a negative frequency. R(t) and I(t) are referred to as real and imaginary, respectively. The more-familiar real-valued sinusoid:

is an analytically more-complicated function than because it comprises both a negative and positive frequency component. In fact, is sometimes called the analytic representation of When working with individual, real-valued sinusoids, a negative frequency has an equivalent positive-frequency form; i.e. so it is often sufficient to consider every frequency a positive one.

-ωt or ωt are called phase shifts, and not negative or positive frequency.

Frequency Negative !!!!! In the calculation negative frequency comes for complex variable but can someone measure it in Spectrum Analyser or Oscilloscope ?

Disagreement in the last contributions comes from the fact that the term "frequency" is used as a concept of physics by some of the contributors, and as a concept of mathematics by some others. Make clear what you are talking about, then you will avoid misunderstandings.

Dear Ahmed Abdul Salam, the ω in -ωt or ωt is 2*pi*f, so -ω is a negative frequency. It comes from the choice of writing the FT in terms of the exponential form rather than the less elegant sine and cosine form.

Nishad T M wrote correctly that exponential terms with negative frequencies are necessary to balance those that exhibit positive frequencies. This is due to the fact that by Fourier series, real-valued signals are approached by sin- and cosine terms, which may be written as a superposition of two exponential terms.

However, I do not agree with his saying "Simply put, negative frequencies represent forward traveling waves, while positive frequencies represent backward traveling waves." For a description of waves, you need to talk about arguments like wt - k r and wt + k r, where k is the wave vector and where r is the poition vector and k r is their inner product. (w represents angular frequency, t represents time). In Fourier approaches of _oscillations_, the space argument does not appear. In Fourier series approaches of _waves_ , you need to balance exponential terms with argument wt - kr by other exponential terms with argument -wt + kr to get a wave travelling parallel to k-direction, while waves travelling antiparallel to k-direction are described by exponential terms with arguments wt + kr and -wt - kr that must be superimposed to end up in real-valued terms.

Sorry, but I cannot understand your comment. What do you mean by "wave propagation is not in the same approach"? It was you who talked about a wave traveling forward or backward. It is an essential criterion for a wave that it depends on a function whose argument depends on wt - kr. Don't you agree?

It may be worth to remember those interested at this topic that a related discussion was earlier initiated at this forum by Lutz von Wangenheim ─ «Do negative frequencies really exist?»: https://www.researchgate.net/post/Do_negative_frequencies_really_exist

As Stephen So well said, negative frequencies come to light when one uses the complex exponential form for a more elegant representation of periodic functions and Fourier transforms with cos (x) = (exp(jx)+exp(-jx))/2 and sin(x) = exp(jx)-exp(-jx)/2j.

....and - as properly formulated already in the original question:

It is a "concept" only without physical relevance.

("Frequencies " do not exist at all as a physical quantity. There exist only periodic events that can be counted, resulting in the definition of the term "frequency". Counting for positive times gives only positive numbers). 

Fernando Soares Schlindwein.

What asymmetry you will get for complex samples? Please give an example . Whether real or complex if the samples are symmetric in the time domain then FT will maintain symmetry. Symmetry in time domain is a sufficient condition and not a necessary condition. IFT will also be symmetric that should be the same as the original time domain sample.

Thanks..

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Dear Mala Mitra

The spectrum for negative values of f is only a mirror image of the spectrum for positive frequencies if the signal in time domain is real.

If frequency is defined as an absolute number then there can be no negative frequencies, but if it is a real number negative frequencies can exist. I made Doppler measurements and at the radar frequency a target moving towards the radar had a higher frequency than the transmitted radar signal (positive Doppler frequency) and a target moving away from the radar had a lower frequency (negative Doppler frequency). When mixed down to base-band (using a quadrature mixer, for instance) these two signals can be discriminated if the signal is in cos and sine (I and Q) channels depending on whether the sine leads or lags the cos. One is a positive frequency (target moving towards radar) and the other is a negative frequency (target moving away from the radar).This corresponds exactly to the complex representation of positive and negative frequency. You do need two signals to distinguish between positive and negative frequencies, the same as you need two numbers to define a complex number. A single wire always has equal amplitude positive and negative frequencies, there is only one number, so it can't describe a complex number.

Hello everyone ...... tried reading everyone'z opinion, and each is very much appreciated. My reply is more like refreshing the question. If there is no such thing as -ve frequency, then what happens in the amplitude modulation when due to the -ve half of the spectrum of the original signal, the required bandwidth is 2B when the original signal had a bandwidth B?

Hello everyone ...... tried reading everyone'z opinion, and each is very much appreciated. My reply is more like refreshing the question. If there is no such thing as negative frequency, then what happens in the amplitude modulation when due to the negative half of the spectrum of the original signal, the required bandwidth is 2.B when the original signal had a bandwidth B?

Negative frequency is real when it occurs in things like Doppler. The frequency actually drops when the source is going away from you. The change in frequency is negative. Negative frequency is also real in the description of AM and FM, where there are frequencies above and below the carrier frequency. If the signal is mixed down to baseband the difference between negative and positive frequencies disappears unless I and Q channels are used to get complex representation of the signal. If this is done then the difference between positive and negative frequencies really matters and means something (contains information) - you can still use this to find out if a source is moving towards you or away from you, so there really are negative frequencies.

In the last answer, replace the word "frequency" by "relative frequency shift". It is not frequency (interpreted as periods per second) that may be negative, it is the relative shift of frequencies in a spectrum.

Yes - I completely agree with the last contribution. A frequency reduction from 500 Hz to 450 Hz must not interpreted as a "new frequency" of -50 Hz.

Negative frequencies are never "real". They are nothing else than a mathematical tool for simplifying calculations ore visualizing spectral properties.

If you start off with 1 MHz, apply a Doppler signal of + or - 10 Hz to it, then mix down to baseband (with 1 MHz) then you will have a 10 Hz signal. If you use a quadrature mixer you will be able to tell whether this was from the Doppler signal at +10 Hz or -10 Hz. You are able to distinguish between positive and negative frequencies. The + or - 10 Hz is relative to the 1 MHz signal, but when it is mixed down it then stands in its own right as a positive or negative frequency, and can be applied to another carrier to appear as a positive or negative sideband on that carrier frequency. It is as real as the numbers used to represent it are real.

Malcolm, as you say: you mix two (or more) signals by a nonlinear process, then you filter the result (i.e. you downconvert the original signal). However, as you inevitably use a nonlinear process to perform the mixing process, you do no longer use the original signal and its frequencies. What you are doing is to interpret the downconverted signal. Mathematically spoken, you map the function representing the original signal by a (very complex) operation. Therefore, it is not frequency in a sense of periods per second, which you are interpreting. The number of periods is a natural number. The time interval (one second) is represented by a positive real number. Periods per second are represented by non-negative numbers, thus. This is different as compared to what you interpret after the down-conversion. There, you do not observe periods per second of the original signal, but a mathematical construct, being generated by an operator and mapped from the space of real-valued functions to another function space.

It is a one to one mapping of a negative or positive frequency difference to a negative or positive frequency. Semantically there may be a significant difference, but not practically. The frequency is not periods per second, but radians per second, and can be negative. The I and Q define a direction of rotation. It is very annoying getting it wrong on a radar (by having I and Q mixed up) and having targets going away rather than approaching.

I and Q define a vector that rotates on the IQ plane. The rate of rotation in radians per second is either positive or negative (or zero) and defines a positive or negative frequency.

If the problem is reduced/simplified to the question "Do negative frequencies exist in reality ?" - the answer must be "No".

At first, we must realize that "frequency" is nothing else than a property of a physical real existing quantity: It is simply the number of events (cycles) of a periodic (sinusoida) event (electrical, mechanical,...).

From this, it is obvious that it can be a positive number only.

That is the background of the definition for the term "frequency".

Hence, neither any "mapping" nor "I and Q signals" nor a "rotating vector" is invoved in the definition.

A "rotating vector" does not exist in reality - it serves only one purpose: It helps to visualize the term "angular frequency" and helps to explain the artificial concept of "negative" frequencies, which are introduced only to simplify the mathematical manipulations of sinusoidal signals.