No apologies necessary on your part. My apologies for not being clear.

My opinion is that there are circumstances in which negative frequencies exist just not in a form or perception that we are capable of. Your example of the dark energy hypothesis I see as supporting this statement or at least how I am thinking.

 

My opinion is that there are circumstances in which negative frequencies exist just not in a form or perception that we are capable of.

djsfantasi, may I ask you if you have any reason for this opinion or is it rather something like a "feeling" without any real background?

In general, of course I agree that there may exist many natural phenomenons we cannot percept because of the limited capabilities of human beings - but in this specific case? Remember, a "frequency" is not a natural phenomenon but a term that was defined to describe a periodic sequence based on a running time variable.

In case something similar exist with a negative sign anywhere in the world around us - perhaps we should find another name for it.

 

Are you sure you know how it was defined? There are lots of things that, at first glance, seem that they were defined one way when in reality they weren't defined that way at all, but rather we tend to assume that the original definition as we first learned about it must have been the original definition altogether. Often the working definition we are familiar with is actually a special case of a broader and more general definition. Also, definitions do change and evolve over time, often as a consequence of broader and more general definitions being discovered/devised. I'm not saying that is or isn't the case here, but just that it's something to keep in mind.

 

Hi WBahn - yes, I totally agree with you.
My contribution regarding definition of the term "frequency" is based on my present knowledge, of course. If you - or any other contributor to this thread - know that is a
special case of a "broader and more general definition" - correct me, please.
It was my primary intention to initiate an answer from djsfantasi .

 

Actually, I think there is a broader and more general definition, namely that frequency is the time-rate of change of phase.

You pretty much have to have a broader definition to be able to describe signals that are anything other than pure, fixed sinusoids. As soon as they become dynamic then the notion of frequency being the number of cycles per second becomes very ambiguous, but the notion of frequency being the time-rate change of phase does not.

An example where this becomes very apparent is simple FM modulation.

 

Since we are referring to signals and electronics I can assume that frequency is derived from the formula ω =2πf, where ω is angular velocity.

In physics and mathematics, the model used is that of a point moving counter-clockwise with angular velocity ω along the circumference of a circle, as on an x-y plot in Cartesian coordinates. The same point moving in a clockwise direction would have an angular velocity of -ω.

This is consistent with the concept of time-rate of change of phase.

 

Actually, I think there is a broader and more general definition, namely that frequency is the time-rate of change of phase.

You pretty much have to have a broader definition to be able to describe signals that are anything other than pure, fixed sinusoids. As soon as they become dynamic then the notion of frequency being the number of cycles per second becomes very ambiguous, but the notion of frequency being the time-rate change of phase does not.

An example where this becomes very apparent is simple FM modulation.

WBahn, I don`t know if it makes sense to go deeper into this particular point - however, I am not quite sure if the expression w=d(phi)/dt can be regarded as a "definition" in its original sense.
This expression (in bold) must be used for all cases where phi can be an arbitrary function of time. As mentioned by you, this is the case for an FM modulated signal.
However, according to my knowledge this expression for the momentary angular frequency w is mathematically derived using the definition for frequency f=1/T (T=period).

On the other hand, my primary interest is still if we can come to a common agreement regarding the question "do signals with a negative frequency exist in the real world"? I say: No.

 

LvW - I didn't know if I was going to post this (I cancelled my last three attempts) because I found myself taking too personally your characterization of my posts as a 'feeling' with no background. I have a degree in Applied Mathematics.

My original post was simply philosophical. If you drive your car backward, are you traveling with negative velocity? Perhaps. If you overdraw you bank account, is there a negative balance? Are these not real? To the bank, the latter is real because you owe them real money.

As several others have pointed out in earlier posts, the concept of negative frequencies is used in some calculations (ok, so here I don't have the background but am trusting the others). Also, based on the definition you presented, negative frequencies would exist in negative time. Not time travel, but negative time. Mathematically, that makes sense to me.

In researching my response, I came across a reference to Brian Greene's book, "The Fabric of the Cosmos", which devotes much of the book to string theory, the duality of light as a wave/particle (from the famous double slit experiment of the 1800s) and negative time.

I am a Mathematician, not a Physicist nor an Engineer. (There are many jokes used to illustrate the difference in thinking styles amongst these groups.) But the possibility of negative frequencies in one of the 7 or 8 additional quantum dimensions, I am not dismissing.

Perhaps our differences do not lie in the definition of frequency, but referring to your summarization of the question, it lies in the definition of the "real world". You asked, "Do signals with a negative frequency exist in the real world." If I have to refer to a quantum universe, that reference is different (a superset) from the 3D plus time universe that we perceive, then maybe we do not disagree.

 

Hi djsfantasi, thanks for your answer and the clarification.
However, I think I also have something to clarify.

When you read my post#42 again you will notice that I didn`t characterize your post as a "feeling". You have mentioned that it was your "opinion" that there might be "circumstances" in which negative frequencies exist - and my only question was if you can give reasons for this opinion or if it was more or less a feeling.
To me, your statement was a bit "vague" - and I have asked you for clarification.
May be I have expressed myself not good enough (due to limited knowledge of the language we have to use for communication).
In no case I wanted to become "personally". If this was your impression - please excuse me.
Now to the subject of our discussion. I agree with you that it seems we are approaching the problem under discussion from different sides (with different sights).
Regarding the "real world":
When I (as an engineer) ask if a certain quantity does exist in the real electronic world, I mean if it can verified simply by measurements.
With respect to the term under discussion, this means:
* Is a negative value for the inverse of a period in accordance with the agreed definition of the term "frequency"?
* Can we produce a signal with a negative frequency? How?
* Can we proof by measurement the existence of a signal with a negative frequency?

Of course, I appreciate that you - with a degree in Mathematics - will have another (and broader) view and I hope you can understand my (perhaps limited) view.
Thank you
LvW

 

LvW, I think I now understand your point of view (as an engineer as you claim to be).
I have two degrees, one in engineering and another in Physics.

Engineers work with the "real" world. Engineers are trained with a given set of tools and they work with these tools. Hence they have the tendency to think and work within the box to which they have been accustomed.

The point I have made twice in this thread is that for certain people there are certain things that they can comprehend and things which they cannot comprehend. If the human mind is not able to provide an explanation for a certain concept and there is a lack of "real" evidence then it could not exist , from their perspective.

No point in arguing any further. I am willing to leave the discussion at that point.

 

WBahn, I don`t know if it makes sense to go deeper into this particular point - however, I am not quite sure if the expression w=d(phi)/dt can be regarded as a "definition" in its original sense.

Insisting that something's definition has to be regarded in its original sense is a pretty thorny road. The original definition of the yard was based on the inch being "three grains of barley, dry and round". The meter was in terms of the distance from the equator to the north pole with the distance estimated by measuring the distance between a particular belfry in Dunkirk and a castle in Barcelona. Does that mean that the expression that 1yd = 0.9144m or that 1m is the distance traveled by light in a vacuum in 1/299,792,458 if a second can't be regarded as definitions?

This expression (in bold) must be used for all cases where phi can be an arbitrary function of time. As mentioned by you, this is the case for an FM modulated signal.
However, according to my knowledge this expression for the momentary angular frequency w is mathematically derived using the definition for frequency f=1/T (T=period).

Actually, you have it backwards. When you want the momentary (a.k.a., instantaneous) angular frequency you pretty much have to use the derivative because just in saying that you want the frequency at a particular moment in time means that it shouldn't depend on what the value of the waveform is going to be at some point in the future. But to use the "definition" involving the period, T, you have to know when the period in which the present moment is located is going to end in the future. You can only do that (or at least claim with some confidence the ability to do that) in the special case where the derivative is a constant (i.e., fixed frequency).

On the other hand, my primary interest is still if we can come to a common agreement regarding the question "do signals with a negative frequency exist in the real world"? I say: No.

And it would appear we can't, come to common agreement that is. I say, "Yes," because the very notion of "frequency" is mathematical in nature and a means of describing real phenomena and there are too many real phenomena that can be well-described by the use of negative frequencies.

So we will have to agree to disagree.

 

Are you sure you know how it was defined? There are lots of things that, at first glance, seem that they were defined one way when in reality they weren't defined that way at all, but rather we tend to assume that the original definition as we first learned about it must have been the original definition altogether. Often the working definition we are familiar with is actually a special case of a broader and more general definition. Also, definitions do change and evolve over time, often as a consequence of broader and more general definitions being discovered/devised. I'm not saying that is or isn't the case here, but just that it's something to keep in mind.

That's a good example of saying something that is generally true, but has little relevance to how true your base point is.

Actually, I think there is a broader and more general definition, namely that frequency is the time-rate of change of phase. ...

Total BS. You can't use a specific situation of one element from a subset to prove a "broad definition" of the entire set!

"Frequency" is derived from one concept; how "frequent" an event is. The units are Hz, which is defined as "events per second".

If someone knocks on your door 3 times a second that is more frequent and at a higher frequency than if they knocked 2 times a second. No sinewaves are needed.

If you actually want "a broader and more general definition" then events per second it is. Phase and waveshape are quite irrelevant to frequency in the vast majority of cases. ie; Someone can knock on your door at a frequency of 3 times per second, and it will remain 3Hz no matter what the phase or shape of their knocking is.

 

That's a good example of saying something that is generally true, but has little relevance to how true your base point is.



Total BS. You can't use a specific situation of one element from a subset to prove a "broad definition" of the entire set!

"Frequency" is derived from one concept; how "frequent" an event is. The units are Hz, which is defined as "events per second".

If someone knocks on your door 3 times a second that is more frequent and at a higher frequency than if they knocked 2 times a second. No sinewaves are needed.

If you actually want "a broader and more general definition" then events per second it is. Phase and waveshape are quite irrelevant to frequency in the vast majority of cases. ie; Someone can knock on your door at a frequency of 3 times per second, and it will remain 3Hz no matter what the phase or shape of their knocking is.

I've already said that we are going to have to agree to disagree -- so could we just do that?

Or are you going to continue to insist that everyone must accept the only interpretation that you want to latch onto by continuing to use special case examples (someone knocking on your door at a fixed, unchanging frequency of 3 times per second) as the basis for your definition of frequency? What if the person is knocking at your door slowly at first and then gets quicker and then gets slower? What is the frequency at which he is knocking on your door then?

 

Hi WBahn, probably you are right that we must come to the conclusion that we "agree to disagree". But why?
From my (engineeering) point of view this is a rather unsatisfying situation, because the term "frequency" is a technical parameter and, thus, it should be possible to answer two simple questions:
* Can signals with a negative frequency generated? How?
* How can negative frequencies verified by measurements? (In this context, I don`t think that a signal analyzer with a computer onboard - realizing Euler´s formula - would be a good answer).

Up to now, I didn`t get any answer to these questions.

Finally, two comments:

Quote:Actually, you have it backwards. When you want the momentary (a.k.a., instantaneous) angular frequency you pretty much have to use the derivative

Yes - I never have argued against this. That`s exactly what I have said in my former contribution. But WHY must we use the derivative? We have to DERIVE this rule. I think, it is not sufficient just to state that the derivative is needed to compute the instantaniuos frequency. We have to proove this!
And - as I have mentioned the corresponding mathematical derivation uses the definition f=1/T (T=Period)
Comment: T is the difference beween two discrete times and, thus, positive because time is continuously running in positive direction (even if due to shifting of the reference point both discrete times are negative, as claimed earlier in this thread).

Quote: the very notion of "frequency" is mathematical in nature and a means of describing real phenomena and there are too many real phenomena that can be well-described by the use of negative frequencies.

Fully agreed. In one of my former contributions I have mentioned the well-known fact that the introduction (introduction of a concept based on Euler`s formula!) of negative frequencies allows a simple description of some "real phenomena". No question about it.
But this does not answer the question if signals with negative frequencies really "exist" (can be produced and verified by measurements).
________
Finally, I confess that this is just my engineering view. People with another background (philosophy, mathematics, modern physics) may come to another conclusion (for example, based on a negative-time concept and n-dimensional space).

LvW

 

Hi WBahn, probably you are right that we must come to the conclusion that we "agree to disagree". But why?
From my (engineeering) point of view this is a rather unsatisfying situation, because the term "frequency" is a technical parameter and, thus, it should be possible to answer two simple questions:
* Can signals with a negative frequency generated? How?
* How can negative frequencies verified by measurements? (In this context, I don`t think that a signal analyzer with a computer onboard - realizing Euler´s formula - would be a good answer).

Up to now, I didn`t get any answer to these questions.

Part of the problem is that you seem to be insisting that everyone has to be in lock-step agreement on the definition of technical terms. Perhaps in a utopian world, but not in the real world. There are lots of technical terms in which different groups of people use the same terms and they mean very different things by them. Sadly, it is the case that sometimes the terms and meanings are close enough that when the two groups talk to each other they think they are using the same meanings and communicating ideas between each other but, in fact, aren't.

We ran headlong into this in my research at the Academy when it became apparent to us that we were working with two different communities, namely computer science and electrical engineering, that both used the term "key" in the context of secure communications and both communities often talked at each other using this term and not batting an eye when the other side used it. Yet it turns out that in the vast majority of cases they are meaning two very different things and it turns out that both sides are assuming that the other side has already solved a central part of a really big problem and that their part of the solution relies on the other side having done so.

Finally, two comments:

Quote:Actually, you have it backwards. When you want the momentary (a.k.a., instantaneous) angular frequency you pretty much have to use the derivative

Yes - I never have argued against this. That`s exactly what I have said in my former contribution.
[/QUOTE]

I went back and reread what you wrote and I agree with you. I was reading your response a bit too fast and thought you were saying something different from what you said. My apologies.

But WHY must we use the derivative? We have to DERIVE this rule. I think, it is not sufficient just to state that the derivative is needed to compute the instantaniuos frequency. We have to proove this!

Okay. While this probably doesn't qualify as a rigorous proof (how do you prove a definition?) I think it is reasonably compelling and is very simple.

When we say we have a sinusoidally varying voltage, why do we use the term "sinusoidal"? It would seem that we are taking the sine of something, right? Well, what is it that we are taking the sine of? It is NOT time! The trigonometric functions are transcendental functions and, as such, require dimensionless arguments. In the case of the trig functions, the argument is in radians (which is distance/distance or some other quantity related to the circumference of a circular arc and the radius of that arc) and, hence, dimensionless as required.

Thus, if we have a sinusoidal voltage we have a voltage that is related to the sine of a real angle in which that angle varies with time.

v(t) = Vm sin(θ(t))

Just as the time-derivative of position is a sufficiently useful quantity that we give it a name (namely velocity) and it has units of distance/time, so it is that the time-derivative of angle is a sufficiently useful quantity and we give it a name (namely angular velocity or, in other contexts, angular frequency) and it has units of 1/time.

As is often the case when it is helpful to distinguish average and instantaneous measures, we use the qualifiers average and instantaneous. S the instantaneous frequency is

ω(t) = dθ(t)/dt

In the special case that voltage varies linearly with time, then

θ(t)=kt

and

ω(t) = k

This, in turn, allows us to write, for this very special case,

θ(t)=k*t

ω(t) = dθ(t)/dt = k = ω

Note the distinction between ω(t) and just ω, which is a constant.

θ(t) = ωt

v(t) = Vm sin(ωt)

Take note of two things:

1) The ω we all know, love, and use without thinking twice comes as a direct result of using a general definition of angular frequency and applying it to a special case.

2) We have placed no constraint on the value of ω (other than that it be real to satisfy the constraint that the angle be real). Specifically, we haven't said that it has to be positive.

So what if it is negative? Does that all of a sudden mean that we have a signal that is not physically realizable?

Furthermore, note that this is NOT a general result, but only applies to the case of a signal in which the angle is changing linearly with time. Specifically, it is NOT true that

v(t) = Vm sin(ωt)

generalized to

v(t) = Vm sin(ω(t)*t)

when ω(t) is NOT a constant.

In other words, the general definition reduces to the special case, but the special case does not generalize back simply by noting that there is a time variation of the frequency.

And - as I have mentioned the corresponding mathematical derivation uses the definition f=1/T (T=Period)

Okay. What's good for the goose and all. You have "mentioned" it. That doesn't make it so. So prove that the corresponding mathematical derivation of instantaneous frequency being the time-derivative of the angle uses the definition f=1/T.

Comment: T is the difference beween two discrete times and, thus, positive because time is continuously running in positive direction (even if due to shifting of the reference point both discrete times are negative, as claimed earlier in this thread).

That's a matter of perspective, but I won't argue it.

Quote: the very notion of "frequency" is mathematical in nature and a means of describing real phenomena and there are too many real phenomena that can be well-described by the use of negative frequencies.

Fully agreed. In one of my former contributions I have mentioned the well-known fact that the introduction (introduction of a concept based on Euler`s formula!) of negative frequencies allows a simple description of some "real phenomena". No question about it.
But this does not answer the question if signals with negative frequencies really "exist" (can be produced and verified by measurements).
[/QUOTE]

I've already given such an example. If you IQ modulate a baseband signal onto a carrier and your baseband signal has a positive frequency it will move the carrier up in frequency while if it has a negative frequency it will move it down. The only difference between the two signals is the sign of the frequency. I can receive the signal and IQ demodulate it and tell you if the baseband signal was +1kHz or -1kHz.

The reason is that the IQ signals that are generated from the baseband signal are in quadrature relationship to the baseband signal that thus are sensitive to the difference between positive and negative frequencies.

Finally, I confess that this is just my engineering view. People with another background (philosophy, mathematics, modern physics) may come to another conclusion based on a negative-time concept and n-dimensional space.

I don't think that you have to base things on negative time or n-dimensional space or general relatively or string theory or anything similar.

 

part of the problem is that you seem to be insisting that everyone has to be in lock-step agreement on the definition of technical terms.

I do not „insist“ on a definition, but I rather assume that we are using a commonly accepted definition. What should I do instead of? In case I am wrong it is necessary to correct me. I think, that is the only possible way to discuss with each other on a common technical basis.

Take note of two things:
1) The ω we all know, love, and use without thinking twice comes as a direct result of using a general definition of angular frequency and applying it to a special case.

2) We have placed no constraint on the value of ω (other than that it be real to satisfy the constraint that the angle be real). Specifically, we haven't said that it has to be positive.
So what if it is negative? Does that all of a sudden mean that we have a signal that is not physically realizable?

You have formulated a question which follows directly from your calculations (and, thus, is logical). But what is your answer?

May I give you my answer? (Sorry, it is not a short one):

Of course I can fully agree to all your calculation as well as your conclusions - in particular, that a special case must not be generalized. This also applies to many other examples.

You have started with the expression ω(t) = dθ(t)/dt
which for the special case θ(t)=k*t
leads to the final result
v(t) = Vm sin(k*t).

It is true, that - up to now - no constraint regarding the sign of k was necessary.
Note that - up to now - I didn`t use the term „frequency“ (although you already have used the notation k=ω).

In summary: Starting point for your calculation was the time derivative of an angle, and for the special case of a rising phase (linear with time) you arrived at the classical sinusoidal signal description. Only now, we realize that the constant k is identical to the angular frequency ω.
This confirms the correctness of the start expression - which, therefore, also can serve as a definition for the term „frequency“.
Now - this „new“ definition must be in accordance with the „old“ definition which undoubtly is f=1/T. Otherwise, we must find another new name for the constant k (why not?).
But when we try to cope with the „old“ definition and set k=ω we must at the same time require that k=ω>0 (because f=1/T>0).

So prove that the corresponding mathematical derivation of instantaneous frequency being the time-derivative of the angle uses the definition f=1/T.

OK - I can do it. I will create a pdf-attachement within the next days.

If you IQ modulate a baseband signal onto a carrier and your baseband signal has a positive frequency it will move the carrier up in frequency while if it has a negative frequency it will move it down.

...if it has a negative frequency...?
But this is the main question: Where does the baseband signal with a negative frequency comes from?

 

"Frequency" is derived from one concept; how "frequent" an event is. The units are Hz, which is defined as "events per second".

If someone knocks on your door 3 times a second that is more frequent and at a higher frequency than if they knocked 2 times a second.

That's pretty much how I see it. I abhor the idea that there can be a negative frequency, except as a useful mathematical "trick" that we tolerate while we look for the "real" answer. Like i.

But to be my own devil's advocate, suppose your door contains a sophisticated noise canceling technology that can mitigate any sound from one knock every second. So a visitor knocks 3 times and the home owner hears only 2. I think you could argue that the door is producing knocks at -1Hz.

Our brains prefer to think of the door producing "anti-knocks" at a positive frequency of 1 Hz. But I think that says more about our brains than it does about the phenomena that are occurring.

 

I do not „insist“ on a definition, but I rather assume that we are using a commonly accepted definition. What should I do instead of? In case I am wrong it is necessary to correct me. I think, that is the only possible way to discuss with each other on a common technical basis.

I didn't mean that you are insisting on a particular definition, but rather that you are insisting (assuming is probably a better word) that one exists. It's a natural and reasonable expectation. Alas, but were it the case. Certainly there are some things that there is wider agreement on than others. Just as certainly, the notion of negative frequency is not among them.

In summary: Starting point for your calculation was the time derivative of an angle

Actually, the starting point for my calculation what the argument of the sine function that is being used to describe something that is "sinusoidal".

This confirms the correctness of the start expression - which, therefore, also can serve as a definition for the term „frequency“.
Now - this „new“ definition must be in accordance with the „old“ definition which undoubtly is f=1/T. Otherwise, we must find another new name for the constant k (why not?).
But when we try to cope with the „old“ definition and set k=ω we must at the same time require that k=ω>0 (because f=1/T>0).

Why? Let's consider something else that has this same flavor. Now, I'm just guessing, but I'd be willing to bet that the original definition of an "exponent" went something like this: The value x with an exponent of n means that x is multiplied by itself n times. Hence, 9^3 means 9x9x9 which is 729.

At some point someone noted that this could be generalized by "filling in the gaps" along the curve connecting, for instance, 5^2, 5^3, 5^4 and so on. Along the way they leant meaning to 9^2.5, which is 243. Does this mean that we can't call 2.5 the exponent since it is not "in accord" with the original definition that the exponent is the number of times a number is multiplied by itself? Or can we call this new concept an exponent and accept that we have genralized the notion of an exponent to a broader meaning and, in doing so, have rendered the original definition to the staus of a special case.

OK - I can do it. I will create a pdf-attachement within the next days.

Looking forward to seeing it. I'm sure it can be done, but it seems a bit tricky to me.

...if it has a negative frequency...?
But this is the main question: Where does the baseband signal with a negative frequency comes from?

It is negative relative to the frequency of the IQ modulator's mixing frequency. (I hope I've got that right... it's been a couple years since I've looked at this stuff).

 

...
But to be my own devil's advocate, suppose your door contains a sophisticated noise canceling technology that can mitigate any sound from one knock every second. So a visitor knocks 3 times and the home owner hears only 2. I think you could argue that the door is producing knocks at -1Hz.
...

So your point is that adding a negative frequency to a larger total reduces the total? That's a relative argument, which is the problem I have with some people's blind acceptance of "negative frequency".

What if instead of adding a negative 1Hz to the door, you are just subtracting a positive value of 1Hz from a larger positive value of 3Hz? The process is a subtraction, but that is not proof that a real world negative value exists, any more than "T - 5 minutes" proves that negative time exists.

Here are the facts that have surfaced so far;
1. relative negative frequency exists
2. negative frequency doesn't exist
(also relevant for time, distance etc)

 

Here are the facts that have surfaced so far;
1. relative negative frequency exists
2. negative frequency doesn't exist
(also relevant for time, distance etc)

The how is it that I can take a signal at a frequency Fm and park it at an offset relative to a carrier frequency, Fc+Fm, and I get different results depending on if Fm is positive or negative? Specifically, if Fm is positive I get a result that is at a higher frequency than the carrier and if Fm is negative I get a result that is at a lower frequency than the carrier?

Those are "facts" only because you have declared them to be so. Fine. If that is the worldview you want to constrain yourself to, that's fine. But you don't get to constrain the rest of us to it.