Quote:
Originally Posted by WBahn
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.
___________________________________________________________


Here it comes (please, see attachement):



As mentioned several times in this thread, the angular frequency can be expressed as

wo=d(phi)/dt.

However, as I do not consider this expression as the original DEFINITION of the term „frequency“ the attached pdf file shows how this expression can be derived from the relation f=1/T , which I consider as the initial definition (T and f both being positive).


Please note that the derivation makes no use of the angular frequency wo and it`s visualization as a rotating pointer (vector). This is important when we speak about the sign of the frequency.
Instead, from the beginning an unknown parameter k is used (unit [1/s]).
At the end of the derivation it is shown that k is identical to a quantity which is commonly known as angular frequency wo.

Result: As the derivation is based on positive quantities „f“ and „T“ also the resulting expression wo=d(phi)/dt must be considerd as positive.
Of course, one can imagine that the direction of a rotating pointer (visualisation of wo) is reversed. However, this case is not in agreement with the initial condition w=2*Pi*f>0.

 

So your point is that adding a negative frequency to a larger total reduces the total?

Not really. I agree that "frequency" means a count of events in a time period. The units are the dimensionless count of the events, over time, or inverse time. The event must be defined, e.g. door knocks or zero-crossings.

And I also more-or-less agree that events don't un-happen so long as time runs forward. (And I believe it always does.) The lowest count of door knocks you can have is none, not some negative number. You cannot have negative door knocks unless you redefine the event you are counting to include an "un"-knock, for example as I alluded to previously.

The trouble is, in the door example, there are two observers with two non-identical definitions of the knock event. The knocker counts his knuckles hitting the door 3 times. The home owner hears only 2 from his easy-chair. A 3rd party scientist that has only the two pieces of data from the observers might say there is something happening at -1 Hz between the knocker and the listener. Once the scientist discovers the mechanism at work, he would instead report the active noise canceling as an event occurring at 1 Hz.

 

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?
...

Becasue they are RELATIVE to the carrier. Some are relatively negative (compared to the carrier) some relatively positive.

...
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.

Someone needs to be unbiased and simplify/summarise other people's points to something like a conclusion.

The points FOR negative frequency all rely heavily on relative frequencies or relative processes.

The points AGAINST negative frequency rely on the concept that frequency itself can only be >0, apart from LvW whose point is still unfolding (and seems to be a proof that eliminates the relative factor or redistributes it to be relatively posiitve).

 

Becasue they are RELATIVE to the carrier. Some are relatively negative (compared to the carrier) some relatively positive.

The carrier doesn't enter into it until you upconvert it. At baseband, you simply have a signal at a frequency that can be positive or negative. So how can you have distinguishable positive and negative frequencies BEFORE you introduce a carrier into the mix? If I gave you the baseband signal it might have a signal at +500Hz and one at -300Hz. Please describe that signal using the positive-only frequencies (i.e., remove the relative frequency stuff that you say is there).

There IS an answer to those questions that don't require the concept of negative frequencies, but you aren't even close to it.

Someone needs to be unbiased and simplify/summarise other people's points to something like a conclusion.

The points FOR negative frequency all rely heavily on relative frequencies or relative processes.

I gave a development that started from the premise that a sinusoidal signal involves the sine of something. How did that rely heavily on relative frequencies or relative processes? I pointed out how IQ modulation distinquishes between positive and negative frequencies and describe the effect that it has. I did not describe how the IQ modulation itself makes the distinction (and it has nothing to do with the carrier that the signal is eventually empressed upon).

The points AGAINST negative frequency rely on the concept that frequency itself can only be >0, apart from LvW whose point is still unfolding (and seems to be a proof that eliminates the relative factor or redistributes it to be relatively posiitve).

Do you even begin to appreciate how circular your reasoning is? Basically you just said that the proof that negative frequencies don't exist relies on the concept that negative frequencies don't exist.

Put that one right up with an argument that Santa Claus exists being based on the concept that Santa Claus exists.

 

... At baseband, you simply have a signal at a frequency that can be positive or negative....

WBahn - because I really try to understand your reasoning and your position I kindly ask you to explain the above statement.

How can you say "..simply have a signal at a frequency that can be positive or negative" ? Simply?
In this thread (up to now 65 cotributions) we exchange pro/con arguments to find something like a proof - and you simply state that "at baseband" we have signals with negative frequencies. I am confused about this logic. Can you help me?

 

WBahn - because I really try to understand your reasoning and your position I kindly ask you to explain the above statement.

How can you say "..simply have a signal at a frequency that can be positive or negative" ? Simply?
In this thread (up to now 65 cotributions) we exchange pro/con arguments to find something like a proof - and you simply state that "at baseband" we have signals with negative frequencies. I am confused about this logic. Can you help me?

The "simply" was in a different context than you would use it in a proof.

I haven't played with IQ modulation in well over a year and I've never played with it too much in depth, so I am hesitant to try to explain it in the kind of detail that would be needed. But I'll try -- and keep in mind that this is just from memory.

In IQ modulation, you take an information signal and you modulate it by a two sine waves that are in quadrature with the output of one being the I-component (the in-phase component) and the other being the Q-component (the quadrature or out-of-phase component). These are then added and sent to the RF modulator.

But you can also generate the I and Q components directly. For instance, you can set

I(t) = Asin(ωt)
Q(t) = Acos(ωt).

You can also put two different information signals into the I and Q channels, but let's not go there.

Clearly, I can pick ω to be either positive or negative. I am not picking it to be negative relative to anything, I am just picking it to be negative. If I give you those I and Q signals, you can tell me if the frequency is positive or negative.

Now, I mentioned before that you can describe this without using negative frequencies, and you can. But it makes things very ugly.

 

......
But you can also generate the I and Q components directly. For instance, you can set

I(t) = Asin(ωt)
Q(t) = Acos(ωt).

Clearly, I can pick ω to be either positive or negative. I am not picking it to be negative relative to anything, I am just picking it to be negative. If I give you those I and Q signals, you can tell me if the frequency is positive or negative.

I know what IQ modulation is - and, more than that, I am sure that both signals (I + Q) have nothing to do with a "negative frequency" . Its just a phase shift.
But - what do you mean with "I am just picking it to be negative." ?
I am completely lost because I do not understand the meaning of this statement.
Perhaps somebody else can?

 

I know what IQ modulation is - and, more than that, I am sure that both signals (I + Q) have nothing to do with a "negative frequency" . Its just a phase shift.

But notice in the I/Q components, a positive frequency introduces no phase shift into either the I or the Q while a negative frequency introduces a phase shift only into the I and not into the Q. So it's not just as simple as a phase shift.

But - what do you mean with "I am just picking it to be negative." ?
I am completely lost because I do not understand the meaning of this statement.
Perhaps somebody else can?

If I pick ω to be +10rad/s I am picking a positive frequency. If I pick ω to be -10rad/s I am picking a negative frequency.

But fine. I have offered before to agree to disagree. That apparently isn't good enough for anyone.

So fine. You win. There is no such thing as negative frequency. Just like conventional current, it is a meaningless construct that has no place in the real world and any mention of it should be barred from education.

Satisfied?

I'm done.

 

If I pick ω to be +10rad/s I am picking a positive frequency. If I pick ω to be -10rad/s I am picking a negative frequency.

But fine. I have offered before to agree to disagree. That apparently isn't good enough for anyone.

So fine. You win. There is no such thing as negative frequency. Just like conventional current, it is a meaningless construct that has no place in the real world and any mention of it should be barred from education.

Satisfied?

I do not want to "win" or to be "satisfied". I want to find out the truth. That´s all.
But, an agreement to "disagree" - as far as technical definitions are concerned - is, indeed, unsatisfying. Because such an outcome from a discussion between engineers should be restricted to very few cases only.
Why? Because we have the opportunity to measure and to verify. Thus, disagreements regarding the existence of a technical quantity are not satisfying at all.

WBahn - to be honest, I do not understand your reaction.
The main question was if "negative frequencies" are a real quantity that can be verified by measuremnets or if it is only a mathematical tool (Euler`s famous formula) that was introduced for practical reasons only.

And in this respect, the concept (concept!) of negative frequencies is of great importance and has its place in teaching system theory. And it is surely NOT a "meaningless construct" as you have stated (ironically?).
Therefore, I can and will not folllow your recommendation that "any mention of it should be barred from education. ".

I am sure you are able to distinguish between
* a theoretical concept (as another example, take the introduction of the complex frequency variable "s") that is used for well-known purposes, and
* values, properties, parameters that can be verified by measurements (to proove their existence in the "real world" of engineers).

LvW

 

Another WBahn crash and burn? I'm going to stop conversing with him, as it never remains civil.

Reminds me of "schoolteacher syndrome". I've met a few people like that, they spend all day every day with students far inferior in ability and intellect to themselves, so they end up assuming everyone they meet is also inferior in ability and intellect. It makes for a very unfriendly and intolerant discussion.

 

... you can describe this without using negative frequencies, and you can. But it makes things very ugly.

And that's the crux of it. We tolerate concepts like imaginary numbers and negative frequencies when, and usually only when, they provide a simpler or more elegant mathematical solution to a real physical problem. The solution might be reached in other ways, but a temporary "suspension of disbelief" is tolerated because we hope/know we can make it go away when the solution is finally reached. It's a shortcut.

 

And that's the crux of it.......

I am not really sure what you mean (may be a a linguistic problem only).
For my opinion, the "crux" is that we always should know what we are doing - and why!
I mean: If we are using some artificial quantities because of some good reasons or if we speak about physical realities.
This may be, in particular, a problem for students and other newcomers.
Take for example, the complex frequency variable "s".
Rather often I was faced with questions like this:
*What is the meaning of a pole of a transfer function?
*Why does the transfer function magnitude at the pole not approach "infinity"
*In contrary - why even drops the magnitude by 3 dB at the pole frequency?

 

Is negative frequency real? To answer this question you need both a definition of frequency ( which has been discussed a good deal here ) and a definition of real. I will accept that the answer will depend upon the definition of frequency ( you could for example say that full periods must be counted using the positive integers and then divide by the time for that count, in this case you could never get a negative answer ). But you can define it many other ways including the idea of phase change per unit time ( perhaps multiplied by a constant ) in which case you may get negative answers. So what do you mean by real? There seems to be some idea on the forum that there is math that is real and there is some unreal stuff called pure math. If it is math then it is pure math. Imaginary numbers are no more or less real than real numbers ( despite the somewhat unfortunate names ). At various times people thought that 1 was not a “real” number, also 0, the negative numbers, the irrational numbers, the imaginary numbers. They are all equally real and/or unreal. ( think about it, have you ever seen the number 5, not the numeral 5, not 5 things, rather just 5. all numbers are abstractions which are just constructs following a set of rules, axioms, which at least some people find interesting )

So when we use math to do electronics in some sense nothing is real, “is it real?” is not even a good question. There are a couple of good questions however:

1) Is there an interpretation of the math that makes physical sense?
2) Is the mathematical formulation useful?

I would maintain that negative frequencies ( and even imaginary frequencies which can help elucidate exponential decay ) can pass both these tests, and having an ability to work with these concepts gives electronics designer more ability to deal with systems which are real in the sense you can look down on your workbench and see them.

 

Good point, well taken. Therefore, even frequency is an abstraction.

 

There are a couple of good questions however:

1) Is there an interpretation of the math that makes physical sense?
2) Is the mathematical formulation useful?

I think you have it backwards. Math is an abstraction created by human interpretation of our "physical sense". It structure says a lot about US, and perhaps a little about the world.

 

...( you could for example say that full periods must be counted using the positive integers and then divide by the time for that count, in this case you could never get a negative answer ).
...
But you can define it many other ways including the idea of phase change per unit time ...
...

They are not equal or equally interchangable. Phase change per unit time is very specific and is of limited validity or use, it applies only to a subset of what "frequency" is (ie sinewaves, some calcs) and some things we do with frequencies.

As an example of the many things that have "frequency" in the real world, let's say you wash your hands 20 times a day. That is one of the very common "frequencies" ie "20 events per day" (or even 0.000231 Hz if you like) BUT the length of handwashing per event may vary, likewise the period between events might be totally different for each event! How is "phase change per unit time" even applicable or useful in working with this frequency?

Events per second (or events per unit time) is always valid, and far outweighs phase change per unit time in being a broad generalisation or best understanding of what frequency is and whether it can be negative.

Even IF your point is correct that phase change per unit time can be a proof of a negative frequency, it is still trumped by the vast majority of real world "frequency" examples where there is little to no "phase change" (or nothing at all happening; no signal, no phase) between the frequent "events".

 

russ_hensel,
I agree that one can argue about the meaning of "real". However, in this specific case, I think the engineering view (rather than a philosophical one) is most appropriate.
And therefore: A technical quantity is considered as "real" if it´s existence can be verified by measurements. (Up to now, I never have heard about such a verification as far as negative frequencies are concerned).
And, of course, you are right that one "can define it many other ways". Everybody can use it´s own definition - however, this makes a technical discussion nearly impossible. Thus, I think it is a good practice to rely on a commonly agreed definition.
As far as the present example is concerned, a formulation like "phase change per unit time" is based on a process of visualization (rotating phasor) only and cannot be considered as a valid and commonly agreed definition. More than that, in a pdf attachement to a former post I have shown how the expression d(phi)/dt can be mathematically derived from the classical definition f=1/T.

 

I think you have it backwards. Math is an abstraction created by human interpretation of our "physical sense". It structure says a lot about US, and perhaps a little about the world.

I do not quite see how this makes what I said backwards, although I do not disagree entirely with what you say here. Math originally was strongly tied to our sense of the physical world but has, I think, moved far beyond that so that now our physical sense may come from the math rather than the other way around: an example -> differential non-euclidean geometry as the "correct" discription of space rather than euclidean traditional understanding ( as in general relativity ). In physics ( which underlies electrical engineering ) the math often leads the insight into a physical system.

 

russ_hensel,
I agree that one can argue about the meaning of "real". However, in this specific case, I think the engineering view (rather than a philosophical one) is most appropriate.
And therefore: A technical quantity is considered as "real" if it´s existence can be verified by measurements. (Up to now, I never have heard about such a verification as far as negative frequencies are concerned).
And, of course, you are right that one "can define it many other ways". Everybody can use it´s own definition - however, this makes a technical discussion nearly impossible. Thus, I think it is a good practice to rely on a commonly agreed definition.
As far as the present example is concerned, a formulation like "phase change per unit time" is based on a process of visualization (rotating phasor) only and cannot be considered as a valid and commonly agreed definition. More than that, in a pdf attachement to a former post I have shown how the expression d(phi)/dt can be mathematically derived from the classical definition f=1/T.

I think I could devise a circuit that would give positive frequency readings against some signals and negative readings against others. This would verify the "realness" under your definition.

As for multiple definitions we are awash in them, in any particular situation it is important to know which one is in use ( and among the set of all definitions you should be using a "useful definition" ) and then apply it consistently throughout the discussion.

Take resistance: it is common to recognize negative resistance, but this makes sense only in the differential interpretation of V/I. Sometimes we use one definition sometimes another. Using more than one does not mean that everyone gets his own, but anyone is open to advancing his own, and will prevail if it is useful enough in understand some phenomena. It is not easy to come up with a new useful definition so normally we will stay within a small set of accepted definitions.

 

I think I could devise a circuit that would give positive frequency readings against some signals and negative readings against others. This would verify the "realness" under your definition.

* Yes - I also could imagine a measuring device that allows readings against an arbritrary reference point (on a spectrum analyzer, for instance) leading to the impression that there is a negative frequency. However, the question is if this arbritrary reference point makes sense.
Example: A 100 kHz carrier is amplitude modulated with 1kHz. Of course, if I allocate the 0 frequency reference to the 100 kHz signal, the lower sideband appears at -1kHz. Question: Is this a proof? Is this a realistic measurement ?
No - I don`t think so, because I have shifted the whole frequency scale.

* I can agree to all other parts of your contribution.