periodic function foureir transform in practice

[perchick]

its seams like a very simple question but after 2 hours online i still don't understand this.
for function Asin(2∏*f1t) the Fourier transporm is A/2j[δ(f+f1)-δ(f-f1)].
how do i look at this in realty? i mean, the delta function has no physical meaning and there is no negative frequency's. the meaning of the function is that my power is in the frequency f1 and the amplitude is A. how do i get this result? if i look at the negative frequency's as positive and taking the absolute value of the fourier transform (|F|, δ(f-f1)=δ(f+f1)) I still get just 1/2 of the amplitude not to mention i cant get rid of the delta function which has no meaning seen that its discontinuity at f1.
i need to explain results i got in the lab and i cant find the mathematical tools (i dont want to use fourier series, i want to explain everything with the transform)

 

[steveB]

i mean, the delta function has no physical meaning and there is no negative frequency's.

Since you have already made up your mind about these two opinions, which you state as facts, we have no way to help you. When you are willing to open your mind and consider that the Dirac delta has physical meaning (that's why Dirac invented it) and that there is meaning to negative frequency (just as there is meaning to negative distance) please let us know and we may be able to help you.

 

[MrAl]

Since you have already made up your mind about these two opinions, which you state as facts, we have no way to help you. When you are willing to open your mind and consider that the Dirac delta has physical meaning (that's why Dirac invented it) and that there is meaning to negative frequency (just as there is meaning to negative distance) please let us know and we may be able to help you.

Hi Steve,

Please try not to be so harsh You know these kinds of questions come up all the time when students transition from the purely physical to the edge of physical reality. Think back to when you first encountered this kind of thing.

I would also like to hear your take on this as im sure it would be informative and interesting. Assume he didnt actually mean to say "physical meaning" as much as he meant "physical reality", as in not being able to actually generate that which isnt really possible but still being able to theorize about it as if it were.

 

[perchick]

Hello Steve
Your response seems kind of harsh. Let me point out that English is not my native language so maybe you just misunderstood what I’ve tried to say.

I was under the impression this is Homework help forum. I think most of the questions here are from students and by coming here and asking question we acknowledge that we do not know everything, otherwise we wouldn’t ask those questions here. As MrAl pointed out, what I meant by that was that the function and negative frequency’s has no physical reality. I would have thought it is pretty obvious that I meant to say that seeing that I pointed out that I’m asking this question to explain what I got in the lab.

I hope that you won’t answer this way to other students. Most of them will feel that their question is stupid after this type of answer and won’t come back to ask again. If you don’t remember how it’s like studding so much new material ill remind you that occasionally you have to make assumption in order to progress with your learning, so even if I did mean what you thought I meant, your response is still not very helpful.

let me ask the same question in a different way: how does the Fourier transform of sine function represent the amplitude in the next signal: **broken link removed**
i need to get rid of the delta function some how and the negative frequency's and make sure i get A amplitude. does the Fourier transform represents only the signal in terms of mathematical instrument for calculating the power? if so, then i have to take square of the delta and integrate it (parseval theorem)? for periodic functions the parseval theorem is a little different and is calculated by the Fourier series coefficient, isn't it?

hope this clarify the question i have
Have a nice day
perchick

 

[steveB]

OK, I apologize if that sounded harsh. That is not my intent at all. My purpose was two-fold.

First, i want to get you in the right mindset for learning, which is to be open minded. Often, there are mental blocks in the way, that must be moved before a concept can be understood. - (kind of like what Yoda said: "you must unlearn what you have learned" )

Second, I normally don't put much time into a first time poster because they often don't even come back to look at the answer, and that wastes my time. So, a quick answer to set the stage allows me to test that you are sticking around. It seems you are, and I'm glad that you didn't run away even though you thought my intent was to be harsh. Learning takes perseverance in all forms, and it seems you have that.

I will look at this a little later today to see if I can contribute something useful to the thread.

Thanks for your understanding.

 

[steveB]

So, let's first deal with the concept of negative frequency.

The best example I can think of where negative frequency shows up is with motors. Just as we do mechanical analysis of velocity of a particle in one dimension and consider both positive and negative values for speed, we can talk about positive and negative speed (frequency) of a motor. Spinning in one direction is positive speed and the other direction is negative speed, with units of radians/second typically.

I like the motor analogy because the complex vector represented by A exp(j w t) rotates in the complex plane. The vector A exp(j w t) is a positive rotation (positive frequency) and A exp(-j w t) is a negative rotation (negative frequency). Now, what happens if we add these two complex numbers? The sum results in a real value that oscillates on the real number line. Mentally, you can visualize this as two vectors started pointing to the right, in the complex plane, at time t=0. As time progresses, the vectors rotate in opposite directions and the imaginary components sum to zero, while the real component sums to double the value.

We can also do this mathematically using the Euler relation exp(j w t)=cos(wt) + j sin(wt). Note that A exp(j w t) + A exp(-j w t)=2A cos(wt), as you can work out for yourself.

The above should allow you to see how a Fourier series works. By adding a positive frequency and a negative frequency for complex sinusoidal rotations, we can get a real valued cosine function. Now, your example is a sine function, but this is handled by using imaginary values for A and then subtracting the counter-rotating vectors.


Now, let's deal with this confusing Dirac delta functions. One reason for using Fourier series, rather than Fourier transforms for periodic signals, is to avoid the delta function altogether. The series form is very simple as a summation with simple coefficients, while the transform is an integral calculation with delta functions acting like the coefficients.

Basically, the simple way to look at this is to note that the main purpose of the Dirac delta function is to operate under an integral and convert the integral to a summation. This is the famous sifting property of the Dirac delta function.

 

[perchick]

thank you Steve, your explanation was a lot of help
about your explanation with frequency's as vectors i found this java applet which is pretty cool https://ptolemy.eecs.berkeley.edu/eecs20/week10/negativefreqs.html
about the amplitude and dirak function i simply used the Fourier coefficient aκ and a-κ in absolute value to show that the amplitude is A.
i did get into some trouble explaining triangular periodic wave with the foureir transform.the wave is 4Vp-p 1KHz. first i raised the dc level to get triangular wave that goes from 0 to 4V. which does not change the amplitude in the frequency domain. then i got the foureir transform which is T1*sinc^2(ωT1/2) and from that transform i got the coefficient using that any periodic function is represented as X(ω)=Ʃ2πaκδ(ω-ω0κ) so i got aκ=(T1/2π)*(sin(0.5πκ)/(0.5^2π^2k^2). i know that the sine function is correct because i got that for even K the amplitude is 0. the problem i got is that everything else doesn't much what i got (and others) in the lab. i should get something like 0.89/k^2*sin((0.5πκ)) but i get something a lot smaller like 3.22*10^-5/k^2 which make no sense seen that this number is so small. i guess i got something wrong in the transform but i cant seem to find it. when i tried to directly calculate the transform i didnt get anything like the sinc function. if you can find what i did wrong that will be great
thank you again for all your help

 

[MrAl]

Hi,

I have a question for you perchick, what are you measuring in the lab, what parameters and what values?

Also, i assume that by 'physical' you mean something that can be measured in some way or at least being able to theorize that it could be measured some how.

e^(i*w*t) is a helix rotating one way (say clockwise) and e^(-i*w*t) is also a helix rotating the other way (say counterclockwise). This doesnt require a reversal of time either so i suppose we might be able to attribute that negative sign to apply to the angular frequency. Slightly different but similar is the product under the integral, still a helix. The integral is what looks like a 0.5*sin(x)^2 waveform existing only in a plane at 45 degrees to the time axis and real axis.

So even with geometrical interpretation we still havent answered the question: "Does negative frequency have any real physical representation". And by that i mean some observed process that can only be measured using negative frequency, or at least considering that negative frequency really exists in nature as demonstrated in some physical process, and as revealed with the Fourier Transform without having to arbitrarily ignore the negative part of the result (as is the common way to handle this).

 

[steveB]

i found this java applet which is pretty cool https://ptolemy.eecs.berkeley.edu/eecs20/week10/negativefreqs.html

This is a nice applet that exactly shows what I was trying to describe. These things are not easy to explain in words and work better at a blackboard, but it seems you understand the concept, which is good.

about the amplitude and dirak function i simply used the Fourier coefficient aκ and a-κ in absolute value to show that the amplitude is A.

I'm not sure I follow you here. The coefficients are generally used as they are, not with absolute value. Maybe I'm just not understanding what you are saying.

i did get into some trouble explaining triangular periodic wave with the foureir transform.the wave is 4Vp-p 1KHz. first i raised the dc level to get triangular wave that goes from 0 to 4V. which does not change the amplitude in the frequency domain. then i got the foureir transform which is T1*sinc^2(ωT1/2) and from that transform i got the coefficient using that any periodic function is represented as X(ω)=Ʃ2πaκδ(ω-ω0κ) so i got aκ=(T1/2π)*(sin(0.5πκ)/(0.5^2π^2k^2). i know that the sine function is correct because i got that for even K the amplitude is 0. the problem i got is that everything else doesn't much what i got (and others) in the lab. i should get something like 0.89/k^2*sin((0.5πκ)) but i get something a lot smaller like 3.22*10^-5/k^2 which make no sense seen that this number is so small. i guess i got something wrong in the transform but i cant seem to find it. when i tried to directly calculate the transform i didnt get anything like the sinc function. if you can find what i did wrong that will be great
thank you again for all your help

Without seeing the whole thing worked out carefully, I can't say what the issue is. Usually, it's just a simple error in doing the calculations. It is very easy to make errors in these types of calculations and personally I have to do them out very carefully and double check them to get them right. Then, I'll use a symbolic processor to check it all again. If you post a file with your work, I can try to identify the particular error.

 

[steveB]

"Does negative frequency have any real physical representation".

Without trying to directly answer this question, I have a good example to mention from some real-world work I did a few years ago.

I was involved in designing a position measurement system, and assigned the task of developing a digital algorithm to demodulate position information that was encoded in a high frequency carrier. Basically, the position information modulated the carrier directly in phase. Others had tried to develop a phase locked approach to lock into the carrier frequency and to then demodulate, but they were unable to keep a good lock, and reliability was critical since it involved transportation of passengers. The main reason for needing to know the carrier is that frequency drift in the carrier would look like a position change and it would be impossible to discriminate between frequency drift and position information.

When I looked at the problem, I eventually realized that it was possible to make a detection scheme that was very tolerant to frequency drift and did not require exact knowledge of the carrier frequency. This was done by assuming a value for the carrier frequency that was near the correct value. Then two demodulation calculations were performed, one at the carrier frequency and one at the negative of that frequency. Then these two demodulation calculations were added so that any frequency drift cancelled out and the position phase information was doubled. The main constraint was that the frequency drift specification had to ensure that the carrier was within the bandwidth of the demodulator, which is quite easy to do, and a phase correction on the digital filter was needed to correct the phase error induced by the demodulations filter.

I suppose this result is open to interpretation, but to me this shows that our notion that frequency is positive is really just a convention, or a definition, that we get used to. Also, not everything associated with the positive and negative frequency is symmetrical, since in this case, phase information and carrier drift had opposite symmetry relative to the positive and negative carrier.

 

[Winterstone]

Also, i assume that by 'physical' you mean something that can be measured in some way or at least being able to theorize that it could be measured some how.

So even with geometrical interpretation we still havent answered the question: "Does negative frequency have any real physical representation". .

For my opinion, in this context we have to be very exact with wordings and with definitions in order to answer such a question. What I mean is the following:
Is the term "frequency" a physical quantity? With other words: Does something like a "frequency" really exist?
I don`t think so.
Rather, it is nothing more than a property of a real periodic physical process, which can exist in the world of mechanics, electronics and optics.
And as everybody knows - it is defined simply as the number of periodic repetitions per second.
Thus, we have to answer the question: Can the number of repetitions of such a real physical process be negative? To me, the answer is clear: No.

Therefore, the introduction of "negative frequencies" is a very helpful tool to simplify formulas, but it is nothing else than a pure theoretical idea.

 

[MrAl]

Hi again,


Steve:
That's interesting, i'd have to see more of what that was actually about to understand fully however. If you had to use the concept of negative frequency to arrive at a solution and you also had to measure this frequency that could mean something.

Winterstone:
You mean to say that you do not want to acknowledge even 'regular' frequency as being something that actually exists? That's an interesting idea, but we can measure it right? This idea may border on the purely philosophical however much like a hole in the ground (is it real or not?). For example, paint a small dot on the end of a motor shaft and every time that dot passes through 0 degrees (relative to horizontal with the origin at the very center of the shaft) we count 1 more increment, then take the ratio of the count divided by the time, which would give us revolutions per second, and since the dot made one cycle around the circle we might call this cycles per second. It's also interesting that doing it that way, if the shaft was rotating in the other direction, that we would still measure the same thing, events per unit time. So for this discussion i think we are 'accepting' the 'normal' frequency to actually exist and then make a comparison to what the Fourier Transform tells us is 'negative' frequency.
So we can measure it, but i guess we can also ask is there some standard way to measure frequency other than events per unit time. Does the reverse rotation really represent what we might call negative frequency in the same sense that it does in the Fourier Transform.
With cos(wt) for example, we cant tell if it is rotating clockwise or counter clockwise:
cos(wt)=cos(-wt)
so that tells us that we can not distinguish clockwise rotation from counterclockwise rotation just knowing cos(wt). But we know from the geometrical view of the exponentials that one helix is rotating clockwise and one helix counterclockwise with time. So maybe the concept of frequency with the Fourier Transform is different than the events per unit time concept.

But also interesting is that at least from the Fourier Transform we can calculate the one ohm energy per unit bandwidth, or simply the total one ohm energy in a signal by integrating it's Fourier Transform from negative infinity to positive infinity. This gives us a sort of physical interpretation, and if we only integrate one side (like from zero to positive infinity) we only get 1/2 the total energy. This gives credence to the fact that both sides are somehow 'real', but unfortunately even that isnt the last word because after all we used the Fourier Transform to arrive at that energy so it could still be just a mathematical requirement rather than a physical phenomenon.

There is a physical phenomenon known as negative frequency in the stimulation of some materials with light. The energy of the light causes something to happen in the material where the atoms of the material can emit an extra amount of light that would not be present unless there was what they are calling negative photon frequency. Unfortunately i dont know enough about this process to discuss it in depth, and i cant say that it's the same thing as a rotating shaft anyway which appears to rotate in only one direction at a time wheres the Fourier Transform of cos(wt) has both positive and negative frequency components even as we seem to observe the shaft rotating in only one direction.

Also interesting is that if we use a strobe light to observe the end of the shaft we can make the moving dot appear to rotate counter clockwise even though when observed with ordinary light it appears to rotate clockwise. Could it have something to do with the way we observe or measure frequency in general that always seems to lead us to a single rotation unlike the Fourier Transform.

I was tempted to suggest that the negative frequency comes from the fact that the cos(wt) wave does not start at t=0, but actually starts at minus infinity. But there are transforms of other functions that also have negative frequency where their amplitude is actually zero up until t=0, so that's not an option.

 

[Winterstone]

Interesting discussion, isn`t it?

Hello again, MrAl - I like to continue the discussion and I will try to give some comments to your explanations - just some thoughts that came into my mind reading your reply.
At first, I apologize for being „pedantic“, but I think in the discussed context (positive/negative) - as expressed already in my former posting - it is important to be very exact (pedantic?) and to discriminate between definitions and physical realities.

Quote MrAl: You mean to say that you do not want to acknowledge even 'regular' frequency as being something that actually exists?

I know that in the world of electronics and communication it is common to use the word „frequency“ if one means a (mostly) sinusoidal wave (voltage, current, electro-magnetic,..).
Thus, I think even „regular“ frequencies do not „exist“ as a physical reality because they are just defined as a property of another physical reality - and that`s a periodic wave (voltage, current, ...)
Similarly, does the rise time (or the duty cycle) of a squarewave „exist“? No - both are only definitions to describe a physically existent quantity.

That's an interesting idea, but we can measure it right?
Of course - properties can be measured.

For example, paint a small dot on the end of a motor shaft and every time that dot passes through 0 degrees (relative to horizontal with the origin at the very center of the shaft) we count 1 more increment, then take the ratio of the count divided by the time, which would give us revolutions per second, and since the dot made one cycle around the circle we might call this cycles per second. It's also interesting that doing it that way, if the shaft was rotating in the other direction, that we would still measure the same thing, events per unit time. So for this discussion i think we are 'accepting' the 'normal' frequency to actually exist ....

As stated above: We are measuring a property of the mechanical plant ...
Moreover, I think, this involves a new definition (or perhaps an extension of the original definition for the term „frequency“) because now a direction of the physical process is considered. The original definition for „frequency“ is a pure positive number that can be counted - without any direction.
On the other hand, I agree that the parameter „revolution per minute“ can be, of course, defined as positive or negative. Therefore, I think we shouldn`t mix the terms „frequency (in Hz) “ and „revolutions (in rpm)“.
In this context, I consider it also as necessary to allocate the unit „Hz“ to the frequency f only and to measure the angular frequency w=2*Pi*f in rad/s only. This is the only correct unit and helps to avoid misinterpretations in formulas, graphs, etc..

....and then make a comparison to what the Fourier Transform tells us is 'negative' frequency.
So we can measure it, but i guess we can also ask is there some standard way to measure frequency other than events per unit time.

Yes - of course. But has the measurement method a retroactive effect on the definition?

... the Fourier Transform of cos(wt) has both positive and negative frequency components ...

...if we use Euler´s (artificial) mathematical description. The real measurement of the spectral distribution gives only one single „line“ with the correct amplitude A (computation of the artificial two-sided spectrum gives two lines with A/2).

Finally, the introduction of negative frequencies, of course, has many advantages (e.g. simplification of formulas) and I admit - it can help to better explain some particular observations. I think, the same applies to complex frequencies, which also have been invented as another extension of the term „frequency“. And everybody knows about their advantages - but that´s not a proof for their real physical existence.
In contrary - for my opinion it is very important to know that complex frequencies do not "exist" (which means: They cannot be generated technically). In this context, I remember some questions from students: "Why is it not possible to measure an infinite amplitude at the pole frequency"?

W.

 

[steveB]

If you had to use the concept of negative frequency to arrive at a solution and you also had to measure this frequency that could mean something.

This is an interesting point. In a sense, we are saying that a property that has meaning in an abstract complex space is somehow manefesting an effect in the real world. Basically, our model considers the complex values, but reality does not. Yet, somehow the meaningful information from our "complex" theory is embedded in things we can measure. So, does this make negative frequency real or unreal? The case of a spinning motor has real direction, so I'm tempted to say that case has real meaning for negative frequency, and the direction information is given by the sign. However, time varying signals don't have real rotations and only can consider abstract rotations in a complex plane, which makes the question more difficult. In a sense, a real oscillation has both a positive rotation frequency and a negative rotating frequency, that sums to something that does not rotate.

I'm reminded of something analogous in Quantum Mechanics. Wavefunctions are complex valued objects, but we only observe real valued things as measumements or probability distributions (the magnitude squared of the wave function). Since we do not directly measure the complex valued wavefunction, we can question whether it is something real, or just an artifact of a theory. Yet, there are experiments that seem to show some meaning to the phase information embedded in the wavefunction.

Honestly, I dont' know how to answer these types of questions, but they do provide for interesting discussions.

 

[MrAl]

Hi again,

I'm posting an image that shows the frequencies after transforming f(t)=cos(wt). Look to the far right.

Steve:
Yes interesting

Winterstone:
Yes, i think i understand what you are saying. But we are still looking for a physical explanation of the impulse shown in the attachment that exists in the negative frequency domain. Look to the far right to the image that has the two impulses. One is on the positive frequency side (first quadrant) and the other is on the negative frequency side (second quadrant). We assume we know what the positive frequency impulse is, but maybe we dont even understand that in it's entirety either. Integrating either side (squared) from zero to positive infinity gives us only half the total 1 ohm energy, while integrating both sides gives us the total energy. So actually neither side explains the whole picture
So we're looking for an explanation. I tend to agree however that it's just a mathematical tool. We also use "imaginary numbers" to arrive at real life conclusions, but we dont expect to find imaginary apples sitting on the shelf at the supermarket

 

[Winterstone]

MrAl, hello again.
I suppose, you will not be surprised that I know the mathematics behind the two-sided spectral representation of a signal (as shown in your attachement).

But we are still looking for a physical explanation of the impulse shown in the attachment that exists in the negative frequency domain.

Yes- we can try to find a physical explanation. But - for me - it would be not a surprise if we fail in some cases. Can we always expect to find a physical explanation for a phenomenon that results from a mathematical manipulation only?

Look to the far right to the image that has the two impulses. One is on the positive frequency side (first quadrant) and the other is on the negative frequency side (second quadrant). We assume we know what the positive frequency impulse is, but maybe we dont even understand that in it's entirety either. Integrating either side (squared) from zero to positive infinity gives us only half the total 1 ohm energy, while integrating both sides gives us the total energy. So actually neither side explains the whole picture

But remember: It is a known fact that the one-sided spectrum (which results from the real form of the FOURIER transform) contains amplitude that are twice as high if compared with the two-sided spectrum. Thus, there is no problem to interpret it correctly.

So we're looking for an explanation. I tend to agree however that it's just a mathematical tool. We also use "imaginary numbers" to arrive at real life conclusions, but we dont expect to find imaginary apples sitting on the shelf at the supermarket

As explained in my example with the complex frequency (s=sigma+jw) I have no problems to accept that we are using from time to time mathematical tools, which lead to results without direct physical "existence".
Nevertheless, we normally don`t create new names but we use words and units that come from the known real physical quantities. This may be one reason for misunderstandings and misinterpretations.
(see also comment at the end of my post)

As a counterexample - I like to mention the group delay. Some people think (and even some textbooks claim) that group delay and group velocity cannot be negative. I agree, that - at first sight - one could have problems to understand the meaning of negative values. However, negative group delay/velocity has been experimentally proven to exist.

With regards
W.

Comment (added later): In this context (old words for new phenomena) I remember that there is (at least) one textbook author who does NOT use the phrase "frequency" in conjunctiuon with the two sided spectrum (with the aim to avoid "negative frequencies"). Instead he uses the PHASOR: Q+=exp(jwt) and Q-=exp(-jwt).

 

[MrAl]

Hi again,


I had no doubt that you understood the calculation itself, that was never a question.

Now that i think about it, that could be the whole key. We're trying to understand frequency itself but all we really have to do is understand that the transform is sampling the real waveform of a function using both positive and negative time. Thus it may turn up components both positive and negative.
Looking at the end of the motor shaft again with the dot painted near one edge of the shaft on the end, if we look at it in positive time we see it rotating one way, and if we look at it in negative time we see it rotating the other way. Since physics admits that there is no preference, we get both as valid results.