spectra for FS and FT

[PG1995]

Hi

Could you please help me with these queries? Thanks a lot.

Regards
PG

 

[steveB]

For Q1, you may be having trouble because you wrote the answer for the angle in an ambiguous way. There is only one angle, but you wrote ±. It seems to me that you need to clarify this and include a (-1)^n to allow the value of n to select the correct sign for the particular coefficient. Then you will get the alternating pattern you showed in the solution using the complex exponential form.

By the way, you should try to get comfortable with the complex exponential form. In the long run, it is easier and more standard to use this form.

For Q2, I don't have time to determine if the answer is correct. When i have more time, I may look at it. However, you should try to invent your own ways to double check your answers. Often, it is easy to take an answer and verify it is correct. In this case you have an answer worked out in the exponential form and you have conversion formulas that translate between the forms. You can check your answer and verify that you know how to use the conversion formulas at the same time. Checking is a helpful part of learning and a necessary part of working. I do understand that (as a practical matter during exam period) time can be short for you and it is helpful if someone else can check, but in this case, time is short on my side too. (blame the SuperBowl on this one )

 

[PG1995]

Hi

Could you please help me with this query? Thank you.

Regards
PG

 

[steveB]

PG,

I would question the book you got that example from. It looks to me they are either using a different definition of the coefficients (off by a factor or 2 from what you are used to), or they made a mistake.

 

[PG1995]

Thank you.

The example was taken from this PDF (page #5) and I don't think they are using different definition of the coefficients because in another PDF (page #16), the same author has plotted correct plots for trigonometric and exponential forms of Fourier series for a function where exponential amplitude spikes are half in length that of trigonometric spikes. Therefore, I believe they made a mistake in that case. Kindly correct me if I have it wrong. Thanks.

Please help me with this query too. Many thanks.

Regards
PG

 

[steveB]

I can't say for sure, but it looks to me they made a mistake. To be sure I would need to put more time in, but I'm confident you have the correct understanding, so there is no need to check it out.

For your other question, isn't the angle the same for n=0 too? So, I think what they wrote is correct.

 

[PG1995]

Thank you.

I can't say for sure, but it looks to me they made a mistake. To be sure I would need to put more time in, but I'm confident you have the correct understanding, so there is no need to check it out.

It's okay if you think my version is correct but, anyway, I think I'm going to use that example in another thread so we can assume for the sake of discussion there that both versions are correct.

Edit: Won't use this example.

For your other question, isn't the angle the same for n=0 too? So, I think what they wrote is correct.

But don't we have a DC component at n=0? Having a phase spike for a DC doesn't make sense. For example, you can see here that trigonometric FS starts at n=1. Where do I have it wrong? Please let me know. Thanks.

Regards
PG

 

[steveB]

But don't we have a DC component at n=0? Having a phase spike for a DC doesn't make sense. For example, you can see here that trigonometric FS starts at n=0. Where do I have it wrong? Please let me know.

Well yes, you have a point. It's like a zero vector does not really have a direction, or the θ component in polar coordinates has no meaning when r=0.

 

[PG1995]

Thank you, Steve.

I don't recall I have ever asked you this question before. Why does Fourier series or transform use sinusoidal functions for representation of a given function? What is so special about these sinusoids? Why aren't, say, parabolic functions used? I have a faint recollection that I have read somewhere that the reason for this is that sine and cosine are really easy to handle mathematically and their derivatives and integrals are also sinusoids. Is this the only reason to it for using sinusoids? Perhaps, Fourier was too much into water waves (which looks very similar to sine waves) that he came up with this idea! Thanks.

Regards
PG

 

[steveB]

Thank you, Steve.

I don't think I have ever asked you this question before. Why does Fourier series or transform use sinusoidal functions for representation of a given function? What is so special about these sinusoids? Why aren't, say, parabolic functions used? I have a faint recollection that I have read somewhere that the reason for this is that sine and cosine are really easily to handle mathematically and their derivatives and integrals are also sinusoids. Is this the only reason to it for using sinusoids? Perhaps, Fourier was too much into water waves (which looks very similar to sine waves) that he came up with this idea! Thanks.

Regards
PG

That's a very good question. I didn't find the answer to that until I studied quantum mechanics at the graduate level. I'll see if I can take that explanation and put it in terms you will be able to understand. I'll need to consult some references and refresh my memory first.

 

[steveB]

After reviewing my text books, I think I should not go down the path of explaining what I intended. I think at different experience levels certain explanations are required, such that they are not too simple and not too complicated for the experience we have at any given time. So, I'll give the explanation that suited me at the undergraduate level, and leave aside the details of the explanation what suited me later.

Eventually, you will reach a point where this simpler explanation is not quite sufficient, and perhaps that is even now, but this is the best I can offer. Later, I would recommend studying quantum mechanics from the modern approach typically given in grad level classes. Don't mistake that to mean that I'm talking about difficulty level, but instead I'm talking about the foundation and approach used. Actually, this modern approach is easier, from many point of view, than the the old wave-mechanics approach. In wave mechanics you can view Fourier theory as a type of assumption for the theory, whereas the modern approach starts with basic assumptions about Hilbert spaces, and then Fourier theory seems to fall right out of the theory in the relevant cases for symmetry. In particular I would eventually refer you to "Modern Quantum Mechanics" by J.J. Sakurai. The very first chapter lays the ground work and derives Fourier theory as a consequence of basic assumptions about Hilbert space and symmetry properties related to position and momentum. I view this as the proper way to answer your question, but you are not ready to absorb it now. Consider Sakurai's words near the end of this first chapter. "This pair of equations (he was referring to the Fourier transforms between position-space and momentum-space wave functions) is just what one expects from Fourier's inversion theorem. Apparently the mathematics we have developed somehow "knows" Fourier's work on integral transforms.".

For now, let's keep it simple.

First, sine/cosine waves are not the only possible functions, but are the simplest to consider. Basically, you are looking for natural eigenfunctions for the space and boundary conditions you are dealing with. Often cylindrical harmonics (using Bessel Functions) , or spherical harmonics are used when the boundary conditions favor cylindrical or spherical symmetry, respectively. But, this is something you can study in the future. The key point is that Fourier analysis is not the only way, but seems to be the most natural.

So, why are sine and cosine waves for Fourier Series and Fourier Transforms, the natural eigenfunctions? Well, consider a violin or guitar string. It is tied at each end with a boundary condition of zero movement and zero velocity at each end. The natural solutions are sine/cosine waves at the fundamental fequency and integer multiples of the fundamental frequency. Then we can think of superposition and say that any general solution must be a sum of all the possible eigenfunctions for this system. This then directly leads to Fourier Series.

Now, to get to Fourier Analysis, simply take the limit as the length of the string goes to infinity. The fundamental frequency goes to zero and then waves at all frequencies are viewed as eigenfunctions of the continuous space without end boundaries. This is why Fourier Analysis works, in a nutshell.

So Euclidean space, or the time-dimension (which have the same symmetry properties) have sine/cosine waves as natural eigenfunctions.

OK, this is very crude and imprecise words here, but I hope it gives a clue to you. Again, the rigorous way involves mathematics (which is not terribly difficult, by the way) that you are probably not ready to delve deeply into yet.

 

[PG1995]

Thank you very much, Steve.

It was really nice of you that you took all the trouble of reviewing your text books just to help me. I was able to understand some of the parts from your reply and you are right that I need a certain level of experience and knowledge to fully understand it. Once again, many thanks.

Best regards
PG

 

[PG1995]

Hi

I was still thinking about it. Fourier was born in 1768, almost 41 years after Newton's death. At that time quantum mechanics was not around. Then, what went in Fourier's mind that he chose sinusoidal functions. I'm sure we can't get into his mind but I still wonder that what really inspired his work in that particular direction. In Fourier's days Taylor series, which is based on derivatives, would have been quite popular. Thanks.

Regards
PG

 

[steveB]

Hi

I was still thinking about it. Fourier was born in 1768, almost 41 years after Newton's death. At that time quantum mechanics was not around. Then, what went in Fourier's mind that he chose sinusoidal functions. I'm sure we can't get into his mind but I still wonder that what really inspired his work in that particular direction. In Fourier's days Taylor series, which is based on derivatives, would have been quite popular. Thanks.

Regards
PG

I read up on this many years ago, but I can't remember the details now. Certainly, in Fourier's time, sinusoidal solutions were cropping up for various interesting problems that were being researched at that time. You can do historical research to gain some insight about this. I seem to remember that, at the time, others were skeptical of the mathematical rigor of what Fourier proposed, but obviously that is no longer an issue.

Actually, it is not Quantum Mechanics, per se, that is relevant, but the work of Hilbert (and others) helps set the better framework for answering this question. It just so happens that the mathematics of Hilbert Spaces forms most of the essential postulates of Modern Quantum Mechanics, and engineers tend do deal better with math in a physics context, rather than in an abstract math context.

Below, I've provided a link to a reference that talks a little bit about this, but I think this question requires going beyond internet information and tracking down books that focus on the history. Also, perhaps a question posed at a math-forum could help you get directed to good information.

**broken link removed**

 

[PG1995]

Thank you, Steve.

You are right that to find an answer to it requires us to get down into the history of mathematics. The study of history of science and mathematics is an informative experience. I'm saying so because I have had a little experience of it. I was once interested in knowing that how Avogadro reached the conclusion that equal volumes of gases at the same temperature and pressure contained equal number of particles. I searched this topic extensively and read through some science history books on Google Books. At that time, I can say, I had found almost 40% of the answer but then my Firefox crashed and all the opened tabs were not recoverable and it simply killed my appetite for finding the answer! Someday, I will find the answers to both these issues, related to Avogadro and Fourier, and post them here.

Regards
PG

 

[PG1995]

Hi

Could you please help me with this query? Thanks a lot.

Regards
PG

PS: Please note that this attachment is used in post #18.

 

[steveB]

I haven't checked your work carefully, but it looks like you approached it correctly. I don't think your reasoning on why the answer is wrong is necessarily correct. The average value is 1/2, so your value of a0=1 is correct because it gives a DC term of 1/2. Then the shape of the function indicates that only sine and not cosine functions should be used in the Fourier series. This is clear because once the DC term is subtracted off, the function is antisymmetric.

So, I'm guessing you did it correct, but even if it is wrong, the correct answer will have the same property that f(0)=1/2 for any finite sum approximation of the Fourier series.

EDIT: This webpage shows that your answer is correct. The only difference between this answer and your answer is that your answer has a DC offset of 1/2 and an amplitude of 1 peak to peak, while the Wolfram example has zero offset and an amplitude of 2 peak to peak.

https://mathworld.wolfram.com/FourierSeriesSquareWave.html

 

[PG1995]

DC component etc.

Hi

Yes, you are right that the answer is correct. I believe what led me to think that there is an issue with the answer is my not-very-good understanding of the DC component. So, I think I should work on it.

Please note that Q2 and Q3 are very much related to each other. I have separated them so that you can refer to the referenced examples easily. The question Q5 is a totally different one and an important one and I really need your help with it. Thank you.

I understand that I'm asking too many questions in this post. But I believe asking all these questions together might help you to better understand where I'm going wrong. You can take your time and address the queries when you have time.

Q1:
The concept of DC component is inherent to Fourier series and not to Fourier transform. Am I correct? Kindly let me know.



Q2:
When finding FS of a function, the FS looks at the function in symmetric terms along the x-axis. In other words, the FS tries to symmetrize the function along the x-axis. Then, the DC component tells us that what constant value should be added to symmetrized FS version so that the function gets raised to its actual position.

Let me elaborate on what I said above. I could have told you the value of DC components for these two examples, Example 3.4 and Example 3.5, by just looking at the given figures. For instance, in Example 3.4, you can easily notice that a square wave has been lifted by a constant value of "1/2". In other words, you need to subtract "1/2" from the given version of the function to get symmetric version. Likewise, we can easily notice that the triangular wave for Example 3.5 is already symmetrical along x-axis therefore DC value is zero.

Do I have the concept right?



Q3:
Now I myself going to point out some loopholes in my own understanding of the DC component. Please look at this example about raised cosine function which, according to the answer given, has DC component of "1/2". According to what I said above the DC component should have been "1" because "1" should be subtracted from the given function to get a symmetric version. Simply stated, the given function is a raised cosine function using a constant value of "1". Where am I going wrong with this example?

Likewise, in this example about exponential function, the given DC component is "0.504". I don't see how we can find a "symmetrized version" in this case because I don't see any symmetry along x-axis in this case. This means that the DC component is not all about symmetry along x-axis.

Now it is clear that my understanding of DC component is flawed. Please guide me.



Q4:
Please have a look here.



Q5:
Many a time, there are differences between a mathematical model of something and the real thing which that model represents. The mathematical model is only there to help us do the calculations easily. My question to you: Is mathematical FS or FT representation of some function an exact replicate of that function? For example, if you have connected a wire to a source supplying an electric current in form of a square wave, then do you think that electric current is flowing in form of pulses (i.e. square wave) or in form of odd harmonics of sine wave? Putting it differently, do you think electrons are moving form of pulses (i.e. square wave) or odd harmonics of a sine wave?

I don't know your answer to the above question. But I suspect that you don't think that the current exists in form of odd harmonics and you are of the the opinion that Fourier representation is just a mathematical tool in this regard. But still in many topics, such as filters, we talk and practically it looks like as if Fourier representation is as much real as it could be. For instance, it is often said that this filter can reject low frequencies etc. I can give you more examples here to state my confusion but I'm sure you already understand where I'm having difficulty. Please help me. Thank you.

Regards
PG

 

[steveB]

I'll start with questions Q2 and Q3.

Where did you get the idea about needing a symmetric function when the DC component is subtracted? It's not even clear what your definitions of symmetric and symmetrize are.

There exists definitions of symmetric functions and antisymmetric functions, and this can be related to the FS. I wouldn't say the FS tries to do anything along those lines, but thinking in terms of symmetric and anti-symmetric sometimes helps. There is a theorem that says any (typically behaved) function can be represented as a sum of a symmetric function and an anti-symmetric function. This is useful for FS because the sine portion of the FS will be used to generate the ant-symmetric portion of the signal. The symmetric portion can also be broken down into a DC component plus a symmetric function that has zero DC offset. This is also useful because the DC portion becomes Ao/2 and the symmetric part that remains is represented by the cosine portion of the FS.

I'm not sure if what I described is related to what you are talking about in Q2 and Q3, but that's the only comment I can make about what you said. Otherwise, what you said doesn't strike me as something useful. However, I could be missing your point.

By the way, often "even" and "odd" are used to describe functions that are symmetric or antisymmetric.

 

[steveB]

For the remaining questions.

Q1: Practically speaking, yes. You could say the DC value is zero for aperiodic signals that we deal with in practice. Still, you can define some strange functions that are aperiodic and still have nonzero average values.

Q4: This is a tricky question and concerns like this are what originally made people doubt the rigor of Fourier's work, but it has held up. There is another odd thing about the Fourier series of a square wave. Notice how the overshoot at the transitions is always the same amplitude, no matter how many terms you add up. This also seems to be an issue. However, if you correctly and rigorously take limits, the FS representation is identical. Your concern is similar to the idea that 0/0 or 0*∞ are indeterminate. Well, yes they are, until you specify the details of how they go to zero or infinity. Then, it is possible to say more about the limiting value. Sometimes 0/0 goes to zero. Sometimes it goes to infinity. And, sometimes it goes to a finite constant value. For the FS, you can't add up an infinite number of zeros and say the answer is zero. You have to be more rigorous than that.

Q5: Tricky question!. I view the mathematics as correct and exact. But I view all physics theories we know as an approximation. Even quantum mechanics is well known to be only approximately correct. The interesting thing is that non-relativistic quantum mechanics and electromagnetic field theory in vacuum are truly linear theories. That is, they are exactly linear in the mathematical sense, and hence the Fourier theory is an exactly correct and equivalent viewpoint within the theory. However, we know those theories are only approximate representations or models of reality. So, you have to make your own interpretation here. I think the question is not provable one way of the other, and it is a viewpoint we might hold on to. I don't have a strong opinion one way or the other.