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understanding fourier series and fourier transform

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PG1995

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Hi

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

Regards
PG
 

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Q2:
Fundamental idea of Fourier transform is that any function can be modeled with sin and cos IF you REPEAT it.

Without repeating the function, it can't be modeled.
 
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Hi,

Q1: 2L is the period if f(x)=f(x+2*L), which they say it is.

Q2: Periodic functions are often described in terms of just the first period. The previous and next periods (cycles) are assumed to occur the same such that f(x)=f(x+T) where T is the period.
 
Thank you, vlad, MrAl.

Q1: 2L is the period if f(x)=f(x+2*L), which they say it is.

Yes, they do mention it. But you see there they are talking about f(x) as piecewise continuous function so I thought perhaps 2L could only be the period of a piecewise continuous function. Well, you can say I was confirming it. Thank you.

Q2: Periodic functions are often described in terms of just the first period. The previous and next periods (cycles) are assumed to occur the same such that f(x)=f(x+T) where T is the period.

I think I kind of get your point. But the definitions given there work because the graphs are symmetric along the y-axis. For instance, if there is a shift of 180° in Fig. 23-1 then the definition needs to be changed.

Regards
PG
 
Hi, again,

I have read that fourier series is used for periodic signals and for aperiodic signals fourier transform is to be used. Can fourier transform also work with periodic signals in addition to aperiodic? If that's true then what's the need for fourier series when two jobs can be done with a single tool? Could you please let me know. Thanks.

Regards
PG
 
The Fourier Transform is general enough to include Fourier series, but Fourier Series is a limiting case. You are probably aware that taking limits as things go to infinity or zero can get conceptually difficult. The Fourier series is just easier to work with when you have a periodic signal. The limiting process has already been done for you and you don't need to struggle with how the continuous function, which is the Fourier Transform, changes into a discrete series of impulse functions when the function becomes periodic.

Also, you can argue the other way too. The Fourier series can be converted to the Fourier transform as a limiting case. But, again, the mathematics of the change over is not convenient for practical work.
 
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Thanks a lot, Steve.

Could you please help me with this query?

Regards
PG
 

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I disagree about what you say about memorizing versus conceptual understanding. Both are important, but understanding takes precedence. I think this is particularly true for this subject. Also, if you understand the concept of what you are doing with Fourier Analysis/Series, you can often deduce a formula that you forgot, or at least get the major pieces correct. Also, looking up a formula you forgot is much easier than trying to learn concepts you need when you have to solve a problem. More importantly, if you don't know the concepts, you might not even recognize when it is appropriate to apply a theoretical tool.

One can also ask about memorizing the concepts themselves. It is very hard to memorize concepts in a useful way. You may be able to repeat the concepts when you memorize them, but you will not truly understand concepts without practice and application of those concepts. By applying concepts, you internalize them and make them part of your thinking mind. Over time, you might forget a concept, but it is never truly gone, because once you quickly refresh your memory about the concept, all the wiring in your brain is still there on how to use it. Somehow it is like a skill, such as riding a bicycle or playing a musical instrument. You never forget it, but only get rusty when not practiced.

The exponential, sine and cosine forms are essentially equivalent. It's hard to say one is more useful, but I personally like the exponential form. I find it a little more elegant mathematically and easier to apply and manipulate from a practical point of view. It is also easier to bridge over to Laplace transforms from the exponential form.

As far as the Fourier integral theorem, it is definitely helpful to memorize one form of the theorem. Personally I know it in the exponential form, and not in the form you show. However, I could easily deduce the form shown and might only struggle with getting the scaling constants correct. However, these I would quickly find by doing a simple example. So again, concepts beat memorization.

Note that this subject of Fourier Theory is very powerful and extremely useful to engineers, particularly electrical engineers. Even if you don't do anything quantitative, there is great value in qualitatively understanding the concepts. Many problems are solved by logical deduction rather than actual calculations. The concepts in Fourier theory are very powerful when making deductions.
 
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Thank you very much, Steve.

I really appreciate your concern and your advice. Personally I have always be of the opinion that conceptual understanding is important because that is supposed to be the main purpose of education, and that's the reason I also bug you and MrAl so much! :). But I still insist that the formulae should be at one's finger tips especially during an exam. If you ask me I don't think it would be really an easy task to derive a formula even if one has good conceptual understanding the reasons being time constraint, pressure of the exam, etc. In my view, in an exam one is tested more for one's memory rather than one's understanding of the concepts. But very important factor which we haven't included is that this varies from one institution to another. Now I'm more of the opinion that one should have to maintain a balance between conceptual understanding and memorization. Nonetheless, I completely agree with what you say.

As far as the Fourier integral theorem, it is definitely helpful to memorize one form of the theorem. Personally I know it in the exponential form, and not in the form you show.

But look under **broken link removed** "Equivalent Forms of Fourier's Integral Theorem", the formula #33.4, it shows exponential form.

Once again, thanks.

Best wishes
PG
 
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In my view, in an exam one is tested more for one's memory rather than one's understanding of the concepts. But very important factor which we haven't included is that this varies from one institution to another. Now I'm more of the opinion that one should have to maintain a balance between conceptual understanding and memorization.

I agree with all you are saying. The situation you deal with on tests in school is quite different from what we deal with in a practical work environment. In real work, concepts drive everything we do, and memory is based on the person (some people remember well and others dont) and our familiarity with the subject at hand (things we use often, we remember, and things we rarely use, we are more likely to forget).

You are right that testing strategy varies with institution and also with the subject and with the teacher. I've taken many courses where we were allowed to use all our notes and our books too. Most of my graduate level classes had take home exams where any (non-human) resources could be used. These types of tests heavily stress the creative use of concepts. Then there are other situations where no books or notes are allowed and you are expected to know by memory. I think for engineering/physics subjects, the former is better than the latter since it reflects what people do in the real world.

One of the best arguments I can make for memorizing formulas is based on pride rather than utility. Even though I can look up or derive any formula I would need, it is a little embarassing to not know a formula I should know. For example, let's say an entry level engineer comes to me for advice about magnetics at work and I don't know Maxwell's equations off the top of my head. That does not make a good impression on my co-worker or my boss. So, not knowing formulas will make me ashamed, but not being able to use concepts will get me fired. :D I'm exaggerating a little to make a point, but hopefully you understand what I'm trying to say.
 
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Thank you for the reply.

In real work, concepts drive everything we do, and memory is based on the person (some people remember well and others dont) and our familiarity with the subject at hand (things we use often, we remember, and things we rarely use, we are more likely to forget).

For this case, I would make a similar statement as I did before that it differs from one place to another. I took some courses from PhD instructors in the past. If you ask me they didn't know much. Their minds were stagnant and so were they. I don't know how things work out in the US but it seems you guys are always in touch with the subjects related to your field. You are right in saying that concepts drive everything in practical work and I completely agree with this. In many countries, as you know, senior engineers are given managerial types of jobs where their main role is oversee the project and utilize other junior engineers' and assistants' talents.

You are right that testing strategy varies with institution and also with the subject and with the teacher. I've taken many courses where we were allowed to use all our notes and our books too. Most of my graduate level classes had take home exams where any (non-human) resources could be used. These types of tests heavily stress the creative use of concepts. Then there are other situations where no books or notes are allowed and you are expected to know by memory. I think for engineering/physics subjects, the former is better than the latter since it reflects what people do in the real world.

I wish I had the same privilege! Where I study they don't even let you use a graphic calculator and during an exam you can't even share a simple scientific calculator with some other guy. The worse of all you can't even to a washroom! :) I think now you can see the truth in my statement that if it wasn't for persons like you then I would have said goodbye to the studies long ago!

By the way, was it the same exponential form given under **broken link removed** "Equivalent Forms of Fourier's Integral Theorem" that you had in mind?

Kindly also help me with **broken link removed** when you have time. The Document #1 used in the attachment was taken from this PDF.

Thanks a lot.

Best wishes
PG
 
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By the way, was it the same exponential form given under **broken link removed** "Equivalent Forms of Fourier's Integral Theorem" that you had in mind?

Yes, that is the one. The complex exponential is a very useful substitute for sine and cosine functions. The resulting math is quite beautiful and extremely practical. As far as I can tell, it is the standard form used in electrical engineering.

Kindly also help me with **broken link removed** when you have time. The Document #1 used in the attachment was taken from this PDF.

Q1. When you use the sine/cosine form, you are capturing the phase information with the relative amplitudes of the sine and cosine. You see the formula that relates the phase as θ=atan(-b/a).

Also, in the case of the complex exponential form, the phase is captured by the complex value of the coefficient and the phase is atan(imag(c)/real(c)).

Basically, this all relates to representations in rectangular coordinates and in polar coordinates. Each frequency component needs two values, and you are free to think of x,y or r,θ when you consider a two dimensional space. It just so happens that the math gets a little more elegant if you use a complex space, which also captures two dimensions.

So basically, to answer your question directly, it is important to have amplitude, frequency and phase correct for each component in a Fourier sum.

Q2 I would call it a DC term, not a factor. Terms are things we add, while factors multiply. This term is the DC offset or the average value of the signal.

Q3. The answer is no. The integration for the function is from 0 to 2∏, but the function is zero from ∏ to 2∏. Hence, you must integrate a value of 1 only from 0 to ∏.
 
Thank you, Steve.

The Wikipedia article says:
The frequency spectrum of a time-domain signal is a representation of that signal in the frequency domain. The frequency spectrum can be generated via a Fourier transform of the signal, and the resulting values are usually presented as amplitude and phase, both plotted versus frequency.

I had thought that along the x-axis frequency is plotted and y-axis is used for the representation of amplitude. Could you please tell me how the y-axis incorporates both amplitude and phase information simultaneously? I couldn't find any such graph on the google. I'm sure I was not using a proper search phrase. Thank you.

The Wikipedia article says:
The Laplace transform is related to the Fourier transform, but whereas the Fourier transform expresses a function or signal as a series of modes of vibration (frequencies), the Laplace transform resolves a function into its moments.

I have been through many different webpages but each one confused me more. I feel Fourier series and transform are intuitively easier to understand than the Laplace. What is this "moment" thing in this context? I know the 'moment' from physics as the force multiplied by perpendicular distance from the turning point. Kindly help me if possible. Thnaks

Regards
PG
 
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The Wikipedia article says:

I had thought that along the x-axis frequency is plotted and y-axis is used for the representation of amplitude. Could you please tell me how the y-axis incorporates both amplitude and phase information simultaneously? I couldn't find any such graph on the google. I'm sure I was not using a proper search phrase.

Obviously, you need two plots to specify the two different pieces of information. Usually, you see separate amplitude and phase plots versus frequency. However, you could also plot the real and imaginary components separately.


I feel Fourier series and transform are intuitively easier to understand than the Laplace. What is this "moment" thing in this context? I know the 'moment' from physics as the force multiplied by perpendicular distance from the turning point. Kindly help me if possible. Thnaks

I wouldn't try to understand Laplace from the point of view of moments. That seems very abstract and non-intuitive to me.

You are correct that Fourier is easyier to grasp. However, the switch to Laplace can be made very intuitive if you realize that Fourier requires exciting the system with a simple sine wave, while Laplace requires exciting the system with a decaying/growing sine wave.

The frequency variable s is often written as σ+jω, which is a complex valued frequency. Fourier drives the system with functions that look like exp(jωt), which are complex sine waves cos(jωt)+j sin(jωt). However, Laplace drives the system with functions that look like exp(st), which are complex growing/decaying sine waves exp(σt) (cos(jωt)+j sin(jωt)). The incorporation of the exp(σt) creates either an exponentially growing or exponentially decaying sine wave, depending on the sine of σ, and σ=0 results in the Fourier transform. So, the idea of Laplace theory is to make a more general input function to work with.
 
Hi

Q1:
Do you find what I say here correct? The discrete version of the Fourier is comparatively difficult. I went through three different books and collected the formulas.

Q2:
In almost all the formulas the main variable is Ω which has units of rad/sample. When a discrete function is transformed into its Fourier form, the y-axis shows magnitude and x-axis is labelled with Ω. What does the graph say in this case? I don't get especially the labeling of x-axis. For example, in case of Fourier series for periodic continuous time functions, the y-axis shows magnitude/amplitude and x-axis is labelled with frequency. Suppose, the frequency is 50 Hz and magnitude is 10, then it means that you need a sinusoid with magnitude 10 and frequency 50 Hz. If you think my question is not clear, then please let me know. I will try to put it differently. Thank you.

Regards
PG
 
Q1:
Do you find what I say here correct? The discrete version of the Fourier is comparatively difficult. I went through three different books and collected the formulas.
Overall it seems correct. I haven't checked every detail, but I trust you transferred the formulas correctly.


Q2:
In almost all the formulas the main variable is Ω which has units of rad/sample. When a discrete function is transformed into its Fourier form, the y-axis shows magnitude and x-axis is labelled with Ω. What does the graph say in this case? I don't get especially the labeling of x-axis. For example, in case of Fourier series for periodic continuous time functions, the y-axis shows magnitude/amplitude and x-axis is labelled with frequency. Suppose, the frequency is 50 Hz and magnitude is 10, then it means that you need a sinusoid with magnitude 10 and frequency 50 Hz. If you think my question is not clear, then please let me know. I will try to put it differently. Thank you.

Transforms tell you their meaning directly in their formulas. Look at the inverse transform to see what the frequency domain function means. In all these cases, the transform is the magnitude and phase of a complex exponential as a function of frequency. This is true for discrete time and for continuous time. Whether the inverse transform is a sum or an integral, you are adding up a bunch of complex exponentials, which are essentially sinusoids.
 
Z-transform

Hi

Could you please help me with this query? Though it seems what I'm saying is correct, it's still always better to confirm it. Thank you.
 

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I would draw an analogy with a different types of screw drivers: phillips head type and regular straight type. We use the correct too for the job, generally. Likewise, z-transform versus discrete Fourier transforms, as well as Laplace Transforms versus Fourier Transforms are all useful tools, and each is suited for particular jobs.
 
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