Fourier Series coefficient question

[okami11408]

I'm learning about the Fourier Series and have a few question.

First, we have 2 methods to find the Fourier coefficient of the Fourier Series.

1. Use the formula Cn below.

note that: x=2∏ft

2. Expand the function.

For Example I have y(t)=3cos(2∏9t)+4cos(2∏8t), let use the Expansion method

y(t)=3cos(2∏9t)+4cos(2∏8t) <---- Already expand

we have

y(j2∏9t)=3 at f=9 Hz Fourier coefficient=3/2
y(j2∏8t)=4 at f=8 Hz Fourier coefficient=4/2

Note:I devided by 2 for two side analysis.

This method makes sense to me.

but when it comes to use of formula Cn, it becomes nonsense.

note that: x=2∏ft

My question is how come multiply the whole function with sine & cosine at some

particular frequency and integrate it. it turns into the Fourier coefficient.

I mean why is that actually work? It's an area, how can that related to the coefficient.

I want some intuition.

Thank you!

 

[p.0774023179]

I hope i'll get it clear enough.
Once you fix a frequency for your Fourier decomposition, the sin-waves (of that decomposition) are harmonics of that frequency. So if your frequency is f, then the sin-waves will be of frequency f, 2f, 3f, up to infinity (in theory). Now this is the hint: the sin-waves used for the decomposition (i.e. the sins of the frequencies above) behave like "a vector basis over a vector-space"! If you multiply any two (e.g. sin(f)*sin(nf) for instance) of them and integrate the result is null -- they are orthonormal. If you multiply one with itself (and integrate) the result is non-null. And most important any signal can be written as a weighted sum of the vector basis in that vector space .... this is known as the Fourier transform.

The intuition is as simple as that!

It actually works in the same way for the wavelet theory. There again the (decomposing) wavelets are a vector-base over the vector-space of all signals ...

To answer your precise question " how come multiply the whole function with sine & cosine at some particular frequency and integrate it. it turns into the Fourier coefficient."
It is simple: when multiplying (a whatever signal) with a sin-wave, all the sin-waves (in that signal's decomposition as a sum of sin-waves) that are not of the same harmonic as the sin-wave will null-out by the considerents presented above. Only the one with the same frequency will stay ... and return the Fourier coeficient.

Hope this was clear ...

I did a quick search for you and found this **broken link removed**
Check section 3 ...

Good luck!

 

[okami11408]

Thank you so much!

That changes my life!

I never know a signal behave like a vector!

But I still confused at some point.

1. If a signal has frequency f1 f2 f3 f4... Is it the same thing as vector has i j k component?

2. Is there any theory behind multiply and integrate? Why multiply and integrate behave like the dot product?

3. Why we have to divided the integral result by the period(2pi)?

I know if we don't do that the coefficient won't come, but why?

And If I take limit T-->infinity and integral from -infinity--->+infinity.

why the equation become zero not a Fourier transform?

This is the most difficult topic I've encountered.

Please help again! Thank you!

 

[p.0774023179]

You're welcome.
I suggest you search for "Hilbert space" at this link: https://en.wikipedia.org/wiki/Fourier_series .... you will see that:
1) the 2pi comes from a definition
2) that the vector space I was refering to actually has a name
3) that the Fourier transform is just an instance in a larger theory

And be careful with the language! You cannot say that if a signal has some "frequency f1, f2, etc ... is the same thing as vector i j k components".
You can say that harmonic sin-waves of frequency f, 2f, 3f, etc. behave like the i j k components. Because, like for i, j and k, multipliying any two (not the same) of them returns zero and the multiplication by itself is not zero. Also, and most important while for a vector v=a*i + b*j + c*k the a,b and c are unique (the proof is simple) so in the case of Fourier the a, b and c are the Fourier coeficients -- which are also unique.

There is no theory behind the multiply and integrate. It just happens that in the case of harmonic sin-waves the multiply and integrate behaves nice. I miss clear words here to get any further ... Multiply and integrate behaves like a dot-product in this particular case of sin-waves!! It also behaves like this for some other classes of waves -- recall the wavelets I mentioned in my first email. In the end mathematics is abstraction ...

Last, the integral is from -pi to pi because the Fourier transform supposes a periodic function and -pi to pi is a period for the signal.

I sugest with this intuition you try to look in some math/engineering text books. The effort to clearly express these matters is not small

hope this helps.

 

[okami11408]

That helps a lot, I finally got it! :-o

And I just found the derivation of the equation too! https://cnx.org/content/m10733/latest/

Thank you so much! Thank you!