Fourier Series Expansion Question

Other Math and Science
1

Having a little trouble understanding this math. Hopefully someone here can help.

In the Fourier Series Expansion as shown below:

http://www.engr.trinity.edu/~paul/fourier/fourier/img25.gif

I noticed the Ao term is divided by two, where in other texts I see it like more this:

where the Ao term is not divided by two. What is the reason for this? Is there also a difference in the way Ao is defined between texts? I have plotted a Fourier Series expansion of a function, over the function itself, without dividing Ao by two, and the fourier series expansion fits nicely over top of it. If I divide Ao by two, the fourier expansion sits way too low on the plot. Is Ao supposed to be divided by two or not?

Also, on finding the Fourier Series Expansion Coefficients, is taking the integral from zero to T (one period) just the same as taking the integral from -T/2 to +T/2?

Finally, to someone experienced in such things, ignoring the slight roughness (didn’t plot with a small enough time interval), does this look mostly correct, for n=8 terms?

http://4true.org/images/fourier_n8.JPG

Thanks.

2

This site may help. http://en.wikipedia.org/wiki/Fourier_series
I did fourier analysis is college(Electrical Engineering Technologies), using it you can represent a non-sinasoidial(saw-tooth) waveform using a series of sinasoidial waveforms.(sin wave)

3

Isn’t this what you use to make music tuners?

4

A Fourier series is just one small example in my favorite (and as far as I’m concerned, the most interesting and exciting) branch of mathematics.

If you REALLY want to understand Fourier series expansion, I HIGHLY recommend you take a course in Linear Algebra, and maybe a Linear Systems class, usually given in the control systems area of engineering.

For all of you that have had basic matrix algebra and calculus 3, you have seen vector projections. That is, you can project any vector onto another vector to determine its projection (like a shadow) onto that vector. Taking this a little further, you then learn how to project any vector onto a set of “basis” vectors - this gives you the “components” of the original vector. In calc 3 you learned the (i,j,k) basis vectors, which more or less breaks down to the (x,y,z) coordinates in 3-D space.

Sorry for this up until this point, but if you’ve followed along so far, hopefully you get this next part…

What most people don’t realize, is that linear algebra can be further extended from basis vectors into basis functions.

Just like the (i,j,k) are orthogonal and normal vectors (i.e. orthonormal vectors), functions can be made that are also orthonormal, making them “basis” functions (just like the “basis” vectors). Then, any arbitrary function can be projected onto this set of orthonormal “basis” functions, giving you the “components” of the original functions onto the basis functions.

In the Fourier series, the sin and cos terms ARE orthonormal basis functions, and the coefficients that you are calculating are simply the projections of the original functions onto the basis functions.

Once you understand this, the possibilities are endless: you can figure out all kinds of cool basis functions to project upon and make great approximations for many different applications and solve all kinds of cool problems. If you’ve ever heard of or have used any of the following tihngs, you’ve seen this in action:

  • Fourier series
  • Frequency domain analysis / signal processing
  • Finite Element Analysis
  • Computational Fluid Mechanics
  • Wavelet transforms
  • Element Free Analysis
  • cubic spline approximations
    … and many more.

Anyway, linear algebra and linear systems is something to get truly excited about. I always thought math was cool, but after taking these courses, all of those weird things that never quite made sense (like Fourier series) all of the sudden clicked and seemed like second nature. I highly recommend you look into it.

Lastly, I want to apologize for my poor attempt to explain this stuff. I’m a bit busy now and have other stuff on my mind, so I couldn’t put a lot of time into the explanation. Hopefully I’ll get some time soon and come back to this post and edit it into a more understandable explanation.

5

Thanks. That aspect of the transform theory has always confused me to no end. I could always get over the fact that Im working in a domain that has no relevance to time because as Oliver Heaviside once said, “Why should I refuse a good dinner simply because I don’t understand the digestive processes involved?” Convolution intergral or multiplication? Convolution integral or multipication? The answer is self explanatory.:slight_smile:

6

Ao is the DC component… You can check your formula by assuming f(x) = constant = c and integrate over the period. You will probably find that A0 = 2*C so it is necessary that f(x) = A0/2 + … unless your formula for An has a 2 compensation built in.