Complex Fourier Series Coeffcients; what are they?

[CE Trainee]

Complex Fourier Series Coeffcients; what are they? what do they represent?

Homework Statement

I'm not sure if this is the right place for this but it seems appropriate. I just started an intro signals and systems course at my university at the beginning of this semester. We started Complex Fourier Series wednesday and after the lecture, I was confused about a couple of things.

I took an electrical engineering math class last semester that introduced the fundamental math that EE's should know. It didn't go into too much detail but it covered the basics. My problem is in both classes, the professors introduced complex Fourier series and talked about finding the coefficients. They never said what the complex coefficients actually are or what they represent or what they are used for. Thats my question. What are the Fourier series coefficients? What do they represent and why are they important? I suppose I could mindlessly chug through the formulas but I like to understand things in their entirety.

Also, if anyone can recommend some supplemental reading and exercises that really helps explain the material, it would be much appreciated.

Thanks for the help.

 

[ystael]

Are you asking what the Fourier series of a function is or means? The Wikipedia entry on Fourier series provides a good introduction. They start out with real sine and cosine series because in that case it's a little easier to visualize what's going on, but that doesn't really matter -- the exponential series is just a much nicer way to keep track of all the same information.

 

[HallsofIvy]

A "complex Fourier series" is of the form $f(x)= \sum_{n=-\infty}^\infty a_n e^{inx}$. Essentially, you can think of the functions $e^{inx}$ as an orthonormal basis for an infinite dimensional vector space (where the inner product is defined as [/itex]<f, g>=1/(2\pi i) \int_{-\infty}^\infty f(x)\overline{g}(x)dx[/itex], where the $1/(2\pi i)$ "normalizes" the "basis vectors". From that, then, the coefficients are given by $a_n= 1/(2\pi i)\int_{-\infty}^\infty} f(x)e^{inx}dx$.

 

[CE Trainee]

A "complex Fourier series" is of the form $f(x)= \sum_{n=-\infty}^\infty a_n e^{inx}$. Essentially, you can think of the functions $e^{inx}$ as an orthonormal basis for an infinite dimensional vector space (where the inner product is defined as [/itex]<f, g>=1/(2\pi i) \int_{-\infty}^\infty f(x)\overline{g}(x)dx[/itex], where the $1/(2\pi i)$ "normalizes" the "basis vectors". From that, then, the coefficients are given by $a_n= 1/(2\pi i)\int_{-\infty}^\infty} f(x)e^{inx}dx$.

What are the "coefficients"? What do they represent? Are they amplitudes of the sinusoids? Thats what I don't understand. The professor asks to find the coefficients but doesn't explain what they are.

 

[ystael]

What are the "coefficients"? What do they represent? Are they amplitudes of the sinusoids? Thats what I don't understand. The professor asks to find the coefficients but doesn't explain what they are.

Yes, exactly the amplitudes of the sinusoids; the coefficients are the quantities $$a_n$$ in HallsOfIvy's post above.

 

[vela]

If you expand a real function $f(x)$, you can show that $c_n = c_{-n}^*$. If you write $c_n = (a_n + i b_n)/2$, you can then combine the $e^{inx}$ and $e^{-inx}$ terms and get

$$c_ne^{inx}+c_{-n}e^{-inx} = a_n\frac{e^{inx}+e^{-inx}}{2}+ib_n\frac{e^{inx}-e^{-inx}}{2} = a_n\cos(nx)-b_n\sin(nx)$$

So you can think of the complex series as shorthand for the sine and cosine series.

Another way of thinking about $c_n$ is that it contains all of the information about the nth frequency component, both amplitude and phase.