Fourier series: relation of coefficients

[divB]

Hi,

The Fourier series can (among others) expressed in terms of sines and cosines with coefficients $a_n$ and $b_n$ and solely by sines using amplitudes $A_n$ and phase $\phi_n$.

I want to express the latter using $a_n$ and $b_n$. Using

$$a_n = A_n \sin(\phi_n) \\ b_n = A_n \cos(\phi_n)$$

I quickly found $A_n$ by expressing the arccos and arcsin. For $\phi_n$ I would get

$$\phi_n = \arccos \frac{a_n}{\sqrt{a_n^2 + b_n^2}}$$

However, according to the German Wikipedia (http://de.wikipedia.org/wiki/Fourier-Reihe) this seems not so trivial. One option there is (for n \neq 0):

$$\phi_n = 2 \arctan \frac{b_n}{A_n + a_n}$$

or using arctan and signum function. What I am missing or is my approach also correct?

Thanks,
divB

 

[Simon Bridge]

You are starting with the two styles of the Fourier transforms:
$$\sum_n a_n\cos nx + b_n\sin nx = \sum_m A_m\sin(mx+\phi_m)$$
... and you want to relate $\{a_n,b_n\}$ to $\{A_m,\phi_m\}$ ... is that correct?

 

[divB]

yes, exactly. And I relate them with $\sin(a+b) = \sin(a)\cos(b) + \cos(a)\sin(b)$ to get:

$$A_0 = a_0/2\\ a_0 = 2A_0$$

$$a_n = A_n \sin\phi_n \\ b_n = A_n \cos\phi_n$$

For represeting $A_n$ I get using both equations above:

$$A_n = a_n/\sin\phi_n \\ cos\phi_n = b_n/A_n \\ \phi_n = \arccos b_n/A_n = \arcsin \sqrt{1-b_n^2/A_n^2} \\ A_n = \frac{a_n}{\sqrt{1-\frac{b_n^2}{A_n^2}}} \\ \dots \\ A_n = \sqrt{a_n^2 + b_n^2}$$

I think this is correct, right?
In the same spirit, I can derive

$$\phi_n = \arccos\frac{b_n}{A_n} = \arccos\frac{b_n}{\sqrt{a_n^2 + b_n^2}}$$

... but according to the German Wikipedia this is wrong. Why?

 

[Simon Bridge]

That works ...

To put it another way - you can expand each term in the sin-only expansion in terms of the sin-cos expansion ... i.e.

$$A_n\sin(nx+\phi_n)=\sum_m a_m\cos(mx)+b_m\sin(mx)$$

Then you apply the identity.
Since you are expanding an arbitrary sine in terms of sines and cosines, the sum should only have one term in it... (for each value of n on the RHS.)

Leads you to:

$A_n^2=a_n^2+b_n^2$

$\phi_n=\arctan(a_n/b_n)$

... sure. Nicely done.

 

[blue_raver22]

Hi divB,

Your approach is correct and can be used to find the amplitude and phase of the Fourier series coefficients. However, the other expressions you mentioned from the German Wikipedia are also valid and may be used in different contexts or for different purposes. The important thing to remember is that the coefficients a_n and b_n represent the amplitude and phase of the corresponding sine and cosine terms in the Fourier series, and your method is one way of finding these values. Keep in mind that there may be alternative methods and expressions for calculating these coefficients, but as long as they result in the correct amplitude and phase values, they are all valid.