Alternate expressions of Fourier series formula

[Esran]

Homework Statement

Show that the Fourier series formula F(t)=\frac{1}{2}a_{0}+\sum^{\infty}_{n=1}(a_{n}cos(nwt)+b_{n}sin(nwt)) can be expressed as F(t)=\frac{1}{2}a_{0}+\sum^{\infty}_{n=1}c_{n}cos(nwt-\phi_{n}). Relate the coefficients c_{n} to a_{n} and b_{n}.

Homework Equations

We have the usual equations for the coefficients of a Fourier series.

The Attempt at a Solution

I'm really just checking the integrity of my solution here. I want to be sure I did not misunderstand the nature of \phi_{n} or anything else.

Let n\in Z^{+}. Pick \phi_{n} such that \frac{a_{n}}{cos(\phi_{n})}=\frac{b_{n}}{sin(\phi_{n})}. We know we can do this since we could just choose \phi_{n}=nw_{0}t, where w_{0}=2 \pi f, the fundamental frequency of the Fourier series. Let c_{n}=\frac{a_{n}}{cos(\phi_{n})}=\frac{b_{n}}{sin(\phi_{n})}.

F(t)=\frac{1}{2}a_{0}+\sum^{\infty}_{n=1}(a_{n}cos(nwt)+b_{n}sin(nwt))
F(t)=\frac{1}{2}a_{0}+\sum^{\infty}_{n=1}(c_{n}cos(\phi_{n})cos(nwt)+c_{n}sin(\phi_{n})sin(nwt))
F(t)=\frac{1}{2}a_{0}+\sum^{\infty}_{n=1}(c_{n}cos(nwt-\phi_{n}))

Which completes the problem.

 

[diazona]

Your proof seems fine, but the one thing I would point out is that φ_n is supposed to be a constant. So you shouldn't pick φ_n = nω₀t, because that would make the phase constant a linear function of time. Instead, just solve the equation
\frac{a_n}{\cos\phi_n} = \frac{b_n}{\sin\phi_n}
for φ_n, getting
\phi_n = \tan^{-1}\frac{b_n}{a_n}
(I'll leave the investigation of the case a_n=0 to you ) Using that value of φ_n, you can define c_n the same way you did, and the rest of the proof should be unchanged.

 

[Esran]

Thanks!