What is the Fundamental Frequency in the Fourier Series of cos4t + sin8t?

[helderdias]

Hi everyone,

So I was trying to calcule the Fourier Series of x(t) = cos4t + sin8t, but I'm a little bit confused. What would be ω₀ in this case since I have a combination of two functions with different frequencies?

Thank you in advance.

 

[HallsofIvy]

I don't know what you mean by \omega_0 here. A Fourier series is a sum
\sum_{n=0}^\infty A_n sin(nt)+ B_n cos(nt)
Here, obviously A_8= 1, B_4= 1 and all other coefficients are 0.

 

[helderdias]

I thought the series was the sum of An*cos(nw0t) + Bn*sen(nw0t)

 

[theorem4.5.9]

Typically Fourier series are presented as HallsofIvy posted. However, there's no reason not to include a frequency term \omega_0. This can simplify some expressions, and you are free to choose any \omega_0 you like. It's important to remember that any choice of \omega_0 changes the periodicity to \frac{2\pi}{\omega_0}.

For your example, it's clearly best to choose \omega_0=1