A question in my lecture was:
Determine the Fourier series coefficients of the signal x(t) = sin3(πt)
And a hint was that I didn't need to evaluate any integrals (the Fourier analysis functions) to obtain my answer.
I know A0 = 0 because this is the average over a period of a sine function. But how do I evaluate any other Ak by intuition?
Thanks in advance.
[Math] Fourier series coefficients without integrals
fourier analysisfourier seriessignal processing
Best Answer
Using \begin{align} \sin^{3}(x) &= - \frac{1}{8 i} \, \left( e^{i x} - e^{-i x} \right)^{3} \\ &= - \frac{1}{ 8 i} \, ( 2 i \sin(3 x) - 6 i \sin(x)) \\ &= \frac{3}{4} \, \sin(x) - \frac{1}{4} \, \sin(3 x) \end{align} then it is determined that there are only two non-zero coefficients of a Fourier Sine series expansion.
Since the Fourier Sine series is given by $$F_{s}(x) = \sum_{n=1}^{\infty} B_{n} \, \sin(n x) = B_{1} \, \sin(x) + B_{2} \, \sin(2 x) + B_{3} \, \sin(3 x) + \cdots$$ then for $F_{s}(x) = \sin^{3}(x)$ it is determined that $B_{2} = 0$, $B_{n} = 0, n \geq 4$, and $B_{1} = \frac{3}{4}$, $B_{3} = - \frac{1}{4}$.