Fourier series from the graph

Question

I am trying to find the power spectral density, autocorrelation function, and power content of a periodic signal. In order to do this I need to find the Fourier series coefficients.

I am trying to solve the problem analytically, but when I find the closed form of the signal x(t), I found j dependency in the signal. Whatever I do, I cannot get rid of the complex part. Can a real signal have j dependency? How can I find x(t) for this signal analytically from its Fourier series?

Answer 1

This is an antisymmetric signal x'(t) with period T=3, shifted up by 1 unit:

The formula for the coefficents b is:

$$ b_n = \frac{2}{T} \int_{-T/2}^{T/2} x'(t) dt$$

At the end you have

$$x(t) = \sum_k b_k \sin (k \frac{2 \pi}{T}\cdot t) + 1$$