What could be the mathematical equation of the given signal?

Question

We know that Fourier series for periodic signal $y(t)$ is given by

$$ y(t) = \sum\limits_{m=0}^{+\infty} a_m \cos(w_m t) + \sum\limits_{m=0}^{+\infty}b_m \sin(w_m t). \quad (2)$$

Now,I want to find Fourier series of a periodic sinusoidal signal $y(t)$ given below. But I don't understand how should I decide what are the harmonics present and how to calculate Fourier coefficients.

So what could be the mathematical equation of the given signal ?

Note: if there is any way than Fourier series to express the given signal in the form of mathematical equation ,you can explain with other way also.

Answer 1

You can calculate it's Fast Fourier Transform, but choose to only calculate the frequencies you need.

To express Fourier series in terms of a continous Fourier transform, we need to introduce Theory of distributions as the fourier transform of sines and cosines are sums of Dirac distributions. If we are fine with considering the discrete fourier transform (DFT) then fourier series and the DFT coincide.

Combine this with the fact that the Fourier transform is linear and you will be done.

Answer 2

Just looking at the signal, it seems to have components only up to the fourth harmonic. You can read the $y(t)$ values at intervals of $\frac T8$ from the plot-I would pick the peaks of all the obvious waves, then use the orthogonality of the Discrete Fourier Transform to compute the coefficients. You can do that in Excel easily.

Added: As points out, the function is odd so only sines will be involved. It appears four terms will be enough to get close, so the form would be $$y(t)=\sum_{i=1}^4 a_i \sin \frac {2 \pi i}T$$ You can just pick off the first four peaks from the graph, which seem to be at $\frac T{16}, \frac {3T}{16}, \frac {5T}{16}, \frac {7T}{16}$ and solve the four simultaneous equations for the $a_i$. The FFT is easier if you learn how to do it, as it gives you each coefficient directly.