Fourier Series
Among the most valuable techniques in signal analysis is the Fourier series. It enables the transformation of any periodic signal into a combination of sinusoids. The resultant signal is the summation of three sinusoids, each with a frequency that is a multiple of the fundamental frequency f0. Therefore, we can simplify a complex periodic signal into sinusoids [4].
A. Definition :
Jean-Baptiste Fourier, a French mathematician, discovered that he could express any periodic signal as a sum of sinusoids. Therefore, for any periodic function f(t), Fourier demonstrated the following equivalence :
\(f\left(t\right)=a_v+\sum_{n=1}^{\infty}{a_ncos\left(n\omega_0t\right)+b_nsin\left(n\omega_0t\right)}\)
Where \(a_v\),\( a_n\) and \(b_n\) are the Fourier coefficients, and \(\omega_0\) is the fundamental frequency. Frequencies that are integer multiples of \(\omega_0\) (such as \({2\omega}_0\) , \({3\omega}_0\) , etc.) are referred to as harmonics. For instance, \({2\omega}_0\) is the second harmonic, \({3\omega}_0\) is the third harmonic, and so on.
B. Fourier Coefficients :
The Fourier coefficients are obtained according to the following equations :
\(a_v=\frac{1}{T}\int_{t_0}^{t_0+T}f\left(t\right)dt\;\) \(\;\) \(\;\)(2)
\(a_n=\frac{2}{T}\int_{t_0}^{t_0+T}f\left(t\right)cos\left(n\omega_0t\right)dt\;\) \(\;\) \(\;\)(3)
\(b_n=\frac{2}{T}\int_{t_0}^{t_0+T}f\left(t\right)sin\left(n\omega_0t\right)dt\;\) \(\;\) \(\;\) (4)
Note that av represents the average value of the signal.
C. Property :
If f(t) is even \(\implies\) \(b_n=0\ \ \ for\ \ \ \ \forall\ t\)
If f(t) is odd \(\implies\) \(a_n=0\ \ \ for\ \ \ \ \forall t\)
When we add a constant to f(t), it alters the average value, not \(a_n\) and \(b_n\)