Fourier Series and Fourier Analysis of a Signal
Most of the phenomena studied in the domain of Engineering and Science are periodic in nature. For instance, current and voltage in an alternating current circuit. These periodic functions could be analyzed into their constituent components (fundamentals and harmonics) by a process called Fourier analysis. A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series, but not all trigonometric series are Fourier series. Fourier series is used to describe a periodic signal in terms of cosine and sine waves. In other words, it allows us to model any arbitrary periodic signal with a combination of sines and cosines.
1. Fundamental Frequency
The fundamental angular frequency of a periodic signal \( x(t) \) with period \( T \) is
\[ \omega_0 = \frac{2\pi}{T} \]
2. Forms of the Fourier Series
A periodic signal can be represented either in trigonometric form or in complex exponential form.
2.1 Trigonometric (Sine–Cosine) Form
\[ x(t) = a_0 + \sum_{n=1}^{\infty} \left[a_n\cos(n\omega_0 t) + b_n\sin(n\omega_0 t)\right] \]
2.2 Complex Exponential Form
\[ x(t) = \sum_{n=-\infty}^{\infty} C_n e^{jn\omega_0 t} \]
3. Fourier Series Coefficients
3.1 Trigonometric Coefficients
DC term:
\[ a_0 = \frac{1}{T}\int_{t_0}^{t_0 + T} x(t)\, dt \]
Cosine coefficients:
\[ a_n = \frac{2}{T}\int_{t_0}^{t_0 + T} x(t)\cos(n\omega_0 t)\, dt \]
Sine coefficients:
\[ b_n = \frac{2}{T}\int_{t_0}^{t_0 + T} x(t)\sin(n\omega_0 t)\, dt \]
3.2 Complex Exponential Coefficients
\[ C_n = \frac{1}{T}\int_{t_0}^{t_0 + T} x(t)e^{-jn\omega_0 t}\, dt \]
4. Example: Fourier Series of \( x(t) = t \) on \([-1,1]\)
\[ \omega_0 = \pi \]
\[ a_0 = 0 \]
\[ a_n = 0 \]
\[ b_n = \frac{2(-1)^{n+1}}{n\pi} \]
Final Fourier series:
\[ x(t)= \sum_{n=1}^{\infty}\frac{2(-1)^{n+1}}{n\pi}\sin(n\pi t) \]
Fourier Series of a Rectangular Wave
5. Definition of the Rectangular Wave
\[ x(t)= \begin{cases} A, & |t| < \frac{T}{2} \\ 0, & \text{otherwise} \end{cases} \]
