Fourier Series and Fourier Analysis of a Signal
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 \([-1, 1]\), \( T = 2 \)
Fundamental frequency:
\[ \omega_0 = \frac{2\pi}{2} = \pi \]
Compute coefficients:
\[ a_0 = \frac{1}{2}\int_{-1}^{1} t\, dt = 0 \]
\[ a_n = \int_{-1}^{1} t\cos(n\pi t)\, dt = 0 \quad (\text{odd-even symmetry}) \]
\[ b_n = \int_{-1}^{1} t\sin(n\pi t)\, dt = \frac{2(-1)^{n+1}}{n\pi} \]
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
\[ x(t)= \begin{cases} A, & |t| < \frac{T}{2}, \\ 0, & \text{otherwise} \end{cases} \]
This pulse repeats every \( T_0 \). Define:
- Pulse width: \( T \)
- Period: \( T_0 \)
- Duty cycle: \( D = \frac{T}{T_0} \)
- Fundamental frequency: \( \omega_0 = \frac{2\pi}{T_0} \)
6. Fourier Series (Complex Exponential Form)
The coefficients are
\[ C_n = AD \, \mathrm{sinc}(n\pi D) \]
Explicitly:
\[ C_n = \frac{AT}{T_0} \frac{\sin\left(n\pi \frac{T}{T_0}\right)}{n\pi \frac{T}{T_0}} \]
The DC component:
\[ C_0 = AD \]
7. Fourier Series (Trigonometric Form)
Since the rectangular wave is even:
- \( b_n = 0 \)
\[ x(t) = a_0 + \sum_{n=1}^{\infty} a_n\cos(n\omega_0 t) \]
\[ a_0 = AD \]
\[ a_n = 2AD\,\mathrm{sinc}(n\pi D) \]
8. Special Case: Square Wave (50% Duty Cycle)
\( D = \frac{1}{2} \)
The Fourier series becomes
\[ x(t) = \frac{4A}{\pi} \left[ \cos(\omega_0 t) + \frac{1}{3}\cos(3\omega_0 t) + \frac{1}{5}\cos(5\omega_0 t) + \cdots \right] \]
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.
The common form of the Fourier series
Sinusoidal functions are periodic over 2Ï€ angular distance.For a periodic function f(x), (let’s assume a function other than sinusoidal function i.e., a square wave with a period of 2Ï€)
** I’ve often used the terms ‘cos(nx)’ and ‘sin(nx)’ throughout the article. They actually indicates that cos(nx) and sin(nx) are periodic on the interval 2Ï€ for any integer n
e.g., for n = 1, cosx has a period of 2Ï€/1 = 2Ï€
for n = 2, cos2x has a period of 2π/2 = π and so on.
The signal's direction of propagation is indicated by the letter ‘x’. It denotes the signal is propagating along x-axis. This axis can also be used as a ‘time’ axis.
In the above equations, we assumed that the periodic signal f(x) has a period of 2Ï€. So now, if a function is periodic on the interval [-L, L], we'll talk about how to expand it into a Fourier series.
