Fourier Series Formula

The Fourier series represents a periodic function \(f(x)\) as an infinite sum of sines and cosines or as an infinite sum of complex exponentials. It can be expressed in both trigonometric and exponential forms:

\[f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left[ a_n \cos\left(\frac{2\pi n}{T}x\right) + b_n \sin\left(\frac{2\pi n}{T}x\right) \right]\]

Trigonometric Form:

\[\begin{aligned} a_0 &= \frac{1}{T} \int_{0}^{T} f(x) \, dx \\ a_n &= \frac{2}{T} \int_{0}^{T} f(x) \cos\left(\frac{2\pi n}{T}x\right) \, dx \\ b_n &= \frac{2}{T} \int_{0}^{T} f(x) \sin\left(\frac{2\pi n}{T}x\right) \, dx \\ T &= \text{Period of the function } f(x) \end{aligned}\]

Where:

\[f(x) = \sum_{n=-\infty}^{\infty} c_n e^{i\frac{2\pi n}{T}x}\]

Exponential Form:

\[c_n = \frac{1}{T} \int_{0}^{T} f(x) e^{-i\frac{2\pi n}{T}x} \, dx\]

Where:

Properties of Fourier Series

The Fourier series has several properties:

Linearity: The Fourier series is a linear operation. The series of a linear combination of functions is the same as the linear combination of the individual series.
\[\frac{1}{T} \int_{0}^{T} |f(x)|^2 \, dx = \sum_{n=-\infty}^{\infty} |c_n|^2\]
\(f(x)\)Parseval’s Theorem:
Periodicity: The Fourier series repeats with the fundamental frequency \(\frac{1}{T}\), where \(T\) is the period of the function.
Convergence: The Fourier series converges to \(f(x)\) under certain conditions, such as piecewise continuity and boundedness of \(f(x)\).
Orthogonality: The sines and cosines in the trigonometric form of the Fourier series are orthogonal functions, making it suitable for analyzing periodic signals.
Complex Exponentials: The exponential form of the Fourier series simplifies calculations involving complex signals and is often preferred in engineering applications.

These properties make the Fourier series a powerful tool in signal processing, mathematics, and engineering.