Decoding Periodic Signals: Your Formula Handbook on Fourier Series

Introduction to Fourier Series

The Fourier series is a powerful mathematical tool that decomposes periodic functions into an infinite sum of sine and cosine functions (or complex exponentials). Named after French mathematician Jean-Baptiste Joseph Fourier (1768-1830), this technique has revolutionized signal processing, physics, and engineering.

Key Concept

Any periodic function that satisfies certain conditions (Dirichlet conditions) can be represented as a sum of simple sinusoidal waves of different frequencies, amplitudes, and phases.

Fourier Series Formulas

General Trigonometric Form

For a periodic function $f(x)$ with period $T$:

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

Fourier Coefficients:

$$\begin{aligned} a_0 &= \frac{2}{T} \int_{0}^{T} f(x) \, dx \\[10pt] a_n &= \frac{2}{T} \int_{0}^{T} f(x) \cos\left(\frac{2\pi n}{T}x\right) \, dx \\[10pt] b_n &= \frac{2}{T} \int_{0}^{T} f(x) \sin\left(\frac{2\pi n}{T}x\right) \, dx \end{aligned}$$

Alternative Form (Angular Frequency)

Using angular frequency $\omega_0 = \frac{2\pi}{T}$:

$$f(t) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left[ a_n \cos(n\omega_0 t) + b_n \sin(n\omega_0 t) \right]$$ $$\begin{aligned} a_0 &= \frac{2}{T} \int_{-T/2}^{T/2} f(t) \, dt \\[10pt] a_n &= \frac{2}{T} \int_{-T/2}^{T/2} f(t) \cos(n\omega_0 t) \, dt \\[10pt] b_n &= \frac{2}{T} \int_{-T/2}^{T/2} f(t) \sin(n\omega_0 t) \, dt \end{aligned}$$

Different Forms of Fourier Series

1. Amplitude-Phase Form

$$f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} A_n \cos\left(\frac{2\pi n}{T}x - \phi_n\right)$$

Where:

$$A_n = \sqrt{a_n^2 + b_n^2}, \quad \phi_n = \arctan\left(\frac{b_n}{a_n}\right)$$

2. Complex Exponential Form

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

Complex Coefficients:

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

Relationship with Trigonometric Coefficients:

$$c_0 = \frac{a_0}{2}, \quad c_n = \frac{a_n - ib_n}{2}, \quad c_{-n} = \frac{a_n + ib_n}{2} = c_n^*$$

3. Period 2π Form

For functions with period $2\pi$:

$$f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} [a_n \cos(nx) + b_n \sin(nx)]$$ $$a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) \, dx, \quad b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx) \, dx$$

Properties of Fourier Series

1. Linearity

If $f(x)$ and $g(x)$ have Fourier series, then:

$$\mathcal{F}[af(x) + bg(x)] = a\mathcal{F}[f(x)] + b\mathcal{F}[g(x)]$$

2. Parseval's Theorem

Energy conservation in frequency domain:

$$\frac{1}{T} \int_{0}^{T} |f(x)|^2 \, dx = \frac{|a_0|^2}{4} + \frac{1}{2}\sum_{n=1}^{\infty} (a_n^2 + b_n^2)$$ $$= \sum_{n=-\infty}^{\infty} |c_n|^2$$

3. Time Shifting

Shifting in time domain:

$$f(x - x_0) \leftrightarrow c_n e^{-i\frac{2\pi n}{T}x_0}$$

4. Differentiation

If $f(x)$ is continuous and differentiable:

$$f'(x) \leftrightarrow i\frac{2\pi n}{T}c_n$$

5. Integration

Integration of Fourier series:

$$\int f(x) dx \leftrightarrow \frac{T}{i2\pi n}c_n \text{ (for } n \neq 0\text{)}$$

6. Orthogonality

Sine and cosine functions are orthogonal:

$$\int_{0}^{T} \cos(n\omega_0 t)\cos(m\omega_0 t) dt = \begin{cases} 0 & n \neq m \\ T/2 & n = m \end{cases}$$

Convergence Conditions

Dirichlet Conditions

A periodic function $f(x)$ can be represented by a convergent Fourier series if it satisfies the Dirichlet conditions:

Condition 1: $f(x)$ must be periodic with period $T$
Condition 2: $f(x)$ must have a finite number of discontinuities in any period
Condition 3: $f(x)$ must have a finite number of maxima and minima in any period
Condition 4: $f(x)$ must be absolutely integrable over one period: $$\int_{0}^{T} |f(x)| \, dx < \infty$$

Convergence at Discontinuities

Gibbs Phenomenon

At points of discontinuity, the Fourier series converges to the average of the left and right limits:

$$\lim_{N \to \infty} S_N(x_0) = \frac{f(x_0^-) + f(x_0^+)}{2}$$

The Gibbs phenomenon causes an overshoot of approximately 9% of the jump magnitude near discontinuities, which doesn't disappear as more terms are added.

Symmetry Properties

Symmetry Type	Condition	Fourier Series Form
Even Function	$f(-x) = f(x)$	Only cosine terms: $b_n = 0$ $a_n = \frac{4}{T}\int_0^{T/2} f(x)\cos\left(\frac{2\pi nx}{T}\right)dx$
Odd Function	$f(-x) = -f(x)$	Only sine terms: $a_n = 0$, $a_0 = 0$ $b_n = \frac{4}{T}\int_0^{T/2} f(x)\sin\left(\frac{2\pi nx}{T}\right)dx$
Half-Wave Symmetry	$f(x + T/2) = -f(x)$	Only odd harmonics: $a_n = b_n = 0$ for even $n$
Quarter-Wave Symmetry	Even + Half-wave	Only odd cosine terms

Applications of Fourier Series

Signal Processing

Analysis of periodic signals, filtering, spectral analysis, audio compression (MP3), and image compression (JPEG).

Electrical Engineering

AC circuit analysis, power system harmonics, communication systems, and antenna design.

Physics

Heat transfer, wave propagation, quantum mechanics, vibration analysis, and acoustics.

Mathematics

Solving partial differential equations, boundary value problems, and studying function spaces.

Control Systems

System response analysis, frequency domain design, and stability analysis.

Data Compression

Basis for many compression algorithms including DCT (Discrete Cosine Transform) used in multimedia.

Common Fourier Series Examples

1. Square Wave

Function definition (period $2\pi$):

$$f(x) = \begin{cases} 1 & 0 < x < \pi \\ -1 & \pi < x < 2\pi \end{cases}$$

Fourier Series:

$$f(x) = \frac{4}{\pi}\sum_{n=1,3,5,...}^{\infty} \frac{1}{n}\sin(nx) = \frac{4}{\pi}\left[\sin(x) + \frac{1}{3}\sin(3x) + \frac{1}{5}\sin(5x) + \cdots\right]$$

2. Sawtooth Wave

Function: $f(x) = x$ for $-\pi < x < \pi$

Fourier Series:

$$f(x) = 2\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}\sin(nx) = 2\left[\sin(x) - \frac{1}{2}\sin(2x) + \frac{1}{3}\sin(3x) - \cdots\right]$$

3. Triangular Wave

Function: $f(x) = |x|$ for $-\pi < x < \pi$

Fourier Series:

$$f(x) = \frac{\pi}{2} - \frac{4}{\pi}\sum_{n=1,3,5,...}^{\infty} \frac{1}{n^2}\cos(nx)$$

4. Full-Wave Rectified Sine

Function: $f(x) = |\sin(x)|$

Fourier Series:

$$f(x) = \frac{2}{\pi} - \frac{4}{\pi}\sum_{n=1}^{\infty} \frac{1}{4n^2-1}\cos(2nx)$$

Important Notes

The Fourier series provides a frequency domain representation of periodic signals
For non-periodic functions, use the Fourier Transform instead
The fundamental frequency is $f_0 = 1/T$, and all other components are integer multiples (harmonics)
More terms in the series provide better approximation to the original function
Computational implementation uses the Discrete Fourier Transform (DFT) and its efficient algorithm, the Fast Fourier Transform (FFT)

Symmetry Type	Condition	Fourier Series Form
Even Function	\(f(-x) = f(x)\)	Only cosine terms: \(b_n = 0\) \(a_n = \frac{4}{T}\int_0^{T/2} f(x)\cos\left(\frac{2\pi nx}{T}\right)dx\)
Odd Function	\(f(-x) = -f(x)\)	Only sine terms: \(a_n = 0\), \(a_0 = 0\) \(b_n = \frac{4}{T}\int_0^{T/2} f(x)\sin\left(\frac{2\pi nx}{T}\right)dx\)
Half-Wave Symmetry	\(f(x + T/2) = -f(x)\)	Only odd harmonics: \(a_n = b_n = 0\) for even \(n\)
Quarter-Wave Symmetry	Even + Half-wave	Only odd cosine terms