Fourier Series | Electrical Energy Technology

1. Introduction to Fourier Series

The Fourier series represents a breakthrough in mathematical analysis that allows any periodic function to be decomposed into a sum of simple oscillating functions (sines and cosines). This concept revolutionized our understanding of complex waveforms and laid the foundation for modern signal processing and analysis.

1.1 Historical Context

In the early 19th century, while studying heat transfer, Jean-Baptiste Joseph Fourier discovered that complicated periodic functions could be expressed as infinite series of trigonometric functions. Initially met with skepticism by mathematicians including Lagrange and Laplace, Fourier's work eventually gained acceptance and has since become fundamental in various fields of science and engineering.

1.2 Significance in Engineering

For electrical engineering students, Fourier series are particularly important because:

They allow complex periodic signals to be analyzed in terms of their frequency components
They form the foundation for understanding filters, communication systems, and power systems
They provide insights into system behavior in both time and frequency domains
They lead to transforms (Fourier, Laplace) that are essential for advanced circuit analysis

Fourier Series Visualization — Animation showing how a square wave can be approximated by adding sine waves

2. Definition and Basic Concepts

2.1 Periodic Functions

A function \(f(t)\) is periodic with period \(T\) if for all \(t\):

\[ f(t + T) = f(t) \]

The fundamental frequency \(\omega_0\) of the function is related to the period by:

\[ \omega_0 = \frac{2\pi}{T} \]

Example:

The function \(f(t) = \sin(3t)\) has period \(T = \frac{2\pi}{3}\), because:

\(f(t + \frac{2\pi}{3}) = \sin(3t + 2\pi) = \sin(3t) = f(t)\)

Its fundamental frequency is \(\omega_0 = \frac{2\pi}{T} = 3\) rad/s

2.2 The Basic Principle

The Fourier series states that any periodic function \(f(t)\) with period \(T\) can be represented as an infinite sum of sines and cosines:

\[ f(t) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left[ a_n \cos(n\omega_0 t) + b_n \sin(n\omega_0 t) \right] \]

This is the trigonometric form of the Fourier series. The coefficients \(a_0\), \(a_n\), and \(b_n\) determine the contribution of each frequency component.

3. Fourier Coefficients

3.1 Calculating Fourier Coefficients

The Fourier coefficients are calculated using the following integrals:

\[ a_0 = \frac{2}{T} \int_{t_0}^{t_0+T} f(t) \, dt \] \[ a_n = \frac{2}{T} \int_{t_0}^{t_0+T} f(t) \cos(n\omega_0 t) \, dt \] \[ b_n = \frac{2}{T} \int_{t_0}^{t_0+T} f(t) \sin(n\omega_0 t) \, dt \]

Where \(t_0\) can be any starting point for the interval of integration, as long as the interval spans exactly one period \(T\). Common choices are \(t_0 = -\frac{T}{2}\) or \(t_0 = 0\).

Example: Square Wave

For a square wave with period \(T = 2\pi\):

\(f(t) = \begin{cases} 1, & 0 < t < \pi \\ -1, & -\pi < t < 0 \end{cases}\)

The Fourier coefficients are:

\(a_0 = 0\) (no DC component)

\(a_n = 0\) (for all \(n\))

\(b_n = \begin{cases} \frac{4}{n\pi}, & \text{for odd } n \\ 0, & \text{for even } n \end{cases}\)

Therefore, the Fourier series is:

\(f(t) = \frac{4}{\pi}\sum_{k=0}^{\infty} \frac{\sin((2k+1)t)}{2k+1}\)

3.2 Physical Interpretation

The Fourier coefficients have important physical interpretations:

\(a_0/2\) represents the DC component (average value) of the function
\(a_n\) and \(b_n\) represent the amplitudes of the cosine and sine components at frequency \(n\omega_0\)
\(\sqrt{a_n^2 + b_n^2}\) gives the amplitude of the \(n\)th harmonic
\(\arctan(-b_n/a_n)\) gives the phase angle of the \(n\)th harmonic

4. Fourier Series Representations

4.1 Trigonometric Form

The standard trigonometric form of the Fourier series is:

\[ f(t) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left[ a_n \cos(n\omega_0 t) + b_n \sin(n\omega_0 t) \right] \]

This form clearly separates the even (cosine) and odd (sine) components of the function.

4.2 Amplitude-Phase Form

The trigonometric form can be rewritten in the amplitude-phase format:

\[ f(t) = \frac{a_0}{2} + \sum_{n=1}^{\infty} A_n \cos(n\omega_0 t + \phi_n) \]

Where:

\[ A_n = \sqrt{a_n^2 + b_n^2} \] \[ \phi_n = -\arctan\left(\frac{b_n}{a_n}\right) \]

This form is particularly useful in engineering applications as it directly shows the amplitude and phase of each harmonic component.

4.3 Complex Exponential Form

Using Euler's formula, the Fourier series can be expressed in complex exponential form:

\[ f(t) = \sum_{n=-\infty}^{\infty} c_n e^{in\omega_0 t} \]

Where the complex Fourier coefficients \(c_n\) are given by:

\[ c_n = \frac{1}{T} \int_{t_0}^{t_0+T} f(t) e^{-in\omega_0 t} \, dt \]

The relationship between the complex coefficients and the trigonometric coefficients is:

\[ c_0 = \frac{a_0}{2} \] \[ c_n = \frac{a_n - ib_n}{2} \text{ for } n > 0 \] \[ c_{-n} = \frac{a_n + ib_n}{2} \text{ for } n > 0 \]

For real-valued functions, we have the property \(c_{-n} = c_n^*\) (complex conjugate).

Example: Periodic Square Wave

For the square wave defined earlier, the complex coefficients are:

\(c_0 = 0\)

\(c_n = \begin{cases} \frac{2i}{n\pi}, & \text{for odd } n \\ 0, & \text{for even } n \end{cases}\)

\(c_{-n} = -c_n\) (for all \(n\))

Therefore, the complex exponential form is:

\(f(t) = \frac{2i}{\pi}\sum_{k=0}^{\infty} \frac{e^{i(2k+1)t} - e^{-i(2k+1)t}}{2k+1}\)

Which simplifies to:

\(f(t) = \frac{4}{\pi}\sum_{k=0}^{\infty} \frac{\sin((2k+1)t)}{2k+1}\)

5. Convergence Properties

5.1 Dirichlet Conditions

A Fourier series converges to the function at points of continuity if the function satisfies the Dirichlet conditions:

The function \(f(t)\) is periodic
The function has a finite number of discontinuities in one period
The function has a finite number of maxima and minima in one period
The function is absolutely integrable over one period: \(\int_{t_0}^{t_0+T} |f(t)| \, dt < \infty\)

5.2 Convergence at Discontinuities

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

\[ \sum_{n=-\infty}^{\infty} c_n e^{in\omega_0 t_0} = \frac{f(t_0^-) + f(t_0^+)}{2} \]

This phenomenon is known as the Gibbs phenomenon, which manifests as overshoot and ringing near discontinuities even as more terms are added to the series.

5.3 Uniform Convergence

A Fourier series converges uniformly if the function is continuous and has a piecewise continuous derivative. Uniform convergence ensures that the series approximates the function equally well at all points.

5.4 Rate of Convergence

The rate of convergence of a Fourier series depends on the smoothness of the function:

For functions with discontinuities: coefficients decay as \(O(1/n)\)
For continuous functions with discontinuous derivatives: coefficients decay as \(O(1/n^2)\)
For functions with \(k\) continuous derivatives: coefficients decay as \(O(1/n^{k+1})\)
For infinitely differentiable functions: coefficients decay faster than any power of \(1/n\)

Example: Convergence Rates

Consider these two functions:

1. Square wave: \(f(t) = \text{sgn}(\sin(t))\)

2. Triangle wave: \(g(t) = \frac{2}{\pi}|\arcsin(\sin(t))|\)

The square wave has discontinuities, so its Fourier coefficients decay as \(1/n\), making the series converge slowly.

The triangle wave is continuous but has discontinuous derivatives, so its coefficients decay as \(1/n^2\), and the series converges more rapidly.

This means we need fewer terms to achieve a good approximation for the triangle wave compared to the square wave.

6. Spectral Analysis

6.1 Frequency Spectrum

The Fourier series decomposes a periodic function into its frequency components, allowing us to analyze its spectrum. The spectrum can be visualized in several ways:

6.1.1 Amplitude Spectrum

The amplitude spectrum shows the magnitude of each frequency component.

\[ A_n = \sqrt{a_n^2 + b_n^2} \]

For the complex form, the amplitude spectrum is given by \(|c_n|\).

Example:

For a square wave, the amplitude spectrum is:

\(A_n = \begin{cases} \frac{4}{n\pi}, & \text{for odd } n \\ 0, & \text{for even } n \end{cases}\)

This shows that a square wave consists only of odd harmonics, with amplitudes inversely proportional to the harmonic number.

6.1.2 Phase Spectrum

The phase spectrum shows the phase angle of each frequency component.

\[ \phi_n = -\arctan\left(\frac{b_n}{a_n}\right) \]

For the complex form, the phase spectrum is given by \(\arg(c_n)\).

Example:

For a square wave centered at the origin, the phase spectrum is:

\(\phi_n = \begin{cases} -\frac{\pi}{2}, & \text{for odd } n \\ \text{undefined}, & \text{for even } n \end{cases}\)

The phase is -90° for all odd harmonics, indicating that the wave consists of sine terms rather than cosine terms.

6.2 Power Spectrum

The power spectrum shows the power distribution across frequencies. For a periodic function, the power in the \(n\)th harmonic is proportional to \(A_n^2\).

In terms of complex Fourier coefficients, the power in the \(n\)th harmonic is:

\[ P_n = |c_n|^2 + |c_{-n}|^2 = 2|c_n|^2 \text{ (for real functions)} \]

The total power in the signal is the sum of the powers in all harmonics:

\[ P_{\text{total}} = |c_0|^2 + 2\sum_{n=1}^{\infty} |c_n|^2 = \frac{a_0^2}{4} + \frac{1}{2}\sum_{n=1}^{\infty} (a_n^2 + b_n^2) \]

Example: Power in a Square Wave

For a square wave with amplitude 1, the power spectrum is:

\(P_n = \begin{cases} \frac{16}{n^2\pi^2}, & \text{for odd } n \\ 0, & \text{for even } n \end{cases}\)

The total power is \(P_{\text{total}} = 1\), which equals the mean square value of the square wave.

The first harmonic (n=1) contains \(\frac{16}{\pi^2} \approx 81.06\%\) of the total power.

6.3 Spectrum Visualization

There are several ways to visualize the spectrum of a periodic signal:

Line spectrum: Vertical lines at each harmonic frequency with heights proportional to the amplitude or power
Amplitude vs. frequency plot: Shows how the amplitude varies with frequency
Phase vs. frequency plot: Shows how the phase varies with frequency
Waterfall plot: 3D visualization showing how the spectrum evolves over time

These visualizations are essential tools for understanding the frequency content of signals in electrical engineering applications.

7. Parseval's Theorem

7.1 Statement of Parseval's Theorem

Parseval's theorem establishes a relationship between the energy of a signal in the time domain and its energy in the frequency domain. For a periodic function with period \(T\), the theorem states:

\[ \frac{1}{T} \int_{t_0}^{t_0+T} |f(t)|^2 \, dt = |c_0|^2 + 2 \sum_{n=1}^{\infty} |c_n|^2 \]

Using the trigonometric form, this becomes:

\[ \frac{1}{T} \int_{t_0}^{t_0+T} |f(t)|^2 \, dt = \frac{a_0^2}{4} + \frac{1}{2} \sum_{n=1}^{\infty} (a_n^2 + b_n^2) \]

Example: Parseval's Theorem for a Square Wave

For a square wave with amplitude 1 and period \(2\pi\):

The mean square value is \(\frac{1}{2\pi} \int_{0}^{2\pi} f^2(t) \, dt = 1\)

Using Parseval's theorem:

\(1 = \sum_{n=1,3,5,...}^{\infty} \frac{8}{\pi^2 n^2}\)

This confirms the known result that \(\sum_{n=1,3,5,...}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{8}\)

7.2 Energy Conservation

Parseval's theorem is a statement of energy conservation: the energy in the time domain equals the energy in the frequency domain. This principle is fundamental to signal processing and system analysis.

7.3 Practical Applications

Parseval's theorem has numerous practical applications in engineering:

Calculating the power or energy of a signal from its spectrum
Determining how much energy is contained in specific frequency bands
Evaluating the effectiveness of signal compression techniques
Analyzing the energy efficiency of communication systems

Example: Approximation Error

If we approximate a function \(f(t)\) using the first \(N\) terms of its Fourier series, we get:

\(f_N(t) = c_0 + \sum_{n=1}^{N} (c_n e^{in\omega_0 t} + c_{-n} e^{-in\omega_0 t})\)

The mean square error of this approximation is:

\(\frac{1}{T} \int_{0}^{T} |f(t) - f_N(t)|^2 \, dt = 2 \sum_{n=N+1}^{\infty} |c_n|^2\)

This allows us to determine how many terms are needed to achieve a desired accuracy.

8. Applications in Electrical Engineering

8.1 Power Systems Analysis

In power systems, non-sinusoidal voltages and currents can be analyzed using Fourier series to determine:

Harmonic content in power systems
Power factor and power quality
Total harmonic distortion (THD)
Losses due to harmonic currents

Example: Non-linear Load Analysis

Consider a non-linear load drawing current:

\[ i(t) = 10\sin(\omega t) + 3\sin(3\omega t) + 1.5\sin(5\omega t) \text{ A} \]

The THD can be calculated as:

\[ \text{THD} = \frac{\sqrt{3^2 + 1.5^2}}{10} \times 100\% = 33.5\% \]

This high THD indicates significant harmonic distortion that may require filtering or other power quality improvements.

8.2 Filter Design

Fourier series is fundamental to the design and analysis of filters, helping engineers:

Understand frequency response of circuits
Design filters to remove unwanted frequency components
Analyze filter performance in different frequency bands
Implement signal processing algorithms

Example: Low-Pass Filter

If a square wave with Fourier series:

\[ f(t) = \frac{4}{\pi}\sum_{k=0}^{\infty} \frac{\sin((2k+1)\omega_0 t)}{2k+1} \]

passes through a low-pass filter with cutoff frequency \(\omega_c = 3\omega_0\), the output will be approximately:

\[ y(t) \approx \frac{4}{\pi}\left[\sin(\omega_0 t) + \frac{\sin(3\omega_0 t)}{3}\right] \]

This shows how the filter removes higher harmonics while preserving the lower frequency components.

8.3 Communication Systems

Fourier series is essential in the analysis and design of communication systems:

Bandwidth requirements for transmitting periodic signals
Signal-to-noise ratio calculations
Modulation techniques (AM, FM, PCM)
Spectrum utilization and channel capacity

Example: AM Modulation

A message signal \(m(t) = \cos(\omega_m t)\) modulates a carrier \(c(t) = \cos(\omega_c t)\) to produce:

\[ s(t) = [1 + m(t)]c(t) = [1 + \cos(\omega_m t)]\cos(\omega_c t) \]

Using trigonometric identities, this can be expressed as:

\[ s(t) = \cos(\omega_c t) + \frac{1}{2}\cos((\omega_c + \omega_m)t) + \frac{1}{2}\cos((\omega_c - \omega_m)t) \]

The Fourier spectrum shows the carrier at \(\omega_c\) and two sidebands at \(\omega_c \pm \omega_m\), illustrating the bandwidth requirements.

8.4 Control Systems

In control systems, Fourier series helps analyze:

System response to periodic inputs
Frequency response and transfer functions
Stability criteria in the frequency domain
Harmonic distortion and system linearity

Example: Steady-State Response

For a linear system with transfer function \(H(j\omega)\), the steady-state response to a periodic input \(x(t) = \sum_{n=-\infty}^{\infty} c_n e^{jn\omega_0 t}\) is:

\[ y(t) = \sum_{n=-\infty}^{\infty} c_n H(jn\omega_0) e^{jn\omega_0 t} \]

This allows engineers to predict how a system will respond to complex periodic inputs by analyzing each frequency component separately.

9. Problem-Solving Strategies

9.1 General Approach

When solving problems involving Fourier series, follow these steps:

Identify the period and fundamental frequency: Determine the period \(T\) and calculate \(\omega_0 = 2\pi/T\)
Choose the appropriate form: Select the trigonometric or complex exponential form based on the problem
Calculate the Fourier coefficients: Use the appropriate integrals for the chosen form
Simplify and analyze: Look for patterns in the coefficients, possible simplifications, and physical interpretations
Verify: Check your solution by substituting it back into the original problem or using properties like Parseval's theorem

9.2 Symmetry Properties

Exploiting symmetry can greatly simplify the calculation of Fourier coefficients:

Even Functions

For an even function \(f(-t) = f(t)\):

\(b_n = 0\) for all \(n\) (no sine terms)
\(a_0 = \frac{4}{T} \int_{0}^{T/2} f(t) \, dt\)
\(a_n = \frac{4}{T} \int_{0}^{T/2} f(t) \cos(n\omega_0 t) \, dt\)

Example: \(f(t) = \cos(t)\), \(f(t) = |t|\)

Odd Functions

For an odd function \(f(-t) = -f(t)\):

\(a_n = 0\) for all \(n\) (no cosine terms)
\(a_0 = 0\) (no DC component)
\(b_n = \frac{4}{T} \int_{0}^{T/2} f(t) \sin(n\omega_0 t) \, dt\)

Example: \(f(t) = \sin(t)\), \(f(t) = t\) (over \([-\pi, \pi]\))

Half-Wave Symmetry

For functions with half-wave symmetry \(f(t + T/2) = -f(t)\):

All even harmonics are zero: \(a_{2n} = b_{2n} = 0\) for all \(n\)
Only odd harmonics contribute to the series

Example: \(f(t) = \sin(t)\), square wave

Quarter-Wave Symmetry

Combining even/odd symmetry with half-wave symmetry further simplifies calculations

9.3 Common Pitfalls

Avoid these common mistakes when working with Fourier series:

Incorrect period identification: Ensure you correctly identify the fundamental period of the function
Integration limits: The integration should be performed over exactly one period
Neglecting discontinuities: Remember that the series converges to the average value at discontinuities
Orthogonality confusion: The orthogonality properties of sines and cosines are key to deriving the coefficient formulas
Sign errors: Be careful with the signs when working with odd functions or negative arguments

9.4 Computational Tools

Several computational tools can assist with Fourier series calculations:

MATLAB: Provides built-in functions for Fourier analysis, such as fft and ifft
Python: Libraries like NumPy and SciPy offer Fourier transform capabilities
Maple/Mathematica: Symbolic computation software can handle the integrals for Fourier coefficients
Specialized software: Signal processing and circuit simulation software often include Fourier analysis tools

10. Examples and Exercises

Example 10.1: Square Wave

Find the Fourier series for a square wave with period \(T = 2\):

\[ f(t) = \begin{cases} 1, & 0 < t < 1 \\ -1, & -1 < t < 0 \end{cases} \]

Solution:

First, we identify that \(\omega_0 = 2\pi/T = \pi\).

Since the function is odd, we know \(a_0 = 0\) and \(a_n = 0\) for all \(n\).

For the sine coefficients:

\[ b_n = \frac{2}{2}\int_{-1}^{1} f(t)\sin(n\pi t) \, dt = \int_{-1}^{0} (-1)\sin(n\pi t) \, dt + \int_{0}^{1} 1 \cdot \sin(n\pi t) \, dt \]

After integration:

\[ b_n = \frac{1 - (-1)^n}{n\pi} \cdot 2 = \begin{cases} \frac{4}{n\pi}, & \text{for odd } n \\ 0, & \text{for even } n \end{cases} \]

Therefore, the Fourier series is:

\[ f(t) = \frac{4}{\pi}\sum_{k=0}^{\infty} \frac{\sin((2k+1)\pi t)}{2k+1} \]

Example 10.2: Triangle Wave

Find the Fourier series for a triangle wave with period \(T = 2\):

\[ f(t) = \begin{cases} t+1, & -1 < t \leq 0 \\ 1-t, & 0 < t \leq 1 \end{cases} \]

Solution:

First, we identify that \(\omega_0 = 2\pi/T = \pi\).

The function has even symmetry around the point \(t=0\), so \(b_n = 0\) for all \(n\).

For the DC component:

\[ a_0 = \frac{2}{2}\int_{-1}^{1} f(t) \, dt = \int_{-1}^{0} (t+1) \, dt + \int_{0}^{1} (1-t) \, dt = \frac{1}{2} + \frac{1}{2} = 1 \]

For the cosine coefficients:

\[ a_n = \frac{2}{2}\int_{-1}^{1} f(t)\cos(n\pi t) \, dt \]

After integration:

\[ a_n = \frac{4}{(n\pi)^2}((-1)^n - 1) = \begin{cases} -\frac{8}{(n\pi)^2}, & \text{for odd } n \\ 0, & \text{for even } n \end{cases} \]

Therefore, the Fourier series is:

\[ f(t) = \frac{1}{2} - \frac{8}{\pi^2}\sum_{k=0}^{\infty} \frac{\cos((2k+1)\pi t)}{(2k+1)^2} \]

Example 10.3: Full-Wave Rectified Sine

Find the Fourier series for a full-wave rectified sine wave:

\[ f(t) = |\sin(\omega_0 t)| \]

Solution:

The period is \(T = \pi/\omega_0\) (half the period of \(\sin(\omega_0 t)\)).

The function has even symmetry, so \(b_n = 0\) for all \(n\).

For the DC component:

\[ a_0 = \frac{2}{T}\int_{0}^{T/2} |\sin(\omega_0 t)| \, dt = \frac{2\omega_0}{\pi}\int_{0}^{\pi/2} \sin(t) \, dt = \frac{4}{\pi} \]

For the cosine coefficients:

\[ a_n = \frac{2}{T}\int_{0}^{T} |\sin(\omega_0 t)|\cos(2n\omega_0 t) \, dt \]

After integration:

\[ a_n = \frac{2}{\pi}\frac{2}{1-(2n)^2} = -\frac{4}{\pi}\frac{1}{4n^2-1} \text{ for } n \geq 1 \]

Therefore, the Fourier series is:

\[ f(t) = \frac{2}{\pi} - \frac{4}{\pi}\sum_{n=1}^{\infty} \frac{\cos(2n\omega_0 t)}{4n^2-1} \]

Practice Exercises

Test your understanding by working through these exercises:

Find the Fourier series for the periodic function with period \(T = 2\) defined by:

\[ f(t) = t^2 \text{ for } -1 < t \leq 1 \]
Find the Fourier series for the sawtooth wave:

\[ f(t) = t \text{ for } -\pi < t \leq \pi \]
A square wave with amplitude 1 and period \(2\pi\) is passed through a low-pass filter that keeps only the first three non-zero harmonics. What is the expression for the output signal?
Using Parseval's theorem, calculate the power in a triangle wave with amplitude 1 and determine what percentage of the power is contained in the first harmonic.
A periodic signal is given by \(f(t) = 2 + 3\cos(100\pi t) + 4\sin(100\pi t) + \cos(300\pi t) + 2\sin(500\pi t)\). Find its fundamental frequency, period, average value, RMS value, and power.

Conclusion

Fourier series provide a powerful mathematical framework for analyzing periodic signals and systems. By decomposing complex waveforms into simple sinusoidal components, they enable engineers to understand frequency content, design filters, analyze power systems, and develop communication systems.

For electrical engineering students, mastering Fourier series is essential for a deeper understanding of signals, systems, and their behavior in both time and frequency domains. The concepts introduced here form the foundation for more advanced topics such as Fourier transforms, Laplace transforms, and spectral analysis.

By applying the principles and techniques covered in this chapter, you'll be equipped to analyze complex periodic phenomena in various electrical engineering applications and develop innovative solutions to engineering challenges.

Glossary of Terms

Fourier Series: A method of representing a periodic function as an infinite sum of sines and cosines
Fundamental Frequency: The lowest frequency component in a periodic signal, equal to \(\omega_0 = 2\pi/T\)
Harmonic: A frequency component that is an integer multiple of the fundamental frequency
Fourier Coefficients: Constants (\(a_0\), \(a_n\), \(b_n\), or \(c_n\)) that determine the amplitude of each frequency component
Amplitude Spectrum: A plot showing the magnitude of each frequency component in a signal
Phase Spectrum: A plot showing the phase angle of each frequency component in a signal
Parseval's Theorem: A statement of energy conservation between time and frequency domains
Gibbs Phenomenon: The overshoot and ringing that occurs in a Fourier series approximation near discontinuities
Total Harmonic Distortion (THD): A measure of the sum of all harmonic components relative to the fundamental component
Even Function: A function where \(f(-t) = f(t)\), resulting in only cosine terms in its Fourier series
Odd Function: A function where \(f(-t) = -f(t)\), resulting in only sine terms in its Fourier series
Half-Wave Symmetry: A property where \(f(t + T/2) = -f(t)\), resulting in only odd harmonics in the Fourier series