5. Fourier Concepts

Fourier Concepts

Introduction to Fourier Concepts

Many patterns in nature repeat over time. Sound waves, ocean tides, vibrating strings, alternating electrical currents, and even some biological rhythms all display periodic behaviour.

However, these patterns are not always simple sine or cosine waves. They may have complicated shapes, contain sharp changes, or combine several different frequencies.

Fourier analysis is a mathematical technique that allows complicated signals and functions to be represented as combinations of simpler sinusoidal waves.

The central idea is that a periodic function can be built from sine and cosine waves with different amplitudes, frequencies, and phases.

This principle connects trigonometry, calculus, physics, engineering, and digital technology.

Understanding Periodic Functions

A periodic function is a function that repeats its values after a fixed interval.

Mathematically:

f(t + T) = f(t)

where T is the period.

The period is the smallest positive interval over which the complete pattern repeats, when such a smallest interval exists.

Examples of periodic functions include:

• Sine waves.

• Cosine waves.

• Repeating electrical signals.

• Vibrations of musical instruments.

• Rotational motion.

• Regular mechanical oscillations.

Period and Frequency

The period and frequency are related by:

f = 1/T

where:

• f is frequency in hertz (Hz).

• T is period in seconds.

For example, if a wave repeats every 0.02 seconds:

f = 1/0.02

f = 50 Hz

The wave completes 50 cycles every second.

Angular Frequency

Angular frequency is represented by ω.

ω = 2πf = 2π/T

Its SI unit is radians per second.

A basic sinusoidal function can be written:

y(t) = A sin(ωt + φ)

where:

• A is amplitude.

• ω is angular frequency.

• φ is phase angle.

• t is time.

These quantities determine the wave's height, repetition rate, and horizontal position.

Combining Periodic Functions

One of the most important principles in Fourier analysis is superposition.

Superposition means that two or more wave functions can be added together to produce a resultant waveform.

For example:

y₁(t) = sin t

y₂(t) = 0.5sin(3t)

The combined function is:

y(t) = sin t + 0.5sin(3t)

The first wave has amplitude 1 and angular frequency 1 rad/s.

The second wave has amplitude 0.5 and angular frequency 3 rad/s.

The combined waveform is no longer a simple sine wave, but it remains periodic.

Worked Example: Combining Two Waves

Consider:

y₁(t) = 3sin(2πt)

y₂(t) = sin(4πt)

The combined signal is:

y(t) = 3sin(2πt) + sin(4πt)

At t = 0.25 s:

y₁ = 3sin(π/2) = 3

y₂ = sin π = 0

Therefore:

y = 3

At t = 0.125 s:

y₁ = 3sin(π/4) ≈ 2.121

y₂ = sin(π/2) = 1

Therefore:

y ≈ 3.121

The combined wave can have a greater displacement than either individual wave at certain times.

This illustrates how overlapping sinusoidal components can produce more complicated patterns.

What Is Fourier Analysis?

Fourier analysis is the process of examining a function or signal in terms of its frequency components.

Instead of looking only at how a signal changes over time, Fourier analysis investigates which frequencies contribute to the signal.

There are two important ways to describe a signal.

Time domain

Shows how the signal changes with time.

Frequency domain

Shows the frequencies present in the signal and their relative amplitudes.

For example, a musical sound may contain a fundamental frequency of 200 Hz along with additional components at 400 Hz, 600 Hz, and 800 Hz.

In the time domain, these components combine to form a complicated waveform.

In the frequency domain, they appear as separate frequency components.

Why Fourier Analysis Is Useful

Fourier analysis helps mathematicians, scientists, and engineers:

• Identify frequencies within complicated signals.

• Separate useful information from unwanted noise.

• Analyze vibrations and oscillations.

• Understand musical tones and sound quality.

• Process images and audio.

• Investigate electrical signals.

• Study periodic physical systems.

It provides a way to understand complicated patterns by examining their simpler components.

Fundamental Frequency and Harmonics

A periodic signal can contain several sinusoidal components whose frequencies are whole-number multiples of a fundamental frequency.

The fundamental frequency is the lowest positive frequency in the harmonic series.

It is commonly written as f₀.

The harmonics are:

• First harmonic: f₀

• Second harmonic: 2f₀

• Third harmonic: 3f₀

• Fourth harmonic: 4f₀

• Fifth harmonic: 5f₀

For example, if the fundamental frequency is 100 Hz:

The relative amplitudes and phases of these harmonics determine the overall shape of the signal.

Worked Example: Identifying Harmonics

A periodic signal is represented by:

y(t) = 4sin(2π100t) + 2sin(2π200t) + sin(2π300t)

Identify the harmonic components.

The frequencies are:

100 Hz, 200 Hz, and 300 Hz.

Therefore:

• Fundamental frequency = 100 Hz.

• Second harmonic = 200 Hz.

• Third harmonic = 300 Hz.

The amplitudes are:

4, 2, and 1 respectively.

The fundamental has the greatest amplitude.

This means the signal is strongly influenced by its lowest-frequency component, while the higher harmonics modify its shape.

Fourier Series

A Fourier series represents a periodic function as a sum of sine and cosine functions.

A general Fourier series can be written as:

f(t) = a₀/2 + Σ[aₙcos(nω₀t) + bₙsin(nω₀t)]

where the sum runs from n = 1 to infinity and:

• a₀/2 is the constant or average component.

• aₙ are cosine coefficients.

• bₙ are sine coefficients.

• n identifies the harmonic number.

• ω₀ is the fundamental angular frequency.

For a periodic function with period T:

ω₀ = 2π/T

The coefficients determine how much of each sine and cosine component is present.

Not every periodic function can be represented pointwise by a Fourier series without qualifications. For many commonly studied piecewise-smooth functions, the series converges to the function at continuous points and to the midpoint of the left- and right-hand limits at jump discontinuities.

Fourier Approximation

In practice, we often use only a limited number of terms.

For example:

f(t) ≈ sin t + (1/3)sin 3t + (1/5)sin 5t

This approximation combines three sinusoidal components.

Adding more appropriate terms can improve the approximation to a target periodic function.

The quality of the approximation depends on the function, the selected harmonics, and the number of terms included.

Building a Square Wave

A square wave alternates between two constant values.

It contains abrupt transitions that cannot be produced by a single sine wave.

However, a square wave can be approximated by adding odd harmonics.

For a square wave with values +1 and −1:

f(t) = (4/π)[sin t + (1/3)sin 3t + (1/5)sin 5t + ...]

This Fourier series uses only odd sine harmonics.

First Approximation

Using only the fundamental:

f₁(t) = (4/π)sin t

This is a sine wave, not a square wave.

Second Approximation

Add the third harmonic:

f₂(t) = (4/π)[sin t + (1/3)sin 3t]

The waveform begins to flatten near its maximum and minimum values.

Third Approximation

Add the fifth harmonic:

f₃(t) = (4/π)[sin t + (1/3)sin 3t + (1/5)sin 5t]

The approximation becomes more square-like.

As additional odd harmonics are included, the flat regions and sharp transitions become more clearly defined.

At the discontinuities, the Fourier series converges to zero, the midpoint between +1 and −1.

The Gibbs Phenomenon

When a Fourier series approximates a function with a sudden jump, small oscillations may appear near the discontinuity.

This behaviour is called the Gibbs phenomenon.

Even when more harmonics are added, the overshoot near a jump does not completely disappear.

Instead:

• The oscillations become concentrated into a narrower region.

• The approximation improves away from the discontinuity.

• The maximum overshoot approaches approximately 9% of the jump height.

This phenomenon is important in signal processing because sharp transitions can produce unwanted oscillations or ringing in reconstructed signals.

Fourier Approximations of Other Waveforms

Different combinations of harmonics can produce different repeating wave shapes.

Triangle Waves

A triangle wave rises and falls linearly.

Its Fourier series contains odd harmonics whose amplitudes decrease rapidly.

One common form is:

f(t) = (8/π²)[sin t − sin 3t/9 + sin 5t/25 − ...]

The harmonic amplitudes decrease in proportion to 1/n² for odd n.

This relatively rapid decrease means that a small number of harmonics can provide a good approximation.

Sawtooth Waves

A sawtooth wave rises or falls steadily before suddenly jumping back to its starting value.

One common sawtooth series is:

f(t) = 2[sin t − sin 2t/2 + sin 3t/3 − sin 4t/4 + ...]

This represents f(t) = t for −π < t < π, extended periodically, with the sign convention shown.

The sawtooth wave contains both even and odd harmonics.

Comparing Waveforms

These differences help explain why waves with the same fundamental frequency can have very different shapes.

Interpreting Fourier Approximations

A Fourier approximation can be interpreted by examining its harmonic components.

Consider:

f(t) = 3sin t + sin 2t + 0.5sin 4t

The components have:

The fundamental component has the largest amplitude.

The higher-frequency components introduce additional variation.

What Happens When Harmonics Are Added?

Adding harmonics can:

• Change the shape of the waveform.

• Introduce sharper features.

• Produce additional peaks or valleys.

• Improve the approximation of a complicated signal.

• Change the distribution of energy across frequencies.

However, adding harmonics does not necessarily increase the maximum displacement, because different components can reinforce or cancel one another.

Fourier Analysis and Sound

Sound is one of the clearest real-world applications of Fourier analysis.

A musical note is rarely a pure sine wave.

Instead, it usually contains a fundamental frequency and additional harmonics.

These harmonics help determine the timbre, or characteristic sound quality, of an instrument.

For example, a violin and a flute may play the same note at the same fundamental frequency but sound different because their harmonic content differs.

Worked Example: Musical Harmonics

A musical note has a fundamental frequency of 220 Hz.

Determine the frequencies of its first four harmonics.

First harmonic:

f₁ = 220 Hz

Second harmonic:

f₂ = 440 Hz

Third harmonic:

f₃ = 660 Hz

Fourth harmonic:

f₄ = 880 Hz

These harmonics may combine to create a more complex musical tone.

The relative strengths of the harmonics influence how the instrument sounds.

Fourier Analysis in Digital Audio

Digital audio systems record sound as a sequence of numerical samples.

Fourier analysis can identify the frequencies present in those samples.

This is useful for:

• Equalizing bass and treble.

• Removing background noise.

• Detecting musical pitches.

• Compressing audio.

• Recognizing speech.

• Analyzing recordings.

A digital signal can be analyzed using the Discrete Fourier Transform (DFT).

A computationally efficient method for calculating many DFTs is the Fast Fourier Transform (FFT).

Fourier Analysis in Medical Imaging

Fourier methods are used in several medical imaging technologies.

For example, magnetic resonance imaging (MRI) collects signals that contain spatial-frequency information.

Mathematical transformations are used to reconstruct images from the measured data.

Fourier analysis helps convert the collected signals into spatial information that can be displayed as an image.

Fourier methods are also important in other imaging and signal-processing applications.

Fourier Analysis in Electrical Engineering

Electrical signals frequently contain multiple frequency components.

An alternating-current signal may contain a fundamental frequency together with unwanted harmonics.

For example:

V(t) = 100sin(2π50t) + 10sin(2π150t)

The signal contains:

• A 50 Hz fundamental component.

• A 150 Hz third harmonic.

The additional harmonic distorts the waveform.

Fourier analysis helps engineers identify these components and design filters or other systems to reduce unwanted distortion.

Fourier Analysis in Vibrations and Structures

Buildings, bridges, machines, and vehicles can vibrate at different frequencies.

Fourier analysis can help identify dominant vibration frequencies.

This is useful for:

• Detecting mechanical faults.

• Monitoring structural vibrations.

• Investigating resonance.

• Improving machine design.

• Reducing unwanted noise.

• Evaluating structural performance.

For example, if a machine begins producing unusually strong vibrations at a particular frequency, engineers may investigate whether a component is damaged or operating near resonance.

Connections to Euler's Formula

Fourier analysis is closely connected to Euler's Formula:

e^(iθ) = cos θ + i sin θ

This formula allows sine and cosine functions to be represented using complex exponentials.

For example:

cos θ = [e^(iθ) + e^(−iθ)]/2

sin θ = [e^(iθ) − e^(−iθ)]/(2i)

These relationships allow Fourier series to be expressed in exponential form.

A complex Fourier series can be written:

f(t) = Σ cₙe^(inω₀t)

where the sum runs over all integers n and cₙ are complex Fourier coefficients.

This representation is especially useful in advanced mathematics, physics, and engineering because exponential functions are often easier to manipulate algebraically and differentiate.

Common Mistakes

Assuming every periodic signal is a single sine wave

Many periodic signals contain several harmonics.

Confusing period and frequency

Period measures time per cycle, while frequency measures cycles per second.

Assuming harmonics have arbitrary frequencies

Harmonic frequencies are integer multiples of a fundamental frequency.

Thinking higher harmonics always have larger amplitudes

In many Fourier series, higher harmonics have smaller amplitudes.

Confusing the time domain with the frequency domain

The time domain shows how a signal changes over time. The frequency domain shows its frequency components.

Assuming Fourier approximations are always exact

A finite Fourier series usually approximates a complicated waveform rather than reproducing it perfectly.

Ignoring phase

Two waves with the same frequency and amplitude can combine differently depending on their phase difference.

Assuming adding waves always increases displacement

Waves can interfere constructively or destructively.

Forgetting the Gibbs phenomenon

Fourier approximations of discontinuous functions can produce oscillations near sharp transitions.

Key Terms

• Periodic function: A function that repeats after a fixed interval.

• Period: The time or interval required for one complete repetition.

• Frequency: The number of cycles occurring per unit time.

• Angular frequency: The rate of change of phase, measured in radians per second.

• Amplitude: The maximum displacement of a sinusoidal component from its equilibrium position.

• Phase: A quantity describing the position of a wave within its cycle.

• Superposition: The principle that wave functions can be added to form a resultant wave.

• Fourier analysis: A mathematical method for studying functions and signals in terms of frequency components.

• Fourier series: A representation of a periodic function using a sum of sine and cosine terms.

• Fundamental frequency: The lowest positive frequency in a harmonic series.

• Harmonic: A sinusoidal component whose frequency is an integer multiple of the fundamental frequency.

• Harmonic component: An individual sine or cosine wave contributing to a periodic signal.

• Fourier coefficient: A value describing the contribution of a particular sinusoidal component.

• Fourier approximation: A finite sum of harmonic components used to approximate a function.

• Time domain: A representation showing how a signal changes with time.

• Frequency domain: A representation showing the frequencies contained in a signal.

• Frequency spectrum: A display of the frequency components and their magnitudes or other properties.

• Square wave: A periodic waveform alternating between two constant levels.

• Triangle wave: A periodic waveform that rises and falls linearly.

• Sawtooth wave: A periodic waveform with a gradual linear change and a sudden jump.

• Gibbs phenomenon: Oscillatory overshoot near discontinuities in Fourier approximations.

• Timbre: The characteristic quality of a sound influenced by its harmonic content.

• Discrete Fourier Transform: A mathematical transformation used to analyze discrete signals in terms of frequency components.

• Fast Fourier Transform: An efficient algorithm for computing a discrete Fourier transform.

• Signal processing: The analysis and modification of signals to extract information or improve their usefulness.

Key Takeaways

• Fourier analysis shows how complicated periodic functions can be represented as combinations of simpler sinusoidal waves.

• Periodic functions repeat after a fixed interval called the period.

• Frequency and period are related by f = 1/T.

• The principle of superposition allows multiple waves to combine into a resultant waveform.

• A Fourier series expresses a periodic function as a sum of sine and cosine components.

• The fundamental frequency is the lowest positive frequency in a harmonic series.

• Harmonics occur at integer multiples of the fundamental frequency.

• Different harmonic amplitudes and phases produce different waveform shapes.

• Square waves can be approximated using odd harmonics.

• Triangle waves contain odd harmonics with amplitudes that decrease rapidly.

• Sawtooth waves contain both even and odd harmonics.

• Adding more Fourier terms often improves an approximation, although discontinuities produce special behaviour.

• The Gibbs phenomenon describes oscillatory overshoot near sharp jumps.

• Fourier analysis connects time-domain signals with their frequency-domain representations.

• Fourier methods are important in sound analysis, digital audio, electrical engineering, medical imaging, vibration analysis, and communications.

• Euler's Formula provides an important mathematical connection between Fourier analysis and complex exponential functions.

Check Your Understanding

Recall and Understanding

1. What is a periodic function?

2. Explain the purpose of Fourier analysis.

3. What is the difference between a fundamental frequency and a harmonic?

4. Explain the difference between the time domain and the frequency domain.

5. Why can a complicated periodic waveform be represented using multiple sine and cosine functions?

Application

6. A wave has a period of 0.01 seconds. Calculate its frequency.

7. A periodic signal has a fundamental frequency of 60 Hz. Determine the frequencies of its second, third, and fourth harmonics.

8. Identify the frequencies and amplitudes in the signal y(t) = 3sin(2π50t) + sin(2π150t).

9. Calculate the combined displacement of y₁ = 2sin t and y₂ = sin 2t at t = π/2.

10. Explain how adding a third harmonic changes the shape of a simple sine wave.

Analysis and Challenge

11. Compare the harmonic components of square, triangle, and sawtooth waves.

12. Explain why adding more Fourier terms improves the approximation of a square wave but does not completely eliminate overshoot near its discontinuities.

13. Two musical instruments produce the same fundamental frequency but sound different. Explain how Fourier analysis can help explain this difference.

14. An electrical signal contains components at 50 Hz, 150 Hz, and 250 Hz. Identify the harmonic numbers and explain why an engineer might investigate the higher-frequency components.

15. Describe how Euler's Formula connects complex exponential functions with the sine and cosine components used in Fourier analysis.

Final Summary

Fourier analysis provides a powerful way to understand complicated periodic functions by representing them as combinations of simpler sine and cosine waves.

By identifying fundamental frequencies, harmonics, amplitudes, and phases, mathematicians and scientists can analyze the structure of signals that may initially appear complicated.

Fourier approximations also demonstrate how increasingly accurate representations of complex waveforms can be constructed by adding suitable harmonic components.

These concepts form an important foundation for advanced mathematics, physics, electrical engineering, digital signal processing, medical imaging, and the study of waves and vibrations.