Chapter Contents

• Fourier Series
• Helpful Revision
• 1. Overview of Fourier Series
• 2. Full Range Fourier Series
• 3. Fourier Series of Even and Odd Functions
• 4. Fourier Series of Half Range Functions
• 5. Harmonic Analysis
• 6. Line Spectrum
• 7. Fast Fourier Transform
• Fourier Series Problem Solver

• Comments, Questions?

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5. Harmonic Analysis

Recall the Fourier series (that we met in Full Range Fourier Series):

We can re-arrange this series and write it as:

+ (a₁ cos t + b₁ sin t) + (a₂ cos 2t + b₂ sin 2t) + (a₃ cos 3t + b₃ sin 3t) + ...

The term (a₁ cos t + b₁ sin t) is known as the fundamental.

The term (a₂ cos 2t + b₂ sin 2t) is called the second harmonic.

The term (a₃ cos 3t + b₃ sin 3t) is called the third harmonic, etc.

Odd Harmonics

The Fourier series will contain odd harmonics if f(t + π) = - f(t).

Example:

In this case, the Fourier expansion will be of the form:

a₀/2 + (a₁ cos t + b₁ sin t) + (a₃ cos 3t + b₃ sin 3t) + (a₅ cos 5t + b₅ sin 5t) + ...

All of the harmonics are odd.

Even Harmonics

The Fourier series will contain even harmonics if f(t + π) = f(t).
(That is, it has period π.)

In this case, the Fourier expansion will be of the form:

a₀/2 + (a₂ cos 2t + b2 sin 2t) + (a₄ cos 4t + b₄ sin 4t) + (a₆ cos 6t + b₆ sin 6t) + ...

All of the harmonics are even.

EXAMPLES

Determine the existence of odd or even harmonics for the following functions.

(a)

f(t) = f(t + 2π).

Answer

(b)

f(t) = f(t + π).

Answer