Engineering Math Series 18: Fourier Series

The core message of Fourier series is surprisingly simple: even complex periodic functions can be decomposed into sums of sines and cosines. For anyone learning signal processing and PDEs, this single sentence is virtually the starting point.

Why Decompose Periodic Functions

It is easier to analyze a complex waveform as a sum of simple oscillations rather than handling it as-is. Each oscillation has a frequency and amplitude, so the structure of the function can be understood as "how much of each frequency component is mixed in."

The Fourier series of a function with period $2L$ is usually written as

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

Why Sines and Cosines

Sines and cosines are orthogonal to each other. This means different frequency components separate cleanly. Thanks to this, the amount of each frequency contained in the function can be extracted through integration.

The Meaning of Coefficients

The coefficients are computed as follows.

$a_n = \frac{1}{L}\int_{-L}^{L} f(x)\cos \frac{n\pi x}{L}\,dx$

$b_n = \frac{1}{L}\int_{-L}^{L} f(x)\sin \frac{n\pi x}{L}\,dx$

This is the process of projecting how much $f(x)$ resembles cosine or sine of a specific frequency.

Worked Example

On the interval $[-\pi, \pi]$, consider

$f(x)=x$

Since this function is odd, the cosine coefficients are 0 and only sine coefficients remain.

Computing gives

$b_n = \frac{2(-1)^{n+1}}{n}$

so

$$x = 2\left(\sin x - \frac{1}{2}\sin 2x + \frac{1}{3}\sin 3x - \cdots \right)$$

The important point here is that even a linear function, when periodically extended, can be expressed as a sum of oscillation components.

Understanding Convergence

If the function is sufficiently well-behaved, the Fourier series converges to the original function. In cases with discontinuities, there are detailed rules such as converging to the average of the jump, but at the introductory level the fact that "a good function can be recovered as a sum of frequency components" is more important.

Engineering Applications

Signal Processing

Audio, communication signals, and sensor data are all frequently analyzed in terms of frequency components.

Heat Equation

When solving PDEs with boundary conditions, the strategy of expanding the solution as sine/cosine series appears as a fundamental approach.

Compression and Filtering

The perspective of keeping only important frequency components or removing specific bands originates here.

Common Mistakes

Memorizing formulas and missing the meaning of orthogonality

The coefficient formulas are understood much more deeply through "projection" than through memorization.

Not using odd/even function properties

Exploiting symmetry greatly simplifies computation.

Expecting perfect pointwise agreement for discontinuous functions

Fourier series are very powerful, but near jumps, separate convergence analysis is needed.

One-Line Summary

Fourier series is a tool for decomposing periodic functions into sine and cosine components of multiple frequencies.

Next Post Preview

In the next post, we will go one step beyond periodic functions and look at the Fourier transform and frequency-domain perspective for handling non-periodic signals.

References

• Erwin Kreyszig, Advanced Engineering Mathematics, 10th Edition
• Ronald N. Bracewell, The Fourier Transform and Its Applications
• Alan V. Oppenheim, Alan S. Willsky, Signals and Systems