What is a Fourier Series?

Possibly the most important question...

The theory of Fourier series lies in the idea that most signals, and all engineering signals, can be represented as a sum of sine waves. Yes, including square waves and triangle waves. In fact, they're possibly the most-used examples. This has great implications for engineering.

Let's see how a square wave is built up. Don't worry about the actual content of the waves yet; we'll deal with that in a moment.

Start with a sine wave:

Add another, with an amplitude 1/3 of the original and a frequency 3 times that of the first. (This is known as the 3rd harmonic.)

Add another, with an amplitude 1/5 of the original and a frequency 5 times that of the first. (5th harmonic.)

If I carry on until the 15th harmonic, you should see a pattern emerging:

Despite it looking quite noisy, it still bears more than a passing resemblance to the square wave. In fact, if you add more and more harmonics, you get closer and closer to a square wave.

If you want to play around with harmonics and see what waves you can get, you can download my Wave Shaper program here. It allows you to add sine waves together, invert their phase and generally mess around. You can also hear what you create by clicking on the piano keyboard...

Why bother?

Sine waves have lots of interesting properties. Notably, many natural operations deal with a set of differing frequency sine waves as if they were processed individually (they are linear with regard to frequency). We can therefore look at applying these operations to a single sine wave and merely add and multiply to look at the effect on the entire signal.

Fourier series give you a great picture of the kind of content of your signal. For example, a sharp transition in your data generally results from a high-frequency sine wave (since only high-frequency sine waves have the fast-changing edge required), and so by cutting out the low frequencies, you can pick out the edges. This is particularly useful in image processing.

At the other end of the scale, if your data has noise at a particular frequency (e.g. 50 or 60Hz mains), you can pick apart the data to its constituent parts, remove the noise frequency, then stitch the rest back together to get a signal without the noise. This kind of notch filtering is very common in audio.

Alternatively, if you have high-frequency noise, you can just choose to turn down the high frequencies.

Since high frequencies correspond to higher pitches in our ears, you can alter the character of a sound by filtering its frequency content; this is how the EQ or tone controls on your audio system work.

They're very useful in image processing. It's possible to look at an image and distil it down to small blocks, then look at each line:

You can then treat each line as a 1D signal. It's then easy to see how you would apply the same techniques to the data that you would with any other signal.

Next, we will look at how to generate these component sine waves from periodic waveforms - Fourier series.