Wave Theory - Part 1

Apologies for the delay here. I started writing a number of things into this post, however it became a long, boring, technical marathon. So I'm going to break up these types of articles and post some interstitial things as well. I originally wanted to look at Quantization Noise, but there were so many concepts involved that I decided to start with the "Basics".


Superposition

Science and nature detest corners. Curves, even constantly varying ones, are much easier to deal with. This is because curves add together nicely through a principle called "Superposition." Long story short, curves can be added together and taken apart without any difficulty. 

If you've ever looked at music on an oscilloscope, you'll see a random squiggle that never seems to stop changing. Whilst it can make for an interesting visual effect, it's not very helpful. However, due to Superposition, we can break that seemingly random signal down into the component inputs. This is usually called the Fourier Function Transform, but you might be more familiar with the term "Spectrum Analyser." (I strongly advise not looking too deeply into it unless you really like maths.)

If you run a signal though a Spectrum Analyser, you change the random squiggle into a series of columns. Each column represents a frequency range; the height of each column denotes the amount that each particular frequency range is contributing to the original signal.

Input signals (Left) and the resultant Spectrum Analysis (Right)
Thanks Wiki Commons

Let's have a look at some pictures; because they are easier to understand than words.
The top picture is your standard sine-wave. Since there is only one frequency, the Spectrum Analysis shows us that information; one tall column. All of the power in this signal is contained within that narrow frequency range.

The second set of images shows static; a totally random signal at low level. If we look at the Spectrum Analysis, you'll see that the power is spread randomly across all frequencies. We'll get into the origins of static one of these days. Just not today.

The last set of images shows these two signals superimposed onto each other. At each point, the "height" of the two signals is added together to make the bottom left image. It's a bit hard to see in the image, but you'll note that the curve is no longer smooth. However, this is nearly impossible to tell by looking at the input signal.
However, when we look at the Spectrum Analysis, we can clearly see that most of the energy is still in that main frequency range, but there is energy spread out across the other frequencies.

This effect is normally called "Noise".

Repeating Patterns
There is one other cool thing about superposition.

Basically, any repeating pattern can be built up with the right combination of sine waves. Take, for instance, a square wave:

Once again, Thanks Wiki!

If you have a look at the above image, you'll see three lines.
The Red line is a "true" square wave. The Green-dashed line shows a Fourier Approximation of the square wave using 5 component waves. The blue-dashed line uses 15 component waves. These waves are superimposed onto each other, like this:

The left-hand images shows our four component waves and their relative powers. Just by looking at the left-hand side, we can see that the "Fundamental" frequency, which is the same as the frequency of the square wave, has the most power.

The second column shows the superposition, but without adding the waves together. The third column shows the resultant, superimposed wave. As we go down the list, it starts looking more and more like a square wave.

The right-hand column, again, is what you'd see if you put the signal into a Spectrum analyser. There's a couple of points to note here:

•  For a square wave of a fundamental frequency F, the frequency of the component waves (f) is as follows - f = (2n+1)F, where n starts at 0 and goes to infinity
•  The power of each wave drops off significantly as n gets bigger (or, to put it another way, as the frequency of the component wave goes up, the power of that component goes down).

It turns out that once you get past n=16 or so, the power in the higher frequencies is so low that it no longer matters if you include them or not.



Next Time
It is very tempting to plough ahead here and talk about why I just wrecked your mind with superposition, but I won't.

Here's a hint though, it has to do with square waves and noise, and why digital and analogue aren't all that different.

As always, please feel free to post questions in the comments section!