There are a lot of different subjects already posted on interpolation, but I wanted to find some really interesting use of interpolation other than just graphics and image transformations. I saw one study talking about interpolation being used to detect heart deficiencies (based upon normal heart rates), but that was a little too complicated and a sparse example. Then it occurred to me that if interpolation could be applied to images and graphics, couldn’t it be applied to music and sound?

Interpolation and sound is actually a very important relationship because of a very simple, physical reason: sound waves are analog waves, and anything stored on a computer is digital. This means that when recording sound, we have to record the sound with thresholds of analog values, so we can store the sound files digitally. This means that CD players, computers, MP3 players, and anything else that plays digital music has some sort of interpolation algorithm to be able to play the sound waves without making them sound very different from the original recordings.  The term used for this process is called ‘oversampling’.  That’s why occasionally we’ll see CD players that boast ‘8x oversampling’ as it means that you’ll get 8 times more sample points than what is recorded on the CD.

This is an example of a sound being recorded and stored digitally (it’s the C above high C). The analog sound wave is the underneath while the digital sound wave is the red line (sampled at 32 khz). It vaguely resembles the original wave right? You may wonder why not just sample at a higher rate so we can get more data points that will more likely resemble the line. The problem is that it’s a fine line between storage space, storing all of these points, and having a less curvacious line.

This is where interpolation comes in as instead of storing and sampling at a higher rate, why don’t we repeat the sample between the original ones, and thus get a higher sampling rate that should look closer to the original wave, and sound a lot better than if we played the non-interpolated wave. That’s the idea to where by using some interpolation method, we can store less, and have the quality of sound that hopefully no one will notice the difference to.

Here we have the sound of a guitar being plucked. The left most is the original wave, the middle is the wave after a linear interpolation algorithm has been run over it, and the right one is using cubic interpolation. One can see how there are some peaks and valleys which are missed such as the first valley and the last entirely seen peak, so this isn’t a perfect replacement.

Another interesting aspect that I’m not really going to go into here though is now applying different filters to the sound first, and then re-interpolating to hopefully get an even more accurate solution. Or one can imagine putting a filter (these filters I should say are just algorithms that adjust the digitally sampled data to try to hone in on a certain sound - like a low-pass filter which tries to reduce the reflections of sounds so we can just get the wave itself and not the slight different copies of itself) to make it sound like we’re in a large hall or a live stage.

Who would have thought: Interpolation in the aural world us!

Sources:

http://www.alpha-ii.com/Info/AudioInt.html

http://www.earlevel.com/Digital%20Audio/Oversampling.html