rotation in function space

what's going on ?

When you have a scalar product, you can build a rotation application.

So what about messing with rotation applications on spaces we don't often see rotations on, such as function space ?

In the graph here, the red curve (originally the purple sinus) is turning in the plane defined by the two functions yellow and green.

how to implement ?

One way would be to use a function basis and do traditionnal transformations with huge matrices.

This works and it's what you're seeing on the animated graph here.

But is terribly inefficient when considering arbitrary audio files.

A smarter approach is to consider a weighted combination of the projected and rejected functions (weights beeing $sin$ and $cos$ as well as some eventual normalizations)

This better implementation has been done in scilab :

scilab code

You can try my implementation of wave files rotation with scilab

You can modify the script to load any sound you like.

conclusion

Well the result does look nice, but doesn't sound very interesting.

It just sound like we surperposed the two sounds with a weight, which you can feel when you look at the actual calculation. (it's not obvious from an abstract rotation in function space though)

However, since it seems like rotation is a fun way to mix function in the regular space, maybe adding functions in the fourier space might sound more interesting.

So I tried convolution of a sound with another one and it sounded really cool !

Scilab implementation of such a thing is almost trivial :

snd1 = loadwave("/path/to/sound1.wav")
snd2 = loadwave("/path/to/sound2.wav")
snd3 = squarewave(1:100,50) // if you wanna try things

snd1 = snd1(1,:)            // extract one channel
snd2 = snd2(1,:)            // extract one channel

out = convol(snd1,snd2)     // convolution !
out = out/max(out)          // normalize

playsnd(out)                // check it out