Sine Generation Tutorial
Version 3.1, English.
Done two weeks before my final exams.
School suxx.
Contents
- Introduction
- Combined sine/cosine generation
a) using vector rotation or complex numbers
b) using derivations
- Recursive sine/cosine synthesis
- The End
Introduction
Here it is, after the original (German) version from 1997, I've decided to rewrite my small tutorial in English and to add some more tricks which might be helpful to you!
Special thanks fly out to fontex^kolor for his supportive feedback!
Another reason was that lots of people told me that they're using sine tables for sound generation and they were thinking this is the best way, but in my eyes it really isn't.
Well, just read this and think over it!
Combined sine/cosine generation
a) Okay. Before generating any sine/cosine stuff we need to get some basic formulas, in the current case these are the sum-formulas for sine and cosine.
If you draw a unit circle (circle with r = 1) and you draw two triangles lying over each other, you can find out the sum-formulas, which are:
[ cos(a +/- b) = cos(a)*cos(b) -/+ sin(a)*sin(b) ]
[ sin(a +/- b) = sin(a)*cos(b) +/- cos(a)*sin(b) ]
By the way, this is also where you can derive the 2d and 3d z-rotations from, just substitute
cos(a) = x and
sin(a) = y
where:
cos(a +/- b) = new x
sin(a +/- b) = new y
z = z
... and you got the z-rotation. - But this is not the main topic here so let's move back to our topic!
Alternatively, we could set b = 1, and we get:
[ cos(a +/- 1) = cos(a)*cos(1) -/+ sin(a)*sin(1) ]
[ sin(a +/- 1) = sin(a)*cos(1) +/- cos(a)*sin(1) ]
In fact, if you look at this as a 2d-vector, this is just a very simple vector rotation, which always rotates your vector by 1 degree.
When seen as a complex number, it's just a complex multiplication by e^(+/-j2pi/frq) which is exactly the same just in another notation.
Now we could already use this for generating a sine and a cosine table, but it isn't really impressive, and it is rather slow ...
Nevertheless this technique is often used in FFT algorithms because it is still a lot faster than the FPU implementations fsin/fcos/fsincos. So here is some C-style pseudo-code:
sine [ 0 ] = 0.0;
cosine[ 0 ] = 1.0;
sine [ 1 ] = sin( 2*M_PI / length );
cosine[ 1 ] = cos( 2*M_PI / length );
for ( int x=2; x < length; x++ ) {
sine [ x ] = sine [ x-1 ] * cosine[ 1 ] +
cosine[ x-1 ] * sine [ 1 ];
cosine[ x ] = cosine[ x-1 ] * cosine[ 1 ] -
sine [ x-1 ] * sine [ 1 ];
}
For usage in FFT algorithms etc. this is quite okay, but for real-time sound this is too slow and too unflexible. As an example, as soon as you want to change the amplitude or the frequency while calculating the sine/cosine wave you run into troubles, sooner or later ...
So, let's see if we can find something better.
b) By doing some simulations for electronic LRC filter networks I got the idea to create sine/cosine waves by using their derivates, let's get those first:
cos'(w * x) = - w * sin(w * x)
sin'(w * x) = w * cos(w * x)
Hey, that looks pretty simple, doesn't it?
Let's put it into pseudo-code, here we go:
double sin = 0.0,
cos = 1.0,
frq = 2.0 * M_PI / length;
for ( 1 .. length-1 ) {
cos -= sin * frq;
sin += cos * frq;
}
That really kicks ass, doesn't it? And it's awesome fast! - The cool thing is, the initial value of cosine doesn't really need to be 1.0! For instance if you want values [ -127..+127 ] just define 'cos = 127.0' - that's all!
The function also stays stable when changing the amplitude or frequency during your generation, thus it is perfect for sound generation!
- That's why I use this one very often!
Recursive sine/cosine synthesis
Sometimes not only speed is important but space. In this case I usually use some other recursive synthesis methods.
Here, we're looking at our generator that I've described in the first section, the one with the vector-rotation (etc...).
It is possible to simplify the process so you don't need to calculate both sine and cosine.
After looking through a very handy math book (Taschenbuch "mathematischer Formeln", Bartsch) I found a useful approach for solving the problem, the magic trick lies within the _product_terms_, they are:
[ cos(a)*cos(b) = 1/2 * (cos(a - b) + cos(a + b)) ]
[ sin(a)*sin(b) = 1/2 * (cos(a - b) - cos(a + b)) ]
[ sin(a)*cos(b) = 1/2 * (sin(a - b) + sin(a + b)) ]
If you use the upper equations and substitute them in the right way, you get:
[ cos(x) = cos(x - 1)*cos(1) - sin(x - 1)*sin(1) ]
[ cos(x) = cos(x - 1)*cos(1) - 1/2*(cos(x - 2) - cos(x)) ] *2
[ 2*cos(x) = 2*cos(x - 1)*cos(1) - cos(x - 2) + cos(x) ] -cos(x)
[ sin(x) = sin(x - 1)*cos(1) + cos(x - 1)*sin(1) ]
[ sin(x) = sin(x - 1)*cos(1) + 1/2*(sin(x - 2) + sin(x)) ] *2
[ 2*sin(x) = 2*sin(x - 1)*cos(1) - sin(x - 2) + sin(x) ] -sin(x)
So our simplified results are:
[ cos(x) = 2*cos(x - 1)*cos(1) - cos(x - 2) ]
[ sin(x) = 2*sin(x - 1)*cos(1) - sin(x - 2) ]
Damn that really kicks ass, doesn't it?
Here is some pseudo C-code for cos(x):
cosine[ 0 ] = 1.0;
cosine[ 1 ] = cos( 2*M_PI / length );
double cos2 = 2.0 * cosine[ 1 ];
for ( int x=2; x < length; x++ ) {
cosine[ x ] = cosine[ x-1 ] * cos2 - cosine[ x-2 ];
}
... and here is some pseudo C-code for sin(x):
sine [ 0 ] = 0.0;
sine [ 1 ] = sin( 2*M_PI / length );
double cos2 = 2.0 * cos( 2*M_PI / length );
for ( int x=2; x < length; x++ ) {
sine[ x ] = sine[ x-1 ] * cos2 - sine[ x-2 ];
}
Using one of those two formulas you can calculate sine/cosine tables in about 30-40 bytes, only if properly implemented in Assembler, of course!
For static sine/cosine tables this is perfect, but it isn't always useful for sound-code, since if you change cos(1) or sin(1) during generation the function gets unstable.
But, it is possible to rewrite the cos() formula and generate a digital band-resonator, the intuitive result will be exactly the same as if you would have defined your filter-poles within the z-domain!
On first sight this is quite amazing!
The End
Phew, *if* you ever reached the end of this file, I really hope I could give you some new ideas that may be helpful for your future projects.
If you've got any comments, or anything else, just drop me an e-mail! - I'm also very glad to get mails that just say "Hey, I've read your document", so I know it was not completely useless, otherwise I'm gonna stop wasting time writing such useless shit. ;-)
If you know other cool (e.g. no taylor rows!) methods for creating sine or cosine, please tell me!
Yours,