Hello,
When I was playing with fb_fcomb(), I accidentally noticed this
trigonometric identity (thanks to maxima):
cos(x)*sin(n*x) = (sin((n+1)*x)+sin((n-1)*x))/2
and it gave me idea to write the following:
harmonics(vs, c0,s0)
= c0*vn(0),s0*vn(0), 1,c0, 0,s0
: seq(n, outputs(vs)-1, add(vn(n+1)))
: _,_, !,!, !,!
with {
vn(n) = ba.take(n+1, vs);
add(vn, co,so, cn_2,cn_1, sn_2,sn_1) =
co+cn*vn, so+sn*vn, cn_1,cn, sn_1,sn
with {
cn = 2*c0*cn_1 - cn_2;
sn = 2*c0*sn_1 - sn_2;
};
};
What does it do. For example, for any x
cos(x), sin(x) : harmonics((10,20,30))
outputs
10*cos(x) + 20*cos(2*x) + 30*cos(3*x),
10*sin(x) + 20*sin(2*x) + 30*sin(3*x);
(of course, modulo floating point issues)
IOW,
qo(f) = os.oscrq(f); // or any other complex oscillator
process = qo(f) : harmonics((10,20,30));
acts as
// pseuodo code, x * qo(...) won't compile
process = 10*qo(f) + 20*qo(2*f) + 30*qo(3*f);
but it is more simple/cheap.
-------------------------------------------------------------------------------
Note that it does not use the
[ c0 -s0 ]
[ s0 c0 ]
rotation matrix, and this means that
- we save one multiplication in the calculation of each harmonic
- more importantly, the calculations of cn and sn do not depend
on each other.
So if you only need the "sines" part you can do
qo : harmonics(volumes_list) : !,_;
and compiler will discard the half of the calculations.
-------------------------------------------------------------------------------
Questions:
- does it make sense to add this function to oscillators.lib ?
- is it a known method? I am just curious.
-------------------------------------------------------------------------------
Just in case... perhaps the more "generic" version makes more sense:
harmonics_parallel(N, c0,s0)
= c0,s0, 1,c0, 0,s0
: seq(n, N-1, (si.bus(2*(n+1)),add))
: si.bus(2*N), !,!, !,!
with {
add(cn_2,cn_1, sn_2,sn_1) =
cn,sn, cn_1,cn, sn_1,sn
with {
cn = 2*c0*cn_1 - cn_2;
sn = 2*c0*sn_1 - sn_2;
};
};
so that
qo(f) : harmonics_parallel(3);
outputs
qo(f), qo(2*f), qo(3*f);
Oleg.
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