at the last AES in NYC, i was talking with some other folks (that likely
hang out here, too) about this family of soft clipping curves made outa
polynomials (so you have some idea of how high in frequency any
generated images will appear).
these are odd-order, odd-symmetry polynomials that are monotonic from -1
< x < +1 and have as many continuous derivatives as possible at +/- 1
where these curves might be spliced to constant-valued rails.
the whole idea is to integrate the even polynomial (1 - x^2)^N
x
g(x) = integral{ (1 - v^2)^N dv}
0
you figger this out using binomial expansion and integrating each power
term.
normalize g(x) with whatever g(1) is so that the curve is g(x)/g(1) and
splice that to two constant functions for the rails
{ -1 x <= -1
{
f(x) = { g(x)/g(1) -1 <= x <= +1
{
{ +1 +1 <= x
you can "hard limit" (at +/- 1) before passing through this soft clipper
and it still works fine. but it has some gain in the "linear" region
which is g(0).
if you want to add some "even harmonic distortion" to this, add a little
bit of
(1 - x^2)^M
to f(x) for |x| < 1 and it's still smooth everywhere, but there is a
little DC added (which has to be the case for even-symmetry distortion).
M does not have to be the same as N and i wouldn't expect it to be.
you can think of f(x) as a smooth approximation the "sign" or "signum"
function
sgn(x) = lim f(a*x)
a -> +inf
or
sgn(x) = lim f(x)
N -> +inf
which motivates using this as a smooth approximation of the sgn(x)
function as an alternative to
(2/pi)*arctan(a*x) or tanh(a*x). from the sgn(x) function, you can
create smooth versions of the unit step function and use that for
splicing. as many derivatives are continuous in the splice as possible.
and it satisfies conditions of symmetry and complementarity that are
useful in our line of work.
u(x) = 1/2 * (1 + sgn(x)) =approx 1/2*(1 + f(x))
you can run a raised cosine (Hann) through this and get a more flattened
Hann. in some old stuff i wrote, i dubbed this window:
w(x) = 1/2 + (9/16)*cos(pi*x) - (1/16)*cos(3*pi*x)
as the "Flattened Hann Window" but long ago Carla Scaletti called it the
"Bristow-Johnson window" in some Kyma manual. i don't think it deserves
that label (i've seen that function in some wavelet/filterbank lit since
for half-band filters). you get that window by running a simple Hann
through the biased waveshaper:
1/2 * ( 1 + f(2x-1) )
with N=1. you will get an even more pronounced effect (of smoothly
flattening the Hann) with higher N.
below is a matlab file that demonstrates this as a soft clipping function.
BTW, Olli N, this can be integrated with that "splicing theory" thing we
were talking about approximately a year ago. it would define the
"odd-symmetry" component. we should do an AES paper about this. i
think now there is nearly enough "meat" to make a decent paper. before
i didn't think so.
________________________________________________________________________
FILE: softclip.m
line_color = ['g' 'c' 'b' 'm' 'r'];
figure;
hold on;
x = linspace(-2, 2, 2^18 + 1);
x = max(x, -1); % hard clip for |x|> 1
x = min(x, +1);
for N = 0:10
n = linspace(0, N, N+1);
a = (-1).^n ./ (factorial(N-n) .* factorial(n) .* (2*n + 1));
a = a/sum(a);
y = x .* polyval(fliplr(a), x.^2); % stupid MATLAB puts the coefs
in the wrong order
plot(x, y, line_color(mod(N,length(line_color))+1) );
end
hold off;
________________________________________________________________________
have fun with it, if you're so inclined.
L8r,
--
r b-j r...@audioimagination.com
"Imagination is more important than knowledge."
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