This reminds me of experimenting with polynomials as an amplitude enveloping function for a soft synthesiser. There was something rather alluring about the idea of a one-line-of-code amplitude envelope - unfortunately it made creating the envelopes pretty tiresome, and when I thought about the performance I realised it was probably slower to do that maths than the few conditionals you would use for a standard piecewise envelope. So basically a rubbish idea unless you have some strange need to have your amplitude envelope be a single line of code :I
-----Original Message----- From: music-dsp-boun...@music.columbia.edu [mailto:music-dsp-boun...@music.columbia.edu] On Behalf Of robert bristow-johnson Sent: 29 October 2013 01:56 To: music-dsp@music.columbia.edu Subject: [music-dsp] family of soft clipping functions. 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." -- dupswapdrop -- the music-dsp mailing list and website: subscription info, FAQ, source code archive, list archive, book reviews, dsp links http://music.columbia.edu/cmc/music-dsp http://music.columbia.edu/mailman/listinfo/music-dsp -- dupswapdrop -- the music-dsp mailing list and website: subscription info, FAQ, source code archive, list archive, book reviews, dsp links http://music.columbia.edu/cmc/music-dsp http://music.columbia.edu/mailman/listinfo/music-dsp
