How to calculate Leaky Integrator cutoff frequency?
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How to calculate Leaky Integrator cutoff frequency?
by Steven Cook-2
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Hi,
My subject says it all really - what is the correct way (if any) of calculating the cutoff frequency of a Leaky Integrator. I'm currently using the following formula, but I've no idea where I got it from or if it is correct, so I'm starting to doubt that it is... cutoff = pow (e, -fc / sampleRate * PI * 2.0); ...where 'fc' is the desired cutoff frequency and 'e' is the well-known mathematical constant. Regards, Steven Cook. http://www.spcplugins.com/ +44(0)1271 867288 -- 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 |
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Re: How to calculate Leaky Integrator cutoff frequency?
by robert bristow-johnson
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it's nice to see music-dsp happening again. i thought it was falling silent. On Aug 22, 2009, at 12:25 PM, Bogac Topaktas wrote: > > y(n) = alpha * x(n) + (1 - alpha) * y(n-1) > > or re-arrange to save one multiplier: > > y(n) = y(n-1) + alpha * ( x(n) - y(n-1) ) > in my opinion, the simplest way to look at this is to recognize that a leaky integrator is the simpliest 1-pole digital low-pass filter (not to be confused with the 1-pole, 1-zero LPF that you would get from bilinear transforming a simple analog 1-pole LPF). here, the pole, p, is 1-alpha y[n] = (1-p)*x[n] + p*y[n-1] Y(z) = (1-p)*X(z) + p*(z^-1)*Y(z) Y(z)/X(z) = H(z) = (1 - p) / (1 - p*z^-1) = (1-p)*z / (z - p) H(e^jw) = (1-p)/(1 - p*cos(w) + j*sin(w)) the magnitude squared frequency response: |H(e^jw)|^2 = (1-p)^2 / ( (1 - p*cos(w))^2 + (sin(w))^2 ) = (1-p)^2 / (1 + p^2 - 2*p*cos(w)) the -3dB corner frequency, w0, (a.k.a. "half-power frequency") occurs when |H(e^jw)|^2 = 1/2 or the denominator of |H(e^jw)|^2 is twice as large as the numerator: 1 + p^2 - 2*p*cos(w0) = 2*(1-p)^2 = 2*(1 - 2*p + p^2) = 2 - 4*p + 2*p^2 cos(w0) = -(1 - 4*p + p^2)/(2*p) = 2 - (1/2)*(p + 1/p) = 1 - ((1-p)^2)/(2*p) (choose your favorite form) then w0 is w0 = 2*pi*f0/Fs = arccos( 2 - (1/2)*(p + 1/p) ) = arccos( 2 - cosh(log(p)) ) plug in 1-alpha in for p, and it looks slightly less pretty. you can see that since the magnitude to the argument to the arccos() function must be no greater than 1, then 1 <= cosh(log(p)) = (1/2)*(p + 1/p) <= 3 or (remember that log(p) < 0 so we take the left-hand arccosh) 0 <= -log(p) <= arccosh(3) = log(3 + sqrt(3^2-1)) = 1.762747174 or e^(-arccosh(3)) = 0.171572875 <= p <= 1 . we know that p < 1 for stability, i think the meaning of restricting 0.171572875 <= p (clearly p can be set closer to zero and you still have a working 1-pole filter) is that there is no -3 dB frequency if p is lower. -- r b-j rbj@... "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 |
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decibels
by Didier Dambrin
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Not that it's any important in practice, but something has always puzzled me with the dB scale. Too often I read "+6dB" talking about a 2x gain. But I like square things thus I never quite liked that dB scale where +6dB means *1.995something. Example: the definition of brown noise says -6dB/octave. But if I was to generate brown noise, I'd probably rather divide by 2 per octave, it sounds more "logical", as I wouldn't see where those x1.995/octave would come from. So, the dB scale being 10^(x/20) & not 2^(x/6), is it only for historical reasons or something I'm missing? & is it common to interpret dB values as 2^(x/6)? Even just to display better rounded values to users? & in fact, why aren't people using a simple 2^x linear scale for gain, just as for octaves? -- 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 |
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Re: decibels
by Richard Dobson
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Didier Dambrin wrote:
> Not that it's any important in practice, but something has always puzzled me > with the dB scale. > Too often I read "+6dB" talking about a 2x gain. But I like square things > thus I never quite liked that dB scale where +6dB means *1.995something. > That's just a matter of where you do your casual rounding. It is just as likely you will divide (or multiply) by 2, and the proper dB value (20 * log10(x)) is +-6.02. Nobody wants so say "six point oh two" all the time, unless you want to be thought of as a COMPLETE NERD, so we say "sixdeebee" and leave it at at that. Which is still pretty nerdy. Better to say "a fifth" than "702 Cents". Even though that is what an equal-tempered fifth is. > Example: the definition of brown noise says -6dB/octave. Engineers will say 20dB/decade. Take your pick! But if I was to > generate brown noise, I'd probably rather divide by 2 per octave, it sounds > more "logical", as I wouldn't see where those x1.995/octave would come from. > The dB range is nothing more than a convenience, with respect to our ears. We hear logarithmically, and -60dB (colloquially speaking) is the "threshold of silence", rather more convenient and expressive than saying "point oh oh one". We need ~something that (a) compares things and (b) give us a tidy scale with which to discuss the huge dynamic range we can hear (and measure). It's a convention; like feet and inches. When you scale by 0.05, is that a lot? - not very much ? medium? How much more is it that 0.03? > So, the dB scale being 10^(x/20) & not 2^(x/6), is it only for historical > reasons or something I'm missing? Been around a very long time. See no reason to change it. Though you ~could~. > & is it common to interpret dB values as 2^(x/6)? Even just to display > better rounded values to users? > > & in fact, why aren't people using a simple 2^x linear scale for gain, just > as for octaves? > We can't really deal with more than about ten octaves; and a semitone is quite a large difference sometimes. But with intensity, the range is (hand-waving) 0.0001 to 10000 (or so), and the smallest difference we can detect is pretty big - about a dB. Golden-ears will claim a half-dB. The electrical engineers will no doubt have their own perspective on it. All about comparing voltages v powers, etc. Too late at night to be more lucid, sorry! Richard Dobson -- 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 |
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Re: decibels
by Didier Dambrin
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Well you'd be surprised. For years I've been asked for "more precision" in
my volume faders or meters, people wanna see all dB decimals. I think people like rounded numbers in general (I thought that was a programmer's disorder), no one can stand "4.11" when it can be a rounded 4. So maybe it would have been better to use a *6 dB scale, where +1 doubles the amplitude. > But with intensity, the range is > (hand-waving) 0.0001 to 10000 (or so), and the smallest difference we > can detect is pretty big - about a dB. Golden-ears will claim a half-dB. > > The electrical engineers will no doubt have their own perspective on it. > All about comparing voltages v powers, etc. > > Too late at night to be more lucid, sorry! > > Richard Dobson > 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 |
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Re: decibels
by Richard Dobson
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Didier Dambrin wrote:
> Well you'd be surprised. For years I've been asked for "more precision" in > my volume faders or meters, people wanna see all dB decimals. I think people > like rounded numbers in general (I thought that was a programmer's > disorder), no one can stand "4.11" when it can be a rounded 4. > So maybe it would have been better to use a *6 dB scale, where +1 doubles > the amplitude. > Hmm, not really - as I wrote, the smallest difference of level most people can distinguish (somewhat frequency-dependent) is 1dB. Digital volume control chips are usually quantized in half-dB steps (and plenty of audio engineers will say that that is the minimum acceptable). I don't know if my Denon DM30 has that; but I can hear each step as the volume is changed. Of course, what the typical punter thinks is "half as loud" is very much cruder than that - I don't know the references, but IIRC listening tests showed a wide variation in such a subjective thing. A half-dB step is educational. Sounds like you are really looking for a VU meter scale - -20 to +3. OK for overall power monitoring, but not for getting the hi-hat to sit just right in the mix. I lopoked at good-ol' wikipedia again, and this says it most succinctly: "A change in power ratio by a factor of 10 is a 10 dB change. ". Most of the time, that is what we are interested in, not voltages but powers (ratio of powers is 10*log10(p1/p2)); so the decibel scale gets you a direct map of what is actually going on, power-wise. There could be a case, I suppose, for nominating a 3dB change ("half-power point") as a unit. but mastering engineers routinely set EQ changes of fractional dB. Funnily enough, 1.5dB (or thereabouts) seems a popular increment. Besides, your +1 scale sounds dangerously close to the guitar amp discussion in "This is Spinal Tap" : "This goes up to eleven!". Richard Dobson -- 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 |
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Re: How to calculate Leaky Integrator cutoff frequency?
by Steven Cook-2
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> y(n) = alpha * x(n) + (1 - alpha) * y(n-1)
> > or re-arrange to save one multiplier: > > y(n) = y(n-1) + alpha * ( x(n) - y(n-1) ) Thanks for these answers. My code (for a BLIT-based synthesizer) looks more like: y(n) = y(n-1) * alpha + x(in) Does that make a difference? Steven Cook. http://www.spcplugins.com/ -- 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 |
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Re: decibels
by Didier Dambrin
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Yes but that's what I like about it: often a gain knob must allow increase &
reduction, but generally you allow more space for reduction. So I like the idea of a 0..11 gain, 10 being the 0dB point. At the same time I'd never make a gain control linear in dB's since it'd have to go down to -infinite. But I like the 0..10..11 as -inf..0dB..6dB. I in fact often calibrate my volume controls to -inf..0dB..6dB but more in a 1..4..5 (then 1..10..12) range, unless I need to allow more gain. I like this 75-25% split, which also roughly matches the usual "100" MIDI 7bit default for velocity or volume. (I seem to remember the top 25% was around +5dB in the MIDI group standardization) So gain controls in software are (I assume) never linear in power nor in dB, yet we have no standard linear scale that the user would understand. > Besides, your +1 scale sounds dangerously close to the guitar amp > discussion in "This is Spinal Tap" : "This goes up to eleven!". > > Richard Dobson > -- 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 |
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Re: decibels
by Richard Dobson
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Didier Dambrin wrote:
> Yes but that's what I like about it: often a gain knob must allow increase & > reduction, but generally you allow more space for reduction. > So I like the idea of a 0..11 gain, 10 being the 0dB point. > At the same time I'd never make a gain control linear in dB's since it'd > have to go down to -infinite. What's the problem there" At the end of the day you have a finite word-length. No need to compute anything less than -144dB! Drawing -inf on a display is just that - a picture. But I like the 0..10..11 as -inf..0dB..6dB. That's (among other things) a calibration issue (and an issue of where on that scale you put 0dBFS). It depends on whether compatibility with industry practice is important to you. See e.g. the Bob Katz K-metering system, which systemizes digital headroom ~and the fader law for different requirements: http://www.digido.com/level-practices-part-2-includes-the-k-system.html Richard Dobson -- 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 |
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Re: How to calculate Leaky Integrator cutoff frequency?
by Lubomir I. Ivanov
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----- Original Message -----
From: "Steven Cook" <stevenpaulcook@...> >> y(n) = alpha * x(n) + (1 - alpha) * y(n-1) > Thanks for these answers. My code (for a BLIT-based synthesizer) looks > more > like: > > y(n) = y(n-1) * alpha + x(n) > > Does that make a difference? for one pole (p0) and requirement of unity gain: p0 = 0.7 (this is a filter that has a zero at z = 0) -- a0 = g (gain) = 0.3 b0 = 1 b1 = -p0 = -0.7 difference equation: y[n] = a0*x[n-1] - b1*y[n-1] = 0.3 * x[n-1] + 0.7 * y[n-1] -------------------- but there is a relation: (a0+...+a[n]) = (b0+...+b[n]) if we add a zero (z0) to the z-plane: p0 = 0.7 z0 = -0.5 -- coefficient calculation: a0 = g; a1 = -g*z0 = g*0.5 b0 = 1 b1 = -p0 = -0.7 from unity gain requirement, (a0 + a1) must be equal to (b0 + b1 = 1 - 0.7) or: a0 + a1 = 0.3; g + g*0.5 = 0.3 g = 0.2; therefore: a0 = 0.2; a1 = 0.1; and the difference equation is: y[n] = a0*x[n] + a1*x[n-1] - b1*y[n-1] = 0.2*x[n] + 0.1*x[n-1] + 0.7*y[n-1]; ----------- regards lubomir -- 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 |
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Re: decibels
by Didier Dambrin
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But IMHO a software gain control should always go down to 0. For the
user -144dB would probably fine, but with floating point mixing, it makes a difference whether the signal is inaudible or really silent. If it's really silent you can skip further processing as an optimization. And even if you stopped at -144dB, you still wouldn't use a linear scale in dB for a gain control, as half of the control's range wouldn't be much useful. > Didier Dambrin wrote: >> Yes but that's what I like about it: often a gain knob must allow >> increase & >> reduction, but generally you allow more space for reduction. >> So I like the idea of a 0..11 gain, 10 being the 0dB point. >> At the same time I'd never make a gain control linear in dB's since it'd >> have to go down to -infinite. > > What's the problem there" At the end of the day you have a finite > word-length. No need to compute anything less than -144dB! Drawing -inf > on a display is just that - a picture. > > > But I like the 0..10..11 as -inf..0dB..6dB. > > That's (among other things) a calibration issue (and an issue of where > on that scale you put 0dBFS). > > It depends on whether compatibility with industry practice is important > to you. See e.g. the Bob Katz K-metering system, which systemizes > digital headroom ~and the fader law for different requirements: > > http://www.digido.com/level-practices-part-2-includes-the-k-system.html > > > Richard Dobson > 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 |
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Re: How to calculate Leaky Integrator cutoff frequency?
by robert bristow-johnson
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On Aug 23, 2009, at 4:41 AM, Steven Cook wrote: >> y(n) = alpha * x(n) + (1 - alpha) * y(n-1) >> >> or re-arrange to save one multiplier: >> >> y(n) = y(n-1) + alpha * ( x(n) - y(n-1) ) > > Thanks for these answers. My code (for a BLIT-based synthesizer) > looks more > like: > > y(n) = y(n-1) * alpha + x(in) > > Does that make a difference? only in nomenclature and in a constant gain that does not affect the -3 dB frequency because what we *mean* by the "-3 dB frequency" is the frequency where the gain is 3 dB lower than at DC, no matter what the gain there is. your "alpha" is precisely the same as "p" in what i wrote. but, using your alpha, the model i describe is y[n] = (1-alpha)*x[n] + alpha*y[n-1] the only difference is that my input is scaled by (1-alpha) and yours is not. my DC gain is 1 (or 0 dB). that means that your DC gain is 1/(1-alpha) which is bigger than 1. the -3 dB frequency is still the same: w0 = 2*pi*f0/Fs = arccos( 2 - (1/2)*(alpha + 1/alpha) ) = arccos( 2 - cosh(log(alpha)) ) if your alpha is less than e^(-arccosh(3)) = 0.171572875, then you don't have a -3 dB frequency (the LPF attenuation never exceeds 3 dB). if your alpha is equal to 1, you have a non-leaky integrator and if your alpha is greater than 1, your LPF will blow up. one last notational issue;\: i would recommend sticking with the now common notation that you see in DSP and discrete-filtering texts and most of the IEEE (or AES) lit. for discrete-time or discrete- frequency sequences, we use brackets: x[n], y[n], h[n], X[k], Y[k], H[k] for continuous-time and continuous-frequency functions, we use parenths: x(t), X(f), or maybe X(omega) or X(w) or sometimes (for consistency with the Laplace transform), X(jw) in older textbooks, they used the standard subscript x or x_n instead of "x[n]" n but sometimes, in continuous-time signals, we would get a *vector* of signals { x_1(t), x_2(t), x_3(t), ... x_m(t) } where, if this was converted to discrete time, it was cumbersome to have two subscripts; one for which element in the vector and another for discrete time. so the present notation was evolved { x_1[n], x_2[n], x_3[n], ... x_m[n] } and widely adopted. of course this notation doesn't work with MATLAB, but they screw up the x[0] issue anyway (the bastards). the main reason is that if you see an argument in brackets or as a subscript, we're talking about a discrete sequence where the argument better the hell be an integer rather than a continuous function where the argument is not restricted to or generally assumed to be an integer. -- r b-j rbj@... "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 |
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Re: decibels
by Vadim Zavalishin
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Hi Didier and Richard
Thought, I'd drop in some thoughts. > Hmm, not really - as I wrote, the smallest difference of level most > people can distinguish (somewhat frequency-dependent) is 1dB. Digital > volume control chips are usually quantized in half-dB steps (and plenty > of audio engineers will say that that is the minimum acceptable). I > don't know if my Denon DM30 has that; but I can hear each step as the > volume is changed. I would guess, that the 1dB minimum audible difference is referring to the context of two signals being played one after another with some gap between them. If you quickly change a level of a continuously playing signal you might be able to recognize a smaller change. Further on, I would expect that in the case of mixing several tracks together, smaller gain changes for a given track might be able to produce an audible difference (not neccesarily explicitly perceived as a loudness change!) due to psychoacoustic effects of the signal interaction. Plus, the subtleties are very important in the mix, and audio engineers should have a good ear for that. Regards, Vadim -- Vadim Zavalishin Senior Software Developer | R&D Tel +49-30-611035-0 Fax +49-30-611035-2600 NATIVE INSTRUMENTS GmbH Schlesische Str. 28 10997 Berlin, Germany http://www.native-instruments.com Registergericht: Amtsgericht Charlottenburg Registernummer: HRB 72458 UST.-ID.-Nr. DE 20 374 7747 Geschaeftsfuehrung: Daniel Haver (CEO), Mate Galic -- 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 |
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