Book: The Art of VA Filter Design 2.1.2
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Music Engineer Music Engineer https://www.kvraudio.com/forum/memberlist.php?mode=viewprofile&u=15959
- KVRAF
- 4264 posts since 8 Mar, 2004 from Berlin, Germany
much thanks! great stuff indeed. this, together with mystran's code in the other thread, this will probably finally get me to jump on this TPT/ZDF train, too. by now, i'm roughly halfway through the book.
what i didn't quite understand, though, is why on page 9, we can forget about the DC term. i mean, if the non-constant term (1/s)*e^(s*t) is quickly decaying away with t (i.e. the real part of s might be large and negative), after some time, only the constant term will persist - or not? what did i get wrong here?
what i didn't quite understand, though, is why on page 9, we can forget about the DC term. i mean, if the non-constant term (1/s)*e^(s*t) is quickly decaying away with t (i.e. the real part of s might be large and negative), after some time, only the constant term will persist - or not? what did i get wrong here?
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- KVRAF
- Topic Starter
- 1606 posts since 12 Apr, 2002
The constant term will not decay to zero if the integrator is used alone. However, if used inside a stable filter, it will.Robin from www.rs-met.com wrote:what i didn't quite understand, though, is why on page 9, we can forget about the DC term. i mean, if the non-constant term (1/s)*e^(s*t) is quickly decaying away with t (i.e. the real part of s might be large and negative), after some time, only the constant term will persist - or not? what did i get wrong here?
Maybe I should rewrite this passage, perhaps you're right, I didn't put this clearly enough into words therethe book wrote:Suppose further that the filter is stable
- KVRAF
- 7868 posts since 12 Feb, 2006 from Helsinki, Finland
If you mean DC term y(t0) then one can forget it on the basis that the circuit state is initially assumed zero (ie capacitors are discharged).Z1202 wrote:The constant term will not decay to zero if the integrator is used alone. However, if used inside a stable filter, it will.Robin from www.rs-met.com wrote:what i didn't quite understand, though, is why on page 9, we can forget about the DC term. i mean, if the non-constant term (1/s)*e^(s*t) is quickly decaying away with t (i.e. the real part of s might be large and negative), after some time, only the constant term will persist - or not? what did i get wrong here?
As for the stable-filter: it will (eventually) decay inside any filter that has a finite DC gain (ie no self-oscillating resonance at DC). This is not strictly the same as "stable filter" once you introduce non-linearities, because you can have a useful filter that has some persistent DC even when it's still filtering in as if it was a stable filter.. but it's not very important for analysis.
- KVRAF
- 7868 posts since 12 Feb, 2006 from Helsinki, Finland
Oh and for your appendix for two-pole filters, you might want to include (I couldn't find it) the usual modification that maintains "constant bandwidth":
I hope I got those right, but the point is: in all these we can pick a (w,Q) point on s-plane, then move poles and zeroes in opposite directions such that their geometric mean stays at the desired point. For peaking we move them on the "Q-axis" (which ofcourse is not a straight line on Euclidean s-plane) and for shelving we move them on the "w-axis". For asymmetric shelves you can do both of these: difference on "w" modifies the asymptotic gain, where as difference on "Q" modifies the mid-point gain.
edit: it should be noted that these two then are independent in such an asymmetric shelf: if you map one parameter on the w log-difference and another on the Q log-different, then another two on nominal w and Q, you can tweak any of them without affecting any of the rest; cutoff gives the center point, nominal Q the bandwidth, the shelving gain does what it should and the mid-point gain (if I'm not mistaken) references to the symmetric midpoint gain (ie half the shelving gain).
edit2: and if you slave the midpoint gain of an asymmetric shelf to be a scalar multiple of the asymptotic shelving gain, you get an asymmetric shelf that maintains it's shape for different shelving gains and is complementary (ie equal boost and cut cancel).. it should be noted that when there is a peak/and or dip, these are at the nominal cutoff only when shelving-gain is unity
Code: Select all
eg peaking:
s^2 + s*(K/Q) + 1
H(s) = ------------------- where K = sqrt(gain)
s^2 + s/(K*Q) + 1
and low-shelf:
s^2 + s * (sqrt(K)/Q) + K
K * ----------------------------- again K = sqrt(gain)
K*s^2 + s * (sqrt(K)/Q) + 1
and high-shelf:
K*s^2 + s * (sqrt(K)/Q) + 1
K * ----------------------------- again K = sqrt(gain)
s^2 + s * (sqrt(K)/Q) + K
edit: it should be noted that these two then are independent in such an asymmetric shelf: if you map one parameter on the w log-difference and another on the Q log-different, then another two on nominal w and Q, you can tweak any of them without affecting any of the rest; cutoff gives the center point, nominal Q the bandwidth, the shelving gain does what it should and the mid-point gain (if I'm not mistaken) references to the symmetric midpoint gain (ie half the shelving gain).
edit2: and if you slave the midpoint gain of an asymmetric shelf to be a scalar multiple of the asymptotic shelving gain, you get an asymmetric shelf that maintains it's shape for different shelving gains and is complementary (ie equal boost and cut cancel).. it should be noted that when there is a peak/and or dip, these are at the nominal cutoff only when shelving-gain is unity
Last edited by mystran on Wed May 23, 2012 3:03 pm, edited 1 time in total.
- KVRAF
- 4129 posts since 11 Aug, 2006 from Texas
Thanks for the suggestion, I use and love calibre. I even tried converting it before posting but the complex mathematical equations did not convert from pdf well at all. Considering it looks like the source material is LaTeX there are several tools that will handle the conversion from .tex -> .epub/.mobi much better than trying to do so from PDF.whyterabbyt wrote:Try this:bmrzycki wrote:Vadim, is it possible to get the book in a portable reader format like epub and mobi?
http://calibre-ebook.com/
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Music Engineer Music Engineer https://www.kvraudio.com/forum/memberlist.php?mode=viewprofile&u=15959
- KVRAF
- 4264 posts since 8 Mar, 2004 from Berlin, Germany
yes, but there was also this other component in the DC term: -(1/s)*e^(-s*t0). but OK, when considering the integrator only in the context of a finite DC-gain filter - i can see now why the DC component dies away. but i was thinking that we may also forget about this, when we consider the integrator in itself. actually, the integrator itself could be considered as a limiting case of a stable filter - i.e. borderline stable, conditionally stable or however this is commonly called. not trying to nitpick, but that somehow confused me.mystran wrote:If you mean DC term y(t0) then one can forget it on the basis that the circuit state is initially assumed zero (ie capacitors are discharged).Z1202 wrote:The constant term will not decay to zero if the integrator is used alone. However, if used inside a stable filter, it will.Robin from www.rs-met.com wrote:what i didn't quite understand, though, is why on page 9, we can forget about the DC term. i mean, if the non-constant term (1/s)*e^(s*t) is quickly decaying away with t (i.e. the real part of s might be large and negative), after some time, only the constant term will persist - or not? what did i get wrong here?
- KVRAF
- 4129 posts since 11 Aug, 2006 from Texas
I'm only asking in case the conversion requires minimal effort from LaTeX. Please don't waste a lot of time on this. Heck, if you're interested (and willing) you can send me the .tex source and I'll try to convert them for you.Z1202 wrote:As for putting out a MOBI or EPUB version of the book, I guess one needs something like a PDF to EPUB converter, not sure which ones are out there and how well can they work for this kind of PDF, but you could try them out. Maybe later I'll check the options of building EPUB directly from latex, no promises though as to when or whether, as I'm pretty sure this will require at least some hand-tuning if being possible at all. I also wonder how well would this kind of text scale across different reader devices.
- KVRAF
- 7868 posts since 12 Feb, 2006 from Helsinki, Finland
Urgh, if you mean replacing the integrator with something that doesn't have a pole at DC, then sure.. but your filter then will have some losses at low frequencies. When the integrator is "borderline stable" you are not limited at low-cutoffs: things work all the way down to the limit.Robin from www.rs-met.com wrote:yes, but there was also this other component in the DC term: -(1/s)*e^(-s*t0). but OK, when considering the integrator only in the context of a finite DC-gain filter - i can see now why the DC component dies away. but i was thinking that we may also forget about this, when we consider the integrator in itself. actually, the integrator itself could be considered as a limiting case of a stable filter - i.e. borderline stable, conditionally stable or however this is commonly called. not trying to nitpick, but that somehow confused me.mystran wrote:If you mean DC term y(t0) then one can forget it on the basis that the circuit state is initially assumed zero (ie capacitors are discharged).Z1202 wrote:The constant term will not decay to zero if the integrator is used alone. However, if used inside a stable filter, it will.Robin from www.rs-met.com wrote:what i didn't quite understand, though, is why on page 9, we can forget about the DC term. i mean, if the non-constant term (1/s)*e^(s*t) is quickly decaying away with t (i.e. the real part of s might be large and negative), after some time, only the constant term will persist - or not? what did i get wrong here?
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- KVRAF
- Topic Starter
- 1606 posts since 12 Apr, 2002
The term in question is a combination of the initial state and the "initial phase mismatch". The latter is typically nonzero. Anyway, these two terms basically constutute the transient response and as such will decay to zero in a stable LTI system.mystran wrote:If you mean DC term y(t0) then one can forget it on the basis that the circuit state is initially assumed zero (ie capacitors are discharged).Z1202 wrote:The constant term will not decay to zero if the integrator is used alone. However, if used inside a stable filter, it will.Robin from www.rs-met.com wrote:what i didn't quite understand, though, is why on page 9, we can forget about the DC term. i mean, if the non-constant term (1/s)*e^(s*t) is quickly decaying away with t (i.e. the real part of s might be large and negative), after some time, only the constant term will persist - or not? what did i get wrong here?
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- KVRAF
- Topic Starter
- 1606 posts since 12 Apr, 2002
I assume peaking in your terminology is the one I refer to as band-shelving on p.81? There is a bandwidth-based formula for the damping. I guess it should match yours, if neither of us did a mistake. Can't compare them without a piece of paper thoughmystran wrote:Oh and for your appendix for two-pole filters, you might want to include (I couldn't find it) the usual modification that maintains "constant bandwidth":
I hope I got those right, but the point is: in all these we can pick a (w,Q) point on s-plane, then move poles and zeroes in opposite directions such that their geometric mean stays at the desired point. For peaking we move them on the "Q-axis" (which ofcourse is not a straight line on Euclidean s-plane) and for shelving we move them on the "w-axis". For asymmetric shelves you can do both of these: difference on "w" modifies the asymptotic gain, where as difference on "Q" modifies the mid-point gain.Code: Select all
eg peaking: s^2 + s*(K/Q) + 1 H(s) = ------------------- where K = sqrt(gain) s^2 + s/(K*Q) + 1 and low-shelf: s^2 + s * (sqrt(K)/Q) + K K * ----------------------------- again K = sqrt(gain) K*s^2 + s * (sqrt(K)/Q) + 1 and high-shelf: K*s^2 + s * (sqrt(K)/Q) + 1 K * ----------------------------- again K = sqrt(gain) s^2 + s * (sqrt(K)/Q) + K
edit: it should be noted that these two then are independent in such an asymmetric shelf: if you map one parameter on the w log-difference and another on the Q log-different, then another two on nominal w and Q, you can tweak any of them without affecting any of the rest; cutoff gives the center point, nominal Q the bandwidth, the shelving gain does what it should and the mid-point gain (if I'm not mistaken) references to the symmetric midpoint gain (ie half the shelving gain).
edit2: and if you slave the midpoint gain of an asymmetric shelf to be a scalar multiple of the asymptotic shelving gain, you get an asymmetric shelf that maintains it's shape for different shelving gains and is complementary (ie equal boost and cut cancel).. it should be noted that when there is a peak/and or dip, these are at the nominal cutoff only when shelving-gain is unity
As for 2-pole low- and high-shelf I basically only mention this possiblity as a footnote on the same page. IIRC, I had some weird amplitude response shapes from 2-pole low- and high-shelves (maybe at high resonance, don't remember anymore). Still unsure, if it's a good idea to discuss those.
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- KVRAF
- Topic Starter
- 1606 posts since 12 Apr, 2002
Actually I want to stay with a single source file, not different versions for different targets. And AFAIU, longer equations will need different manual formatting, there is no automatic line breaking for equations. I might check this later (but not right now), also I have a friend who has a kindle But thanks for the suggestion anyway.bmrzycki wrote:I'm only asking in case the conversion requires minimal effort from LaTeX. Please don't waste a lot of time on this. Heck, if you're interested (and willing) you can send me the .tex source and I'll try to convert them for you.Z1202 wrote:As for putting out a MOBI or EPUB version of the book, I guess one needs something like a PDF to EPUB converter, not sure which ones are out there and how well can they work for this kind of PDF, but you could try them out. Maybe later I'll check the options of building EPUB directly from latex, no promises though as to when or whether, as I'm pretty sure this will require at least some hand-tuning if being possible at all. I also wonder how well would this kind of text scale across different reader devices.
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- KVRAF
- Topic Starter
- 1606 posts since 12 Apr, 2002
I don't think the limiting case here works, at least not intuitively, since the DC component will decay to zero in stable filters, but for the integrator it doesn't decay at all (decays with infinite characteristic time).Robin from www.rs-met.com wrote:actually, the integrator itself could be considered as a limiting case of a stable filter - i.e. borderline stable, conditionally stable or however this is commonly called. not trying to nitpick, but that somehow confused me.
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Music Engineer Music Engineer https://www.kvraudio.com/forum/memberlist.php?mode=viewprofile&u=15959
- KVRAF
- 4264 posts since 8 Mar, 2004 from Berlin, Germany
hmm, actually i meant considering the integrator itself as stable filter vs considering only the larger structures containing the integrator (such as the lowpass) as stable filters. the integrator is exactly on the border between (strictly) stable and (strictly) unstable and i considered it within the set whereas Vadim didn't. that was the reason for the confusion, i guess.mystran wrote:Urgh, if you mean replacing the integrator with something that doesn't have a pole at DC...
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- KVRAF
- Topic Starter
- 1606 posts since 12 Apr, 2002
I guess for me the intuitive definition of stable is BIBO. And the integrator isn't BIBO. I always considered marginal stability as being not really stable In practice at high resonance settings (poles close to imaginary axis) the filter (especially the digital model thereof) could get unstable in the time-varying case even if it's formally stable (poles in the left semiplane).Robin from www.rs-met.com wrote:hmm, actually i meant considering the integrator itself as stable filter vs considering only the larger structures containing the integrator (such as the lowpass) as stable filters. the integrator is exactly on the border between (strictly) stable and (strictly) unstable and i considered it within the set whereas Vadim didn't. that was the reason for the confusion, i guess.mystran wrote:Urgh, if you mean replacing the integrator with something that doesn't have a pole at DC...
Last edited by Z1202 on Wed May 23, 2012 3:58 pm, edited 1 time in total.
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Music Engineer Music Engineer https://www.kvraudio.com/forum/memberlist.php?mode=viewprofile&u=15959
- KVRAF
- 4264 posts since 8 Mar, 2004 from Berlin, Germany
well, yeah, that was the reason of confusion - the distinction between stable, strictly stable, conditionally stable etc. so i guess, "stable" means indeed "strictly stable", which i then had wrong.Z1202 wrote:since the DC component will decay to zero in stable filters, but for the integrator it doesn't decay at all (decays with infinite characteristic time).