Book: The Art of VA Filter Design 2.1.2

DSP, Plugin and Host development discussion.
Post Reply New Topic
RELATED
PRODUCTS

Post

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?
My website: rs-met.com, My presences on: YouTube, GitHub, Facebook

Post

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?
The constant term will not decay to zero if the integrator is used alone. However, if used inside a stable filter, it will.
the book wrote:Suppose further that the filter is stable
Maybe I should rewrite this passage, perhaps you're right, I didn't put this clearly enough into words there :)

Post

Z1202 wrote:
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?
The constant term will not decay to zero if the integrator is used alone. However, if used inside a stable filter, it will.
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).

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.

Post

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":

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

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
Last edited by mystran on Wed May 23, 2012 3:03 pm, edited 1 time in total.

Post

whyterabbyt wrote:
bmrzycki wrote:Vadim, is it possible to get the book in a portable reader format like epub and mobi?
Try this:
http://calibre-ebook.com/
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.

Post

mystran wrote:
Z1202 wrote:
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?
The constant term will not decay to zero if the integrator is used alone. However, if used inside a stable filter, it will.
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).
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.
My website: rs-met.com, My presences on: YouTube, GitHub, Facebook

Post

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.
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. :)

Post

Robin from www.rs-met.com wrote:
mystran wrote:
Z1202 wrote:
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?
The constant term will not decay to zero if the integrator is used alone. However, if used inside a stable filter, it will.
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).
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.
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.

Post

mystran wrote:
Z1202 wrote:
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?
The constant term will not decay to zero if the integrator is used alone. However, if used inside a stable filter, it will.
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).
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.

Post

mystran 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":

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

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
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 though :)
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.

Post

bmrzycki wrote:
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.
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. :)
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.

Post

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.
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).

Post

mystran wrote:Urgh, if you mean replacing the integrator with something that doesn't have a pole at DC...
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.
My website: rs-met.com, My presences on: YouTube, GitHub, Facebook

Post

Robin from www.rs-met.com wrote:
mystran wrote:Urgh, if you mean replacing the integrator with something that doesn't have a pole at DC...
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.
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).
Last edited by Z1202 on Wed May 23, 2012 3:58 pm, edited 1 time in total.

Post

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).
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.
My website: rs-met.com, My presences on: YouTube, GitHub, Facebook

Post Reply

Return to “DSP and Plugin Development”