Hi All, I have, maybe trivial, question. What is the passivity condition for a transfer function described in z domain? How do I determine if a given transfer function is passive (so its gain is <=1 ?) from its pole-zero location or value of numerator and denominator coefficients? I can't really find any meaningful answers on the web. Many thanks in advance Regards Pawel
Z-transfrom passivity condition
Started by ●March 20, 2009
Reply by ●March 20, 20092009-03-20
On 20 Mar, 12:50, Pawel <prulikow...@gmail.com> wrote:> Hi All, > > I have, maybe trivial, question. What is the passivity condition for a > transfer function described in z domain? How do I determine if a given > transfer function is passive (so its gain is <=1 ?) from its pole-zero > location or value of numerator and denominator coefficients?If you think of 'active' vs 'passive' filters, the terms only make sense in the context of analog electronics. In that case, 'active' filters contain amplifiers, which insert energy to the system, while 'passive' filters only contain components that consume energy (RLC components, maybe others), never insert energy.> I can't > really find any meaningful answers on the web.That's because the question (if that is your question) doesn't make sense in discrete-time domain. You can always amplify a signal by scaling it with a gain factor > 1. Rune
Reply by ●March 20, 20092009-03-20
On Mar 20, 8:10�am, Rune Allnor <all...@tele.ntnu.no> wrote:> On 20 Mar, 12:50, Pawel <prulikow...@gmail.com> wrote: > > > Hi All, > > > I have, maybe trivial, question. What is the passivity condition for a > > transfer function described in z domain? How do I determine if a given > > transfer function is passive (so its gain is <=1 ?) from its pole-zero > > location or value of numerator and denominator coefficients? > > If you think of 'active' vs 'passive' filters, the terms > only make sense in the context of analog electronics. In > that case, 'active' filters contain amplifiers, which > insert energy to the system, while 'passive' filters > only contain components that consume energy (RLC components, > maybe others), never insert energy. > > > I can't > > really find any meaningful answers on the web. > > That's because the question (if that is your question) > doesn't make sense in discrete-time domain. You can > always amplify a signal by scaling it with a gain > factor > 1. > > RuneThe usual method of determining the magnitude of your gain H(z) still applies. Your transfer function as written has poles p_i and zeroes z_k. At a point x in C (complex plane), the gain is the product of the magnitudes (x - z_i ) divided by the product of the magnitudes (x - p_k);. You're likely interested in points x on the unit circle. But as Rune points out, you get extra energy "for free" in the digital domain because the concept of signal power inside a CPU or RAM buffer is all in your head :-) HTH, - Kenn
Reply by ●March 20, 20092009-03-20
On Mar 20, 12:35�pm, kennheinr...@sympatico.ca wrote:> On Mar 20, 8:10�am, Rune Allnor <all...@tele.ntnu.no> wrote: > > > > > On 20 Mar, 12:50, Pawel <prulikow...@gmail.com> wrote: > > > > Hi All, > > > > I have, maybe trivial, question. What is the passivity condition for a > > > transfer function described in z domain? How do I determine if a given > > > transfer function is passive (so its gain is <=1 ?) from its pole-zero > > > location or value of numerator and denominator coefficients? > > > If you think of 'active' vs 'passive' filters, the terms > > only make sense in the context of analog electronics. In > > that case, 'active' filters contain amplifiers, which > > insert energy to the system, while 'passive' filters > > only contain components that consume energy (RLC components, > > maybe others), never insert energy. > > > > I can't > > > really find any meaningful answers on the web. > > > That's because the question (if that is your question) > > doesn't make sense in discrete-time domain. You can > > always amplify a signal by scaling it with a gain > > factor > 1. > > > Rune > > The usual method of determining the magnitude of your gain H(z) still > applies. Your transfer function as written has poles p_i and �zeroes > z_k. At a point x in C (complex plane), the gain is the product of the > magnitudes (x - z_i ) divided by the product of the magnitudes (x - > p_k);. You're likely interested in points x on the unit circle. > > But as Rune points out, you get extra energy "for free" in the digital > domain because the concept of signal power inside a CPU or RAM buffer > is all in your head :-) > > HTH, > > �- KennHi, Thanks for the explanations. However I would argue with You that conecpt of passivity makes only sense in analog domain. You can say the same about stability - how digital filter (or revoked CPU) can become unstable - what would happen would CPU blow up?! (Yes it can burn but not as the result of numerical instability) Surely if one is only interested in purely theoretical procedures than OK those are only numbers and as such cannot be stable, passive nor minimum-phase etc. They only make sense if we relate them to something i.e. transfer function of the real circuit. I would probably need to rephrase my question: How to ensure that z- transfer function that describes real passive system fulfills passivity criteria - what are those criteria? Regards Pawel Pawel
Reply by ●March 20, 20092009-03-20
On 20 Mrz., 15:40, Pawel <prulikow...@gmail.com> wrote:> I would probably need to rephrase my question: How to ensure that z- > transfer function that describes real passive system fulfills > passivity criteria - what are those criteria?What's your definition of "real passive system"? Cheers! SG
Reply by ●March 20, 20092009-03-20
On Mar 20, 3:19�pm, SG <s.gesem...@gmail.com> wrote:> On 20 Mrz., 15:40, Pawel <prulikow...@gmail.com> wrote: > > > I would probably need to rephrase my question: How to ensure that z- > > transfer function that describes real passive system fulfills > > passivity criteria - what are those criteria? > > What's your definition of "real passive system"? > > Cheers! > SGReal - one that exist in our world and can be characterized through the measurements. Passive - that one that do not inject extra energy to the system, gain equal or less than 1 (or 0dB). Pawel
Reply by ●March 20, 20092009-03-20
On 20 Mrz., 16:24, Pawel <prulikow...@gmail.com> wrote:> On Mar 20, 3:19�pm, SG <s.gesem...@gmail.com> wrote: > > What's your definition of "real passive system"? > > Real - one that exist in our world and can be characterized through > the measurements. Passive - that one that do not inject extra energy > to the system, gain equal or less than 1 (or 0dB)....and by "gain<=1" you mean (a) the magnitude response for all frequencies are equal or less than one, or (b) the gain factor "K" you get by converting the numerator/ denominator coefficients of H(z) to a pole-zero factorization (see Matlab's tf2zp function) ? Cheers! SG
Reply by ●March 20, 20092009-03-20
On Mar 20, 3:37�pm, SG <s.gesem...@gmail.com> wrote:> On 20 Mrz., 16:24, Pawel <prulikow...@gmail.com> wrote: > > > On Mar 20, 3:19�pm, SG <s.gesem...@gmail.com> wrote: > > > What's your definition of "real passive system"? > > > Real - one that exist in our world and can be characterized through > > the measurements. Passive - that one that do not inject extra energy > > to the system, gain equal or less than 1 (or 0dB). > > ...and by "gain<=1" you mean > (a) the magnitude response for all frequencies are equal or less than > � � one, or > (b) the gain factor "K" you get by converting the numerator/ > denominator > � � coefficients of H(z) to a pole-zero factorization (see Matlab's > � � tf2zp function) > > ? > > Cheers! > SGBy gain I mean the magnitude of the system response. Regards Pawel
Reply by ●March 20, 20092009-03-20
On 20 Mar, 15:40, Pawel <prulikow...@gmail.com> wrote:> Thanks for the explanations. However I would argue with You that > conecpt of passivity makes only sense in analog domain. You can say > the same about stability - how digital filter (or revoked CPU) can > become unstable - what would happen would CPU blow up?! (Yes it can > burn but not as the result of numerical instability) Surely if one is > only interested in purely theoretical procedures than OK those are > only numbers and as such cannot be stable, passive nor minimum-phase > etc. They only make sense if we relate them to something i.e. transfer > function of the real circuit.Wrong. DSP is concerned only with numerics, not electronics. There is no such thing as 'the real cirquit' in DSP, inasmuch as DSP routines only require a set of numeric data to produce a result in terms of numeric data. Any peripherals (ADCs, DACs) would qualify as interfaces between the continuous time and discrete time domains, which are not required for DSP routines to be well-defined. If you want to discuss cirquits, you need to qualify that. Which would formally push the discussion Off Topic wrt comps.dsp (although lots of people here would likely chime in with useful comments).> I would probably need to rephrase my question: How to ensure that z- > transfer function that describes real passive system fulfills > passivity criteria - what are those criteria?It seems you confuse several issues: - Analog electronics with DSP - 'passivity' with stability Basically, your questions don't make much sense in the context of DSP. Rune
Reply by ●March 20, 20092009-03-20
Pawel wrote:> On Mar 20, 3:19 pm, SG <s.gesem...@gmail.com> wrote: >> On 20 Mrz., 15:40, Pawel <prulikow...@gmail.com> wrote: >> >>> I would probably need to rephrase my question: How to ensure that z- >>> transfer function that describes real passive system fulfills >>> passivity criteria - what are those criteria? >> What's your definition of "real passive system"? >> >> Cheers! >> SG > > Real - one that exist in our world and can be characterized through > the measurements. Passive - that one that do not inject extra energy > to the system, gain equal or less than 1 (or 0dB).What if the system injects energy, but not more than it absorbs? Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
