energy conservation in linear-nonlinear systems
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From: fisico32 on 30 Jul 2010 14:32 Hello Forum, say we have a linear system that preserves the total energy of the input signal x(t) in the output signal y(t). That means that the spectral components at particular frequencies in y(t) may have smaller or bigger amplitudes than in the x(t), as long as the sum of the squared amplitudes is the same as the the sum in the input signal x(t). The increase or decrease of amplitude at a particular frequency f is determined by the gain of the linear system. If the energy of y(t) is the same as the enrgy of x(t), does it mean that the energy has possibly been transfered by the systems from some to spectral components to others? Or is this energy transfer between spectral components possible only in nonlinear systems? Do nonlinear systems usually change the energy of the input (increaseing it or decreasing it) or can they also keep it constant? thanks fisico Or is this
From: Tim Wescott on 30 Jul 2010 14:50 On 07/30/2010 11:32 AM, fisico32 wrote: > Hello Forum, > > say we have a linear system that preserves the total energy of the input > signal x(t) in the output signal y(t). > That means that the spectral components at particular frequencies in y(t) > may have smaller or bigger amplitudes than in the x(t), as long as the sum > of the squared amplitudes is the same as the the sum in the input signal > x(t). > > The increase or decrease of amplitude at a particular frequency f is > determined by the gain of the linear system. If the energy of y(t) is the > same as the enrgy of x(t), does it mean that the energy has possibly been > transfered by the systems from some to spectral components to others? > > Or is this energy transfer between spectral components possible only in > nonlinear systems? Do nonlinear systems usually change the energy of the > input (increaseing it or decreasing it) or can they also keep it constant? You ask these really interesting questions that you should be able to answer for yourself, perhaps after going to the local student's bar and knocking back a few beers (eat some Nachos, too, you can drink more beer that way). I think that a "linear system" (and you really mean "time-invariant linear system, by the question that you ask") is such a mathematical abstraction that to try to subject it to basic thermodynamic considerations as energy conservation is to generate so many conundrums as to let you figure out that it ain't real! Why don't you make up a few two-port networks -- I'd suggest an R-C, and one or two flavors of L-C. Draw boxes around them, and terminate them with 50-ohm resistors (75 if you like cable TV). Now write out the equations for the currents at the input and output, as a function of frequency, and calculate the power in and power out. Do they sum up to zero for the L-C? Do they sum up to zero for the R-C? Now calculate internal dissipation -- _now_ do the energies sum up? Now look at your results, and ponder the question: "does thinking about these things as linear systems that transport energy make any sense at all in this context?" -- Tim Wescott Wescott Design Services http://www.wescottdesign.com Do you need to implement control loops in software? "Applied Control Theory for Embedded Systems" was written for you. See details at http://www.wescottdesign.com/actfes/actfes.html
From: fisico32 on 30 Jul 2010 16:00 >On 07/30/2010 11:32 AM, fisico32 wrote: >> Hello Forum, >> >> say we have a linear system that preserves the total energy of the input >> signal x(t) in the output signal y(t). >> That means that the spectral components at particular frequencies in y(t) >> may have smaller or bigger amplitudes than in the x(t), as long as the sum >> of the squared amplitudes is the same as the the sum in the input signal >> x(t). >> >> The increase or decrease of amplitude at a particular frequency f is >> determined by the gain of the linear system. If the energy of y(t) is the >> same as the enrgy of x(t), does it mean that the energy has possibly been >> transfered by the systems from some to spectral components to others? >> >> Or is this energy transfer between spectral components possible only in >> nonlinear systems? Do nonlinear systems usually change the energy of the >> input (increaseing it or decreasing it) or can they also keep it constant? > >You ask these really interesting questions that you should be able to >answer for yourself, perhaps after going to the local student's bar and >knocking back a few beers (eat some Nachos, too, you can drink more beer >that way). > >I think that a "linear system" (and you really mean "time-invariant >linear system, by the question that you ask") is such a mathematical >abstraction that to try to subject it to basic thermodynamic >considerations as energy conservation is to generate so many conundrums >as to let you figure out that it ain't real! > >Why don't you make up a few two-port networks -- I'd suggest an R-C, and >one or two flavors of L-C. Draw boxes around them, and terminate them >with 50-ohm resistors (75 if you like cable TV). Now write out the >equations for the currents at the input and output, as a function of >frequency, and calculate the power in and power out. Do they sum up to >zero for the L-C? Do they sum up to zero for the R-C? Now calculate >internal dissipation -- _now_ do the energies sum up? > >Now look at your results, and ponder the question: "does thinking about >these things as linear systems that transport energy make any sense at >all in this context?" > >-- > >Tim Wescott >Wescott Design Services >http://www.wescottdesign.com > >Do you need to implement control loops in software? >"Applied Control Theory for Embedded Systems" was written for you. >See details at http://www.wescottdesign.com/actfes/actfes.html > I see your point Tim, by the way it is almost happy hour :) I am learning about Volterra series and thinking about the 2 dimensional FT of the 2nd order kernel h(tau1, tau2), H(w1,w2)..... I think this frequency kernel and how it can describe the trasfer energy from two spectral components w1 and w3 to a third w3... This process is not yet clear to me.....
From: Tim Wescott on 30 Jul 2010 16:54 On 07/30/2010 01:00 PM, fisico32 wrote: >> On 07/30/2010 11:32 AM, fisico32 wrote: >>> Hello Forum, >>> >>> say we have a linear system that preserves the total energy of the > input >>> signal x(t) in the output signal y(t). >>> That means that the spectral components at particular frequencies in > y(t) >>> may have smaller or bigger amplitudes than in the x(t), as long as the > sum >>> of the squared amplitudes is the same as the the sum in the input > signal >>> x(t). >>> >>> The increase or decrease of amplitude at a particular frequency f is >>> determined by the gain of the linear system. If the energy of y(t) is > the >>> same as the enrgy of x(t), does it mean that the energy has possibly > been >>> transfered by the systems from some to spectral components to others? >>> >>> Or is this energy transfer between spectral components possible only in >>> nonlinear systems? Do nonlinear systems usually change the energy of > the >>> input (increaseing it or decreasing it) or can they also keep it > constant? >> >> You ask these really interesting questions that you should be able to >> answer for yourself, perhaps after going to the local student's bar and >> knocking back a few beers (eat some Nachos, too, you can drink more beer >> that way). >> >> I think that a "linear system" (and you really mean "time-invariant >> linear system, by the question that you ask") is such a mathematical >> abstraction that to try to subject it to basic thermodynamic >> considerations as energy conservation is to generate so many conundrums >> as to let you figure out that it ain't real! >> >> Why don't you make up a few two-port networks -- I'd suggest an R-C, and >> one or two flavors of L-C. Draw boxes around them, and terminate them >> with 50-ohm resistors (75 if you like cable TV). Now write out the >> equations for the currents at the input and output, as a function of >> frequency, and calculate the power in and power out. Do they sum up to >> zero for the L-C? Do they sum up to zero for the R-C? Now calculate >> internal dissipation -- _now_ do the energies sum up? >> >> Now look at your results, and ponder the question: "does thinking about >> these things as linear systems that transport energy make any sense at >> all in this context?" >> >> -- >> >> Tim Wescott >> Wescott Design Services >> http://www.wescottdesign.com >> >> Do you need to implement control loops in software? >> "Applied Control Theory for Embedded Systems" was written for you. >> See details at http://www.wescottdesign.com/actfes/actfes.html >> > > > I see your point Tim, > by the way it is almost happy hour :) > > I am learning about Volterra series and thinking about the 2 dimensional FT > of the 2nd order kernel h(tau1, tau2), H(w1,w2)..... > I think this frequency kernel and how it can describe the trasfer energy > from two spectral components w1 and w3 to a third w3... > This process is not yet clear to me..... True. And there are some systems (think about a diode harmonic generator, or a diode mixer, or just a plain old LC filter in a radio) that do not themselves dissipate any energy*, yet have some very interesting and useful properties. The problem arises from two places: first, when you try to equate the spectral content of the input and output signals with the energy being transferred _by_ those signals, and second, when you ignore any sources of power that may be connected to the systems in question or dissipative elements that may be inside the systems, turning input energy into heat. Case in point, that plain old LC filter that I cite above: generally when you buy or build an LC filter for a radio you specify it's frequency response when it is terminated with a 50-ohm load and driven from a 50-ohm source. You don't specify the spectrum of the output (into that 50-ohm load) with respect to the actual voltage on the input of the filter: you specify the output with respect to the drive to the 50-ohm source impedance on the input. Pretty much by definition (because an ideal LC filter is lossless), the energy that doesn't make it to the output load doesn't go away in the filter -- it either gets absorbed in the source resistance, or it never gets generated by the source at all. Similar things happen with nonlinear networks: make that diode doubler that I'm babbling about, and not only will you see significant energy at 2*f at it's _output_, you'll also see some of that energy at it's _input_, or in a well designed one you'll see energy at all harmonics of the carrier _except_ for 2*f. * Or at least don't in the ideal case. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com Do you need to implement control loops in software? "Applied Control Theory for Embedded Systems" was written for you. See details at http://www.wescottdesign.com/actfes/actfes.html
From: maury on 30 Jul 2010 17:09
On Jul 30, 3:00 pm, "fisico32" <marcoscipioni1(a)n_o_s_p_a_m.gmail.com> wrote: > >On 07/30/2010 11:32 AM, fisico32 wrote: > >> Hello Forum, > > >> say we have a linear system that preserves the total energy of the > input > >> signal x(t) in the output signal y(t). > >> That means that the spectral components at particular frequencies in > y(t) > >> may have smaller or bigger amplitudes than in the x(t), as long as the > sum > >> of the squared amplitudes is the same as the the sum in the input > signal > >> x(t). > > >> The increase or decrease of amplitude at a particular frequency f is > >> determined by the gain of the linear system. If the energy of y(t) is > the > >> same as the enrgy of x(t), does it mean that the energy has possibly > been > >> transfered by the systems from some to spectral components to others? > > >> Or is this energy transfer between spectral components possible only in > >> nonlinear systems? Do nonlinear systems usually change the energy of > the > >> input (increaseing it or decreasing it) or can they also keep it > constant? > > >You ask these really interesting questions that you should be able to > >answer for yourself, perhaps after going to the local student's bar and > >knocking back a few beers (eat some Nachos, too, you can drink more beer > >that way). > > >I think that a "linear system" (and you really mean "time-invariant > >linear system, by the question that you ask") is such a mathematical > >abstraction that to try to subject it to basic thermodynamic > >considerations as energy conservation is to generate so many conundrums > >as to let you figure out that it ain't real! > > >Why don't you make up a few two-port networks -- I'd suggest an R-C, and > >one or two flavors of L-C. Draw boxes around them, and terminate them > >with 50-ohm resistors (75 if you like cable TV). Now write out the > >equations for the currents at the input and output, as a function of > >frequency, and calculate the power in and power out. Do they sum up to > >zero for the L-C? Do they sum up to zero for the R-C? Now calculate > >internal dissipation -- _now_ do the energies sum up? > > >Now look at your results, and ponder the question: "does thinking about > >these things as linear systems that transport energy make any sense at > >all in this context?" > > >-- > > >Tim Wescott > >Wescott Design Services > >http://www.wescottdesign.com > > >Do you need to implement control loops in software? > >"Applied Control Theory for Embedded Systems" was written for you. > >See details athttp://www.wescottdesign.com/actfes/actfes.html > > I see your point Tim, > by the way it is almost happy hour :) > > I am learning about Volterra series and thinking about the 2 dimensional FT > of the 2nd order kernel h(tau1, tau2), H(w1,w2)..... > I think this frequency kernel and how it can describe the trasfer energy > from two spectral components w1 and w3 to a third w3... > This process is not yet clear to me.....- Hide quoted text - > > - Show quoted text - Think of it this way. With 2 frequency conponents, the Fourier transform is a function of both exp(i*2*pi*f1) and exp(i*2*pi*f2). Something like X(exp(i*2*pi*f1),exp(i*2*pi*f2)) <=> X(f1,f2) Then, when you do the double integral, you get exp(i*2*pi*[f1 + f2]). This is the mapping of (f1,f2) <=> f3, with f3 = f1+f2. This help? I can't remember if Stephen Boyd addressed this with the Voltarra series, but Pitas and Venetsanopoulos did. Go to the library and look up their book "Nonlinear Digital Filters". They have a chapter on polynomial filters. They also do a power spectral analysis and a bispectral analysis. Maurice Givens |