energy conservation in linear-nonlinear systems
in [DSP]
|
Prev: fft question
Next: Fresh review of an old favorite...
From: fisico32 on 30 Jul 2010 18:21 >On Jul 30, 3:00=A0pm, "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). >> >> =A0 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 i= >n >> >> 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. =A0Draw boxes around them, and terminate them >> >with 50-ohm resistors (75 if you like cable TV). =A0Now 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. =A0Do they sum up t= >o >> >zero for the L-C? =A0Do they sum up to zero for the R-C? =A0Now calculat= >e >> >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)) <=3D> 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) <=3D> f3, with f3 =3D 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 Thanks Maury, I will get the book tomorrow. You say that the two frequencies f1 and f2 are mapped to third frequencies? Does that mean that a third frequency gets generated beside f1 adn f2? thanks >
From: Jerry Avins on 30 Jul 2010 20:15 On 7/30/2010 2:32 PM, 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). You probaly also have time invariance in mind. Certainly, superposition holds then. > 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). But superposition requires that ane energy transferred from one frequency be transferred to another even if the other is not present in the input. That contradicts time-invariant linearity. You should have deduced that for yourself. Lately, it seems that you ask first and think later. ... Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
From: maury on 31 Jul 2010 11:01 On Jul 30, 5:21 pm, "fisico32" <marcoscipioni1(a)n_o_s_p_a_m.gmail.com> wrote: > >On Jul 30, 3:00=A0pm, "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). > >> >> =A0 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 > i= > >n > >> >> 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. =A0Draw boxes around them, and terminate > them > >> >with 50-ohm resistors (75 if you like cable TV). =A0Now 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. =A0Do they sum up > t= > >o > >> >zero for the L-C? =A0Do they sum up to zero for the R-C? =A0Now > calculat= > >e > >> >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)) <=3D> 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) <=3D> f3, with f3 =3D 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 > > Thanks Maury, > I will get the book tomorrow. > You say that the two frequencies f1 and f2 are mapped to third > frequencies? > Does that mean that a third frequency gets generated beside f1 adn f2? > thanks > > > > - Hide quoted text - > > - Show quoted text -- Hide quoted text - > > - Show quoted text - Here's what you need to do. Take the second-order Volterra kernel with frequencies of f1 and f2. Perform the Fourier transform on the kernel and look at what happens to the resultant frequency. When you get Pitas' book it may be clearer, but do the transform exercise first. Math takes "doing" to really understand. Maurice Givens
From: Fred Marshall on 3 Aug 2010 15:13
fisico32 wrote: > > > Thanks Maury, > I will get the book tomorrow. > You say that the two frequencies f1 and f2 are mapped to third > frequencies? > Does that mean that a third frequency gets generated beside f1 adn f2? > thanks > Time to be a bit careful about what you're talking about. Let's assume that you have a linear system, followed by a perfect gain of "1" plus a +1/-1 limiter (a nonlinear element), followed by another linear system. If the input is such that the limits of the limiter are never reached, then the whole thing remains a linear system. It's only when an input is such that the limits of the limiter *are* reached that the nonlinearity comes in. When that happens then multiple other frequencies are introduced/created and, if all the 3 blocks are lossless (in theory at least) then the energy is split amongst all the output frequencies. Case in point. Apply a small sinusoid to the input such that the limiter does nothing. The output will be the gain of the two linear blocks at that one frequency. If they are lossless, then it has to be the same energy as the input - less any losses in the source impedance, etc. etc. Apply a sinusoid of the same frequency to the input such that the limiter generates what is for all intents and purposes a square wave. Do a Fourier Series analysis on that square wave - yielding a bunch of sinusoids that are harmonically related. If two such sinusoids are applied then it just gets more complicated so I've limited this discussion to a single frequency yielding MANY output frequencies .... well, an infinite number theoretically. Does that make it clearer? Now where is the energy? Fred |