Forums

Linear System Properties?

Started by Till Crueger March 26, 2004
Matt Timmermans wrote:
> "Matt Timmermans" <mt0000@sympatico.nospam-remove.ca> wrote in message > news:an59c.40234$A_2.1541424@news20.bellglobal.com... > >>[...] 2) Exponential functions are eigenvalues of all LTI systems [...] > > > er, I mean eigenfunctions. Bob did it first -- I'm sure he meant > eigenfunctions too. ;-) >
Actually no. I don't really understand eigenstuff and was just digging that out of associative memory. Incorrectly as it turns out (as usually happens when you don't understand something.) :-) Bob -- "Things should be described as simply as possible, but no simpler." A. Einstein
Jerry Avins <jya@ieee.org> writes:

> Randy Yates wrote: > >> rhn@mauve.rahul.net (Ronald H. Nicholson Jr.) writes: >> >>>In article <c41jep$g2s$1@f1node01.rhrz.uni-bonn.de>, >>>Till Crueger <TillFC@gmx.net> wrote: >>> >>>>Furthermore I thought about a system which only would double any frequency >>>>component present in a given Signal, and I fail to see how this system is >>>>non-linear. >>> >>>This is not both linear and time-invarient. Shift the input by half the >>>period and the output will shift by a full period. This is equivalent >>>to inverting the gain depending on the starting phase. So either you >>>get non-linear gain, or you lose time invariance. >> The system appears linear to me since >> H(a*x1(t) + b*x2(t)) = a*H(x1(t)) + b*H(x2(T)) > > It's late and I may not be thinking clearly. Are you saying that a > realizable linear system can have as output frequencies that are double > the input frequencies (and to take Till literally, only those > frequencies)? For arbitrary Fn, if the input is F1, F2, f3, ..., then > the output will consist entirely of 2F1, 2F2, 2F3, ...?
Yes, at least it seems so. I have the feeling I'm screwing up somewhere, but I can't see where.
> How?
Do you mean "How can we construct such a system"? Like this: Consider the signal x(t) that is the input to this "linear" system. Then X(f) = F(x(t)), where F(.) denotes the Fourier transform of ".". Then the output, y(t), is y(t) = F^(-1)(X(f/2)), where F^(-1)(.) denotes the inverse Fourier transform of ".".
> I suppose > we could run a tape at double speed, but eventually, we run out of tape.
Yeah, well that'd be true for any input signal that had infinite extent anyway. I'm with Till - I can't see why this system is nonlinear. This is going to the left of this subthread a couple of notches, but I also tried looking at a linear system as a "linear transformation" from a vector space V to V, which is nothing more than a ring homomorphism, for those of us with some abstract algebra. However, I'm having trouble deciding what the vector space V should be here: R^2 or something more like the set of all functions. -- % Randy Yates % "So now it's getting late, %% Fuquay-Varina, NC % and those who hesitate %%% 919-577-9882 % got no one..." %%%% <yates@ieee.org> % 'Waterfall', *Face The Music*, ELO http://home.earthlink.net/~yatescr
Bob Cain <arcane@arcanemethods.com> writes:

> Matt Timmermans wrote: >> "Matt Timmermans" <mt0000@sympatico.nospam-remove.ca> wrote in message >> news:an59c.40234$A_2.1541424@news20.bellglobal.com... >> >>>[...] 2) Exponential functions are eigenvalues of all LTI systems [...] >> er, I mean eigenfunctions. Bob did it first -- I'm sure he meant >> eigenfunctions too. ;-) >> > > Actually no. I don't really understand eigenstuff and was just > digging that out of associative memory. Incorrectly as it turns out > (as usually happens when you don't understand something.) :-)
Hey Bob, could you show me how to design my cache (associative? :) ) like that? That's a pretty good trick! -- % Randy Yates % "Bird, on the wing, %% Fuquay-Varina, NC % goes floating by %%% 919-577-9882 % but there's a teardrop in his eye..." %%%% <yates@ieee.org> % 'One Summer Dream', *Face The Music*, ELO http://home.earthlink.net/~yatescr
In article <smfvgevw.fsf@ieee.org>, Randy Yates  <yates@ieee.org> wrote:
>The system appears linear to me since > > H(a*x1(t) + b*x2(t)) = a*H(x1(t)) + b*H(x2(T))
Do you mean: H(a*x1(t) + b*x2(T)) = a*H(x1(t)) + b*H(x2(T)) where T = t - t0 to combine linearity and time-shift-invariance ? -- Ron Nicholson rhn AT nicholson DOT com http://www.nicholson.com/rhn/ #include <canonical.disclaimer> // only my own opinions, etc.
Randy Yates <yates@ieee.org> wrote in message news:<65crwh5t.fsf@ieee.org>...
> Jerry Avins <jya@ieee.org> writes: > > > Randy Yates wrote: > > > >> Hey Till, > >> You're right - it isn't just sine waves that a linear system will not > >> produce new frequencies of - ANY waveform will be unaltered in > >> frequency by a linear system.
> > The output may contain all the component frequencies > > of the input, but shape isn't necessarily maintained. > > Did I say or imply the _shape_ was maintained? In fact I did not.
I am sure you did not, but the phrasing is somewhat unclear: It is a matter of interpretation what you mean by "[the] waveform will be unaltered in frequency". I suspect Jerry interpreted this as "the magnitude function of frequency spectra can be changed by linear systems", and he is right. I suspect you meant "the linear system does not introduce new frequency components in the spectrum", and you are right. I think you agree, once you find an unambiguous phrasing that both are happy with.
> But beyond the question of what I said or didn't say, your comments seem > to be aimed at how one should explain something, and THAT depends on > style and technique. This is my style. Making the truth sharp, in my > experience, usually dispells bad conclusions and sheds light on wrong > thinking.
I agree. In my experience, that requires some very meticulous attention to phrasing and terminology. But that's a "hobby horse"(?) (Norw. "kjepphest") of mine I'll keep in his stable for now. Rune
Randy Yates wrote:

> Bob Cain <arcane@arcanemethods.com> writes: > > >>Matt Timmermans wrote: >> >>>"Matt Timmermans" <mt0000@sympatico.nospam-remove.ca> wrote in message >>>news:an59c.40234$A_2.1541424@news20.bellglobal.com... >>> >>> >>>>[...] 2) Exponential functions are eigenvalues of all LTI systems [...] >>> >>>er, I mean eigenfunctions. Bob did it first -- I'm sure he meant >>>eigenfunctions too. ;-) >>> >> >>Actually no. I don't really understand eigenstuff and was just >>digging that out of associative memory. Incorrectly as it turns out >>(as usually happens when you don't understand something.) :-) > > > Hey Bob, could you show me how to design my cache (associative? :) ) like that? > That's a pretty good trick!
There's still some work to be done on the ECC code and circuitry. That's getting more and more important as the years go by. About the time I figure it out I expect I won't be able to remember it. :-) Bob -- "Things should be described as simply as possible, but no simpler." A. Einstein
rhn@mauve.rahul.net (Ronald H. Nicholson Jr.) writes:

> In article <smfvgevw.fsf@ieee.org>, Randy Yates <yates@ieee.org> wrote: >>The system appears linear to me since >> >> H(a*x1(t) + b*x2(t)) = a*H(x1(t)) + b*H(x2(T)) > > Do you mean: > > H(a*x1(t) + b*x2(T)) = a*H(x1(t)) + b*H(x2(T)) > where > T = t - t0 > to combine linearity and time-shift-invariance ?
No. I simply mistyped the last t as a "T". I was just focusing on linearity for the moment. -- % Randy Yates % "Ticket to the moon, flight leaves here today %% Fuquay-Varina, NC % from Satellite 2" %%% 919-577-9882 % 'Ticket To The Moon' %%%% <yates@ieee.org> % *Time*, Electric Light Orchestra http://home.earthlink.net/~yatescr
allnor@tele.ntnu.no (Rune Allnor) writes:

> Randy Yates <yates@ieee.org> wrote in message news:<65crwh5t.fsf@ieee.org>... >> Jerry Avins <jya@ieee.org> writes: >> >> > Randy Yates wrote: >> > >> >> Hey Till, >> >> You're right - it isn't just sine waves that a linear system will not >> >> produce new frequencies of - ANY waveform will be unaltered in >> >> frequency by a linear system. > >> > The output may contain all the component frequencies >> > of the input, but shape isn't necessarily maintained. >> >> Did I say or imply the _shape_ was maintained? In fact I did not. > > I am sure you did not, but the phrasing is somewhat unclear: It is > a matter of interpretation what you mean by "[the] waveform will be > unaltered in frequency". I suspect Jerry interpreted this as "the > magnitude function of frequency spectra can be changed by linear > systems", and he is right. I suspect you meant "the linear system does > not introduce new frequency components in the spectrum", and you are > right. I think you agree, once you find an unambiguous phrasing that > both are happy with.
Rune, Jerry, Ronald, and most of all, Till, I apologize. I had my head where the sun doesn't shine yesterday. It's definitely brighter around here today. I stated: You're right - it isn't just sine waves that a linear system will not produce new frequencies of - ANY waveform will be unaltered in frequency by a linear system. Yes, I indeed stated any "wave_FORM_", i.e., any form of wave. Aaahh! That certainly DOES imply shape. I'm an idiot. I was wrong to defend it. Till, I hope I haven't confused you. What I meant (I guess) was that any wave form will come through with the same number of frequencies present, though their amplitudes and phases may be changed (and thus the form of the wave may be changed). Yes, I know - there's also that pesky little case when the response is 0 at some frequency or set of frequencies, thus knocking them completely out. But I think ya'll get what I mean now. I think. -- % Randy Yates % "She's sweet on Wagner-I think she'd die for Beethoven. %% Fuquay-Varina, NC % She love the way Puccini lays down a tune, and %%% 919-577-9882 % Verdi's always creepin' from her room." %%%% <yates@ieee.org> % "Rockaria", *A New World Record*, ELO http://home.earthlink.net/~yatescr
Randy Yates wrote:

   ...

> Do you mean "How can we construct such a system"? Like this: > > Consider the signal x(t) that is the input to this "linear" system. > Then X(f) = F(x(t)), where F(.) denotes the Fourier transform of ".". > Then the output, y(t), is > > y(t) = F^(-1)(X(f/2)), > > where F^(-1)(.) denotes the inverse Fourier transform of ".".
...
> I'm with Till - I can't see why this system is nonlinear.
... How would you go about actually building one of these systems? When I see X(f/2), I worry about realizability and causality. Jerry -- Engineering is the art of making what you want from things you can get. &#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
Ronald H. Nicholson Jr. wrote:

> In article <4065044c$0$3055$61fed72c@news.rcn.com>, > Jerry Avins <jya@ieee.org> wrote: > >>Randy Yates wrote: >> >>>>Furthermore I thought about a system which only would double any frequency >>>>component present in a given Signal, and I fail to see how this system is >>>>non-linear. >>> >>>Could it be that linearity is only a necessary condition for a linear >>>system? >> >>I believe that linearity is both a necessary and a sufficient condition. > > > But linear systems are a superset of systems that are both linear and > time-shift-invariant.
That's as may be, but being linear is clearly a necessary and sufficient condition for being linear. I thought I made a joke. It evidently fell flat. Jerry -- Engineering is the art of making what you want from things you can get. &#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;