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Linear System Properties?

Started by Till Crueger March 26, 2004
Jerry Avins wrote:

> Till wants to know what unique property of sine waves makes them come > through a linear circuit unaltered (presumably in shape) while square > waves do not. However sharp your truths, I don't think your answer to > him was transparent at his level of understanding.
I think it because they are Eigenvalues of a linear, time invariant system even though I'm not sure what that means. :-) Bob -- "Things should be described as simply as possible, but no simpler." A. Einstein
In article <406488a0$0$3077$61fed72c@news.rcn.com>,
Jerry Avins <jya@ieee.org> wrote:

>Subject: Re: Linear System Properties? >From: Jerry Avins <jya@ieee.org> >Reply-To: jya@ieee.org >Organization: The Hectic Eclectic >Date: Fri, 26 Mar 2004 14:46:39 -0500 >Newsgroups: comp.dsp > >Randy Yates wrote: > >> 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. >>> >>>Randy, >> >> >> Jerry, >> >> I hear you (somewhat) but I'm going to play the opposite side here to >> the hilt. >> >> >>>This is true in a sense, >> >> >> This is true absolutely. No "sense" or interpretation required. >> >> >>>but misleading. >> >> >> How can the truth be misleading? >> >> >>>For example, you can't expect >>>the output of a linear system to be a square wave just because the input >>>is excited by one. >> >> >> I do not expect that. What I do expect is that the output >> will not contain any frequencies that weren't in the >> input. >> >> >>>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 thought he asked about shape, at least by implication. > >> 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. > >Till wants to know what unique property of sine waves makes them come >through a linear circuit unaltered (presumably in shape) while square >waves do not. However sharp your truths, I don't think your answer to >him was transparent at his level of understanding.
You need several pieces to make the arguement that sines (and their friends) are uniquely priviledged: 1 - something that will find exactly and only sines. that something is the characterization of the harmonic oscillator as a diferential equation or its discrete anologue. 2 - noticing that that superpostion and time invariance preserves the characterization. superposition preserves the coefficients and time invariance allows for the derivative when there is a useful amount of regularity in the situation. 3 - combining them by the process that says the input satisfies the harmonic oscillation conditions for fixed frequency but undetermined amplitude and phase and the output satisfies the harmonic oscillator conditions for the same fixed frequency but another undetermined amplitude and phase. What gets fixed is the frequency and what is left to be determined is the amplitude and phase of the sine at that frequency. With this arguement you then get fancy and combine many sine waves where each gets its amplitude and phase modified by going through the system. The original question was why sines are special for systems with superposition and time invariance. The answer is that sines come from harmonic oscillators and such systems preserve harmonic oscillator frequencies. And it is really nice that we can do lots of convenient things with the output of harmonic oscillators. It is worth remembering that to get there you used both the superposition and the time invariance assumptions.
>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; >
"Bob Cain" <arcane@arcanemethods.com> wrote in message
news:c42kd911u7t@enews2.newsguy.com...
> Jerry Avins wrote: > > > Till wants to know what unique property of sine waves makes them come > > through a linear circuit unaltered (presumably in shape) while square > > waves do not. However sharp your truths, I don't think your answer to > > him was transparent at his level of understanding. > > I think it because they are Eigenvalues of a linear, time > invariant system even though I'm not sure what that means. :-) >
It means that an LTI system will change only their amplitude, which isn't quite true, because the phase can change as well. To answer the original posters's question takes a few steps. I'm not going to prove them all, but I'll give the steps: 1) An LTI system is completely characterized by its impulse response, and the response of an LTI system to a given input can be found by convolution with its impulse response. 2) Exponential functions are eigenvalues of all LTI systems, because convolving an exponential with anything just changes its amplitude. You can derive that from the formula for convolution, by replacing time shifts with equivalent scaling: A*e^(t-s) = (A/e^s)*e^t. 3) A real sinusoid is the sum of two complex conjugate exponentials. By superposition, the response of an LTI to a sinusoid will be a sum of the same two exponentials, after each has been scaled (weighted) independently. 4) If the LTI has a real impulse response, the weights are complex conjugates, and the result is a real sinusoid of the same frequency as the original, but with a possibly different phase.
"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. ;-)
"Matt Timmermans" <mt0000@sympatico.nospam-remove.ca> writes:
> [...] > 2) Exponential functions are eigenvalues of all LTI systems, because > convolving an exponential with anything just changes its amplitude. You can > derive that from the formula for convolution, by replacing time shifts with > equivalent scaling: A*e^(t-s) = (A/e^s)*e^t.
Huh? A time-shifted complex sinusoid is A*e^(i*(t-s)). Also, by "amplitude" here you must mean something different than usual since, usually, amplitude is real, while the factor e^(-i*s) here is complex. But your point is of course still valid. Nice call, Matt. You cut through to the salient point nicely. -- % Randy Yates % "Remember the good old 1980's, when %% Fuquay-Varina, NC % things were so uncomplicated?" %%% 919-577-9882 % 'Ticket To The Moon' %%%% <yates@ieee.org> % *Time*, Electric Light Orchestra http://home.earthlink.net/~yatescr
"Till Crueger" <TillFC@gmx.net> writes:
> [...] > 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? -- % 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
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)) -- % Randy Yates % "Watching all the days go by... %% Fuquay-Varina, NC % Who are you and who am I?" %%% 919-577-9882 % 'Mission (A World Record)', %%%% <yates@ieee.org> % *A New World Record*, ELO http://home.earthlink.net/~yatescr
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, ...? How? I suppose we could run a tape at double speed, but eventually, we run out of tape. 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;
Randy Yates wrote:

> "Till Crueger" <TillFC@gmx.net> writes: > >>[...] >>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. :-) 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;
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. IMHO. YMMV. -- Ron Nicholson rhn AT nicholson DOT com http://www.nicholson.com/rhn/ #include <canonical.disclaimer> // only my own opinions, etc.