Hi all. I've been thinking a bit about processing of signals that have propagated through an ocean waveguide that change, either because of waves on the surface, tides, or for other oceanographic reasons. In any text on DSP we get aquainted with the Linear Time Invariant (LTI) system model. Which is fine. I have no problems coming up with examples of Nonlinear but still Time Invariant systems. Now, I can't figure out if the concept of a "Linear Time Variant" system makes sense. As far as I know, time variant systems are handeled by adaptive filters, which are nonlinear since their filter responses depend on the data they act on. Is the time variant system itself nonlinear, or is it just that no linear method of dealing with such systems has been found? Any thoughts? Rune
Linear Time Variant systems
Started by ●July 25, 2003
Reply by ●July 25, 20032003-07-25
On 24 Jul 2003, Rune Allnor wrote:> Hi all. > > I've been thinking a bit about processing of signals that have propagated > through an ocean waveguide that change, either because of waves on the > surface, tides, or for other oceanographic reasons. > > In any text on DSP we get aquainted with the Linear Time Invariant > (LTI) system model. Which is fine. I have no problems coming up with > examples of Nonlinear but still Time Invariant systems. > > Now, I can't figure out if the concept of a "Linear Time Variant" system > makes sense. As far as I know, time variant systems are handeled by > adaptive filters, which are nonlinear since their filter responses depend > on the data they act on. > > Is the time variant system itself nonlinear, or is it just that no linear > method of dealing with such systems has been found? > > Any thoughts? > > Rune >hi rune, i think an example might help: y(t) = \sum_{\tau} x(t-tau) h(t;\tau) think of it as convolution, except that at each t, the channel changes slightly. for example, a tapped delay line which tap coefficients change with time. the changes can be independent of the input signal, which extends it beyond the example that you proposed. one way to visualize it is that, to compute the output at each time t, you look at a different channel h_{t}(\tau). for example, fading/wireless channels behave this way, although in most cases it is simplified such that the channel only changes between frames or something. hope that helps, julius -- The most rigorous proofs will be shown by vigorous handwaving. http://www.mit.edu/~kusuma opinion of author is not necessarily of the institute
Reply by ●July 25, 20032003-07-25
How about a multirate filter bank system? It can be linear and time-variant if you don't consider it being periodic invariant. cf Julius Kusuma wrote:> > On 24 Jul 2003, Rune Allnor wrote: > > > Hi all. > > > > I've been thinking a bit about processing of signals that have propagated > > through an ocean waveguide that change, either because of waves on the > > surface, tides, or for other oceanographic reasons. > > > > In any text on DSP we get aquainted with the Linear Time Invariant > > (LTI) system model. Which is fine. I have no problems coming up with > > examples of Nonlinear but still Time Invariant systems. > > > > Now, I can't figure out if the concept of a "Linear Time Variant" system > > makes sense. As far as I know, time variant systems are handeled by > > adaptive filters, which are nonlinear since their filter responses depend > > on the data they act on. > > > > Is the time variant system itself nonlinear, or is it just that no linear > > method of dealing with such systems has been found? > > > > Any thoughts? > > > > Rune > > > > hi rune, > > i think an example might help: > > y(t) = \sum_{\tau} x(t-tau) h(t;\tau) > > think of it as convolution, except that at each t, the channel changes > slightly. for example, a tapped delay line which tap coefficients change > with time. the changes can be independent of the input signal, which > extends it beyond the example that you proposed. > > one way to visualize it is that, to compute the output at each time t, you > look at a different channel h_{t}(\tau). for example, fading/wireless > channels behave this way, although in most cases it is simplified such > that the channel only changes between frames or something. > > hope that helps, > julius > > -- > The most rigorous proofs will be shown by vigorous handwaving. > http://www.mit.edu/~kusuma > > opinion of author is not necessarily of the institute
Reply by ●July 25, 20032003-07-25
Rune Allnor wrote:> Hi all. > > I've been thinking a bit about processing of signals that have > propagated through an ocean waveguide that change, either because > of waves on the surface, tides, or for other oceanographic > reasons. > > In any text on DSP we get aquainted with the Linear Time Invariant > (LTI) system model. Which is fine. I have no problems coming up > with examples of Nonlinear but still Time Invariant systems. > > Now, I can't figure out if the concept of a "Linear Time Variant" > system makes sense. As far as I know, time variant systems are > handeled by adaptive filters, which are nonlinear since their > filter responses depend on the data they act on. > > Is the time variant system itself nonlinear, or is it just that no > linear method of dealing with such systems has been found? > > Any thoughts? > > RuneHi Rune, linear differential equations with time variant coefficients is a subclass of linear time variant systems. I found that in a book of my university professor. The book is written in German and shows a classification tree: linear systems non-linear systems - linear time invariant systems - nonlin. time invariant - linear time variant systems - nonlin. time variant (There's a mathematical derivation there.) Bernhard -- before sending to the above email-address: replace deadspam.com by foerstergroup.de
Reply by ●July 25, 20032003-07-25
Julius Kusuma <kusuma@mit.edu> wrote in message news:<Pine.GSO.4.33L.0307242320240.6766-100000@magic-pi-ball.mit.edu>...> On 24 Jul 2003, Rune Allnor wrote: > > hi rune, > > i think an example might help: > > y(t) = \sum_{\tau} x(t-tau) h(t;\tau) > > think of it as convolution, except that at each t, the channel changes > slightly. for example, a tapped delay line which tap coefficients change > with time. the changes can be independent of the input signal, which > extends it beyond the example that you proposed. > > one way to visualize it is that, to compute the output at each time t, you > look at a different channel h_{t}(\tau). for example, fading/wireless > channels behave this way, although in most cases it is simplified such > that the channel only changes between frames or something. > > hope that helps, > juliusHi Julius, Yes, I've seen those kinds of reasonings before. In sonar problems the slowly varying channel can be represented as perturbations on a base model of the environment. The effects on the signal is usually thrown into a statistical bucket, "coherence", that deals with any random variation in the signal, that additive noise can't account for. The problem is when the change occurs on a time scale faster than the experiment. There have appeared reports in the literature about experiments (data recordings) that lasted for one or two hours where significant variations in medium parameters happens in a matter of minutes. And there is the regular sonar operations that sometimes take place in the vicinity of river outlets. Rune
Reply by ●July 25, 20032003-07-25
Bernhard Holzmayer <holzmayer.bernhard@deadspam.com> wrote in message news:<2642354.sqRcsc7OL3@holzmayer.ifr.rt>...> Rune Allnor wrote: > > > Hi all. > > > > I've been thinking a bit about processing of signals that have > > propagated through an ocean waveguide that change, either because > > of waves on the surface, tides, or for other oceanographic > > reasons. > > > > In any text on DSP we get aquainted with the Linear Time Invariant > > (LTI) system model. Which is fine. I have no problems coming up > > with examples of Nonlinear but still Time Invariant systems. > > > > Now, I can't figure out if the concept of a "Linear Time Variant" > > system makes sense. As far as I know, time variant systems are > > handeled by adaptive filters, which are nonlinear since their > > filter responses depend on the data they act on. > > > > Is the time variant system itself nonlinear, or is it just that no > > linear method of dealing with such systems has been found? > > > > Any thoughts? > > > > Rune > > Hi Rune, > > linear differential equations with time variant coefficients > is a subclass of linear time variant systems. > > I found that in a book of my university professor. > The book is written in German and shows a classification tree: > > linear systems non-linear systems > - linear time invariant systems - nonlin. time invariant > - linear time variant systems - nonlin. time variant > > (There's a mathematical derivation there.) > > BernhardCould you post a reference to the book? It's nearly twenty years since I learned (or rather, was supposed to learn) German. As they say, if you suffer from height anxiety and fear of flying, try parachuting. Reading a maths book in German would have much the same effect on me... Rune
Reply by ●July 25, 20032003-07-25
"Rune Allnor" <allnor@tele.ntnu.no> wrote in message news:f56893ae.0307241918.67e54977@posting.google.com...> Now, I can't figure out if the concept of a "Linear Time Variant" system > makes sense. As far as I know, time variant systems are handeled by > adaptive filters, which are nonlinear since their filter responses depend > on the data they act on.Hi Rune, A non-linear time-invariant system can become linear time-variant if one of the inputs is hidden. An AM radio transmitter, for instance, is linear and time variant when the carrier is considered as part of the system instead of an input. Also, the LTI concepts that DSP is based on become time variant, though still linear, in practice -- the response of a digital system to an input depends on the phase of that input w.r.t the sampling clock, due to the inability to realize perfect sampling and reconstruction filters. Cheers, Matt
Reply by ●July 25, 20032003-07-25
Rune Allnor wrote:>[Skip]> Is the time variant system itself nonlinear, or is it just that no linear > method of dealing with such systems has been found? > > Any thoughts?The system where superposition principle works is called linear system: F(x+y) = F(x) + F(y) That has nothing related to time variance. Vladimir Vassilevsky, Ph.D. DSP and Mixed Signal Design Consultant http://www.abvolt.com
Reply by ●July 25, 20032003-07-25
Rune Allnor wrote:> Hi all. > > I've been thinking a bit about processing of signals that have propagated > through an ocean waveguide that change, either because of waves on the > surface, tides, or for other oceanographic reasons. > > In any text on DSP we get aquainted with the Linear Time Invariant > (LTI) system model. Which is fine. I have no problems coming up with > examples of Nonlinear but still Time Invariant systems. > > Now, I can't figure out if the concept of a "Linear Time Variant" system > makes sense. As far as I know, time variant systems are handeled by > adaptive filters, which are nonlinear since their filter responses depend > on the data they act on. > > Is the time variant system itself nonlinear, or is it just that no linear > method of dealing with such systems has been found? > > Any thoughts? > > RuneTry looking at the first chapter in: Underwater Acoustics: A Linear Systems Theory Approach by Lawrence Ziomek The book is at home at the moment but I think it goes through the time varying linear case in the first chapter. If you look at one of Y. Bar-Shalom's books, assuming he used his course notes as a basis, he develops the Kalman Filter equations for the time varying case. Unfortunately, you have to know a-priori the time variations.
Reply by ●July 25, 20032003-07-25
Matt Timmermans wrote:> > "Rune Allnor" <allnor@tele.ntnu.no> wrote in message > news:f56893ae.0307241918.67e54977@posting.google.com... > > Now, I can't figure out if the concept of a "Linear Time Variant" system > > makes sense. As far as I know, time variant systems are handeled by > > adaptive filters, which are nonlinear since their filter responses depend > > on the data they act on. > > Hi Rune, > > A non-linear time-invariant system can become linear time-variant if one of > the inputs is hidden. An AM radio transmitter, for instance, is linear and > time variant when the carrier is considered as part of the system instead of > an input. > > Also, the LTI concepts that DSP is based on become time variant, though > still linear, in practice -- the response of a digital system to an input > depends on the phase of that input w.r.t the sampling clock, due to the > inability to realize perfect sampling and reconstruction filters. > > Cheers, > > MattMatt, I share Rune's perplexity. It's been so ingrained in me that linearity and the absence of system-generated new frequencies go hand in hand that I can't shake it. In this view, multiplication is a non-linear process, and every modulator is a non-linear device. (A so-called linear modulator being one that allows the recovered signal to be a faithful reproduction of the original. That's a different use of "linear".) We've had this discussion before, and I bow to the collective superior wisdom, but I still don't get it. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
