Sirs, I am going through literature on LTI system and its properties. In order for me to appreciate a little better, can I get some very simple examples of, 1) a non-LTI system that is linear but is time variant (that is, it is non time invariant) 2) a non-LTI system that is time invariaint but non linear Thanks, manish
LTI vs non-LTI
Started by ●April 28, 2013
Reply by ●April 28, 20132013-04-28
On Sun, 28 Apr 2013 04:36:30 -0500, manishp wrote:> Sirs, > > I am going through literature on LTI system and its properties. > > In order for me to appreciate a little better, can I get some very > simple examples of, > > 1) a non-LTI system that is linear but is time variant (that is, it is > non time invariant) > 2) a non-LTI system that is time invariaint but non linear > > Thanks, manishFirst, keep in mind that all real systems, if you look at them long enough are time varying, and all real systems, if push them hard enough (or lightly enough) are nonlinear. So when you say "my system is LTI" what you are really saying is "my system is _close enough_ to LTI that I can assume such for analysis". Having said that: 1: A mixer in a radio is ideally a linear but time varying system. If you have one of those hypothetical mixers that multiply by a sine wave, then the mixer function is: y(t) = A * r(t) * cos(omega * t) If you take A, omega and t as given, then this is linear (because the relationship between y(t) and r(t) obeys superposition). Incidentally, it's also memoryless -- but that doesn't change the fact that it's linear. Filter the RF going into the mixer, filter the IF coming out of the mixer, and now your system is linear, has memory, and is time varying. 2: A shunt-wound motor is nonlinear: the torque developed by the armature of a shunt-wound motor is proportional to the square of the current through the motor: T(t) = k_m * i(t)^2 Again, this expression is memoryless, but in a real system it'd be wrapped with an expression for the dynamics of the load on the motor shaft. 2a: If you're thinking radio, an AGC system is nonlinear: based on the average strength of the signal at the output to the IF, the gain of the IF is varied. I can't be much more specific than that because there's a lot of ways tho arrange an IF, but they're all nonlinear. -- Tim Wescott Control system and signal processing consulting www.wescottdesign.com
Reply by ●April 28, 20132013-04-28
On Sun, 28 Apr 2013 04:36:30 -0500, "manishp" <58525@dsprelated> wrote:>Sirs, > >I am going through literature on LTI system and its properties. > >In order for me to appreciate a little better, can I get some very simple >examples of, > >1) a non-LTI system that is linear but is time variant (that is, it is non >time invariant) >2) a non-LTI system that is time invariaint but non linear > >Thanks, manishHello manish, Upsampling and downsampling are time variant operations. Median filters are nonlinear. Taking the absolute value of a discrete sequence is, in general, nonlinear. I've always thought of mixers as being nonlinear because a mixer's output contains spectral components that did not exist at either of its two inputs. [-Rick-]
Reply by ●April 28, 20132013-04-28
On Sun, 28 Apr 2013 04:36:30 -0500, "manishp" <58525@dsprelated> wrote:>Sirs, > >I am going through literature on LTI system and its properties. > >In order for me to appreciate a little better, can I get some very simple >examples of,First, I like what Tim wrote, that there really aren't any time-invariant systems, but we treat many as such because the time windows that we use for analysis are adequately small that they seem time-invariant within that window. If the window is big enough, analysis may still be useful in making useful conclusions about the system. Likewise many systems that we think of as linear may only be linear under the conditions in which we generally use them. Amplifiers tend to be like this, where we try to apply and use them in their linear region, but if we turn the gain up to much or put in too big of a signal they may become non-linear. Another good example was mentioned from different perspectives by Tim and Rick, namely a mixer. Mixers, when looked at as a whole, are substantially non-linear and often implemented with non-linear devices. However, if we constrain out view to a particular region of interest, say a sub-section of bandwidth, and design it properly, the mixer will behave linearly within that region of interest, which means we may be able to treat and analyze it like a linear function. So, as is often the case, it depends on your definitions of "linear" and "time invariant". ;)>1) a non-LTI system that is linear but is time variant (that is, it is non >time invariant)A FIR filter with tap coefficients that change due to updates or adaptation is a linear but time-variant system.>2) a non-LTI system that is time invariaint but non linearAn amplifier operated in saturation (e.g., a Travelling Wave Tube or any Class C amplifier), a diode mixer, a diode rectifier (actually any rectifier), etc.,etc.>Thanks, manishEric Jacobsen Anchor Hill Communications http://www.anchorhill.com
Reply by ●April 28, 20132013-04-28
On Sun, 28 Apr 2013 09:58:15 -0700, Rick Lyons <R.Lyons@_BOGUS_ieee.org> wrote:>On Sun, 28 Apr 2013 04:36:30 -0500, "manishp" <58525@dsprelated> >wrote: > >>Sirs, >> >>I am going through literature on LTI system and its properties. >> >>In order for me to appreciate a little better, can I get some very simple >>examples of, >> >>1) a non-LTI system that is linear but is time variant (that is, it is non >>time invariant) >>2) a non-LTI system that is time invariaint but non linear >> >>Thanks, manish > >Hello manish, > Upsampling and downsampling are time variant >operations. > >Median filters are nonlinear. >Taking the absolute value of a discrete sequence >is, in general, nonlinear. >I've always thought of mixers as being nonlinear >because a mixer's output contains spectral components >that did not exist at either of its two inputs. > >[-Rick-]Hi manishp, I may have to 'take back' my comment about the absolute value' operation being nonlinear. I'll have to think more about that issue. Also, you might want to take a look at: http://www.dspguide.com/ch5/4.htm [-Rick-]
Reply by ●April 28, 20132013-04-28
On 2013-04-28 19:22, Rick Lyons wrote: [...]> Hi manishp, > I may have to 'take back' my comment about > the absolute value' operation being nonlinear. > I'll have to think more about that issue.You were right, absolute value is, obviously, non linear. abs(ax+by) != a*abs(x)+b*abs(y) in general. For example, for x = -y I guess the simplest example of non linear. time invariant system would be: y(t)=x(t)^2 bye, -- piergiorgio
Reply by ●April 28, 20132013-04-28
On 2013-04-28 19:12, Eric Jacobsen wrote: [...]> First, I like what Tim wrote, that there really aren't any > time-invariant systems, but we treat many as such because the time > windows that we use for analysis are adequately small that they seem > time-invariant within that window. If the window is big enough, > analysis may still be useful in making useful conclusions about the > system.Accordingly to these people, there are: https://en.wikipedia.org/wiki/Time-variant_system They've some interesting examples too. bye, -- piergiorgio
Reply by ●April 28, 20132013-04-28
On 2013-04-28 19:12, Eric Jacobsen wrote: [...]> First, I like what Tim wrote, that there really aren't any > time-invariant systems, but we treat many as such because the timeAch, sorry, I read time-variant... bye, -- piergiorgio
Reply by ●April 28, 20132013-04-28
On Monday, April 29, 2013 2:55:56 AM UTC+12, Tim Wescott wrote:> On Sun, 28 Apr 2013 04:36:30 -0500, manishp wrote: > > > > > Sirs, > > > > > > I am going through literature on LTI system and its properties. > > > > > > In order for me to appreciate a little better, can I get some very > > > simple examples of, > > > > > > 1) a non-LTI system that is linear but is time variant (that is, it is > > > non time invariant) > > > 2) a non-LTI system that is time invariaint but non linear > > > > > > Thanks, manish > > > > First, keep in mind that all real systems, if you look at them long > > enough are time varying, and all real systems, if push them hard enough > > (or lightly enough) are nonlinear. > > > > So when you say "my system is LTI" what you are really saying is "my > > system is _close enough_ to LTI that I can assume such for analysis". > > > > Having said that: > > > > 1: A mixer in a radio is ideally a linear but time varying system. If > > you have one of those hypothetical mixers that multiply by a sine wave, > > then the mixer function is: > > > > y(t) = A * r(t) * cos(omega * t) > > > > If you take A, omega and t as given, then this is linear (because the > > relationship between y(t) and r(t) obeys superposition). > > > > Incidentally, it's also memoryless -- but that doesn't change the fact > > that it's linear. Filter the RF going into the mixer, filter the IF > > coming out of the mixer, and now your system is linear, has memory, and > > is time varying. > > > > 2: A shunt-wound motor is nonlinear: the torque developed by the > > armature of a shunt-wound motor is proportional to the square of the > > current through the motor: > > > > T(t) = k_m * i(t)^2 > >That's for a series wound motor. Series the armature and shunt fields are the same. More generally Torque = k*flux*Ia
Reply by ●April 28, 20132013-04-28
On Sun, 28 Apr 2013 10:56:29 -0700, gyansorova wrote:> On Monday, April 29, 2013 2:55:56 AM UTC+12, Tim Wescott wrote: >> On Sun, 28 Apr 2013 04:36:30 -0500, manishp wrote: >> >> >> >> > Sirs, >> >> >> > >> > I am going through literature on LTI system and its properties. >> >> >> > >> > In order for me to appreciate a little better, can I get some very >> >> > simple examples of, >> >> >> > >> > 1) a non-LTI system that is linear but is time variant (that is, it >> > is >> >> > non time invariant) >> >> > 2) a non-LTI system that is time invariaint but non linear >> >> >> > >> > Thanks, manish >> >> >> >> First, keep in mind that all real systems, if you look at them long >> >> enough are time varying, and all real systems, if push them hard enough >> >> (or lightly enough) are nonlinear. >> >> >> >> So when you say "my system is LTI" what you are really saying is "my >> >> system is _close enough_ to LTI that I can assume such for analysis". >> >> >> >> Having said that: >> >> >> >> 1: A mixer in a radio is ideally a linear but time varying system. If >> >> you have one of those hypothetical mixers that multiply by a sine wave, >> >> then the mixer function is: >> >> >> >> y(t) = A * r(t) * cos(omega * t) >> >> >> >> If you take A, omega and t as given, then this is linear (because the >> >> relationship between y(t) and r(t) obeys superposition). >> >> >> >> Incidentally, it's also memoryless -- but that doesn't change the fact >> >> that it's linear. Filter the RF going into the mixer, filter the IF >> >> coming out of the mixer, and now your system is linear, has memory, and >> >> is time varying. >> >> >> >> 2: A shunt-wound motor is nonlinear: the torque developed by the >> >> armature of a shunt-wound motor is proportional to the square of the >> >> current through the motor: >> >> >> >> T(t) = k_m * i(t)^2 >> >> > That's for a series wound motor. Series the armature and shunt fields > are the same. More generally Torque = k*flux*Ia@#$%!!! Thank you. For some reason I have the words mixed up in my head -- had you been able to see the _drawing_ in my head you would have seen the right schematic. Thanks for correcting. This isn't the first time that I've done that. -- Tim Wescott Control system and signal processing consulting www.wescottdesign.com
