Hello forum, if a time-invariant (zero memory) nonlinear filter is fed with a delta impulse, it will output a certain output. That output will be the same no matter when the impulse is applied due to time invariance.... Superposition (convolution) is not valid for nonlinear system. Is there however some generalized impulse response concept for nonlinear systems? After all, nonlinear system are locally linear....
impulse response of a nonlinear filter
Started by ●June 25, 2010
Reply by ●June 25, 20102010-06-25
On 06/25/2010 09:26 AM, fisico32 wrote:> Hello forum, > > if a time-invariant (zero memory)Do you mean that you are talking about a nonlinear filter that is also zero memory and time-invariant, or are you implying that any time-invariant filter is also zero memory? The first describes a very trivial filter, the second is just plain wrong. Filters need memory to remember the past -- a memoryless element is one that has an instantaneous response to input, and whose impulse response is itself an impulse.> nonlinear filter is fed with a delta > impulse, it will output a certain output.When a sampled-time, shift-invariant nonlinear filter is fed with an impulse _of a certain value_ it will output a certain output.> That output will be the same no matter when the impulse is applied due to > time invariance....That output will be the same _for any other input of the same value_ no matter when the impulse is applied, due to time invariance.> Superposition (convolution) is not valid for nonlinear system. > Is there however some generalized impulse response concept for nonlinear > systems?No, that goes away with linearity.> After all, nonlinear system are locally linear....Not necessarily. Nonlinear systems that happen to be continuous are locally linear, but there are functions, and therefore systems, that are everywhere discontinuous. Granted, they're not usually very interesting for DSP, but they can exist. You _can_ make a linearized model of a nonlinear system. For a nonlinear system that's well behaved enough you could conceivably even probe it with a small-enough impulse and get a clean impulse response out -- assuming that an impulse that's small enough to generate a reasonably linear response isn't so small that the response isn't buried in noise. -- Tim Wescott Control system and signal processing consulting www.wescottdesign.com
Reply by ●June 25, 20102010-06-25
fisico32 wrote:> Hello forum, > > if a time-invariant (zero memory) nonlinear filter is fed with a delta > impulse, it will output a certain output. > > That output will be the same no matter when the impulse is applied due to > time invariance.... > > Superposition (convolution) is not valid for nonlinear system. > Is there however some generalized impulse response concept for nonlinear > systems? > > After all, nonlinear system are locally linear.... >You haven't really revealed your motivation so it's hard to be helpful without guessing. Impulses as inputs aren't usually real, thus not very interesting in practice - aside from all the nice analytical stuff you might do. The one thing that intrigued me long ago, and that I used very usefully, was the notion of "describing functions" for nonlinear system analysis and controls designs. You could look it up. More modern methods may apply but I can't help much with that. The idea is that it linearizes the nonlinear system for analysis purposes - based more on a lowpass assumption of the system rather than "local" linearities I believe. Fred
Reply by ●June 25, 20102010-06-25
On Jun 25, 2:34�pm, Fred Marshall <fmarshallx@remove_the_xacm.org> wrote:> fisico32 wrote: > > Hello forum, > > > if a time-invariant (zero memory) nonlinear filter is fed with a delta > > impulse, it will output a certain output. > > > That output will be the same no matter when the impulse is applied due to > > time invariance.... > > > Superposition (convolution) is not valid �for nonlinear system. > > Is there however some generalized impulse response concept for nonlinear > > systems? > > > After all, nonlinear system are locally linear.... > > You haven't really revealed your motivation so it's hard to be helpful > without guessing. �Impulses as inputs aren't usually real, thus not very > interesting in practice - aside from all the nice analytical stuff you > might do. > > The one thing that intrigued me long ago, and that I used very usefully, > was the notion of "describing functions" for nonlinear system analysis > and controls designs. �You could look it up. �More modern methods may > apply but I can't help much with that. �The idea is that it linearizes > the nonlinear system for analysis purposes - based more on a lowpass > assumption of the system rather than "local" linearities I believe. > > FredI like using solitons in monomode fibers. A "loud" soliton travels faster than a weaker one. You may actually launch a weak one followed by a loud one and the loud one catches up with and passes through the weaker one and leads it from then on. Wierd! Clay
Reply by ●June 25, 20102010-06-25
On 06/25/2010 11:34 AM, Fred Marshall wrote:> fisico32 wrote: >> Hello forum, >> >> if a time-invariant (zero memory) nonlinear filter is fed with a delta >> impulse, it will output a certain output. >> That output will be the same no matter when the impulse is applied due to >> time invariance.... >> >> Superposition (convolution) is not valid for nonlinear system. >> Is there however some generalized impulse response concept for nonlinear >> systems? >> >> After all, nonlinear system are locally linear.... >> > > You haven't really revealed your motivation so it's hard to be helpful > without guessing. Impulses as inputs aren't usually real, thus not very > interesting in practice - aside from all the nice analytical stuff you > might do. > > The one thing that intrigued me long ago, and that I used very usefully, > was the notion of "describing functions" for nonlinear system analysis > and controls designs. You could look it up. More modern methods may > apply but I can't help much with that. The idea is that it linearizes > the nonlinear system for analysis purposes - based more on a lowpass > assumption of the system rather than "local" linearities I believe.I use describing function analysis all the time in control system design. Just because it's way older than me doesn't mean it doesn't work just fine! -- Tim Wescott Control system and signal processing consulting www.wescottdesign.com
Reply by ●June 27, 20102010-06-27
"Fred Marshall" <fmarshallx@remove_the_xacm.org> schrieb im Newsbeitrag news:W8qdnTL8DNNba7nRnZ2dnUVZ_rKdnZ2d@centurytel.net...> fisico32 wrote: >> Hello forum, >> >> if a time-invariant (zero memory) nonlinear filter is fed with a delta >> impulse, it will output a certain output. That output will be the same no >> matter when the impulse is applied due to >> time invariance.... >> >> Superposition (convolution) is not valid for nonlinear system. >> Is there however some generalized impulse response concept for nonlinear >> systems? >> >> After all, nonlinear system are locally linear.... >> > > You haven't really revealed your motivation so it's hard to be helpful > without guessing. Impulses as inputs aren't usually real, thus not very > interesting in practice - aside from all the nice analytical stuff you > might do. > > The one thing that intrigued me long ago, and that I used very usefully, > was the notion of "describing functions" for nonlinear system analysis and > controls designs. You could look it up. More modern methods may apply > but I can't help much with that. The idea is that it linearizes the > nonlinear system for analysis purposes - based more on a lowpass > assumption of the system rather than "local" linearities I believe. >One has a good chance for linearizing non-linear systems. EXAMPLE * http://home.arcor.de/janch/_control/20100627-(non)linear-system/ JCH
Reply by ●June 28, 20102010-06-28
On Jun 25, 11:26�am, "fisico32" <marcoscipioni1@n_o_s_p_a_m.gmail.com> wrote:> Hello forum, > > if a time-invariant (zero memory) nonlinear filter is fed with a delta > impulse, it will output a certain output. > > That output will be the same no matter when the impulse is applied due to > time invariance.... > > Superposition (convolution) is not valid �for nonlinear system. > Is there however some generalized impulse response concept for nonlinear > systems? > > After all, nonlinear system are locally linear....The accepted _generalized_ impulse response _concept_ is the Volterra kernel. Look for work by Stephen Boyd of Stanford Univ. Some work on Volterra kernels has been done lately, but Boyd's work is quite definitive. Maurice Givens
