# impulse response of a nonlinear filter

Started by June 25, 2010
```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....

```
```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
```
```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
```
```On Jun 25, 2:34&#4294967295;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 &#4294967295;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. &#4294967295;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. &#4294967295;You could look it up. &#4294967295;More modern methods may
> apply but I can't help much with that. &#4294967295;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

I 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
```
```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
```
```
"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

```
```On Jun 25, 11:26&#4294967295;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 &#4294967295;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

```