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From: maury on 26 Jul 2010 11:11
On Jul 24, 12:15 pm, Tim Wescott <t...(a)seemywebsite.com> wrote:

>
> Second, if you're facing a nonlinear system, it is much easier to
> represent said nonlinear system in state space form -- there is no such
> thing as a 'nonlinear transfer function', but state-space is the generic
> form for a nonlinear dynamic system.
>

Tim, if the transfer function is the Laplace transform of the impulse
response, and the impulse response of a nonlinear system exists, why
is there no such thing as a 'nonlinear transfer function'?

Maurice
From: Tim Wescott on 26 Jul 2010 12:04
On 07/26/2010 08:11 AM, maury wrote:
> On Jul 24, 12:15 pm, Tim Wescott<t...(a)seemywebsite.com> wrote:
>
>>
>> Second, if you're facing a nonlinear system, it is much easier to
>> represent said nonlinear system in state space form -- there is no such
>> thing as a 'nonlinear transfer function', but state-space is the generic
>> form for a nonlinear dynamic system.
>>
>
> Tim, if the transfer function is the Laplace transform of the impulse
> response, and the impulse response of a nonlinear system exists, why
> is there no such thing as a 'nonlinear transfer function'?

In mathemagic land the general response of a nonlinear system to an
impulse is almost always going to either be infinite*, or zero (consider
a subsystem whose output is the input limited to some finite value --
what's the integral of a finite value for an infinitesimal amount of
time?). For most intents and purposes it'll be useless.

In the real world the response of a system to a true impulse will be a
large explosion. For most intents and purposes this is also useless.

The whole point of a transfer function as an operator is that it works
when superposition works -- and the exclusive essence of the definition
of a linear system is that it is a system where superposition works. A
nonlinear system does not support superposition, so it does not support
a transfer function.

You can _approximate_ a nonlinear system as a linear system, and then
find the transfer function of that _approximation_. When life is going
your way that's often the best way to proceed. But it doesn't work when
you _must_ treat the system as nonlinear.

* Or more honestly mathematically undefined -- what, exactly, is impulse
squared?

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Do you need to implement control loops in software?
"Applied Control Theory for Embedded Systems" was written for you.
See details at http://www.wescottdesign.com/actfes/actfes.html
From: glen herrmannsfeldt on 26 Jul 2010 12:45
Tim Wescott <tim(a)seemywebsite.com> wrote:
> On 07/26/2010 08:11 AM, maury wrote:
(snip)
>> Tim, if the transfer function is the Laplace transform of the impulse
>> response, and the impulse response of a nonlinear system exists, why
>> is there no such thing as a 'nonlinear transfer function'?

> In mathemagic land the general response of a nonlinear system to an

I recently found that our library now has a DVD of a movie I
remember from school so many years ago called "Donald in Mathemagic
Land." Narated by Donald Duck, it shows some of the applications
of mathematics that third or fourth graders (about when I remember
it from) may not yet see. Anyway...

(snip)

> In the real world the response of a system to a true impulse will be a
> large explosion. For most intents and purposes this is also useless.

Well, first, most analog systems are only linear in the small
signal approximation (*). Also, since we can't really make a true
impulse, a sufficiently narrow pulse with appropriate amplitude
is usually used instead.

> The whole point of a transfer function as an operator is that it works
> when superposition works -- and the exclusive essence of the definition
> of a linear system is that it is a system where superposition works. A
> nonlinear system does not support superposition, so it does not support
> a transfer function.

And again, most real systems only support superposition as an
approximation. Fortunately good enough most of the time.
There is the whole field of non-linear optics, though even in
that case it isn't completely non-linear. (**)

> You can _approximate_ a nonlinear system as a linear system, and then
> find the transfer function of that _approximation_. When life is going
> your way that's often the best way to proceed. But it doesn't work when
> you _must_ treat the system as nonlinear.

But also note that the usual digital system is non-linear when
the signal gets larger then the number of bits supplied. And usually
non-linear in ways worse than in the analog world.

> * Or more honestly mathematically undefined -- what, exactly,
> is impulse squared?

About as useful as RMS power, but don't tell the FTC that.

(**) There is in non-linear optics something called a
phase conjugate mirror. You put an optical signal, usually
of a short duration, into one and get out the time reversed
form of the signal. It seems that time reversal is a non-linear
operation even if superposition does apply (for sufficiently
small input signals).

(*) Reminds me of a discussion in this group some time ago
about global warming and the possibility of it being natural
or manmade. It seems to me that we are now way beyond the small
signal approximation in our changes to the environment. It is
easy to consider our own CO2 emissions as small, but forget to
multply by the billions of others on earth, and growing
exponentially. Another important case of non-linearity.

-- glen

From: Tim Wescott on 26 Jul 2010 13:46
On 07/26/2010 09:45 AM, glen herrmannsfeldt wrote:
> Tim Wescott<tim(a)seemywebsite.com> wrote:
>> On 07/26/2010 08:11 AM, maury wrote:
> (snip)
>>> Tim, if the transfer function is the Laplace transform of the impulse
>>> response, and the impulse response of a nonlinear system exists, why
>>> is there no such thing as a 'nonlinear transfer function'?
>
>> In mathemagic land the general response of a nonlinear system to an
>
> I recently found that our library now has a DVD of a movie I
> remember from school so many years ago called "Donald in Mathemagic
> Land." Narated by Donald Duck, it shows some of the applications
> of mathematics that third or fourth graders (about when I remember
> it from) may not yet see. Anyway...

Where do you think I got the name? That's one of my favorite cartoons,
even if I didn't remember just how much it dwells on the mysticism of
the golden mean and how little on other stuff that may be of more import.

>> In the real world the response of a system to a true impulse will be a
>> large explosion. For most intents and purposes this is also useless.
>
> Well, first, most analog systems are only linear in the small
> signal approximation (*). Also, since we can't really make a true
> impulse, a sufficiently narrow pulse with appropriate amplitude
> is usually used instead.
>
>> The whole point of a transfer function as an operator is that it works
>> when superposition works -- and the exclusive essence of the definition
>> of a linear system is that it is a system where superposition works. A
>> nonlinear system does not support superposition, so it does not support
>> a transfer function.
>
> And again, most real systems only support superposition as an
> approximation. Fortunately good enough most of the time.
> There is the whole field of non-linear optics, though even in
> that case it isn't completely non-linear. (**)
>
>> You can _approximate_ a nonlinear system as a linear system, and then
>> find the transfer function of that _approximation_. When life is going
>> your way that's often the best way to proceed. But it doesn't work when
>> you _must_ treat the system as nonlinear.
>
> But also note that the usual digital system is non-linear when
> the signal gets larger then the number of bits supplied. And usually
> non-linear in ways worse than in the analog world.

Well, true -- but my main point is that asking for the transfer function
of a nonlinear system isn't going to bring you joy.

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Do you need to implement control loops in software?
"Applied Control Theory for Embedded Systems" was written for you.
See details at http://www.wescottdesign.com/actfes/actfes.html
From: maury on 26 Jul 2010 14:10
On Jul 26, 12:46 pm, Tim Wescott <t...(a)seemywebsite.com> wrote:
> On 07/26/2010 09:45 AM, glen herrmannsfeldt wrote:
>
> > Tim Wescott<t...(a)seemywebsite.com>  wrote:
> >> On 07/26/2010 08:11 AM, maury wrote:
> > (snip)
> >>> Tim, if the transfer function is the Laplace transform of the impulse
> >>> response, and the impulse response of a nonlinear system exists, why
> >>> is there no such thing as a 'nonlinear transfer function'?
>
> >> In mathemagic land the general response of a nonlinear system to an
>
> > I recently found that our library now has a DVD of a movie I
> > remember from school so many years ago called "Donald in Mathemagic
> > Land."   Narated by Donald Duck, it shows some of the applications
> > of mathematics that third or fourth graders (about when I remember
> > it from) may not yet see.   Anyway...
>
> Where do you think I got the name?  That's one of my favorite cartoons,
> even if I didn't remember just how much it dwells on the mysticism of
> the golden mean and how little on other stuff that may be of more import.
>
>
>
>
>
> >> In the real world the response of a system to a true impulse will be a
> >> large explosion.  For most intents and purposes this is also useless..
>
> > Well, first, most analog systems are only linear in the small
> > signal approximation (*).  Also, since we can't really make a true
> > impulse, a sufficiently narrow pulse with appropriate amplitude
> > is usually used instead.
>
> >> The whole point of a transfer function as an operator is that it works
> >> when superposition works -- and the exclusive essence of the definition
> >> of a linear system is that it is a system where superposition works.  A
> >> nonlinear system does not support superposition, so it does not support
> >> a transfer function.
>
> > And again, most real systems only support superposition as an
> > approximation.  Fortunately good enough most of the time.
> > There is the whole field of non-linear optics, though even in
> > that case it isn't completely non-linear. (**)
>
> >> You can _approximate_ a nonlinear system as a linear system, and then
> >> find the transfer function of that _approximation_.  When life is going
> >> your way that's often the best way to proceed.  But it doesn't work when
> >> you _must_ treat the system as nonlinear.
>
> > But also note that the usual digital system is non-linear when
> > the signal gets larger then the number of bits supplied.  And usually
> > non-linear in ways worse than in the analog world.
>
> Well, true -- but my main point is that asking for the transfer function
> of a nonlinear system isn't going to bring you joy.
>
> --
>
> Tim Wescott
> Wescott Design Serviceshttp://www.wescottdesign.com
>
> Do you need to implement control loops in software?
> "Applied Control Theory for Embedded Systems" was written for you.
> See details athttp://www.wescottdesign.com/actfes/actfes.html- Hide quoted text -
>
> - Show quoted text -

I don't agree with that. There are many systems in nature that are
nonlinear and the transfer function exists. The Volterra kernel, for
example, makes a nonlinear impulse response of nth order, and systyem
identification algorithms exist to measure these kernels. And these
kernels support superposition. Just because a system output is
unstable, doesn't mean it doesn't have a transfer function. Obtaining
the transfer function for a nonlinear system not only brings joy, but
in many instances, is necessary. The Volterra is considered the
general impulse response for nonllinear systems, and the transfer
function does exist using these kernels.

Take a look at the work of Stephen Boyd (Stanford). He wrote quite a
bit on the Volterra series and Volterra kernels in the 1980s.

Also, Stenger's work at the Univ. of Erlangen-Nurnberg, "Adaptive
Volterra Filters for Nonlinear Acoustic Echo Cancellation"

Peddanarappagari (Univ. of Kansas), et al, "Volterra Series Transfer
Function of Single-Mode Fibers"

Reed (Univ. of Essex), et al, "Identification of Discrete Volterra
Series Using Maximum Length Sequences"

Pitas and Venetsanopoulos wrote of Volterra filters and the Volterra
series in their chapter on polynomial filters in their book "Nonlineal
Digital Filters"


Maurice
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