Nomenclature question - what does LTI imply

It's been ~40 years since I took a "Signals and Systems" course and 
never used most of it in the "real world" ;/

Recently I've been thinking -- an admittedly dangerous hobby.

IIRC all the LTI networks I encountered in class were also *PASSIVE*.

But now I'm musing on two port multi node networks with some 
internal couplings being ideal unity gain amplifiers (infinite input 
impedance and zero output impedance).

Are there gotchas?
If so, what is/are keyword(s) for Google search?

TIA

>It's been ~40 years since I took a "Signals and Systems" course and 
>never used most of it in the "real world" ;/

I took Signals and Systems only 38 years ago, so its still fresh in my
mind.

>Recently I've been thinking -- an admittedly dangerous hobby.
>
>IIRC all the LTI networks I encountered in class were also *PASSIVE*.

There's no reason why they should be passive, as long as they are linear.
Of course nothing is perfectly linear in the real world, but within their
output swing range many things can be pretty close.

>But now I'm musing on two port multi node networks with some 
>internal couplings being ideal unity gain amplifiers (infinite input 
>impedance and zero output impedance).

I bet that ideal amp has no limits on its output swing, as well. I wish I
could buy me some of those :-)

>Are there gotchas?
>If so, what is/are keyword(s) for Google search?

Not really, when you are looking at "ideal" device models. In the real
world, of course, everything has a limited output range, even if its super
linear within that range. Even capacitors have a limited signal range - 
the dielectric breaks down at some point.

Steve

steveu wrote:
>> It's been ~40 years since I took a "Signals and Systems" course and
>> never used most of it in the "real world" ;/
>
> I took Signals and Systems only 38 years ago, so its still fresh in my
> mind.
>
>> Recently I've been thinking -- an admittedly dangerous hobby.
>>
>> IIRC all the LTI networks I encountered in class were also *PASSIVE*.
>
> There's no reason why they should be passive, as long as they are linear.
> Of course nothing is perfectly linear in the real world, but within their
> output swing range many things can be pretty close.
>
>> But now I'm musing on two port multi node networks with some
>> internal couplings being ideal unity gain amplifiers (infinite input
>> impedance and zero output impedance).
>
> I bet that ideal amp has no limits on its output swing, as well. I wish I
> could buy me some of those :-)
>
>> Are there gotchas?
>> If so, what is/are keyword(s) for Google search?
>
> Not really, when you are looking at "ideal" device models. In the real
> world, of course, everything has a limited output range, even if its super
> linear within that range. Even capacitors have a limited signal range -
> the dielectric breaks down at some point.
>
> Steve
>

I stay as FAR as possible from 'real' as possible ;)
If I can't get "ideal" right, what chance do I have with "reality"?

Reminds me of a Sophomore lab whose goal was designing a 
oscilloscope horizontal sweep with a specified linearity. I had no 
trouble meeting the goal by a large margin, followed by essential 
this conversation:

INSTRUCTOR: Oh, you used a Miller Run-Up circuit.
Myself:     No, just chose suitable portion of exponential of an RC 
charging curve.

On Mon, 01 Aug 2011 21:39:17 -0500, Richard Owlett wrote:

> It's been ~40 years since I took a "Signals and Systems" course and
> never used most of it in the "real world" ;/
> 
> Recently I've been thinking -- an admittedly dangerous hobby.
> 
> IIRC all the LTI networks I encountered in class were also *PASSIVE*.
> 
> But now I'm musing on two port multi node networks with some internal
> couplings being ideal unity gain amplifiers (infinite input impedance
> and zero output impedance).
> 
> Are there gotchas?
> If so, what is/are keyword(s) for Google search?

Linear and Time Invariant don't depend on passivity (there is, by the 
way, a method of nonlinear systems stability analysis where you try to 
determine if subsystems are effectively passive, and use that to prove 
overall stability).

Linear just means that the system possesses the superposition property, 
and time invariance means that the characteristics of the system are 
independent of time.

The only gotchas are that no real systems are either linear or time 
invariant -- you can only come close to linearity and time invariance, 
and in a full analysis one of your tasks is to determine if you're close 
enough (or at least clearly state your assumptions).

By the way, a sampled-time system is, by definition, time varying.  But 
it can be linear and _shift_ invariant, i.e. the system properties are 
invariant for any integer shift of the sampling interval, even if they 
vary within a sampling interval.

The z transform works for linear shift-invariant systems.

-- 
www.wescottdesign.com

Tim Wescott <tim@seemywebsite.com> wrote:

(snip, someone wrote)
>> IIRC all the LTI networks I encountered in class were also *PASSIVE*.
 
(snip)
> Linear and Time Invariant don't depend on passivity (there is, by the 
> way, a method of nonlinear systems stability analysis where you try to 
> determine if subsystems are effectively passive, and use that to prove 
> overall stability).

It seems to me, though, that passive systems are close to linear
over a wider range.  Note that many dielectrics are close enough
to linear from DC through the optical frequency range.  So close
that non-linear optics is a sub-specialty in optics.  (At really
high optical power levels, the usual optical materials become
non-linear, but that does take a lot of power.)  

Carbon makes nice resistors, linear from DC to GHz, from
nanowatts to tens, hundreds, even thousands of watts.

Air core inductors have enough resistance to be less than 
ideal, though still pretty linear.  Iron or ferrite saturate,
causing some non-linearity.

For transistors, though, from the thermal noise (lower) limit,
to the maximum power isn't so many orders of magnitude.  The 
non-linearity is usually noticable somewhat before both limits.

> Linear just means that the system possesses the superposition property, 
> and time invariance means that the characteristics of the system are 
> independent of time.

It seems to me that we are lucky that nature supplies us with
so many nice materials.  Conductivities range from silver down
to something like silicon dioxide, a wider range than (just about)
any other physical quantity.

-- glen

glen herrmannsfeldt wrote:
> Tim Wescott<tim@seemywebsite.com>  wrote:
>
> (snip, someone wrote)
>>> IIRC all the LTI networks I encountered in class were also *PASSIVE*.
>
> (snip)
>> Linear and Time Invariant don't depend on passivity (there is, by the
>> way, a method of nonlinear systems stability analysis where you try to
>> determine if subsystems are effectively passive, and use that to prove
>> overall stability).
>
> It seems to me, though, that passive systems are close to linear
> over a wider range.  Note that many dielectrics are close enough
> to linear from DC through the optical frequency range.  So close
> that non-linear optics is a sub-specialty in optics.  (At really
> high optical power levels, the usual optical materials become
> non-linear, but that does take a lot of power.)
>
> Carbon makes nice resistors, linear from DC to GHz, from
> nanowatts to tens, hundreds, even thousands of watts.
>
> Air core inductors have enough resistance to be less than
> ideal, though still pretty linear.  Iron or ferrite saturate,
> causing some non-linearity.
>
> For transistors, though, from the thermal noise (lower) limit,
> to the maximum power isn't so many orders of magnitude.  The
> non-linearity is usually noticable somewhat before both limits.
>
>> Linear just means that the system possesses the superposition property,
>> and time invariance means that the characteristics of the system are
>> independent of time.
>
> It seems to me that we are lucky that nature supplies us with
> so many nice materials.  Conductivities range from silver down
> to something like silicon dioxide, a wider range than (just about)
> any other physical quantity.
>
> -- glen


Thank you Tim and Glen. What I was involved in was either inherently 
linear (watching a few thousand protons/second in mass spectroscopy 
{high resolution or isotope ratio} or effectively measuring 
deviations of a ~1 ppm of a 1.4 Tesla field (60 MHz proton NMR)} or 
so nonlinear as measurement wasn't an issue (one phase of a 13 kV 
Niagara Mohawk line going to ground thru a 1 inch wide copper strap 
- interesting photo attached to the report I had to write ;).

For the record, though I don't predate Shannon Cooley and Tukey had 
not made it into my last signals course. My copy of _Signals, 
Systems and Communication_ by Lathi does not mention z-plane.

Tim Wescott wrote:
> [large *SNIP*]
>
> By the way, a sampled-time system is, by definition, time varying.  But
> it can be linear and _shift_ invariant, i.e. the system properties are
> invariant for any integer shift of the sampling interval, even if they
> vary within a sampling interval.
>
> The z transform works for linear shift-invariant systems.
>

I'll have to think about that, but 2 AM is not the time.

On 8/2/11 2:54 AM, Richard Owlett wrote:

>
> For the record, though I don't predate Shannon Cooley and Tukey had not
> made it into my last signals course. My copy of _Signals, Systems and
> Communication_ by Lathi does not mention z-plane.
>

my first book was Van Valkenburg and no z-plane (ca. 1975).

Richard, LTI means L and TI in the mapping, g{} of the output from the 
input.

L:   g{ x1(t) + x2(t) } = g{ x1(t) }  +  g{ x2(t) }  for any x1, x2

TI:  if y(t) = g{ x(t) }  then  y(t-T)  =  g{ x(t-T) } for any x, T

together, those two facts bring you to the convolution operation and 
from that, all of these nice Fourier, Laplace theorems for LTI fall into 
place.

active, passive, electrical, conceptual, DSP, it's all the same.

-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

On Tue, 02 Aug 2011 06:28:21 +0000, glen herrmannsfeldt wrote:

> Tim Wescott <tim@seemywebsite.com> wrote:
> 
> (snip, someone wrote)
>>> IIRC all the LTI networks I encountered in class were also *PASSIVE*.
>  
> (snip)
>> Linear and Time Invariant don't depend on passivity (there is, by the
>> way, a method of nonlinear systems stability analysis where you try to
>> determine if subsystems are effectively passive, and use that to prove
>> overall stability).
> 
> It seems to me, though, that passive systems are close to linear over a
> wider range.  (snip)

Diodes are passive.

Perhaps it's safe to say that it's easier to get linear operation over a 
wider range with passive (electronic) components than with active ones.

-- 
www.wescottdesign.com

On 8/2/11 12:51 PM, Tim Wescott wrote:
> On Tue, 02 Aug 2011 06:28:21 +0000, glen herrmannsfeldt wrote:
>
>> Tim Wescott<tim@seemywebsite.com>  wrote:
>>
>> (snip, someone wrote)
>>>> IIRC all the LTI networks I encountered in class were also *PASSIVE*.
>>
>> (snip)
>>> Linear and Time Invariant don't depend on passivity (there is, by the
>>> way, a method of nonlinear systems stability analysis where you try to
>>> determine if subsystems are effectively passive, and use that to prove
>>> overall stability).
>>
>> It seems to me, though, that passive systems are close to linear over a
>> wider range.  (snip)
>
> Diodes are passive.

so are op-amps when no power is connected to them.

> Perhaps it's safe to say that it's easier to get linear operation over a
> wider range with passive (electronic) components than with active ones.

passivity and linearity (and/or time-invariancy) are orthogonal 
concepts.  R's L's and C's are passive and linear.  nicely biased 
transistors and op-amps are active and, over some range, decently 
linear.  negative feedback using linear components (like resistors) can 
make a decently linear amplifier *very* linear.

"active" means you need to hook on external power from somewhere to make 
it work.  "linear" means the operation performed on a sum of signals is 
the same as the sum of the same operation performed on each signal 
individually.  even though DSP systems are, strictly speaking, "active", 
that goes without saying and "active" vs. "passive" just is not in the 
lexicon for algs running in a DSP.  "linear" vs. "non-linear" surely is. 
  "time-variant" vs. "time-invariant"


-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."