Kevin Aylward wrote:



It is that simple.

Nope. Its not. Linearity has many definitions.

It doesn't have "many definitions" when it comes to the EE profession.
The one definition is wholly consistant across academic texts -- without
known contradiction. Everyone who took courses like Circuits, Fields
and Waves, and on and on, also took a Signals and Systems (or similar
under a different name) course. That some, such as you for example,
didn't learn or understand the definition is notwithstanding.

Look, I have no problem with your example of a particular definition of
linearity. I already explained how such an example is meaningless in
analogue design by giving an example.

A mathematical definition only has meaning if it is useful when it is
applied. In analogue design this definition is useless, so it is not
used.

It is useful to the extent modulation is a linear operation produced by
devices such as gilbert cells biased to Class-A. It is very useful
definition: for example, high data rate modern digital communications
systems routinely utilize linear modulation/demodulation.

It doesn't even matter if the upper transistors in the gilbert cells are
driven to the switch mode (switch mode is not a requirement; class-A
will do) by the LO. It is still linear modulation.

This system is linear:

The System
+---------------+
| |
in | /¯¯¯\ | out
x(t) O--------( X )---------O y(t)
| \___/ |
| | |
| | |
| O |
| cos(w_c·t) |
+---------------+

This one is not:

The System
+---------------------+
| |
in | /¯¯¯\ /¯¯¯\ | out
x(t) O--------( X )--( + )-------O y(t)
| \___/ \___/ |
| | | |
| +--------+ |
| | |
| O |
| cos(w_c·t) |
+---------------------+

Do you know why?

Hint: it has nothing to do with any "exp(x)" or other solid state
issues.

You nor anyone else need take my word for it: it is in *all* the
Signals, Systems, and Communications texts I've ever opened up -- they
are wholly consistant with each other; check for yourself.

I don't care a toss about and communication texts.

Sheesh -- no ****!

I care about
linearity as understood in the solution of the non-linear differential
equations used in Spice.

Like I said early on: you can make the answer come out however you want
if you are permitted to make up the rules and change them as you play.

Look, I don't claim that the particular definition of linearity that you
presence is "wrong" in principle. It is a very well known definition.
However, it is one of many, and is simple not applicable in this
context.

It is the definition for the EE profession.

I clearly said that linearity, as defined in analogue design,
is essentially defined by the absence of any frequencies in the output
not present in the input.

That "definition" is incorrect, as has been pointed out already.

You were off on a roll trying to impress people which
mathematical technicalities that are simple irrelevant
in the context of this original discussion.

Right. I was trying to "impress" by using a couple of trig identities.
LOL

I know all about your definition. I dont disagree that it is a valid
definition in some contexts.

It is the definition for the EE context. If you want an LTI system,
then so be it.

However, it is not applicable to electronic
circuit design that is based on the solution of non-linear differential
equations, with the "non-linear" term having a universally accepted
meaning by the 10,000s of mathematician who actually study such
equations.

It is not a matter of a mathematician's characterization of equations.
It is the EE characterization, and this is largely an EE forum. If you
are a mathematician and not an EE, then I can see why you would believe
what you do.

You arnt wrong in this thread because of you particular claim of
linearity.

It isn't "my claim." It is the standard definition for EE's. That some
EE's didn't quite "get it" is notwithstanding.

The real reason for this disagreement is that you are talking apples and
I am talking fish net stockings. You are applying the term linearity in
a completely different sense than the one I am using. Both are valid in
principle, and are indeed well accepted, under their appropriate
conditions.

Your definition is not the EE definition.

That is y=f(x) is non-linear as defined in any math
text book you care to name.

This is a matter of EE definition. Recall how you sniped about the EE
courses I apparently didn't take "with all due respect."

"A linear system, cannot produce frequencies that are not in the input,
essentially, by definition. With all due respect, I would guess you
don't have an EE B.S. degree. This is all pretty basic stuff really."
-- Kevin Aylward


So you originally sniped about my education in electrical engineering,
and then ignore those very same basics taught in all EE curriculums, and
then turn course and take refuge in a math text. Okay, Kevin, have it
your way.