gwhite wrote:
Kevin Aylward wrote:



"Multipliers"
cannot be generally stated to be either linear or non-linear.

If one input of a multiplier is held constant, the other input has a
linear response. If the other input is a function of time, the
response to the first input is non-linear. That is, it dose *not*
satisfy a(f(t)) = f(at).

No, false, or whatever negation pleases you best.

No its correct. If the second input is time varying the output of the
system is *not* a linear function of the first input. Its that simple.

You were already
given the answer. You confuse time-invariance with linearity.

Not at all.

You
need not take my word for it.

I dont.

Consult any Signals and Sytems text,
any Linear Systems text, or any Communications text. IIRC, the
following was a homework problem in Stremler's text:


So, the book is out to lunch, or your interpretation of it is. So what.

Determine linearity
Determine time-invariance

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


http://www.amazon.com/exec/obidos/tg...glance&s=books

I suppose if you want to make up your own definition of linearity, you
can get whatever anwswer you wish.

Indeed.


Linearity can more easily be expressed as:

a(f(t)) = f(at)

Except that isn't "the" definition (hey, but it is true if a = 1). It
doesn't even meet you own description:

I did not say it was. I was keeping it simple.

"A linear system, cannot produce frequencies that are not in the
input, essentially, by definition." -- Kevin Aylward

I would like to see you apply this "definition."

I do. It makes a reasonably good practical definition.


If that is true, then the system is linear. This can be true for
systems with multipliers.

Nope. A signal being acted on by a multiplier is a non-linear system
if the second input is non constant with time. Your way of base on
this one.

Again, you confuse linearity and time-invariance.

Nope. I agree, if, for example, a second input is constant in time, a
circuit can be linear, however, if a second input changes the gain of
the first signal then the output is no longer a simple gain + offset,
therefore, the system in non-linear. The output is not a "simple"
function of the input.

However, I agree, "I suppose if you want to make up your own definition
of linearity you can get whatever answer you wish."


This system is linear and has a multiplier (it is not time
invariant):


Nope, its not.

Oh, but it is.

It is not...

Some rather trivial math can lead you to understand
what linear means.

Ho hum sniped.


No non-linear circuit was used but yet AM was produced.

Nonsense. Your pretty misguided on this.

Well I won't take you word for it, and you need not take mine.

I don't. Its 101 transistor electronics that the collector current
follows the base voltage by an exponential relation. The class A amp you
showed achieved multiplication because:

gm=40.Ic. because gm=di/dv, directly obtained from I=io.exp(Vb/Vt)

So that Vo=40.Ic.Vc

therefore

Vo = 40.Vc.Vi/Re

So, the modulation is achieved precisely because the transistor is
non-linear. This clearly contradicts your claim that your stated class A
amp is a modulator without using non linear properties.

Please present your detailed, alternative argument to support your claim
that your class A amp does not rely on non linearity to achieve
modulation, and why my 101 analysis, as given above, is false.

You
could make it easy just by applying the linearity test. That is,
linearity as it is defined in every Signals and Sytems text, any
Linear Systems text, or any Communications text. I'm not as
"original" as you, I simply trust the guys who wrote the books.

I don't. Furthermore, I don't trust your claim.


You can not achieve
multiplication without a non-linear circuit.

How about this one:

The System
+---------------+
| |
in | /¯¯¯\ | out
x(t) O--------( X )---------O y(t)
| \___/ |
| | |
| | |
| O |
| 2 |
+---------------+

It looks functionally to be an amp with a gain of two. Is a "gain of
2" circuit non-linear? Isn't that a multiplier in there?

Your grasping at straws here. This is trivially not the point. Of course
a fixed, or constant gain on a signal is linear. Jesus wept dude. No
brownie points for you on this one, I'm afraid. It was clear from the
start that one is referring to varying multiplication.


For example, Gilbert
multipliers use the fact that Id=Is.exp(vd/Vt).

I think that descibes pretty much every bipolar. So I guess
transistor amps cannot be made linear, or at least function
sufficiently linear for the purpose of electronic designers. That is
an "interesting" contention.

Out to lunch again. Of course transistors amps can be made as linear as
desired, I made no such connotation. Again, your making up what I say as
you go along because you arguments are so weak.

My point on the class A, non clipping, amp modulator you showed above,
it that it *relies* *explicitly* on the non-linear transfer function to
achieve multiplication.

Again, present your theoretical argument as to how AM modulation
actually occurs in said amplifier, without using any properties derived
from any non-linear behaviour.


That is it logs, adds
and antilog. Balanced switching mixers use switches. Fet mixers use
their square law response.

Not convenient, but it does dispel the "non-linearity is required"
myth.

Its not a myth. I know of no way whatsoever to generate an analogue
multiplication x product terms without having a device satisfying the
property of a.f(t) != f(at), i.e. a non-linear device. Please feel
free to suggest one, but file your patent first.

Why not dispense with the snidery, and simply prove your contention by
applying the linearity test to this one:

I must confess here I made a small error. Contrary to my claim of
ignorance, I am in fact aware of a modulating technique that dose not
rely, it would appear, on a devices non linearity, and have been so
aware for a considerable time, but it slipped my mind. A light dependant
resistor and a light bulb would seem to satisfy this requirement. I
suspect the claim would have to be modified to a direct electrical
method. I await you providing an example.


{pretentious drivel sniped}

Kevin Aylward

http://www.anasoft.co.uk
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