Re Question For Craig Markwardt

"George Dishman" wrote in message
...

"ralph sansbury" wrote in message
...
George, I know you are a superior EE and that

We are talking about the size of the intermediate frequency
range
relative
to the original range 1MHz is small relative to 200MHz but
not
to 1Hz

The terms "narrow band" and "wide band"

But the size of the intermediate frequency relative to the
original
range is what we are talking about. You seem to have your own
subjective
read of what others say and on what you say without understanding
that words are ambiguous and you have to say out loud what you
mean
or what you think is meant before going off halfcocked.


compare the width
of the equipment to the width of the signal being processed.
Wide band in this context means sufficiently wide that it
does not exclude any frequency of interest or produce any
modification of the signal such as emphasising one frequency
more than another.

sequence of voltages at this difference
frequency.(mixer
and
repeated heterodyne up and down conversion etc is the
jargon and
the engineering details I am trying to avoid).

Instead you are inventing a process that doesn't exist
and describing it in far more (and incorrect) detail
than exists in the published documentation.

As I have detailed above you are misunderstanding
what I
am saying

What you are saying is very different to what is being
done. It may be that this is because you are using terms
in an unconventional manner but you will then hit problems
in referring to your text books.

No. I am using terms and descriptions of mixers as in my
1985 Shrader Electronic Communication text which shows
a (tuned)resonant inductor and capacitor circuit for the
intermediate
and different ones for the sum and the input frequencies.
I dont want to keep arguing this point but what I am saying
is in principle what is being done. You are obscuring the essence
of what is being done which is the use of Fourier's transform to
obtain a Fourier series representation of the noisy received
oscillations
You also seem to have changed your understanding of the nasa
documents to come around to my initial impression


Again it may be clearer but it is wrong. It is not just
the carrier oscillations that are digitised, it is the
whole signal, oscillations plus random thermal noise
and any other sources such as the galactic background.

Your understanding is wrong. I did not say CARRIER
oscillations
You can't change the meaning of
'oscillations' to mean only the
part due to the spacecraft transmitter

"oscillations" means something regular,

Not necessarily. And obviously not in this context






I accept your apology but maybe you are similarly misreading
the nasa documents and that is why you are missing the
essence
of the procedure. You cant see the forest from the trees.

NASA don't talk of 'oscillations', they correctly talk of
the signal.

Nope, the amplitude of _all_ frequencies in the band is
calculated and passed on to the next stage without any
judgement.

I am talking about the final stage

The final stage is the carrier PLL, not the FFT. All
the FFTs are removed from the chain once the PLL locks
on and they play no further part in the process. It is
the PLL that tracks the drifting signal and gives us
the accurate measurement.

The FFT as I was using the term includes the PLL.

The two are entirely diffeent and separate.

Not the way I am using the term. Note I say how I am using the
term.

The point
which you insist on obscuring is that this technique
gets at the right sine frequency starting at the right time
from
the
sum of sine functions of various frequencies equivalent as
Fourier showed to
the noisy oscillations observed.

You said "an Fast Fourier Transform procedure is used to find
the underlying "sine" pattern of 1s and 0s that most closely
fits"

The FFT is not applied to "1s and 0s", it is applied to voltage
samples. The frequency is found and the PLL commanded to start
at
that frequency. The PLL locks on and tracks the carrier and it
uses a digital phase comparator that probably treats the signal
as 1s and 0s.

That is what I thought initially and you said I was wrong.
Evidently
you have changed your mind. It doesn't matter however for the
purposes of showing the essence of the procedure and the
rationale
as to why it is reliable.

There are several levels of processing that you are skipping
over
which are very important in establishing that the signal is
genuine and from the right craft. Ultimately that is your main
concern, isn't it?

Yes. I welcome your pointing this out. But I deplore
the obscure and argumentative way that you are doing it.


and I mentioned that the
movement of the Earth etc requires different patterns to
be
obtained successively but the point is that the FFT
procedure
finds the underlying pattern and it is this that is used
to
compare to the given sequence of 1s and 0s.

No it isn't. The final FFT is only used to set initial
frequency for the carrier PLL. If that locks, the
bandwidth is reduced to improve the signal/noise ratio.


You are saying the same thing that I was saying. I
think
it is clearer to say it without the jargon.

Clearer but completely wrong.
No clearer but not detailed.

The FFT does not compare
Again I did not say this. I said that after the
FFT procedure finds the dominant sine function,
this function is then compared to the observed set
of values which I thought you said earlier was reduced to
a set of 1s and 0s and that this was compared to
the corresponding observed set to get the degree of
error.


anything to a pattern of 1s and 0s. It does not compare
anything to anything else and in this case it does not
work on 1s and 0s.

The output of that is fed to the sub-carrier PLL.
Whatever the details a sequence of 1s and 0s
is obtained that is a digitised intermediate version of the
sky frequency.

No, a series of voltages samples like +0.25, -0.375, +0.112
etc.
is the result of digitising the IF.

Again
that has to lock before the signal can be decoded using
a phase detector. Then it gets decoded through the error
correction scheme. There are many critical steps after
the FFT, and in fact the FFT plays no part in the decoding
process whatsoever.

Again I did not say that it did.

The bottom line is a sine representation of a sum of sine
frequency
represention
of an oscillating pattern made possible by the FFT procedure
essentially
and this includes the phase locked loop procedure perhaps
involving
the recognition of some code modulation of the carrier to
insure that
the fitted frequency starts at the right time.


The fact that this representation is a much smaller
frequency than
the GHz sky frequency is ok because when you look at the
difference
between this and a small frequency representation of the
transmitted
frequency the difference is the same as the difference
between the
original frequencies. And it is this difference that is used
to
get the
Doppler shift.




This is the procedure I understood from your comments
and
various books and links.

You seemed to grasp it at the time, why
have you reverted to this grossly inaccurate
description
of the process?
Again I think you have misunderstood what I have
said. I dont think it is inaccurate if you replace single
intermediate
frequency by small range of frequencies around the single
intermediate frequency where small is relative the
original
frequency.


It is very inaccurate when the DSN document tells you
the analog band is the digitised band is 110MHz wide and
the signals of interest are of the order of 1Hz wide.

You are quoting the wrong document. We are talking about
the
intermediate frequency
being smaller that the original frequency. Is that so hard
for
you to understand.

What matters is how wide the frequency range is compared
to what you are looking at.

No what matters in this context is the size of the
intermediate frequency relative
to the size of the original frequency.



If the equipment only handles
a band that is small in comparison to the signal, the edges
will be chopped off, or if the Doppler shift was more than
expected the signal might be lost entirely. If th system
is 'wide band' then there is no such risk. How wide it is
compared to the original is completely irrelevant. Now this
matters because I know you are rferring to text boks and
those will use "wide band" and "narrow band" as terms relating
the width of the channle to the width of the signal, so if
you look up the text for "narrow band", you are going to get
entirely misleading information.

I am continually amazed that a person of your knowledge and
intelligence has so many
blindspots.

On the contrary, I can see potential mistakes you are
about to make through your unfamiliarity with the jargon
and I am trying to educate you in these terms to avoid
those pitfalls before you reach them.

And if you want to try and describe the digital
version of
the
mixer please do so. It was not clear from your emails.

The mixer is analog. The output is digitised and a baseband
extracted as shown on page 10. The details of the method
of mixing are not given but the principle is simply
multiplication of the incoming signal (including noise) by
the reference sine wave.

V_out = V_in * V_ref

where

V_ref = A * sin(wt)

This makes no sense. Electrical oscillations add by the
law of
superposition;

Yes, which is why it takes a special ciruit to get around that.

They dont multiply.

Dual gate fets and other methods of implementing mixers do

If they do then how do they. All I can see is superposition
and
then various filters to extract the desired frequency or range of
frequencies
Perhaps you have and analogue to digital converter to change
the incoming
frequencies to digital and then multiply them and then convert
this back instead
of the filter part of the mixer I see in my 1985 text.???


because that is their intended function and we poor designers
have to make them do it well. It's what engineers get paid
for (though I personally work on the digital side).

The mathematical fact that a sum of sine
and cosine functions can
be represented as a product of related sine and cosine
functions
has to be mentioned dont you think?

Only if you don't already know it.

Yes. And if I already knew it well we would not be having
this discussion would we?

The circuit multiplies
the two voltages together and since the product is the
same as a combination of sum and difference, you can then
discard all of (say) the sum components and keep all of the
difference components by a simple filter. Tuning is not
required, highly undesirable, and is definitely not included
in that part of the DSN system, it uses filtering instead
and to remove the jargon, that means it doesn't select a
single frequency from a range, it accepts the whole range,
treats it all equally, and only rejects a mirror image of
the range very far away.

You seem to have your own subjective
read of what others say and on what you say without understanding
that words are ambiguous and you have to say out loud what you
mean
or what you think is meant before going off halfcocked.
Ralph

"ralph sansbury" wrote in message
...

"George Dishman" wrote in message
...

"ralph sansbury" wrote in message
...
George, I know you are a superior EE and that

We are talking about the size of the intermediate frequency
range
relative
to the original range 1MHz is small relative to 200MHz but
not
to 1Hz

The terms "narrow band" and "wide band"

But the size of the intermediate frequency relative to the
original
range is what we are talking about. You seem to have your own
subjective
read of what others say and on what you say without understanding
that words are ambiguous and you have to say out loud what you
mean
or what you think is meant before going off halfcocked.

This is your original phrase:

A resonance tuner picks out the difference frequency ..

and my reply:

Wrong, it doesn't pick out one frequency, it passes a
complete band of frequencies to the FFT.

If you had said "By _the_ frequency I meant the IF band, we
would not be arguing. Instead you said:

OK relax. It is a small band of frequencies around the single
difference frequency. This is always understood.

There is no such thing as "the single difference frequency"
and if you start thinking in terms of a single frequency, you
will get entirely the wrong understanding. I realise you are
not familiar with much of this, few people are, but that should
make you more amenable to help instead of fighting to keep
using misleading ideas.


What you are saying is very different to what is being
done. It may be that this is because you are using terms
in an unconventional manner but you will then hit problems
in referring to your text books.

No. I am using terms and descriptions of mixers as in my
1985 Shrader Electronic Communication text which shows
a (tuned)resonant inductor and capacitor circuit for the
intermediate
and different ones for the sum and the input frequencies.

That technique works well for single frequencies. For example
if you were trying to pick out Radio Luxemburg and reject the
rest of the medium wave band, it was ideal. If tuned to the
Luxemburg frequency, it would boost that and reduce all the
other stations.

I dont want to keep arguing this point but what I am saying
is in principle what is being done.

No it isn't. What the DSN is trying to do is exactly the
opposite. Their task is like building a repeater on a hill
to re-broadcast the medium wave band into a valley. In that
case the equipment has to amplify the whole band because
different people want different stations at the same time
and if it boosted the BBC for some people, it would swamp
Luxemburg for others. The aim in this case is to amplify
all frequencies across the band equally. That is what the
DSN early stages do and treating it like a tuned circuit
is entirely inappropriate and misleading.

You are obscuring the essence
of what is being done which is the use of Fourier's transform to
obtain a Fourier series representation of the noisy received
oscillations
You also seem to have changed your understanding of the nasa
documents to come around to my initial impression

Go back and read my emails. I spent about six months trying
to get across to you that there was no resonant circuit in
these stages. I can show you at least a dozen mails where
I said that.

Nothing in my understanding of the documents has changed in
any way.

Again it may be clearer but it is wrong. It is not just
the carrier oscillations that are digitised, it is the
whole signal, oscillations plus random thermal noise
and any other sources such as the galactic background.

Your understanding is wrong. I did not say CARRIER
oscillations
You can't change the meaning of
'oscillations' to mean only the
part due to the spacecraft transmitter

"oscillations" means something regular,

Not necessarily. And obviously not in this context

Elsewhere you talk of a "sequence of voltages". If you stick
with that terminology which is entirely accurate, the
confusion won't arise.

The final stage is the carrier PLL, not the FFT. All
the FFTs are removed from the chain once the PLL locks
on and they play no further part in the process. It is
the PLL that tracks the drifting signal and gives us
the accurate measurement.

The FFT as I was using the term includes the PLL.

The two are entirely diffeent and separate.

Not the way I am using the term. Note I say how I am using the
term.

For everyone else, and in the DSN documents, FFT means Fast
Fourier Transform while PLL means Phase Locked Loop. If you
mean something else by them, you will have to define your
usage, but you cannot expect me to know that unless you say
so.

.. The PLL locks on and tracks the carrier and it
uses a digital phase comparator that probably treats the signal
as 1s and 0s.

That is what I thought initially and you said I was wrong.

Ralph, check the emails. I explained to you how a phase
comparator works and gave you the simplest example of
an exclusive-or gate. I also explained what JPL mean by
type 2 and type 3 comparators.

Evidently
you have changed your mind.

No.

It doesn't matter however for the
purposes of showing the essence of the procedure and the
rationale
as to why it is reliable.

Agreed.

There are several levels of processing that you are skipping
over
which are very important in establishing that the signal is
genuine and from the right craft. Ultimately that is your main
concern, isn't it?

Yes. I welcome your pointing this out. But I deplore
the obscure and argumentative way that you are doing it.

Most of what you say is OK but you still treat the initial
stages as if they worked at a "single frequency" (your phrase)
which will lead you astray later if you don't correct it, and
in these posts you glossed over several of the important later
stages. These are the ones that are responsible for finding
and tracking the signal so need to be dealt with accurately.

and I mentioned that the
movement of the Earth etc requires different patterns to be
obtained successively but the point is that the FFT procedure
finds the underlying pattern and it is this that is used to
compare to the given sequence of 1s and 0s.

No it isn't. The final FFT is only used to set initial
frequency for the carrier PLL. If that locks, the
bandwidth is reduced to improve the signal/noise ratio.


You are saying the same thing that I was saying. I think
it is clearer to say it without the jargon.

Clearer but completely wrong.
No clearer but not detailed.

The FFT does not compare
Again I did not say this.

You said above:
.. the point is that the FFT procedure
finds the underlying pattern and it is this that is used to
compare to the given sequence of 1s and 0s.

I read that as saying the FFT is used to compare a pattern
of bit to a given sequence.

I said that after the
FFT procedure finds the dominant sine function,
this function is then compared to the observed set
of values which I thought you said earlier was reduced to
a set of 1s and 0s and that this was compared to
the corresponding observed set to get the degree of
error.

The FFT finds the frequency with the highest amplitude, that
is correct. The computer then passes that measured frequency
to the PLL which is a completely separate system. It contains
a circuit that generates a known frequency and a phase
comparator. The phase comparator is described as "digital"
and probably uses only the polarity information, treating
the signal as 1 or 0 as you say. I was pointing out that it
is part of the PLL that does this, not the FFT.

Again
that has to lock before the signal can be decoded using
a phase detector. Then it gets decoded through the error
correction scheme. There are many critical steps after
the FFT, and in fact the FFT plays no part in the decoding
process whatsoever.

Again I did not say that it did.

It reads that way since you talk of the FFT comparing against
"the sequence of 1s and 0s".

It is very inaccurate when the DSN document tells you
the analog band is the digitised band is 110MHz wide and
the signals of interest are of the order of 1Hz wide.

You are quoting the wrong document. We are talking about the
intermediate frequency
being smaller that the original frequency. Is that so hard for
you to understand.

What matters is how wide the frequency range is compared
to what you are looking at.

No what matters in this context is the size of the
intermediate frequency relative
to the size of the original frequency.

The band is 110MHz wide. You said "A resonance tuner picks out
the difference frequency" and clarified that as "It is a small
band of frequencies around the single difference frequency."
You cannot treat a 110MHz wide flat band as a "single frequency".

This makes no sense. Electrical oscillations add by the law of
superposition;

Yes, which is why it takes a special ciruit to get around that.

They dont multiply.

Dual gate fets and other methods of implementing mixers do

If they do then how do they.

I thought you wanted to discuss principles? There are many
different techniques but most use some sort of non-linearity.
A Field Effect Transistor for example inherently passes a
current that is proportional to the square of the voltage
because of the underlying physics. Diode mixers use their
exponential relationship between current and voltage.

Your text book should cover "diode mixers". Does it have
a chapter on wide bandwidth designs?

All I can see is superposition and
then various filters to extract the desired frequency or range of
frequencies

Superposition means the voltages add so it does not change
the frequency. BTW, superposition is a term usually applied
to EM waves rather than signals on wires.

Perhaps you have and analogue to digital converter to change
the incoming
frequencies to digital and then multiply them and then convert
this back instead
of the filter part of the mixer I see in my 1985 text.???

No, the frequency is too high to be digitised directly. The
purpose of changing the frequency is to bring it down to
something slow enough for the analogue to digital converter
(ADC) to work with.

because that is their intended function and we poor designers
have to make them do it well. It's what engineers get paid
for (though I personally work on the digital side).

The mathematical fact that a sum of sine
and cosine functions can
be represented as a product of related sine and cosine functions
has to be mentioned dont you think?

Only if you don't already know it.

Yes. And if I already knew it well we would not be having
this discussion would we?

You do know it, we discussed it by email for several weeks so
I felt you didn't need that explanation. Since you introduced
it in this thread I think I guessed correctly.

You seem to have your own subjective
read of what others say and on what you say without understanding
that words are ambiguous and you have to say out loud what you
mean
or what you think is meant before going off halfcocked.

If you use an acronym like FFT to mean something other than
what everyone else means but don't say so, you shouldn't be
surprised. When we are dealing with a band of signal and
noise less than 1Hz wide and you call the original IF which
is 100 million times wider "the single frequency", you must
expect your readers to be confused. Threads drift but look
back at the quotes above and that is what you are now
claiming you meant.

George

"George Dishman" wrote in message
...

"ralph sansbury" wrote in message
...


I can understand the following:

Elsewhere you talk of a "sequence of voltages". If you stick
with that terminology which is entirely accurate, the
confusion won't arise.

.. The PLL locks on and tracks the carrier and it
uses a digital phase comparator that probably treats the
signal
as 1s and 0s.

It doesn't matter however for the
purposes of showing the essence of the procedure and the
rationale
as to why it is reliable.

Agreed.


The FFT finds the frequency with the highest amplitude, that
is correct. The computer then passes that measured frequency
to the PLL which is a completely separate system. It contains
a circuit that generates a known frequency and a phase
comparator. The phase comparator is described as "digital"
and probably uses only the polarity information, treating
the signal as 1 or 0 as you say. I was pointing out that it
is part of the PLL that does this, not the FFT.


I still dont understand what you mean by the multiplication
of voltages arriving at the two gates of dual gate transistor:
My sense of this is that the sum of the voltages produces a
pattern
which contains a frequency which is the difference frequency plus
the sum frequency plus the two input frequencies and that the
filters in the special circuits you refer to produce these
separate
components???.

This makes no sense. Electrical oscillations add by
the law of
superposition;

Yes, which is why it takes a special ciruit to get around
that.





All I can see is superposition and
then various filters to extract the desired frequency or
range of
frequencies

"ralph sansbury" wrote in message
...

"George Dishman" wrote in message
...

"ralph sansbury" wrote in message
...


I can understand the following:

Elsewhere you talk of a "sequence of voltages". If you stick
with that terminology which is entirely accurate, the
confusion won't arise.

Excellent. I'll try to remember to use that too.

I still dont understand what you mean by the multiplication
of voltages arriving at the two gates of dual gate transistor:

You produce the sum and difference frequencies by
making use of this identity:

sin(a) * cos(b) = [ sin(a+b) + sin(a-b) ] / 2

The right hand side contains the sum and difference
so is the output from the circuit. The function needed
on the right hand side is multiplication. The same
method works when using a wideband signal where each
component is shifted in frequency by the same amount.
The example I gave used the DSN bands but assumes it
is a single shift where in reality it is done in a
number of stages:

Suppose the signal is a band from 2265MHz to 2375MHz
and it is multiplied by a pure sine wave of 2000MHz.
The sum is a band from 4265MHz to 4375MHz while the
difference is a band from 265MHz to 375MHz.

My sense of this is that the sum of the voltages produces a pattern
which contains a frequency which is the difference frequency plus
the sum frequency plus the two input frequencies and that the
filters in the special circuits you refer to produce these
separate components???.

If you replace "the sum of the voltages" in the first
line by "the product of the voltages", the paragraph
is perfectly correct. The filters are conceptually
separate from the mixer but often merged in practice.

George

"George Dishman" wrote in message
...

"ralph sansbury" wrote in message
...
I still dont understand what you mean by the multiplication
of voltages arriving at the two gates of dual gate
transistor:

You produce the sum and difference frequencies by
making use of this identity:

sin(a) * cos(b) = [ sin(a+b) + sin(a-b) ] / 2

The right hand side contains the sum and difference
so is the output from the circuit. The function needed
on the right hand side is multiplication.

You mean the left hand side. Yes the summation is equal to
the product. And it is the summation that is produced by adding
the voltages at the dual gate transistor along with the input
frequencies.


The same
method works when using a wideband signal where each
component is shifted in frequency by the same amount.


My sense of this is that the sum of the voltages produces a
pattern
which contains a frequency which is the difference frequency
plus
the sum frequency plus the two input frequencies and that the
filters in the special circuits you refer to produce these
separate components???.

If you replace "the sum of the voltages" in the first
line by "the product of the voltages", the paragraph
is perfectly correct. The filters are conceptually
separate from the mixer but often merged in practice.

I think you are overlooking the basic physics here. You
have no reason
for the filters at the output of the dual gate transistor etc,
and
which are an integral 'part of the mixer but conceptually
separate', unless it
is to take the oscillation of voltage which results from the
addition of the
separate oscillating voltages and to produce the different
component frequencies as
different outputs. The difference frequency of oscillating
voltage is then
taken as the desired output.
Of course the mathematical equivalence between the specified
product
and the specified sum makes it mathematically correct to say that
the black box
circuit produces a product of the two input frequencies as well
as the two
input frequencies.
This is adequate for engineers once the circuit has been
designed. But the first designers of the
circuit had to know the physics and to put in the inductors and
capacitors in
a resonant configuration etc to get the desired output ie.
to design around the problem that voltages at the input can only
add.
Thus it is not quite physically correct to say that the
transistor multiplies the
input voltages
and more correct to say that the transistor adds the incoming
voltages such that
the resulting pattern can be written as the sum of the input
frequencies and the
difference frequency and the sum frequency and that the latter
two frequencies
are mathematically equivalent to the product of the frequencies
etc.
Ralph

"ralph sansbury" wrote in message
...

"George Dishman" wrote in message
...

"ralph sansbury" wrote in message
...
I still dont understand what you mean by the multiplication
of voltages arriving at the two gates of dual gate
transistor:

You produce the sum and difference frequencies by
making use of this identity:

sin(a) * cos(b) = [ sin(a+b) + sin(a-b) ] / 2

The right hand side contains the sum and difference
so is the output from the circuit. The function needed
on the right hand side is multiplication.

You mean the left hand side.

Doh! Yes, the left hand side is the multiplication.

Yes the summation is equal to
the product. And it is the summation that is produced by adding
the voltages at the dual gate transistor along with the input
frequencies.

No. Here is a conceptual block diagram:

multiplier filter(s)
+---+
a(t) ----| | +---+
| * |---------| ~ |-- d(t)
b(t) ----| | c(t) +---+
+---+

The time-varying voltages a(t) and b(t) are multiplied
together to produce voltage c(t) = 2 * a(t) * b(t).
(The factor of 2 makes the text easier to read later.)

If a and b are sine waves with angular frequencies w_a and w_b:

a(t) = sin(w_a * t)
b(t) = cos(w_b * t)

then c(t) can also be expressed as

c(t) = sin((w_a+w_b) * t) + sin((w_a-w_b) * t)

The filter rejects the sin((w_a+w_b) * t) term so you are left
with only the difference frequency sin((w_a-w_b) * t)

Now in the real system, one signal is a pure sine wave while
the other is the continuum of frequencies 110MHz wide but
the same analysis applies to each component so you get a
continuum out of the filter but shifted down the spectrum
by the reference frequency.

The same
method works when using a wideband signal where each
component is shifted in frequency by the same amount.


My sense of this is that the sum of the voltages produces a pattern
which contains a frequency which is the difference frequency plus
the sum frequency plus the two input frequencies and that the
filters in the special circuits you refer to produce these
separate components???.

If you replace "the sum of the voltages" in the first
line by "the product of the voltages", the paragraph
is perfectly correct. The filters are conceptually
separate from the mixer but often merged in practice.

I think you are overlooking the basic physics here. You have no
reason
for the filters at the output of the dual gate transistor etc, and
which are an integral 'part of the mixer but conceptually separate',
unless it
is to take the oscillation of voltage which results from the addition of
the
separate oscillating voltages and to produce the different component
frequencies as
different outputs. The difference frequency of oscillating voltage is then
taken as the desired output.

Again, that is all correct except "which results from the addition
of the" should read "which results from the multiplication of the"

Of course the mathematical equivalence between the specified product
and the specified sum makes it mathematically correct to say that the
black box
circuit produces a product of the two input frequencies as well
as the two input frequencies.

Wrong way round. You have to actually multiply in order to
create something equivalent the sum and difference.

This is adequate for engineers once the circuit has been
designed. But the first designers of the
circuit had to know the physics

I am a designer and have an honours degree in physics,
how about you?

and to put in the inductors and
capacitors in a resonant configuration

Yet again: it cannot be a resonant configuration if you want
all the band to get through without distortion, it must have
a flat passband which is easiest to envisage conceptually as
separate high and low pass filters.

etc to get the desired output ie.
to design around the problem that voltages at the input can only
add.
Thus it is not quite physically correct to say that the
transistor multiplies the
input voltages

Sorry Ralph, it is physically accurate. Read your text
book on how a field effect transistor works and find
out the equation that relates "Id", the current from
drain to source to "Vgs", the voltage between gate and
source, in the saturation region. If it doesn't cover it
there are pages below.

and more correct to say that the transistor adds the incoming voltages
such that
the resulting pattern can be written as the sum of the input frequencies
and the
difference frequency and the sum frequency and that the latter two
frequencies
are mathematically equivalent to the product of the frequencies
etc.

Nope, you need to do some homework. You may be able
to find better than these pages but they will do for
a start. FETs are normally run in saturation which
is to the right of the dotted line on the graph:

http://ece-www.colorado.edu/~bart/bo...ter7/ch7_2.htm

with a more detailed analysis on the next page:

http://ece-www.colorado.edu/~bart/bo...h7_3.htm#7_3_2

Here's another, sorry it is in word document format but
it gives a brief look at the physics:

http://www.eng.abdn.ac.uk/~eng188/EG1567/JW-08-FETs.doc

The key equation is in the "Characteristic Equations" on page 5:

ID ~ (Idss/Vt^2)/(Vgs – Vt)^2 = Idss[(Vgs/Vt)-1]^2

George