Disclaimer (disflamer?): Everything that Roy says is true enough to get you
down the road of radio circuit design -- but:

To be absolutely, mathematically correct, if you hold your mouth right, a
"perfect" mixer with it's driving oscillator, is a linear device. It is
_not_ a time-invariant device. It's linear because the IF signal that
results from putting in the sum of any two RF signals is exactly equal to
the sum of the IF signals that each result from each of the RF signals. If
it were nonlinear then this would not be the case (and it wouldn't be a
useful device for mixing).

What gives a mixer it's "mixerness" is that it is linear but time-varying
(output = input * some function of time). It is very easy to confuse
time-varying linear with non-linear, and even easier in practice because in
order to get the effect you need to use componant non-linearities to get the
job done, just as you do with a class A amplifier. But it's usually harder
to get the nonlinearities out of a mixer than an amplifier, so in real
design you have to pay attention to non-linear effects like blocking and
intermodulation in a mixer to a much greater extent than you do with an
amplifier, and this reinforces the idea that a mixer is fundamentally
nonlinear.

This means that when you're analyzing a mixer (and ignoring real-mixer
things like intermodulation) you can still use all the linear circuit theory
stuff as long as you stay away from anything that depends on
time-invariance. This means that _simple_ Laplace and Fourier analysis is
out, but you can still use _careful_ Fourier analysis to figure out what the
output will be for a given input and oscillator frequency. In fact, that's
exactly what you are doing when you analyze a mixer: all of the desired
behavior of a mixer can be exactly predicted with Fourier analysis.

"Roy Lewallen" wrote in message
...
Your friend is right.

If you simply add or subtract two waveforms, no new frequencies are
created. You end up with only the frequencies you started with and no
more. (Theoretically, you could make one or more disappear if one of the
added waveforms contained a precise negative of one or more frequency
components of the other -- but you can never get any new frequencies.)
That's because addition is a linear process, with linear having a
precise definition that's appeared here a number of times before.
(Subtraction is just addition, with one waveform inverted before
adding.) Multiplication, though, is a nonlinear process by the precise
definition used in circuit analysis, and it does create additional
frequencies. Multiplying the two original signals of 1500 and 1955
generates the two new frequencies of 455 and 3455, for a total of four
frequencies after multiplication. Adding them wouldn't do it.

Most good mixers are actually more like switches than multipliers, but
they're still nonlinear -- very much so -- and don't do anything
remotely like adding the two signals. A doubly balanced mixer produces
the sum and difference frequencies while not letting the original two
frequencies get through to the output.

The generation of the new frequencies by multiplication of the two
originals is easily shown mathematically, as your friend says, with a
short derivation by means of a trig identity. I'll be glad to post the
derivation if you or other readers are interested, although it's widely
available elsewhere.

Roy Lewallen, W7EL

Joer wrote:
I'm trying to settle a debate with a friend, and my knowledge of
mixers is pretty rusty.

Say you have a receiver whose IF is 455 kHz, and it's tuned to a
station at 1500 kHz. If all's working OK, at the output of the mixer
you should have four frequencies:

1500 (original signal)
1955 (oscillator signal - osc. working above the signal freq.)
3455 (sum)
455 (difference)

My question is by what process does the mixer produce the 3455 and 455
frequencies. I say it's an add and subtract process, my friend says
(via mathematics) it's a multiplication process. Who's right?

thanks,

Joe W9TXU

Bill,

You said:
" Any good technician will tell you it's an add and subtract process.
Any good engineer will bore you to tears with complicated mathematical
analysis.
Guess which answer is more useful for your purpose?"

Well of course...if you're only interested in what some of what comes out,
then it's an 'add and subtract process'. But isn't that our initial
definition of what we want a mixer to do? This is circular logic. You're
chasing your own tail.

The original question was more in the vein of 'by what mechanism does a
mixer produce sum and difference frequency components'. The correct answer
is that it implements the mathematical product of the two input signals, and
that product contains sum and difference frequencies in addition to a host
of other frequencies that includes the original frequencies, all their
harmonics, and every conceivable product of those frequencies and their
harmonics. It's not just a simple 'add and subtract'. It just so happens
that we're most interested in the sum and difference, but there is much,
much more going on.

The "answer that is most useful for the purpose" is not necessarily the most
simplistic. Consider the following profound statement from W.E. Deming:
"If you can't describe what you are doing as a process, then you don't know
what you are doing"


Joe W3JDR

Bill,

You said:
" Any good technician will tell you it's an add and subtract process.
Any good engineer will bore you to tears with complicated mathematical
analysis.
Guess which answer is more useful for your purpose?"

Well of course...if you're only interested in what some of what comes out,
then it's an 'add and subtract process'. But isn't that our initial
definition of what we want a mixer to do? This is circular logic. You're
chasing your own tail.

The original question was more in the vein of 'by what mechanism does a
mixer produce sum and difference frequency components'. The correct answer
is that it implements the mathematical product of the two input signals, and
that product contains sum and difference frequencies in addition to a host
of other frequencies that includes the original frequencies, all their
harmonics, and every conceivable product of those frequencies and their
harmonics. It's not just a simple 'add and subtract'. It just so happens
that we're most interested in the sum and difference, but there is much,
much more going on.

The "answer that is most useful for the purpose" is not necessarily the most
simplistic. Consider the following profound statement from W.E. Deming:
"If you can't describe what you are doing as a process, then you don't know
what you are doing"


Joe W3JDR

You're absolutely correct. Production of new frequency components can be
done with either nonlinear or time-variant circuits. A square-law diode
detector is an example of the first; a multiplier is an example of the
second. I stand corrected -- thanks for pointing it out.

Roy Lewallen, W7EL

Tim Wescott wrote:
Disclaimer (disflamer?): Everything that Roy says is true enough to get you
down the road of radio circuit design -- but:

To be absolutely, mathematically correct, if you hold your mouth right, a
"perfect" mixer with it's driving oscillator, is a linear device. It is
_not_ a time-invariant device. It's linear because the IF signal that
results from putting in the sum of any two RF signals is exactly equal to
the sum of the IF signals that each result from each of the RF signals. If
it were nonlinear then this would not be the case (and it wouldn't be a
useful device for mixing).

What gives a mixer it's "mixerness" is that it is linear but time-varying
(output = input * some function of time). It is very easy to confuse
time-varying linear with non-linear, and even easier in practice because in
order to get the effect you need to use componant non-linearities to get the
job done, just as you do with a class A amplifier. But it's usually harder
to get the nonlinearities out of a mixer than an amplifier, so in real
design you have to pay attention to non-linear effects like blocking and
intermodulation in a mixer to a much greater extent than you do with an
amplifier, and this reinforces the idea that a mixer is fundamentally
nonlinear.

This means that when you're analyzing a mixer (and ignoring real-mixer
things like intermodulation) you can still use all the linear circuit theory
stuff as long as you stay away from anything that depends on
time-invariance. This means that _simple_ Laplace and Fourier analysis is
out, but you can still use _careful_ Fourier analysis to figure out what the
output will be for a given input and oscillator frequency. In fact, that's
exactly what you are doing when you analyze a mixer: all of the desired
behavior of a mixer can be exactly predicted with Fourier analysis.

"Roy Lewallen" wrote in message
...

Your friend is right.

If you simply add or subtract two waveforms, no new frequencies are
created. You end up with only the frequencies you started with and no
more. (Theoretically, you could make one or more disappear if one of the
added waveforms contained a precise negative of one or more frequency
components of the other -- but you can never get any new frequencies.)
That's because addition is a linear process, with linear having a
precise definition that's appeared here a number of times before.
(Subtraction is just addition, with one waveform inverted before
adding.) Multiplication, though, is a nonlinear process by the precise
definition used in circuit analysis, and it does create additional
frequencies. Multiplying the two original signals of 1500 and 1955
generates the two new frequencies of 455 and 3455, for a total of four
frequencies after multiplication. Adding them wouldn't do it.

Most good mixers are actually more like switches than multipliers, but
they're still nonlinear -- very much so -- and don't do anything
remotely like adding the two signals. A doubly balanced mixer produces
the sum and difference frequencies while not letting the original two
frequencies get through to the output.

The generation of the new frequencies by multiplication of the two
originals is easily shown mathematically, as your friend says, with a
short derivation by means of a trig identity. I'll be glad to post the
derivation if you or other readers are interested, although it's widely
available elsewhere.

Roy Lewallen, W7EL

Joer wrote:

I'm trying to settle a debate with a friend, and my knowledge of
mixers is pretty rusty.

Say you have a receiver whose IF is 455 kHz, and it's tuned to a
station at 1500 kHz. If all's working OK, at the output of the mixer
you should have four frequencies:

1500 (original signal)
1955 (oscillator signal - osc. working above the signal freq.)
3455 (sum)
455 (difference)

My question is by what process does the mixer produce the 3455 and 455
frequencies. I say it's an add and subtract process, my friend says
(via mathematics) it's a multiplication process. Who's right?

thanks,

Joe W9TXU

You're absolutely correct. Production of new frequency components can be
done with either nonlinear or time-variant circuits. A square-law diode
detector is an example of the first; a multiplier is an example of the
second. I stand corrected -- thanks for pointing it out.

Roy Lewallen, W7EL

Tim Wescott wrote:
Disclaimer (disflamer?): Everything that Roy says is true enough to get you
down the road of radio circuit design -- but:

To be absolutely, mathematically correct, if you hold your mouth right, a
"perfect" mixer with it's driving oscillator, is a linear device. It is
_not_ a time-invariant device. It's linear because the IF signal that
results from putting in the sum of any two RF signals is exactly equal to
the sum of the IF signals that each result from each of the RF signals. If
it were nonlinear then this would not be the case (and it wouldn't be a
useful device for mixing).

What gives a mixer it's "mixerness" is that it is linear but time-varying
(output = input * some function of time). It is very easy to confuse
time-varying linear with non-linear, and even easier in practice because in
order to get the effect you need to use componant non-linearities to get the
job done, just as you do with a class A amplifier. But it's usually harder
to get the nonlinearities out of a mixer than an amplifier, so in real
design you have to pay attention to non-linear effects like blocking and
intermodulation in a mixer to a much greater extent than you do with an
amplifier, and this reinforces the idea that a mixer is fundamentally
nonlinear.

This means that when you're analyzing a mixer (and ignoring real-mixer
things like intermodulation) you can still use all the linear circuit theory
stuff as long as you stay away from anything that depends on
time-invariance. This means that _simple_ Laplace and Fourier analysis is
out, but you can still use _careful_ Fourier analysis to figure out what the
output will be for a given input and oscillator frequency. In fact, that's
exactly what you are doing when you analyze a mixer: all of the desired
behavior of a mixer can be exactly predicted with Fourier analysis.

"Roy Lewallen" wrote in message
...

Your friend is right.

If you simply add or subtract two waveforms, no new frequencies are
created. You end up with only the frequencies you started with and no
more. (Theoretically, you could make one or more disappear if one of the
added waveforms contained a precise negative of one or more frequency
components of the other -- but you can never get any new frequencies.)
That's because addition is a linear process, with linear having a
precise definition that's appeared here a number of times before.
(Subtraction is just addition, with one waveform inverted before
adding.) Multiplication, though, is a nonlinear process by the precise
definition used in circuit analysis, and it does create additional
frequencies. Multiplying the two original signals of 1500 and 1955
generates the two new frequencies of 455 and 3455, for a total of four
frequencies after multiplication. Adding them wouldn't do it.

Most good mixers are actually more like switches than multipliers, but
they're still nonlinear -- very much so -- and don't do anything
remotely like adding the two signals. A doubly balanced mixer produces
the sum and difference frequencies while not letting the original two
frequencies get through to the output.

The generation of the new frequencies by multiplication of the two
originals is easily shown mathematically, as your friend says, with a
short derivation by means of a trig identity. I'll be glad to post the
derivation if you or other readers are interested, although it's widely
available elsewhere.

Roy Lewallen, W7EL

Joer wrote:

I'm trying to settle a debate with a friend, and my knowledge of
mixers is pretty rusty.

Say you have a receiver whose IF is 455 kHz, and it's tuned to a
station at 1500 kHz. If all's working OK, at the output of the mixer
you should have four frequencies:

1500 (original signal)
1955 (oscillator signal - osc. working above the signal freq.)
3455 (sum)
455 (difference)

My question is by what process does the mixer produce the 3455 and 455
frequencies. I say it's an add and subtract process, my friend says
(via mathematics) it's a multiplication process. Who's right?

thanks,

Joe W9TXU

A lot of good stuff here and unfortunately some digression.

It might help here if you think this way. The basic RF or as you say,
superhetrodyne mixer, does what it does by instantaneous voltages
multiplication of the _instantaneous voltages_ of the two signals. There
are many types of these RF mixers, but the one which is modeled with this
concept will only give you the two frequencies as Roy's trig identity shows.
You can take a spreadsheet and put a sine wave of one freq in one column and
another freq in another column (use a pretty small angle step, say 1 degree
or less - be careful the degree/radian issue doesn't mess you up) then make
a formula in a third column which is A*B. You will see that the resulting
"product" column has variations which are not the two input frequencies. It
may not me real clear, but if it wiggles faster or slower that the
originals, then it is a different frequency. (now I gotta do this so I see
what it looks like for my self). However, better yet.....
The neatest example of this is to put the exact same frequency in both
inputs columns! Then the "wiggles" of the sine wave will be very obvious.
There will be no doubt about what is coming out of the multiplication.
God, I love spreadsheets. (sorry here, the beginning of a sentence is
capitalized).

Why does this Mister Wizzard stunt work? Because a circuit is doing
something to the voltages present at any given time, and in this case it is
a product thing.

SO, YES you do get the "Sum" and "Difference" frequencies out, so if you
want to call that addition / subtraction while working "in the frequency
domain" that's ok with me. However, all this other garbage holds, just the
same.

If you make a circuit which gives as its output the product of the two input
voltages, Roy's formula holds and you get the sum and difference frequencies
only. This is what we commonly call a "balanced mixer". The term
"balanced" comes from the concept that in this type, if you get the circuit
set up or "balanced" just right, the two input signals don't appear at the
output and the trig identity holds. I suppose it can be called the ideal
type.

When you get into what is commonly called "modulation", you still have this
type of instantaneous voltage multiplication, but usually, like in a Plate
modulated Tube transmitter, it is not so perfect and some of the original
input signals get through to the output (though the audio can't make it out
to the antenna) and you get carrier (one of the input signals) as well.
(I'm not going to get into the 'does the carrier vary in amplitude' or
sideband arguments here.)

All this talk about many more than the two frequencies is the result of what
we call "higher order" non linearities. This is just a way to describe
distortion that keeps the original sine waves from being perfect sine waves
in a circuit.

Also, the sampling talk will just confuse this basic issue, so I advise
ignoring it for now.

FWIW: the model in my brain can somewhat consider time variant the same as
non linearity since you get out something which ain't a simple scaled
version of the input...
73, Steve K;9;D:C:I

How'm I doin' Roy & Reg?

"W3JDR" wrote in message
...
Bill,

You said:
" Any good technician will tell you it's an add and subtract process.
Any good engineer will bore you to tears with complicated mathematical
analysis.
Guess which answer is more useful for your purpose?"

Well of course...if you're only interested in what some of what comes out,
then it's an 'add and subtract process'. But isn't that our initial
definition of what we want a mixer to do? This is circular logic. You're
chasing your own tail.

The original question was more in the vein of 'by what mechanism does a
mixer produce sum and difference frequency components'. The correct answer
is that it implements the mathematical product of the two input signals,
and
that product contains sum and difference frequencies in addition to a host
of other frequencies that includes the original frequencies, all their
harmonics, and every conceivable product of those frequencies and their
harmonics. It's not just a simple 'add and subtract'. It just so happens
that we're most interested in the sum and difference, but there is much,
much more going on.

The "answer that is most useful for the purpose" is not necessarily the
most
simplistic. Consider the following profound statement from W.E. Deming:
"If you can't describe what you are doing as a process, then you don't
know
what you are doing"


Joe W3JDR

A lot of good stuff here and unfortunately some digression.

It might help here if you think this way. The basic RF or as you say,
superhetrodyne mixer, does what it does by instantaneous voltages
multiplication of the _instantaneous voltages_ of the two signals. There
are many types of these RF mixers, but the one which is modeled with this
concept will only give you the two frequencies as Roy's trig identity shows.
You can take a spreadsheet and put a sine wave of one freq in one column and
another freq in another column (use a pretty small angle step, say 1 degree
or less - be careful the degree/radian issue doesn't mess you up) then make
a formula in a third column which is A*B. You will see that the resulting
"product" column has variations which are not the two input frequencies. It
may not me real clear, but if it wiggles faster or slower that the
originals, then it is a different frequency. (now I gotta do this so I see
what it looks like for my self). However, better yet.....
The neatest example of this is to put the exact same frequency in both
inputs columns! Then the "wiggles" of the sine wave will be very obvious.
There will be no doubt about what is coming out of the multiplication.
God, I love spreadsheets. (sorry here, the beginning of a sentence is
capitalized).

Why does this Mister Wizzard stunt work? Because a circuit is doing
something to the voltages present at any given time, and in this case it is
a product thing.

SO, YES you do get the "Sum" and "Difference" frequencies out, so if you
want to call that addition / subtraction while working "in the frequency
domain" that's ok with me. However, all this other garbage holds, just the
same.

If you make a circuit which gives as its output the product of the two input
voltages, Roy's formula holds and you get the sum and difference frequencies
only. This is what we commonly call a "balanced mixer". The term
"balanced" comes from the concept that in this type, if you get the circuit
set up or "balanced" just right, the two input signals don't appear at the
output and the trig identity holds. I suppose it can be called the ideal
type.

When you get into what is commonly called "modulation", you still have this
type of instantaneous voltage multiplication, but usually, like in a Plate
modulated Tube transmitter, it is not so perfect and some of the original
input signals get through to the output (though the audio can't make it out
to the antenna) and you get carrier (one of the input signals) as well.
(I'm not going to get into the 'does the carrier vary in amplitude' or
sideband arguments here.)

All this talk about many more than the two frequencies is the result of what
we call "higher order" non linearities. This is just a way to describe
distortion that keeps the original sine waves from being perfect sine waves
in a circuit.

Also, the sampling talk will just confuse this basic issue, so I advise
ignoring it for now.

FWIW: the model in my brain can somewhat consider time variant the same as
non linearity since you get out something which ain't a simple scaled
version of the input...
73, Steve K;9;D:C:I

How'm I doin' Roy & Reg?

"W3JDR" wrote in message
...
Bill,

You said:
" Any good technician will tell you it's an add and subtract process.
Any good engineer will bore you to tears with complicated mathematical
analysis.
Guess which answer is more useful for your purpose?"

Well of course...if you're only interested in what some of what comes out,
then it's an 'add and subtract process'. But isn't that our initial
definition of what we want a mixer to do? This is circular logic. You're
chasing your own tail.

The original question was more in the vein of 'by what mechanism does a
mixer produce sum and difference frequency components'. The correct answer
is that it implements the mathematical product of the two input signals,
and
that product contains sum and difference frequencies in addition to a host
of other frequencies that includes the original frequencies, all their
harmonics, and every conceivable product of those frequencies and their
harmonics. It's not just a simple 'add and subtract'. It just so happens
that we're most interested in the sum and difference, but there is much,
much more going on.

The "answer that is most useful for the purpose" is not necessarily the
most
simplistic. Consider the following profound statement from W.E. Deming:
"If you can't describe what you are doing as a process, then you don't
know
what you are doing"


Joe W3JDR

Thanks everyone, in fact I received an e-mail from my friend with
similar trigonometric equations, so I'm absolutely convinced! I now
have a slightly better idea of how a superhet mixer functions ....


Joe W9TXU

Thanks everyone, in fact I received an e-mail from my friend with
similar trigonometric equations, so I'm absolutely convinced! I now
have a slightly better idea of how a superhet mixer functions ....


Joe W9TXU

Ian:

[snip]
"Ian White, G3SEK" wrote in message
news
Roy Lewallen wrote::
:
Part of the confusion is that audio engineers talk about "mixing" where
they actually mean adding. Mixing - as RF engineers use the term - is
precisely what they don't want!
:
73 from Ian G3SEK 'In Practice' columnist for RadCom (RSGB)
[snip]

Mixer, modulator, multiplier, demodulator, detector, switcher, balanced
modulator, adder, subtractor, heh, heh....

The term mixer is overused, or... "overloaded" as the computer scientists
like to say.

Yes indeed, too bad for beginners, but it's part of the mystique of our
trade as well, that there are plenty of examples of misuse,
misappropriation, and the outright abuse of terms and their meanings in our
trade! Keeps gurus in business and nosey outsiders out, as well. :-) Heh,
heh...

Even within the English speaking community, there is often no consistency of
terminology use, for example "tube" versus "valve", etc...

British and American use of the term "mixer" in the television production
equipment business has further confusing examples of overuse and overlapping
meanings. In television production technology the term "mixer" is also used
to describe switching and sepcial effects equipment and the terms are
applied differently on each side of the Atlantic. What you Brits call a
television "mixer" is called a television "switcher" in America, and what's
more... the same names are used for the operators of the said
mixing/switching equipment. [Grass Valley, Ross, Central Dynamics, etc...
are manufacturers of such.] You can often see the equipment operator's names
listed opposite the titles Mixer or Switcher on the TV screen when they roll
the credits at the end of television shows. And to make things worse, the
"function" of an audio "mixer" is again entirely different than a video
"mixer", whilst television video mixers often contain integrated audio
mixers. Impossible for beginners to figure out what experts are talking
about, go figure!

--
Peter K1PO
Indialantic By-the-Sea, FL