Yes, your friend is right, but there is a grain of truth in what you are
saying also.

A mixer multiplies two signals as your friend says and as said in the
posting by W7EL. Signals can be represented by cosines, and the product of
two cosines is:

cosA * cosB = 0.5(cos(A-B) - cos(A+B) )

See product identities on http://www.swt.edu/slac/math/trigrev/trigrev.html
and let A=2*pi*f1 and B=2*pi*f2.

So while the operation of the mixer is that signals are _multiplied_, the
frequencies will _add_ or subtract.


--
Sverre Holm, LA3ZA
---------------------------------
www.qsl.net/la3za

Mixers are amplitude modulators.
One signal modulates the amplitude of the other.
Trigonometrically we have -

2*Sin(A)*Sin(B) = Cos(A-B) - Cos(A+B)

Where the A-B and A+B terms are appropriately described as "the products".

In practice there are a great number of unwanted products output from a
mixer because many harmonics of A and B are generated in the process and all
continue to inter-modulate each other.

The wanted product, the IF, is usually A-B or A+B.
---
.................................................. ..........
Regards from Reg, G4FGQ
For Free Radio Design Software go to
http://www.btinternet.com/~g4fgq.regp
.................................................. ..........

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

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

Yes, your friend is right, but there is a grain of truth in what you are
saying also.

A mixer multiplies two signals as your friend says and as said in the
posting by W7EL. Signals can be represented by cosines, and the product of
two cosines is:

cosA * cosB = 0.5(cos(A-B) - cos(A+B) )

See product identities on http://www.swt.edu/slac/math/trigrev/trigrev.html
and let A=2*pi*f1 and B=2*pi*f2.

So while the operation of the mixer is that signals are _multiplied_, the
frequencies will _add_ or subtract.


--
Sverre Holm, LA3ZA
---------------------------------
www.qsl.net/la3za

Mixers are amplitude modulators.
One signal modulates the amplitude of the other.
Trigonometrically we have -

2*Sin(A)*Sin(B) = Cos(A-B) - Cos(A+B)

Where the A-B and A+B terms are appropriately described as "the products".

In practice there are a great number of unwanted products output from a
mixer because many harmonics of A and B are generated in the process and all
continue to inter-modulate each other.

The wanted product, the IF, is usually A-B or A+B.
---
.................................................. ..........
Regards from Reg, G4FGQ
For Free Radio Design Software go to
http://www.btinternet.com/~g4fgq.regp
.................................................. ..........

In article ,
(Joer) writes:

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?

Neither and both! :-)

It depends on the definitions of "linearity," "mixing," and "adding."

In a perfect linear circuit, two or more signals can exist as separate
entities, none affecting any other. You can "mix" them (inject them)
with the group of signals. Since the circuit is perfectly linear, no new
frequencies are added which are attributed to a sum or difference of
existing frequency signal components.

In sound recording "mixing" and "adding" always refers to operation
with nearly-perfect linear circuits to a recording medium.

"Mixers" in radios are highly non-linear. Some are outright switches
(Tayloe Mixer), some are very nearly on-off switches (diode rings),
and some use gross distortion of normally-linear characteristics
(tubes, particularly pentagrids...and Gilbert Cell double-differential
transistor structures). NON-LINEARITY creates new frequencies.

Mixing (in a mixer circuit) an incoming signal with a local oscillator
creates a mathematical sum and difference of the signal and the
LO frequencies...in addition to the existing signal and LO
frequencies passing through the mixer circuit (balanced mixers will
suppress the LO and double-balanced mixers can suppress the
signal frequency as well).

In this process of mixer circuit mixing, the new frequency products
(using "products" in a very general sense, not just multiplication)
still retain the amplitudes of the original. The signal's amplitude
containing AM sidebands is repeated at the new sum and difference
frequencies. Relative phase is also preserved. If the signal has
modulation components due to FM or PM, those appear on the new
sum and difference frequencies. If the much-stronger LO contains
any AM, FM, or PM, that is repeated on the new sum and difference
frequency components as AM, FM, or PM.

It gets worse. :-) The LO is seldom a pure sinewave so it has
harmonic content. New sum and difference frequencies will exist
as a result of LO harmonics! [most of those are simply filtered
out, dissipated, rejected] Scoping an LO injection waveform on
a wideband oscilloscope might come as a shock... :-)

The "process" is all due to NON-LINEARITY. The mixer output
contains the original signals plus components at frequencies
which are the sum and difference of the original...plus a few more.

Mathematics is used as a way of explaining the non-linear mixing
process. That isn't the full explaination but it is close enough.
Don't get caught up in plus and minus signs on equations and
too much argument over that...nor of the mathematical purists
who play games with term re-arrangements and "hidden meanings."

Non-linearity of all amplifiers will cause heterodyne creations. A
low-level example is the intermodulation distortion values such as
"IP3.".

Len Anderson
retired (from regular hours) electronic engineer person

In article ,
(Joer) writes:

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?

Neither and both! :-)

It depends on the definitions of "linearity," "mixing," and "adding."

In a perfect linear circuit, two or more signals can exist as separate
entities, none affecting any other. You can "mix" them (inject them)
with the group of signals. Since the circuit is perfectly linear, no new
frequencies are added which are attributed to a sum or difference of
existing frequency signal components.

In sound recording "mixing" and "adding" always refers to operation
with nearly-perfect linear circuits to a recording medium.

"Mixers" in radios are highly non-linear. Some are outright switches
(Tayloe Mixer), some are very nearly on-off switches (diode rings),
and some use gross distortion of normally-linear characteristics
(tubes, particularly pentagrids...and Gilbert Cell double-differential
transistor structures). NON-LINEARITY creates new frequencies.

Mixing (in a mixer circuit) an incoming signal with a local oscillator
creates a mathematical sum and difference of the signal and the
LO frequencies...in addition to the existing signal and LO
frequencies passing through the mixer circuit (balanced mixers will
suppress the LO and double-balanced mixers can suppress the
signal frequency as well).

In this process of mixer circuit mixing, the new frequency products
(using "products" in a very general sense, not just multiplication)
still retain the amplitudes of the original. The signal's amplitude
containing AM sidebands is repeated at the new sum and difference
frequencies. Relative phase is also preserved. If the signal has
modulation components due to FM or PM, those appear on the new
sum and difference frequencies. If the much-stronger LO contains
any AM, FM, or PM, that is repeated on the new sum and difference
frequency components as AM, FM, or PM.

It gets worse. :-) The LO is seldom a pure sinewave so it has
harmonic content. New sum and difference frequencies will exist
as a result of LO harmonics! [most of those are simply filtered
out, dissipated, rejected] Scoping an LO injection waveform on
a wideband oscilloscope might come as a shock... :-)

The "process" is all due to NON-LINEARITY. The mixer output
contains the original signals plus components at frequencies
which are the sum and difference of the original...plus a few more.

Mathematics is used as a way of explaining the non-linear mixing
process. That isn't the full explaination but it is close enough.
Don't get caught up in plus and minus signs on equations and
too much argument over that...nor of the mathematical purists
who play games with term re-arrangements and "hidden meanings."

Non-linearity of all amplifiers will cause heterodyne creations. A
low-level example is the intermodulation distortion values such as
"IP3.".

Len Anderson
retired (from regular hours) electronic engineer person