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