The two processes, superposition vs. mixing, remain totally different, with different results - new frequencies generated in one case and not in the other; frequencies lost in one and not in the other.
I'll concede that you can start with a mixed signal and decompose it, and you can start with two signals and mix them. These are different physical processes. But they are described by the same mathematical equations... that's all I'm saying.
The fact I can change one into the other by a linear process shows that mixing
can be described by a linear superposition. The mixing and adding equations are the same kind of mathematical object.
I have produced several experimental situations as well as the math to demonstrate that these are equivalent.
You have provided a number of examples that have only supported my case.
You have provided an authoritative reference in a popular physics text whose authors agree with me.
So it remains with you to demonstrate your assertions.
That the mixing case is different mathematically from the adding case.
That there is no change in frequencies involved in one but there is in the other.
Please be clear about your definitions, and cite references.
Of course we can certainly leave it there :)
(I still think we are in agreement - but are using the words slightly differently giving the
appearance of a disagreement. In which case, we can agree to
agree xD)
I enjoyed our correspondence
Well it's been exercise. At least we have demonstrated how to have a gentlemanly scientific debate without acrimony.
Since you have expressed a desire to let matter lie, I have answered your challenge as an afterward:
Afterward
I would advise not trying to explain (analyze) a superhet stage using addition of the RF and LO signals or you'll be very disapponted in the IF output expression!
A challenge!
In the spirit of illustrating how I am using the language above:
f_{LO}=f_{RF}\pm f_{IF}
Well that was easy.
But you want the messy details I'm guessing ... right. It's not good enough to just do this, I have to demonstrate that this comes from a linear superposition of the waveforms?
<rolls up sleeves>
In general the incoming AM signal will be something like:s_{rf}(t)=A[1+g(t)]\cos(2\pi f_{rf}) ... so g(t) is the data-stream we want to recover.
Using low-side injection - after mixing we get:
s_{if}=s_{rf} s_{lo} = s_{rf}\cos{(2\pi f_{lo})}... this is the multiplicative "mixing". So let's explicitly multiply it out.
s_{if}=A[1+g(t)]\cos(2\pi f_{rf})\cos{(2\pi f_{lo})}... I can describe this equation as the linear superposition of two waves:
s_{if}=A[1+g(t)]\big ( a_1 s_1(t)+a_2 s_2(t) \big ): \qquad a_1 = a_2 = 1/2; \; \; s_1 = \cos{2\pi (f_{rf}+f_{lo})}; \; \; s_2 = \cos{2\pi (f_{rf}-f_{lo})}
i.e. s_{if}=\frac{1}{2}A[1+g(t)]\big ( \cos(2\pi f_0 t)+\cos(2\pi f_{if} t) \big )
This is the IF output expression explained in terms of a linear superposition of two signals.
At this stage no electronics has been used to physically separate these waves - this is entirely a mathematical description of the actual multiplied waveform as a linear superposition. I can use this to predict that I need to design a band-pass filter to exclude f
0 from the IF signal if I want a decent IF to send to demodulation.
iirc: the strategy is to tune f
lo so f
if falls in the region passed by the BPF. The BPF has to be wide enough to pass the desired sideband of the IF signal but narrow enough to exclude the undesired sideband for the entire operating range of the reciever.[2]
This has worked on every circuit I have ever built.
Of course, things are never this ideal - real-life carrier waves are seldom a single frequency for eg. This would be an ideal mixer operating on an ideal signal.
The process by which the IF signal is arrived at in the mixer is non-linear - but the before and after picture is still described by the equations shown. You can see this in the mixer's signal response for eg. It's a bit like how conservation of momentum ignores the fine details of what happens
during a collision - concentrating on the before and after pictures.
I have noticed that the mixing process is often referred in engineering books as "beating" the RF and LO signals.
Accessible example - in the context of RADAR..
-----------------------------------
[1]
http://www.usna.edu/EE/ee354/Handouts/Superhet_Handout.pdf
[2] this is usually considered college level electrical engineering or electro-physics - the cliff-notes version for the non-college student: A classical radio receiver tunes the entire circuit so it's resonant frequency is close to the desired incoming signal: the radio station frequency. The super-het receiver, instead, tunes the incoming signal to something close to the receiver's resonant frequency ... sort-of. The process has application in anything to do with radio - RADAR for eg and radio astronomy. Almost all modern radio communication uses this method, and it is not restricted to AM (that's just the easy case).
). If the math you show here is your interpretation that superposition and mixing are fundamentally the same process, that is certainly your prerogative and we are trapped in an infinite loop! But I would advise not trying to explain (analyze) a superhet stage using addition of the RF and LO signals or you'll be very disapponted in the IF output expression!
