Hi again,
Yes that makes sense and that's a reasonable question. You're asking if we are given the required signal, can we find a simpler way to generate it.
What i was trying to show was that with somewhat simple circuits there are various ways to get the desired signal, but some techniques just wont work simply because they wont produce the right signal. That's usually assuming we dont have control over the receiver, but sometimes we do so we might find other techniques if that end is open to us too, but there are other considerations apart from just generating a signal that we can transmit...namely, the power bandwidth where we want to use as much power as we can without wasting any.
So assuming we dont have any control over the receiver design, we have to generate a signal cos(w1*t+w2*t), and just to note that is also not the same as cos(avg(w1*t)+avg(w2*t)) which would result from averaging the two frequencies over time before creating the signal.
The most common way to do this is to multiply the two signals (not the frequencies per se') and filter out the lower side band. Is there any other way to do this?
The usual method comes from multiplying the two signals and then noting that we get both upper and lower sidebands when we use trigonometric identities to expand or reduce the result. We get one result we want and one we dont want, so we filter out the one we dont want.
But what if we work backwards?
If we start from the signal we want:
sig=cos(w1*t+w2*t)
and use trig identities to expand or reduce we find we can get an equivalent:
sig=cos(t*w1)*cos(t*w2)-sin(t*w1)*sin(t*w2)
And what exactly does this expression tell us? Well, the right term is the original modulation where the two signals are multiplied, but the left term is a little different...it's the two signals shifted by 90 degrees each and THEN multiplied, and of course the subtraction is just the analog subtraction of the two signals.
This basically means that if we shift the phase of the two signals and use these new signals in a modulator, then subtract the modulation of the two signals without phase shift we end up with the upper side band, which is the signal we were after, and we didnt use any filters.
How well would this work in practice? We'd first have to design a circuit to shift the phase of the message signal by a constant 90 degrees.
We'd then have to look at the effects of errors in the modulation or phase shifting circuit and see how they affect the result.
In any case, that's probably the only way get what you are after without using a digital technique which would probably be cheaper these days.
ADDED LATER:
I did a quick search on modulation techniques and found that the technique described above is known and has been used for quite some time already, since the days of vacuum tubes. It's known as "Hartley Modulation" and also has it's advantages and disadvantages.
From this we might note that there is no really simple way to do this except for a purely digital technique when that technology is allowed.