This is more a set of comments than an answer per se, but too long to fit in a comment.
Signal from oscillators: 8Vpp 1~20KHz with an offset of ~10V with a new 9V battery.
So the issue is how to couple this to another stage which can amplify it, but at the same time set an appropriate input voltage DC offset suitable to the next stage.
The obvious solution would be to use any amplifier design with reasonably high input impedance, and AC couple to it via a capacitor, so for example a cap from OSC1 to R8.
"The main problem is, on Q1 base, where the signals meet, there's no signal." Whatever voltage signal is at Q1 base will be quite small because the impedance at Q1 base will be small compared to the 1 Meg input resistors. (Especially for frequencies above the knee of the R5-C7 highpass filter.)
So the voltages at Q1 base may well be only 1/100 or 1/1000 of the signals into R8 and R9. In any case what you are more concerned with is the AC currents through R8 and R9 (and thence into Q1-base).
And probably also of concern is the DC voltage at Q1-base -- is it in a sensible range to bias Q1 to operate in it's active range, say with 3 to 4 V DC at Q1 collector? Since you have a 100k collector resistor on Q1, that suggests you are expecting a DC Ic of around 0.03mA to 0.04mA, and thus a DC voltage of rather precisely 0.03V-0.04V across R5 (and not, for example, 0.08V), but there's nothing to set a suitable voltage on Q1-base to make that happen so far as I can see.
Finally, what is the role of C9, 10nF? In parallel with R11 that appears to create a filter that will attenuate output above 160Hz or so, working to considerably suppress the signals in your range of interest, 1 kHz-20kHz.
It's difficult to say anything about what you wrote after "My mission: be able to make its output signal usable" because you don't show a schematic of what your did and it's hard to guess.
FWIW, if you feed an AC audio signal via a capacitor into a voltage follower (which has a high impedance input, hence shouldn't disrupt the source of the signal), you are going to get an output voltage that follows the input voltage. That's assuming you've set the DC level at the follower input to something reasonable. There's not much that can go wrong there, so we need to see exactly what you did that might have cause this to fail.
Bottom line, it looks like your challenge here may be simply understanding how amplifiers work (either op amps or with discrete transistors) and how to satisfy their input requirements for signal voltage or current, impedance, and DC bias (aka offset). Perhaps reading up on that topic might allow you to navigate more satisfactorily?
I got a response from TI about my question. A senior application engineer said that it is not a well written application note and it is not possible to find impedance matching by just using line calculator and smith chart.
The advice was to simulate the circuit on ADS, Microwave or similar software, as the behaviors of components changes by the brand, part model, size, frequency, etc.. The manufacturers provide simulation libraries for their components for the aforementioned SWs.
I used ADS and found two types of simulation to find the correct impedance matching.
One is to simulate the circuit on the schematic view, then you need to define exact component you use, the trace length, trace angle, property of the PCB, etc. According to another application note of TI, it says you can get the correct result about 10% accuracy.
One can get more precise result on the layout simulation due to interaction with the signals from the other components or traces on the board. ADS is a complicated software for a beginner, I could not find a good tutorial how to do it, therefore I simulated on the schematic part and my problem is solved. I don't have network analyzer, therefore I don't know how correct match I got.
Best Answer
As in your description, the FET source follower works to certain extent but with distortion. The source of the distortion is probably not exactly because the signal amplitude is too large, but a secondary effect to that. You mentioned a 12V power supply and 7V signal (assuming the signal is centered), so at the low end, the signal would be at around 2.5V. Most likely, that is too low to turn on the FET adequately.
So maybe give this a try -- bias the base of the source follower FET to around 8V with couple of resistors. Capacitively coupled the input signal to that.
As to matching the impedance, typically, the requirement to match the impedance is because the amplifier is designed to only work at a certain load impedance. That does not seem to apply in what you are doing, so there is no need to match impedance. The amplifier just needs to have enough drive (low enough impedance) to get the signal level that you want.