Assume we want to generate a signal in the 12m band (24.890...24.990MHz). We could aim for a frequency of 24.915Mhz. This frequency could easily be synthesized by means of cheap computer crystals as follows:
- 24.915 = 20.000 + 4.915 = 2 * 10.000 + 4.915
use a 20.000MHz crystal in an oscillator and mix this signal with the 4.915MHz signal
use a 10.000MHz crystal in an oscillator, frequency double the generated 10.000MHz, resulting in 20.000MHz and mix that with the 4.915MHz.
There are a couple of ways to double the frequency of a radio frequency signal.
One of the most simple ways is using just simple (fast switching) diodes in a sort of rectifier circuit. A rectifier folds the negative valley of an AC signal to positive. We obtained two positive humps per cycle, meaning that the resulting AC signal has twice the frequency of the original signal. Yes, it as easy as that.
There are other methods, e.g. using XOR digital gates and a phase shifter, please check the internet for other solutions.
As a closing remark, frequency doubler can be cascaded. With the occasional amplification, one can easily build multipliers for factors 2^x, e.g. 4=2*2, 8=2*2*2, 16=2*2*2*2....