How to determine the LO frequency or the rf frequency? It's hard to say.

My manager told me a task as follows: if the rf frequency is 4300MHz and the IF frequency is 300MHz, then the LO frequency is sweeping from 3900MHz to 4800MHz, that would get the IF for twice. In other words, the first time is (4300MHz-4000MHz)=300MHz and the second time is (-4300MHz+4600MHz)=300MHz. If I got the IF, then how could I tell the LO frequency? or How could I get the only once IF? He didn't allow me to use any filters, 'cause he say the phase of the LOs in twice IF is out of phase in that way which are not the same therefore could be recognized without filters.

You need an image reject mixer. that has two mixers, quadrature LO signals and some 90 degr. phase shifters and summers.

Search for image reject mixer and you will find some block diagrams.

Yes, a mixer can use either a high side LO or a low side LO to generate your 300 MHz IF. If you are using modulation such as QPSK, FSK, or higher level modulation, the modulation will be "inverted".

If you are trying to figure out if it is a high side or low side LO, you will need to do a test. You could sniff the front end with a spectrum analyzer and see if either 4000 or 4600 MHz is leaking out.

Or You could put in a cw tone, and see what IF is coming out: put in 4000 mhz and then tune to 4001 MHz. If the IF is 300 MHz and tunes to 301, you have a lowside LO. If the IF is 300 MHz and tunes to 299 MHz, then you have a highside LO.