RFICDUDE
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Unfortunately you have a large task ahead of you because in addition to finding out the IC impedance, antenna impedance and designing matching networks (if needed) you also need to make sure the transmission lines for your PCB design are the correct impedance.
All the tasks can be done, but it will be quite a learning experience for you if you are responsible for making sure it all works correctly. It is good practical experience, but it will be a challenge if you don't have equipment and/or simulation tools to help you through the design and evaluation phase.
Unfortunately, I don't have a good suggestion for a cheap VNA; although, your plan for renting one is reasonable.
More importantly you need 1. to find out from TI what the IC impedance is over frequency
2. figure out how to design a 50 ohm microstrip transmission line using the PCB stackup of your vendor
3. find out the impedance of the baluns and other components (switches) you plan to use
4. develop matching networks if necessary to keep the system port impedances matched
5. write down a measurement test plan of how you are going to verify the matches are as expected using you PCB. You may find you want to design you board to make it easier to conduct your measurement plan. For instance, you may want a few test structures like transmission lines of various lengths that you can measure separately from the IC circuit. This would help you verify that your transmission lines impedances are close to the design targets.
6. if matching networks are necessary then you will need some sample kits of inductors and capacitors to use to tweak the match designs after assembly to optimize performance.
Of course you'll need to learn the basics of impedance matching and microstrip transmission line design. All of this is classic information, so you should be able to pick it up and apply it fairly quickly.
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