If you're just starting out a recommended DIY build for such a "monopole" antenna would be something like in this Ground-Plane antenna for 2,4 Ghz photograph. (Don't miss the links to http://www.jensenjensen.com/radio/GP24antenna.htm and http://www.csgnetwork.com/antennagpcalc.html which may be helpful as well.)
I know that the antenna needs to be a multiple of 3.1cm in order to have the maximum signal strength, and have also looked into having ground plane radials (which I don't completely understand the point of)
The size will be somewhere in that ballpark but you may need to trim it a bit to get the best match. This will be a difficult to do unless you have tools to measure how well the antenna is absorbing power after you make it.
The ground "plane" (radials in this case) sort of has two purposes:
For one it helps "complete the circuit" of the antenna. Of course an antenna isn't exactly a "circuit" since current has to just displace in and out rather than going through/around in the "electric circuit" sense, so more accurately the ground radials help balance the antenna. Meaning, when the center conductor of the coax/connector is pushing electrons into the monopole the outer conductor wants something to pull from to keep things even overall.
The other purpose is to "hide" the antenna from being interfered with by some of its surroundings, in your case the cable itself. See https://practicalantennas.com/designs/verticals/gp2/ for some more discussion but it's often easier to use a monopole plus four more radials than to build a dipole (monopole plus another monopole) in the first place simply because it means you don't have to worry about keeping the feeder coax out of the way.
However, one thing I don't understand at all is matching the impedance. I read online that I need to match the antenna impedance to the rest of the system, but I have no idea where to even start. Online it states that SMA connectors have 50 ohm impedance, however, it also states impedance is measured in an imaginary reactance component and a real resistance component.
So 50 Ω impedance of the SMA connector or any coax cable connected is only its "characteristic" impedance. It helps to have it correct but ultimately your transmitter will end up only caring that the impedance of the antenna alone is correct.
Impedance sort of means "how does it absorb energy". A resistive ("real") impedance means the energy simply "goes away". The energy leaves the circuit, either as heat or as radio waves. But there's also reactive impedance, which is what the "imaginary" part of the math is used for. This reactive part represents energy going into a capacitor or inductor. That energy is absorbed too, but it's not really gone — it will bounce back into the circuit sooner or later.
Lastly, how do I go about changing the impedance of the monopole to match the system.
You're trying to get the feedpoint to behave approximately as a 50 Ω resistive impedance, across as much of the frequency range needed. For a quarter-wave ground plane the biggest adjustments will be:
- the length of the main "monopole" element (so it's not acting like much of an inductor or capacitor at the center frequency)
- the angle of the radials (bending them down tends to lower the overall impedance)
- the length of the radials (honestly I'm not sure how much of an effect this would/wouldn't have)
But you'll need some sort of way to measure if the changes you make are working or not. This would be an antenna analyzer of some sort or another: an SWR meter or a VNA or a dip meter or…. The received signal strength at a certain distance might sort of provide insight but won't directly or definitively tell you if you have a good match.