I've no specific experience with Nakagami-$m$ distributions / Nakagami-$m$ fading channels, so I can only comment on some of the "usual" problems faced in such applications/research:
Say you want to define number and locations of RISes in order to optimize outage probabilities of users.
From that arise a whole range of mathematical problems.
- You define an outage, which is usually some measure of the signal, typically power, reaching the receiver. But things can get pretty complicated there, already: These are by their nature multipath channels. Is there a limit on the delay spread (i.e., path length difference between shortest and longest still-significant-power path) that your receivers can work with? I think I saw hypergeo distributions as approximators for the distributions of delay spreads in discrete-distance multipath channels, but I can't really remember in which models specifically, sorry.
I'll let you ponder the question of
in a random geometry with reflector positions following a specific distribution (assuming you know that), what is the distribution of the shortest path and the longest path fulfilling a maximum path loss constraint, including reflections?
- I think you'll find that if you want to write down the total power received by a user equipment randomly located somewhere in a random scenario with passive reflectors and a base station, as well as deliberately positionable RISes (in order to have an expression dependent on these RIS positions, so you can optimize these), you start cascading integrals:
integral over
- density of possible positions of the emitter times density of positions that have line of sight to that emitter position times their path loss, plus
- integral over
- density of positions of the first reflector times density of the of the positions that can see the first reflector times their path loss, plus
- integral over
- density of positions of second reflectors…
Note that "reflectors" are both naturally reflecting surfaces and your RISes. Keep in mind you can only optimize the positions of the RISes.
The usual case here is that you want to optimize something, so analytic expressions are desirable, but typically unattainable. How can you simplify these expressions to be optimizable? Or even locally convex?
You arrive, either through intense massaging of temples and formulas at an (approximative) optimizable function for a single user communicating at a time. Great! Now do it with multiple users, jointly optimizing their rate under boundary conditions (no added outages through multi-access interferecne). Modern urban scenarios where RISes are most (as far as I can tell!) attractive are interference-limited (among other things); meaning that just bringing more sum power into the receiver doesn't help, because the interference from and for other users in the same and other cells also decreases a user equipments and a base stations SINR. In a cocktail party full of people having to shout to have conversations (because everyone is so loud), introducing high-tech sound-reflective walls might or might not solve the problem. It's not obvious to me at all! What relatively intuitively (to me!) seems to help is if people who want to talk to each other "cluster" and speak silently; that's the equivalent of making cells smaller.
You say, no, screw that, we start with a single user. Fine, reasonable restriction. You notice that users and passive reflectors move.
- What's a RIS positioning strategy that offers "the least likelihood of surprise outage" when the user equipment or a passive reflector move ever so slightly? No problem, only do a couple of "find maximum gradient for every point and bound that, and find the global optimum under that constraint", right? Uff.
- Oh, movement (especially at vehicle velocities) means Doppler effect, which degrades your reception, and the relative positioning of emitter, reflectors and emitters influence how large that Doppler shift. So, more integrals! Yay!
The list goes on. In an urban scenario, networks are dense, and a user equipment typically only connects to a single base station. How to choose the optimum base station, not only w.r.t. currently optimal power reception, but also w.r.t. near-future developments, like users joining or leaving cells, or the user moving?
On a less modelling/stochastic side of things, we have a very significant problem with RISes: for them to be "intelligent", they need some knowledge of what they should reflect, and into which direction, right? That would imply someone needs to tell them about the state of the whole scenario. That would imply transporting a lot of the channel state information both the user equipment and the base station have to the RIS. But the amount of energy used to transport channel state information could have as well been used simply to increase transmission power, reducing outage probability that way. Is your RIS strategy sensible from a net mutual information flow perspective?
Then there's the conflict between deploying millions to billions of RISes (expensive!) or millions of additional femtocell base stations (expensive! But also brings the advantage of immediate network densification; see the cocktail party analogy above). RIS aren't base stations, but need at least partial base-station channel state knowledge, which means that adding a RIS means adding some additional communication load, whereas a base station can keep its channel state information to itself (and the users it talks to directly), and only needs to bring the payload data back to a backhaul network.
