Upfront warning: amplifying the output of your wifi device is almost certainly illegal. There's strict legal limits on how much power you might output or how much power density you might emit in a specific direction.
I feel like the simplicity of the idea suggests that I am missing something. I think that amplifier chip could only give me a single signal path, but adding a second amplifier with a pair of multiplexors on the antennas would give me the second direction - again, is that correct?
WiFi is a single-frequency system, meaning that packets for both direction are modulated onto the same frequency. Thus, Wifi chips usually only have one antenna port, which switches between receiving and transmitting operation. That means you'll have one unecessary antenna if you build two ;)
Problematic is that of course you mustn't operate the receive amplifier (LNA) simultaneuosly to the transmit amplifier (PA) if one's output is actually connected to the other's input. Some WiFi chipsets do expose a switch signal automatically to signal TX or RX operation, which you can to switch between these modes.
But that's where it gets a bit ugly: Imagine you have a measurement device close to your antenna. A normal WiFi chipset in Europe might emit up to 20 dBm (100 mW) of power. The receiver is still specified to work with powers in the range of -90 dBm (1 pW). That's a difference of 110 dBm, or a factor of 10¹¹ between transmit and receive power.
A good RF switch has an isolation of say 85 dBm, and components on the same PCB, using the same power supply, rarely get more than 60 dBm attenuation to begin with unless you really start designing things for microwave isolation – with all the nice things like milled enclosures etc. If you're doing that for a single instance of a device, that is a rather expensive hobby. Not to mention that I wouldn't assume my first attempt at a design would work well enough, so that I'd presume I need a second spin...
So, cross-talk will likely kill your application's feasibility, unless you really switch the amplifiers off and on alongside with the RF switch. That sounds easy enough, but comes with a whole can of worms because there's some transient behaviour to be expected the moment you switch on an amplifier.
From a technical point of view: if you're just planning to build a point-to-point link with fixed endpoints, go for high-gain antennas instead of amplifiers. Your dipole would radiate nearly all of the energy in directions where the other end isn't, and only look like a very small "target" for the other end's antenna beam.
There's companies that sell Wifi-directive link equipment.
Re: your three links:
- the arduino thing (the fact that it incorporates an Arduino is really just a sideshow) is a feed-forward amplifier controller that bases amplification on an RSSI estimate. That means you really would only want to use this on a single-purpose band, where there's really only one transmitter allowed (the cell tower), because imagine using that in the 2.4 GHz band, where there can be Bluetooth, arbitrary Wifis, microwave ovens, Zigbee... that all change the receive power estimate. It actually gets pretty hard doing that right if the transmissions are not continuous – and wifi is an extremely bursty system. The thing it was designed for seems to be 832 MHz, so probably US-American GSM; GSM has a continuously transmitted control channel which one could use to build such a controller. A modern phone does about the same as that controller, but knows the signal better – I'd expect the advantage of this system to be marginal compared to attaching a phone to a better antenna directly. Oh, and the total illegality… Oh, and the fact that the power control in the phone and in the base station make the adjustability totally unnecessary, since these devices will reduce their own gain by themselves, so fixing the gain of the VGA shouldn't have any negative effect (on the contrary, at maximum gain it should have the least Noise Figure). Anyway, inapplicable to wifi for burstiness and same-frequency reasons.
- A LimeSDR is a complete Software Defined Radio. It's really a bit of overkill to use it as a "RF bent pipe", but sure, it works, probably (the signal quality on LimeSDR does have some trouble if you don't have nice external filters). Having these filters, one could actually separately amplify the up- and downlink channels separately – no need for the expensive and very complex central piece, the LimeSDR. This system is a demo of a digital signal loopback in the FPGA, not very simple or overly useful for the application. Again, this can't work with WiFi, since WiFi doesn't have separate up- and downlink channels.
- Your antenna basics document is a very high-level one, and has but two formulas – it's really very basic.
All in all: instructables is not a source for RF electronics, really. Before building something like that, I'd recommend getting a bit of RF theory right (the antenna basics document doesn't do any of that, it treats antennas as black boxes with parameters), and transmission line theory.