As far as I know, there is no standard for connecting a WiFi controller to your computer, so special programs must be developed for this.
This special program now also exists, so it is possible.
However, anyone with a ballpark console would be wise to download and install this update.
There is a good chance that the WiFi functionality will not work because the servers required for the connection are no longer working. If the bluetooth update is on then at least you won’t have that hassle and you’ll get a nice bluetooth controller that will probably work until the thing actually breaks.
As for the real reason why WiFi and Bluetooth can’t be used together at the same time: It’s technically possible, but in practice, WiFi and Bluetooth are on the same bandwidth and also work through the same chip. Attempting to use them at the same time may cause radio interference and congestion in the circuits connected to the console processor.
This in turn leads to input lag and this kills some games.
The input lag of a bluetooth connection, for example, ensures that it is impossible for the PS4 to send two commands that happen less than roughly 125ms apart, making it absolutely impossible to communicate more than 8 inputs per second for a game wirelessly on the PS4.
This is relevant, for example, for people playing Tetris Effect in main mode, because blocks after level M15 are already fixed after less than 100ms. The real big gamers then connect a PS4 controller via a USB connection (only possible with V2 controllers btw, V1s don’t support that) so that the delay caused by the bluetooth radio protocol is overcome.
Imagine what happens if the chip in this controller, in addition to all the noise and Bluetooth retries, also has to handle all the signals that come over WiFi and then has to choose. This is just shoddy play.
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