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External GNSS-disciplined rubidium input?
I'm not an expert, but I think the short answer is no.
Forgetting about GPS time-stamping of the IQ samples for the moment, one thing the GPS does is correct for any error in the ADC clock. Improving the characteristics of that clock doesn't really matter. Maybe a clock with an improvement in the short term Allan variance (frequency stability) would be of some use. I don't know. Glenn will have something to say about this since he's done exactly what you're proposing (attaching an external Rb frequency standard).
The Kiwi has no provision for attachment of an external 1 PPS input from a GPS receiver that is specifically designed to be a timing receiver (such a distinction exists). That said, the performance of the positioning (hence timing) of the current code, thanks to Christoph's work, is pretty good. The best way to maximize GPS performance is by using a GPS active timing antenna mounted outside in the clear using good low-loss feed line.
There are many other factors that influence the accuracy and resolution of the TDoA solutions. See Christoph's 2019 SDRA talk for full details: https://www.youtube.com/watch?v=LpFoM_lBgxg
The other important point to realize is that Kiwi TDoA is not a "push button" application. It takes an amount of skill, repetition and luck to get good results, especially at HF. Professional SIGINT personnel go through considerable training to be able to manage the tools they have and interpret the results.
GPS receives lots of satellites, but doesn't get fixes
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Connection lost every day
Yes, question is too short to understand exactly what is being talked about.
But to me it sounds like it's the local (private) ip address that the router is moving around for some reason. E.g. you y initialldiscover it's 192.168.1.101 and make a connection. But then later after trying to reconnect there is no response at that address and you have to go through the rediscovery process again only to find it's now at 192.168.1.107.
This might be caused by the DHCP server in the router configured with a really short "lease time", like minute or hours instead of days or weeks. Ideally the Kiwi's Ethernet MAC address should be bound to a fixed private ip address in the DHCP static ip configuration table. But that takes some effort to setup (that's what I do here). That way the local ip addresses never move.
Make sure this is not the problem before you send him off on the goose chase that is DDNS, lol.