Earlier quoted context omitted.
Hi! Yes, "from scratch" is definitely always a bit of a funny term. I also implemented the receiver in Python, which is quite far from "scratch" =). What I mean by it in this context is that I'm taking a piece of hardware that knows nothing about GPS, and just has the ability to sample the EM field, and building up a receiver from there. Re. slow TTFF, or time-to-first-(position)-fix on older hardware, this essential…
The fast TTFF on a typical modern device e.g. your phone is because the device has the Internet, and so it can obtain all the information it needs from the Internet up front, it isn't magically brute forcing everything needed, that's not practical at all. The 12.5 minutes includes a rough multi-week almanac which you could perhaps brute force given available compute and receive capability (original GPS receivers have…
In this case, I was meaning to refer to brute-forcing the Doppler-shifts and PRN phases of each satellite, not the orbital parameters themselves. The project in the OP is able to get a position fix in less than a minute because, if the subframe timings are convenient, you can retrieve the necessary ephemeris parameters from the subframes in that span (and down to as little as 18 seconds in ideal conditions, if my back-of-the-napkin is right).