Homemade GPS Receiver

You can certainly ignore both limits in your homemade-receivers, but if you ever want to ride faster than 1900km/h you will have to account for an increased doppler shift (I hope I'll not help any terrorists by explaining that ;-) ), and even though I (in the end) realized that this is not as dramatic as thought at first, just let me write down some calculations:

Doppler shift is the change of the percieved frequency when either source or receiver move towards another, or away from another. Canonical example: Sound of a emergency vehicle's horn when passing by.

Normally you never see the satellites ride directly towards you (or away from you) but the path is pretty orthogonal to your line-of-sight. So you only account for a fraction of the 14'000 km/h speed of the GPS satellites to cause frequency shifts of the signal. (Max. example listed on the homepade-GPS page is ±5 KHz of doppler, which is relatively 3.3×10^−6 of the GPS frequency, even though the GPS satellite rides at 1.8×10^−5 the speed of light).

If you'd fly with 1900 km/s (1.8×10^−6 of the speed of light) directly towards the GPS bird, this movement will increase the frequency you receive by 1.8×10^−6 × 1,5 GHz = 2.7 kHz. So please increase your doppler search range from ±5 KHz to ±8 KHz to be able to successfully steer your homemade cruise missiles at that the COCOM limit speed, increase more if faster. This will slow down your time-to-first-fix, obviously.

The other limit of 10km is easier to ignore, though. It would only become relevant once you can no longer exclude some solutions to the formulae that lie outside of the GPS satellite's orbit (imagine four coplanar GPS birds used for calculation). And in practice proabably it's sane to assume a position that is not wider than 100km from the earth's surface, again, if you don't plan to ride your DIY receiver to the ISS, or maybe even geostationary orbit.

Another great project in the area of DIY GPS is http://gnss-sdr.org/project It implements a GNSS/GPS receiver using software defined radio (SDR). They also support the RTL2832U based SDR (http://www.rtlsdr.org/), which is a cheap way to get started.

These paragraphs are very deep, and quite revelatory as to the nature of relativity and the Universe itself:

The speed of light is strange in that it has the same value independent of the relative velocity between the source and the observer. This fact is an experimental one that can only make sense if relative motion changes the relationship between space and time intervals to keep the distance covered by light per unit time the same for all observers.

The fact that space and time must get mixed up to keep the speed of light constant implies that, in some sense, space and time must be the same, despite our habit of measuring space in meters and time in seconds. But if time and space are similar to the extent that they can be converted one into the other, then one needs some quantity to convert the units--namely, something measured in meters per second that can be used to multiply seconds of time to get meters of space. That something, the universal conversion factor, is the speed of light. The reason that it is limited is simply the fact that a finite amount of space is equivalent to a finite amount of time.

You're right, that is by far the more interesting part of the system. Thanks for elaborating, that's really cool.