Question: I want to understand what is the purpose of doing localization while an aircraft is flying? What sensors are used by the instruments that do this localization of an aircraft? Is there more than one way to do the localization? Can we do localization without using a GPS? How is magnetic heading used to do localization?

Answer: 
By first principles, one simply expects the pressure difference to be thrust $T$ divided by disc area $\pi r^2$. Due to the fact that the propeller "sucks" ahead and "blows" behind itself, we'd expect to see about half of this pressure difference as a pressure drop ahead of the propeller and the other half as a pressure increase behind the propeller. The effective compression ratio $n$ in ambient pressure $p\_0$ is then

$$ n=\frac{p\_0+T/(2\pi r^2)}{p\_0 - T/(2\pi r^2)}$$

To see what thrust is created for a given amount of power, you can use [this excellent answer](https://aviation.stackexchange.com/a/8822/4108) by Peter Kämpf.
Welcome.

<https://en.wikipedia.org/wiki/Pratt_%26_Whitney_Canada_PT6>

From Wikipedia, the compression ratio of the PT6 varies between 7 and 10.8, depending on the version. Please note the trend between compression ratio and engine power, due to this, I think the Caravan's engine (-114A) has a compression ratio of 7. The Malibu's engine (-42A) has a compression ratio of 8.
In off-design conditions (part-power, low RPM), the PT6 has a pre-swirl system that uses high pressure bleed air to change the direction of inlet air, thus widening the operational range of the engine. When compressed air is bled back to the inlet, the compression ratio obviously drops.