There are quite a few ways PAS can work. I've got posts with details on some of the ones I've run across or read of in various threads, scattered around the forum; not sure the best way to find them in a search though, so a summary from memory (verify these as they may be wrong or incomplete):
First you have the actual PAS type.
There is cadence, and torque.
Most torque sensors also have a cadence sensor. Some of these are independent outputs, analog for the TS and time pulses for the CS. Some of these are entirely digital, as serial data streams, like the various Canbus types (bafang), or the custom data formats some companies may use.
Cadence-only sensors are usually digital, but not often as serial data streams.
There are multiple types of cadence sensor, which may also apply to the ones integrated iwth TS.
Grin's PAS page has more detailed info, but generally there are types that:
-- simply output a pulse whenever a magnet passes.
--latch to a state when a magnet passes, and unlatch when the next one passes.
--output a pulse whenever a magnet passes *in the correct direction* and no output in the other
--output what amounts to a PWM signal based on the cadence
--output a quadrature signal that encodes both speed and direction (these have two signal wires instead of one)
--most operate from 5v, but some require other voltages.
--some output a standard 0-+V range, some have open-collector output that grounds when active and reuqire a controler-side pullup resistor to +V to complete the signal with an off state (like a typical UVW/ABC motor hall)
Torque sensors vary quite a lot in their design, construction, operation, and output type and range.
--They may output a simple analog voltage range, but even this varies.
--some output a range similar to a throttle, and could, with proper safety features in the system, be connected in place of a throttle.
--some output a range starting around halfway up the voltage range, decreasing or increasing in voltage with applied torque.
--They may require a single voltage to operate, or multiple, and that may be 5v, 10v, or some other voltage.
--scale and voltage range vs torque input varies such that you'd have to refer to the manual for the specific sensor on that.
--some use serial data outputs and are probably only compatibel with the systems they come with
Controllers vary widely in how they implement PAS.
--most only use it as an on/off engage switch (are you pedalling, or not) to activate the assist at whatever the presently-chosen assist level is on the display.
--some may actually read the cadence and provide proportional assist (like the Cycle Analyst can do)
--some may not provide PAS assist except at specific assist levels chosen on the display. When a display is not present they may not respond to PAS at all, or respond only at an unchangeable default level.
--TS-based systems may use a combination of cadence and torque to determine assist amount, or levels in the display, or both, or may ignore the cadence and only use that to be sure you are actaully pedalling and not just resting your feet on the pedals causing a false torque reading.
There's a lot of other possible detail to how they can work....
To troubleshoot many PAS related issues, you have to know how your system is intended to respond to it, and which type of PAS it uses, etc.
others can be tested just like throttles or motor halls.