Sensor to determine approximate location of pet inside?

Hello all and thank you in advance for any assistance.

I am trying to determine if a pet is getting closer or farther from an Arduino Uno. This would be inside the home so the sensor would need to work through walls and objects. The accuracy is not so important as I only need to know if the arduino is getting closer or farther. I was considering a microwave doppler radar, however, it seems to only really determine speed but I may be overlooking something. Otherwise I have considered creating an RF collar for the pet that outputs a pulse and then the UNO would calculate how far between the pulses to determine if it is moving in the right direction. Any suggestions would be extremely appreciated.

Look up "indoor localization", which is an extremely difficult problem.

How about via sound detection? Arduino yells out Marco, pet calls out Polo.
In this case, RF signal to shock collar, home in on the resulting "yelp!"

Infrared transmitter attached to the dog collar.
Infrared receiver in every room.

The issue there is I need the sensors to be centrally located, having beacons would increase the cost exponentially and would increase effort on user's part for setup.

Doppler radar will only work if you have continuous signal reception which in a typical house is not possible due to absorption, reflection and interference.