The FAA says that a DME ground station sends a response exactly 50 microseconds after having received an interrogation from an aircraft. For what reason is there a 50 microsecond delay?
The delay is not 50 milliseconds, but 50 microseconds.
It takes some time for the ground equipment to decode the incoming signal, decide it needs to reply, and fire up its transmitter. In order for the distance measurement to be reliable, this inevitable delay needs to be precisely defined and the same for all DME stations. It also gives the airborne interrogation some minimal time to shut down its transmitter and start listening even when it's right on top of the ground station.
50 µs must have felt like a suitable interval to standardize on, long enough for it to be reasonably possible to engineer the ground stations to meet it (given the technology of the time the design was made), yet short enough that one DME station can service an acceptable number of aircraft without needing to handle overlapping delays between different interrogations.
The standard delay also means that a DME station used for ILS can be located midway between the runway's thresholds and adjusted such that aircraft see the distance to the threshold rather than the distance to the physical antenna, by reducing the delay appropriately. (Such a DME does not give useful readings for aircraft not aligned with the runway).
Note, by the way, that the 50 µs is the time between receiving the second pulse of an interrogation pulse-pair and transmitting the second pulse of the resply. So the ground station actually has to start transmitting the reply less than 50 µs after it knows it has received a complete interrogation. For Y channels where there are 30 µs between the two reply pulses, the ground station actually has only 20 µs to react.
the purpose of the 50 microsecond delay is to eliminate the possibility of uncoordinated operation when the aircraft is very close to the ground station
From AVweb - DME Basics
I have not found a clear explanation for what is meant by "uncoordinated operation" in the context of DME. What follows is speculation:
If you look at patents related to DME, it seems this predictable delay ensured that the initial transmission from the aircraft would have been completely received before the reply was transmitted by the base-station and received at the aircraft.
I believe a DME response has to match the pattern of the DME interrogation so that the receiver in the aircraft can identify which response to apply timing to if multiple DME-equipped aircraft are in range.
The pulse spacing is 12 µs1 and this is the same order of magnitude as the round-trip transmission time at 1 mile. A single DME interrogation contains several pulses. It may be difficult to design ground station equipment that can start transmitting a matching response if it has not yet received the pattern of data it has to reproduce. The alternative would be to start transmitting the response as soon as possible after detecting the end of receipt of a valid DME interrogation.
There are delays in processing within the ground station equipment which may vary (at least from station to station depending on equipment) and which therefore need to be made constant for accurate distances to be calculated by the airborne receiver. Any variations would be more significant at closer distances.
It might also be easier to design airborne equipment if it does not have to start receiving the response before it has finished transmitting the interrogation.
I believe DME was invented around 1944-45 but cannot find any patents that early. There are quite a few patents on this subject.