Electronic – How are severed phone lines, the ones on utility poles, repaired when they are severed

You claim to be familiar with the Fourier analyses.

Take a look at this signal:

Here we have a sinusoidal signal summed together with a DC signal. The AC signal has voltage of 1 V peak to peak, while the dc component is 2 V. For each point on the graph, what happened is that the value of the cosinusoidal signal was summed with the 2 V DC level.

If we do Fourier transform of the signal, we'd get a value at the frequency of 0, which represents the DC component and the value at the frequency of the AC signal itself.

A "pure" AC signal goes from Vpeak to -Vpeak and is centered around zero. If the signal's average value is not zero, then it has "DC bias", that is to say that it can be represented as a sum of a DC signal and an AC signal that is centered around zero.

Next, a bit about phone lines and its cabling: Traditional POTS uses only two lines, one of which is ground and the other is traditionally said to be -48 V DC. The negative voltage then summed with the actual signal which carries the data.

ADSL "basically" works same way as fax modems and voice calls: The -48 V DC is summed together with the AC signal which carries the data. At both telephone central office and at subscriber's location filters are used to separate the voice traffic from ADSL signalling. This is possible since ADSL and telephone work in different frequency bands, so they do not overlap. You can then "simply" filter out different bands and send them to different devices.

EDIT: The way right to think about this is that in reality there's no "DC" and "AC" voltage (in fact, terms DC and AC aren't really all that well defined and the conventions depend on region). Those terms are simply abstractions that are used to simplify circuit analyses.

Let's say that you have grounded line G and line with voltage A. Now imagine that it's possible to measure voltage of the A line extremely quickly and that you associate one voltage measurement with one point on time axis. You'd get something that looks like this:

Each measurement point tells you "exact" voltage at the time of the measurement. You don't get two lines because there's only one actual voltage and its the voltage that's shown.

The extra pulses will simply be absorbed, except for the final digit which will be lost.

The 1st, 2nd etc. group selectors only accept a single digit, then hunt into the bank to find a free outlet to the next group. Once the carriage steps up to the top level it can't go any higher, so it just hammers into the backstop.

Final selectors step up for the penultimate digit, then step into the bank for the final digit. An extra pulse in the final digit will cause the carriage to step all the way around the bank and out the other side, where it will 'drop out' back to level 0 and start stepping up again.

Another selector type was the Discriminating Selector Repeater (DSR) which was designed to drop out when reaching certain levels, in order to absorb digits that weren't need in the local exchange. This monster had 29 relays, some with 2 or 3 independent coils. Here's a schematic - see if you can figure out how it works!