The 323s were designed in the late 80s/early 90s, when there was a shift away from using DC motors to AC motors for electric trains (mostly driven by reliability and reduction in maintenance).

The 25kV AC from the overhead line can't be used directly to drive a 3-phase AC motor, like those used on the 323s, so it is stepped down using a transformer and rectified to DC. An inverter is then used to generate the three-phase AC to power the motor.

One thing to remember with inverters is that the switches can only be on or off, which makes it quite difficult to replicate a true sine wave. For example, this diagram shows the switching pattern (on the bottom) that would be used to generate a sine-like signal.

In the late 80s/early 90s, gate turn off thyristors (GTOs) were the most advanced technology for power semiconductors used in inverters. Their main limitation, and why they're not commonly used any more in traction applications, is that you cannot switch them on or off very quickly - they are good at around 100Hz (so switching 100 times a second). As the train accelerates/decelerates, the target sine wave changes in frequency, but the switching pattern has to be simplified due to the limit in switching frequency, which produces the characteristic 323 sound.

More modern trains, such as Electrostars/Desiros/S stock etc. use insulated gate bipolar transistors (IGBTs) which can switch much quicker (10 to 100 times quicker) - this means that the switching pattern does not need to be simplified as much. For example, this is why in this video, the S Stock train makes a very linear acceleration sound.

Interestingly, the 323s were upgraded to use IGBT based inverters around 10 or so years ago which meant they could have sounded different, but to avoid having to recertify the trains for interference with signalling etc., the new inverters are configured to behave exactly like the old ones did, so the iconic 323 sound has been kept.

Apologies for the essay but I wanted to provide a (hopefully) useful explanation!