All power amplifiers share something in common: they are big valves connecting the amp's power supply to the loudspeaker. How they do this is really all that differs between topologies in output stages: traditional amps use a variable valve that never reaches the maximum supply and class D amps use an on/off valve that always reaches the maximum supply. The former is very inefficient and the latter extremely efficient. The end result is pretty much the same but the importance of the power supply is far more critical in the class D than the traditional amplifier.
All power amplifiers have a big DC power supply. This supply could be from a bank of batteries but most practical amplifiers use an AC to DC converter and storage system to supply the needed energy to move the loudspeaker and make music. Regardless of how the DC voltage is generated and stored, the whole goal of a power amp is to connect the energy in the power supply directly to the speaker in just the right proportions to make music.
Because a traditional amplifier output stage never connects the power supply directly to the loudspeaker (if it did it would be in clipping mode) the quality of the power supply DC is less important than in the amp we're talking about. Also, traditional amplifiers typically use feedback to make up for power supply differences and, class D amps do as well, but we'll soon understand how these differ and what is going on.
Just to be clear, however, power amplifiers in general are very sensitive to power supply quality and most have the worst power supplies of anything in your audio chain. I don't want to minimize the importance of the power supply in either topology of power amplifier - but I do want to make the point that Class D amps are far more sensitive to their power supplies than class A/B amps when it comes to performance in sound quality.
Let's start the discussion by reviewing what the output stage of a class D amp looks like and what it does. It's actually quite simple - just an electronic switch that is either on or off - when it is on, the entire energy and voltage in the power supply is connected directly to the loudspeaker (actually through a filter which we'll cover later) and when off, completely disconnected. The longer it is on, the more energy is transferred to the loudspeaker. Remember that this on and off switch is taking place at very high rates of change - typically one hundred thousand times per second - and just like a movie works, if you do it fast enough you can't tell it's not continuously flowing. In fact, to the loudspeaker, it is indeed a continuous analog flow.
But now imagine that the power supply isn't perfect - that it is varying in voltage and the DC you want isn't as clean as what we get out of a battery. When you connect this less-than-optimal power supply directly to the loudspeaker you aren't getting what you expect and that, by definition, is distortion. As I mentioned, this is true for all types of power amps but even more critical to the class D amps because the full power supply is connected directly to the speaker - and whatever ripple, noise, trouble and less-than-optimal power is on the supply is passed straight to the speaker.
Well designed class D controller eliminate many of these issues with a complex feedback solution that varies the amount of on time of each pulse to compensate for power supply issues, but as we know, the less "turd polishing" one has to do to get it right, the better the design sounds.