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That's right!

But as any technician can tell you, there's a huge difference between being 'all digital' and 'digitally-controlled'!

I play in a band and have a variety of synthesizers. Most are entirely analog but a few are digitally-controlled. However despite the digital control they are still considered analog synths because the oscillators and filters and such are still analog despite the digital control.

Much the same is going on with class D amps. As I've mentioned several times, the switching means that class D uses lends itself quite well to use with a digital input (with no analog input). But to get the output transistors to switch properly from the bit stream of the digital source there is an analog conversion system employed. Further, the usual considerations of power supply voltage, the risetime of the output devices and how long it takes for them to shut off, plus the stripping of the switching frequency from the signal at the output of the amp are all concerns in the analog domain.

Its easy to understand how a person might think that such an amplifier is a 'digital amp' but that moniker is erroneous as that is not how the amp works.

I think your confusion here comes from the fact that analog processes cover a lot more ground than just a simple audio signal! For example if there is a reflection on a transmission line you have to use analog techniques to tame it- if you don't, the digital system in which the transmission line (an analog device in itself BTW) resides will generate errors.

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