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The signal is decoded from the digital domain to the analog domain by an exact inverse of the encoding algorithm. The sampling rate determines the frequency range available.
Put simply as long as the sampling rate is sufficiently high (44.1KHz in the case of CD, an exact replica of of any waveform in the frequency rage 44.1Khz/2 can be reconstituted. Engineering requirements mean that the actual frequency range is limited to 20-20Khz.
Essentially, between any two samples, there exists one and only one possible shape of waveform in the range 20-20Khz that will pass through both sample points.
So, you do NOT get a "join-the-dots" approximation. It is a smooth waveform, as near as makes no conceivable audio difference to an exact replica of the analog input.
In fact, the output waveform is a much closer copy of the input waveform than can ever be produced by any analog process.
This has been shown and demonstrated many times.
Chris
Sorry Chris but you're still not getting the point. This is not relevant to what I'm saying. Digital's superior accuracy and bandwidth is a given in my post and the OP.
Playback/replay is what I'm talking about. Put very simplistically to make the point - sound generated by a vibrating pin may communicate better to vinyl enthusiasts than sound generated by a dac chip. It is more inaccurate / distorted / whatever you like but it is generated in a similar way to all sounds in nature. Our brains are more familiar with this than digital sound which has really only become the norm over the last century.
This is not a commentary on sound quality but is just a response to the OP as to why vinyl 'sounds' as good as some of us believe. No vinyl / digital agenda on my part.