Figure 1. 14kHz sine wave, Fs = ~infinite
Above 14kHz tone when sampled at the CD rate gives:
Figure 2. 14kHz sine wave, Fs = 44.1k
Sampling at 44.1k creates a beat that is outputted with incorrect upsampling. Most digital playback solutions suffer from this problem. Such beat artifacts is sometimes received positively by listeners. Assuming the DAC chip performs linear interpolation, the above waveform is the resulting output. Ladder DAC chips would result in step changes yielding "stepped" voltage changes as well as beats.
Correct upsampling (i.e. Bandlimited Interpolation) from 44.1k would yield new signal amplitudes at 96k as follows:
Figure 3. 14kHz sine wave, Fs = 96k
Beat artifact is significantly reduced. Upsampling further to 192k:
Figure 4. 14kHz sine wave, Fs = 192k
Beat almost diminishes.
The thin green & red lines running across the graphs represent the same 14kHz waveform at -90dbfs (green) and a simulated noise floor at -110dbfs. This becomes visible when we zoom in by a factor of 20000!
Figure 5. 14kHz sine wave, Fs = 192k, Scale [-0.00005, 0.00005]
Resolution loss in digital volume control is largely related to legacy implementation issues (insufficient math precision). Above calculation uses high precision math (64bit float gives 53bit volume resolution)! Signal details between red lines (the DAC chips noise floor) becomes unpredictable and resulting output will vary from ideal. Analogue VCs may offer an advantage when it's SNR exceeds the DAC's SNR. This allows for the "noise portion" to be scaled as well.
Correct upsampling is not only about removing beat artifacts but more importantly its about recreating the original analogue waveform. Recreating a pure high frequency sine wave has been very useful in illustrating this.
Audio solutions based on SHARC 32 bit DSPs would hopelessly fail to achieve such levels of perfection. Not only is correct upsampling computationally intensive, it also requires significant memory bandwidth and much greater bit precision. Fortunately, this is available in ample abundance through modern (and cheap) computing technologies.
