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Friday, 27 May 2022

Hi-res is absolute necessity

People tend to forget the audio is analogue. Some says that only 44kHz is necessary because of Nyquist theorem. However this only proves that there is one-to-one correspondence between the original discrete data set and its Discrete Fourier Transform only if we eliminate the higher frequencies in the inverse Fourier transform. This does not mean that the discretely sampled data guarantees to be recovered as its original analogue wave function by the conventional audio circuit with low-pass filter.

Simple thought experiment reveals the contradiction in the 44kHz-sufficiency believers. 22kHz wave is annihilated in the sampling by 90-deg phase shifts. 11kHz is also only guaranteed to be sampled at least 1/sqrt(2) by amplitude.

In this article, I simulated DAC generating step function signals followed by an analogue first-order low-pass filter with its cut-off at the Nyquist frequency. The plot shows the average energy of the regenerated output analogue signal compared to the original conceptual sine wave to be sampled.

As I argued before, and some of audiophiles believe, the 24/192 satisfies the 1dB distortion threshold. To satisfy 1% (0.08dB) distortion threshold, we need to go 24/768. Remember that the human hearing function is not that attenuated at the higher frequency.

We can display frequency response of the low-pass filter we applied layered over the plot as orange. However, be careful that you cannot divide the attenuation into the contribution from the filter and the theoretical contribution. If you do a little thought experiment, you can easily realise that the theory already says that the maximum attenuation at fs/4 is -3dB, which is close to the total attenuation in the plot. The additional attenuation comes from the shape smoothing from the step-shaped original wave shape.






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