El Nino said:
“No... there's only one set of filter coefficients... at 48kHz, the stopband will begin at 0.54fs, i.e., 26.2kHz. Just as "lazy" in the sense of the word you're describing. Except it isn't lazy... it's an intentional design choice. At 44.1 and 48kHz fs, there is no possible digital filter that simultaneously provides flat frequency response to 20kHz, linear phase, fs/2 stopband, and good time domain performance. Filter designers have to weigh the options and pick a middle ground. This is why the very expensive AK4499 also doesn't include a sinc filter with fs/2 stopband, even though it would be exceptionally easy for the AKM engineers to include.
There's a lot of misinformation about digital filters on this forum unfortunately.”
Agreed. Fortunately, there is a trove of correct information about digital filters available. My personal favorite is Multi-Rate Signal Processing for Communications, fredrick j. harris (SIC), Wiley. I’m sure that there must be articles, more accessible than a textbook, that can be found on the internet to cover this same information. I apologize for the fact that the following is not easily accessible to one not skilled in the art; I offer it in the hope that perhaps someone who is might generously take some time to make this accessible to more members.
There is clearly a significant number of people on this forum who have mastered DSP of sampled data. I hope some of these will take the time to explain that the ideal filter is a brick wall with the wall located at exactly Fs/2 which produces a sinc function in the time domain (TD) with nulls at every other sampling point. If the brick wall is at any other frequency than Fs/2, the other samples will not lie at nulls and we would have inter-symbol-interference (ISI) which will manifest as distortion. Any TD waveform that has zeros at all multiples of Fs other than zero could also be used but the sinc is easiest to describe and would be most familiar. Of course, a brick-wall filter is non-realizable.
In practice, one can take a relatively narrower frequency domain (FD) pulse of width alpha/ Fs/2 and convolve it with the desired theoretical brick wall FD filter response and produce an ideal trapezoidal pulse in the FD. Since convolution in one domain is identical to multiplication in the other domain, the zeros of the sinc function will still be present, and no ISI would be introduced. In the Linear TD, the fact that we can only approximate the sharp transitions of a trapezoid results in slight rounding to a sigmoid shape in the FD. Other shapes are possible that depend on the FD waveform convolved with the Brick-wall filter waveform. I am sorry that I cannot take the time to show plots. I am attempting to describe a classical Nyquist filter for Fs/2. It would have its response at Fs/2 exactly 6-dB down from the peak. The pass-band ripple requirements (low to prevent TD echoes), transition width and stop band attenuation (say 80 dB) desired will then determine the number of taps necessary to implement a Nyquist FIR filter with symmetrical coefficients resulting in linear phase vs. frequency leading to the constant group-delay (GD) necessary for 0 distortion (at least over the signal bandwidth). I focus on over-sampling DACs because the analog output filter, which in an NOS DAC would be entirely responsible for the waveform reconstruction, which involves filtering out the replicants. No analog filter can have perfectly linear phase, but with a higher sampling rate, the replicants are so much farther away (176.4 for 4x oversampling) that the filter transition zone can be much larger which allows it to be simpler and to have much more linear phase over the signal bandwidth.
The large number of taps in the digital FIR may result in an amount of GD that may render a particular filter unsuitable for applications where audio must be precisely time-aligned with video. Perhaps short FIRs, albeit with compromised performance, may be perfectly adequate for such applications.
All the DAC chip manufactures are well-aware of this and as far as I knew until El Nino’s statement above, all over-sampling DAC chips provide at least one filter that at least approximates the classical. Then regardless of the FD shape of the signal they convolve with the FD brick-wall filter, the ISI will be zero, but the filter may be down somewhat more than 6 dB at Fs/2. Variations in the shape of the narrower filter can result in the amplitude being 6 to 12 dB down at Fs/2 and the
In the conversion from say 24/96 recording to redbook CD, the desired data is strictly band-limited 20 KHz. Here, a band-limited signal It is sufficiently attenuated above that frequency so as not to produce any significant images. Since the sampled data inherently has replicants at all integer multiples of Fs, then replicants exist at +/- 44.1 KHz. The band-limited signal is negligible below 24.1 KHz. The filter transition zone is from 20 to 24.1 KHz and therefore all the DAC chip manufacturers at least attempt to have the filter reach full attenuation by 24.1 KHz. I’m sorry that I cannot take time to develop tradeoffs of other filter parameters mentioned vs. the distortion introduced and especially the audibility thereof.
I believe that the DAC chip manufacturers are neither lazy nor misinformed; quite the contrary, in that they provide what the above discussion finds as the proper filter design for sampled data. Incidentally, the exact same filter taps designed for 44.1 KHz data would serve perfectly well for 48 KHz data as long as it is clocked at the correct sample rate.
If I had to reduce all the above to a single conclusion, assuming that redbook CDs or their rips are of interest, I might state:
“If you want an oversampling DAC, look for a filter response that has strong rejection at 24.1 KHz and above and is down by 6-12 dB at 22.05 KHz with a top that appears flat to 20 KHz and always look critically for distortion.”
While there is so much more detail that I don’t have time to provide, e.g., compensation for the DAC’s zero-order hold (ZOH) effects on the associated analog output filter, etc., I hope there is a sufficient skeleton here for others to augment what is provided above. I apologize for not making this more accessible and apologize in advance for not being in position to respond to questions. This is my first, and probably last post. Please feel free to flesh it out so that those members so inclined may have better access to the correct information, at least at a high level, that they need to make informed decisions.
LT1