TD and FD cannot be decoupled, you're right. They are related by Fourier transform. But the relationship between them for minor deviations from perfect implementations, and that is what we are all looking at in audio with sought accuracies of 30ppm or so with thd plots for example, is not at all simple.
Not simple, but not impossible. The invocation of the uncertainty principle isn't appropriate here since we are not looking at an infinitely small sinc or a very long time series. For example, DeltaWave detects timing differences in a real audio waveform down to around 1/100 of a ppm (or 10 ppb) in a 44.1k recording. And it does it in ... the
Frequency Domain. If you are going to tell me that 10 ppb difference is audible, then I would really like you to run some well controlled ABX tests, as this will make for some sensational published research.
The proof of the pudding will be a representative simulation and/or a time aligned reconstruction of the same sound file made with different sinc based reconstruction filters compared in analogue.
Run some matlab models and let's see these large or audible results. It's easy to convolve a digital recording with any sinc based filter. Define your filter, create the convolved files and let's compare them in DeltaWave to the original source file. The differences can be viewed and measured in time and frequency domains. For audibility, run some ABX tests.
Here's an example of a DeltaWave built-in low pass FIR filter (16k taps) being applied to a 44.1k recording, compared to the original:
The interesting parts here are -98dB and -136dBA RMS difference (in time domain) and even better numbers for correlated null depth (also measured in the time domain).
You'll also notice that 99% of samples in the two files match to better than -101dB with and without the filter. While I'm sure it's possible to come up with a really poor reconstruction filter, I don't think any reasonable one would produce a -50dB error.
The filter in this example is FIR, constructed from a Kaiser-windowed sinc function:
And to see the phase performance of the FIR filter, here's the phase difference plot between filtered and unfiltered waveforms: