Interpolation is a guess. It may or may not be right and in fact can be way off. The mathematical approach noted in my post is exact. The hardware implementation of the mathematical approach Chord uses is not exact but is much, much more accurate than interpolation.
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Yes but within the 20-20K frequency you care about which is what I meant by within a bandwidth limited signal, it is correct per the mathematics. And as noted when you move to a real implementation that cannot implement the actual mathematical approach exactly, you do of course have tradeoffs. But my point was in fact missing information can be created. That is a fact.
Any mathematical approach is just a way to interpolate between two data points or a series of data points. You canāt mathematically calculate precisely something that is not there. It is just a guess. But, it might sound better than a hole.
Yes. You can. Within a bandwidth limited signal, mathematically you can absolutely do this. That is the point. You are stating what you believe without actually taking time to study the mathematics.
I stand by what I believe. It is impossible to calculate with certainty something that is not there.
I give up. Nothing I can do to convince someone who does not understand a topic, who refuses to learn and believes unequivocally that what they state is trueā¦
If by chance you do care to learn, read thisā¦
From the paper:
āGiven that itās 35 years since CD went on sale in Europe, you might suppose that most audiophiles
would now have a firm grasp of the basics of sampling theory, but in truth it remains widely misunderstood.ā
āWhat Shannon showed in his famous paper was that any bandlimited continuous signal ā that is, any analogue signal with a strict limit on its maximum frequency ā can be exactly described as a sum of time spaced sinc(x) waveforms. Not only that, if the signal is sampled, ie if its amplitude is measured, at regular intervals, at a rate at least double that of the highest signal frequency, each sample amplitude represents the amplitude of the associated sinc(x) waveform centred on that sampling point. At all other sampling points the value of that particular sinc(x) function is zero, just as the value of the sinc(x) functions centred on each other sampling point are zero here. So sampling the waveform as described extracts all the information necessary to reconstruct it, and to do so with complete accuracy.ā
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Rob Watts, working as a consultant with Chord Electronics, has completely subjugated the whole āupsamplingā argument in digital audio. The Hugo MScaler is a total game-changer, and in combination with the Hugo TT2 is a very close-call to my Linn KDS/3.
I personally agree Martin!
But Iām trying to separate the mathematical certainty of reconstructing an accurate waveform from sampled audio as described by the mathematics that creates 100% accurate audio (and missing information) within the band you can hear from the actual approach of Chord because the mathematical approach cannot be exactly implemented in todayās hardware. The former canāt be argued against because itās mathematically proven (regardless of what some think).
What is game changing about Chord is that they have come closer to implementing this mathematical approach than anyone by leaps and bounds (and with every generation of DAC/scaling technology Chord gets closer and closer)ā¦ Whether their implementation and the tradeoffs they have made are superior in everyoneās or anyones subjective tests is not the point that Iām making even though they are for me.
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I believe that certain other manufacturers, such as Linn, have better analogue output stages from their digital products (specifically the Linn Kilmax range) than Chord, especially where comparisons the TT2 are concerned. Although Iāve yet to hear DAVE 
I think the last sentence you made is very important. Iām sure the mathematical aspect you describe is spot on, but what really matters is how it sounds at the end, which is absolutely subjective for everyone. For the same reason, people should not choose their equipment based on reviews on audioscience that focus strictly on technical measurements. Yet, lots of people still like the sound of the low rated equipment there.
I have been trying various upsampling techniques (HQ player, manually upsampled files, some HW, although never anything from Chord) and it always sounded different - never better or worse per say, but different, but mostly less natural. Thatās why I compared it to a DSP effects earlier from purely listening experience.
Also an excellent article @killdozer. From the articleā¦
" The sampling theorem introduces the concept of a sample rate that is sufficient for perfect fidelity for the class of functions that are band-limited to a given bandwidth, such that no actual information is lost in the sampling process. It expresses the sufficient sample rate in terms of the bandwidth for the class of functions. The theorem also leads to a formula for perfectly reconstructing the original continuous-time function from the samples."
Completely agree @Jakub_Burdych. I started this particular discussion because of the countless times people have claimed that upsampling has no purpose because itās impossible to create anything that is missing. That is not true mathematically. I suppose itās actually more accurate to say that the āmissing informationā is actually not missing, you just have to work to find itā¦
And while there are countless technical measurements for frequency response and noise and the like to compare audio gear, there are no technical measurements to compare or judge how accurately equipment produces transients which are critical to your brain creating an accurate soundstage.
So I agree, you must get beyond just the numbers and listenā¦
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Indeed. All that āinformationā is implicit in the samples. Rather than saying weāre recovering āmissing informationā, itās more accurate to say that weāre re-creating the original waveform.
I still scratch my head in puzzlement when I hear that people are upsampling on their music server, or converting to DSD on the server. Re-creating the analog waveform is the DACās job! I suppose if you have a bad or feeble or incapable DAC, you might want to do this. But shouldnāt you just get a better DAC?
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Lots of people like Vegemite, too.
Not that Iām judging. I myself like SPAM. SPAM Lite, of course ā Iām no fool!
Not to beat a dead horse, but per the mathematics, you are reproducing completely accurate audio.
The reason that chord separates the DAC (which can upscale to a certain level) from the vastly more capable scaling of blu and mscaler is that it can draw tremendous power (mscaler can draw up to 10 amps) to do this and can generate a lot of heat. So by separating the DAC and scaler, both power and heat can be managed and also noise can be minimized. Itās not currently possible to take chords approach in a single DAC so you canāt simple buy a better oneā¦
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So let me ask about what might be the converse of this. Iāve found that various attempts to upsample or use HQPlayer has led to an subjectively āworseā sound feeding my MSB DAC. These DACs will also do some (significant) degree of signal manipulation and filtering.
Does upsampling prior to a DAC make it more difficult for the DAC to do itās job? Basically, are you now feeding it a waveform it will have a hard time (or wonāt be able) to run itās own algorithms on?
IMHO, It depends ā¦ (and all of this is IMHO).
In the case of my Dutch & Dutch 8c speakers, itās better to upsample/downsample to 48kHz prior because the DSP tech the 8Cs uses internally convert anything coming in to 48kHz so you are best served by sending exactly that so you donāt have to do multiple d/a a/d conversions.
My Chord Dave is designed specifically to take the 705 kHz or 768 kHz upsample feed from my Chord Blu (depending on the source sampling rate) and process with optimal filters and algorithms. (the Hugo TT and mScaler are also designed to work optimally in this fashion)
Taking any source material, upscaling using random methods, applying random filters and sending to random DACs can lead to almost any results and often end up sounding worse than just leaving the music as is. I think too many people just assume upsampling to the highest level automatically makes music better but It takes significant work, listening (better yet a/b testing) to find the right magic combo if indeed one is to be found.
Yeah, and I can do my custom software implementation with 32M taps to upsample to 11.2 MHz in software, using power of modern CPUās and GPUās. 
If you are into number games.
I stand corrected sir, forgot you were HQ player! lol. Which is why what you are doing in HQ Player is so well respected!