"The basic precepts of science and engineering mandate that tests must be of a device actually performing a relevant design task, that the test result observation and evaluation be objective not subjective, that the test be repeatable by other persons, and that these other persons cannot be the author himself of the device's design. You violated all these mandates, so your entire engineering approach (and worldwide engineering movement) is actually anti-science and anti-engineering. You are, in objective fact, merely asserting that you subjectively think that your own sketch, of your very own design concept, subjectively looks pretty to your eye - and a sketch of a competing design, authored by someone else, subjectively looks ugly to you. Is that what university taught you, as being the true essence of science and engineering, while you were getting your PhD?"

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By the way, when we conduct our correct shaped pulse response test, it's immediately obvious that only 1 reconstruction filter design, only 1 coefficient function, achieves and provides virtually perfect time domain transient response performance, hence correct reconstruction in the time domain of the original pre-sampled signal waveform. This 1 filter design is the maligned, rejected, old fashioned sinc brickwall filter (which proves how misleadingly wrong the standard single sample version of the so-called impulse response test is, since it seems to show that the sinc brickwall filter has awful impulse response, the very opposite of the truth).

Meanwhile, this correct shaped pulse test shows that all the revisionist digital filter designs, which pretend such devoted fealty to time domain accuracy, actually butcher the time domain signal waveform, and produce very bad time domain distortion in their attempted reconstruction (with MQA being the worst of all).

The 34 years of steady and universal digital engineering drumbeat, that the sinc coefficient function had horrible time domain impulse response, hence horrible time domain transient response performance, had pre-conditioned us to expect the worst, when we first tried testing the sinc coefficient function with our shaped test pulse. So we were stunned by what we saw, and by its huge implications for the entire world of science and engineering.

But our initial amazement, at seeing the sinc brickwall filter deliver perfect time domain transient response when actually performing its reconstruction task, was quickly sobered to anguished incredulity.

Why oh why, oh why, in 34 years of digital engineering's awareness of, and sensitive dedication to, better time domain transient response from digital filters, had no one, in the whole worldwide community of brilliant PhD digital engineers, simply remembered the Goldilocks lesson all of us learned at age 4, that to test a filter design you need to make it actually perform its designed task of convolution?

Why oh why, oh why did none of them remember the age 5 lesson all of us learned, that meaningful addition requires 2 or more numbers, and thus convolution, which to be activated depends on addition, cannot even be activated to perform anything at all, by just 1 number (e.g. the single sample input Kronecker delta impulse test signal)?

Why oh why, oh why did none of them remember the age 6 lesson all of us learned, that multiplying anything by 1 is effectively a null operation that does nothing, hence merely multiplying the filter design's coefficient function (that embodies the digital filter's design) by 1 (the single sample impulse test input signal) does not test the filter design actual performance at all, but instead merely photocopies this coefficient function, merely photocopies the filter designer's own napkin sketch of his own filter design?

Why oh why, oh why did no scientist or engineer in the last 34 years, indeed since the days of Kronecker himself, notice and see and comprehend how digital filters actually work, using convolution that relies on addition - and thus, necessarily a priori (without even having to perform any experiment), he could have alerted the whole scientific and engineering world that a digital so-called 'impulse response test' using the Kronecker delta as its single sample (this 'shortest possible, nearly ideal') test impulse signal, was actually not even a performance test at all, but merely a photocopying of the filter designer's own napkin sketch?

Why oh why, oh why did no scientist or engineer, in all the years since Kronecker, see and understand and warn everyone that the digital impulse response test is worthless as an evaluative test, is actually dangerous as a backwards reverse evaluative misleader, and indeed is not even a test at all? Why did none of them realize that any digital filter engineer can simply sketch on a napkin his subjectively prettiest, most fanciful filter coefficient function profile that he can fantasize about -- and then the so-called 'impulse response test' will simply and merely, every time, 'confirm' to the designer that his pretty function sketch 'actually performs' just as beautifully in the time domain as his napkin sketch looked. Narcissus subjectively loved his own beauty, reflected back to him, but that self-adoring reflection was no evaluative test, and he was no scientist.

Now, it's bad enough that the so-called 'impulse response test' is not even a test at all, and instead is merely a self-congratulatory mirror reflection to the designer's vanity, of his own filter design creation. But this perniciousness is made far worse by the fact that the so-called 'impulse response test' actually reflects back a perversely inverted image of the filter design's true time-domain worth - wherein seeming impulse-shaped accuracy actually portends horrible time domain butchering distortion, but awful ugliness and 'impulse response failure' can actually (in the sinc filter) portend perfect time domain accuracy (i.e. true scientific, objective beauty).

Indeed, the worldwide digital engineering community could have saved itself, and saved all consumers, 34 years of defectively designed digital systems and products, if just one engineer somewhere, sometime had thought to try a test impulse signal employing the simplest possible modification of the Kronecker delta, a test impulse comprising just 2 equal height sample dots, i.e. not even a shaped pulse at all, but simply a slightly wider version of the single sample impulse test signal. He would have, for the first time, obeyed the age 4 Goldilocks rule by actually making the filter under test actually perform its design task of convolution, for the first time.

And, when he performed this actual convolution performance test on the old fashioned widely maligned sinc function, whose coefficient function looks so awful, he would have seen, before his very eyes, an amazing transformation, from a frog to a prince (nay, to a king). By using just 2 sample dots of equal height as his convolution-activating test impulse, he would have merely activated the addition component of convolution (not yet even actively using convolution's multiplication component). But that would be enough.

He would have seen the mere simple addition of two equal sinc coefficient functions, with one being offset from the other by just (and exactly) one sample period/interval. And he would have seen all that violent ringing of the sinc coefficient function magically self-cancel itself out of existence. He would also have seen the long post-response and long pre-response of the sinc coefficient function magically self-cancel itself out of existence.

This of course would have immediately shown and proven that the coefficient function of any digital filter design can look wildly different from its actual time domain transient response performance. This would have been hugely shocking news to the entire world of digital - as it surely is now to all of you around the world. This of course necessarily a priori means that the classic digital 'impulse response test' is utterly useless, and far worse is tragically very misleading to the whole world of digital. The digital impulse response test is utterly invalid, since it merely photocopies the filter designer's own napkin sketch of his coefficient function - and as we now know (as we should have learned from some engineer trying the simple 2 dot test impulse sometime in the past 34 years) the true time domain transient response performance of a digital filter looks nothing at all like its coefficient function.

The sinc coefficient function is a superbly powerful, sophisticated tool for performing reconstruction, and it puts all its power and sophistication into complex techniques for achieving just one goal, perfect reconstruction in the time domain of the original pre-sampled signal waveform (hence perfect time domain transient response). We devote an entire installment of this article Digital Done Wrong to showing and explaining the complex ways in which the sinc coefficient function actually works. It's shameful that, in the entire history of digital engineering (even long before digital audio), apparently no one has ever even tried to peer into and discover, and come to comprehend, the ways that the well-known and widely used sinc function actually works.

Here's just one quick example of the sinc function's sophisticated flexibility and power. Note that ours is a time domain analysis, because that's the domain where we are focusing on final delivered reconstruction filter performance, and also because everything that happens in Nature actually takes place in the time domain.

The sinc coefficient function automatically self-cancels its violent ringing and long duration response when it is not needed, for accurate time domain reconstruction, which is at low frequencies, where adjacent sample dots are equal or similar in amplitude (as they are in this impulse response test using a test pulse consisting of 2 equal amplitude adjacent sample dots).

But to correctly reconstruct higher frequencies, both the sinc's ringing pattern and its long response are crucially required, in order to correctly boost and accurately reconstruct between sample dots the original high amplitude of the original pre-sampled signal waveform, that inevitably has its high peaks unluckily sampled, with the sample dots being well below the true peak amplitude value of the high frequency peak between them. The sinc function automatically 'recognizes' these high frequencies by 'sensing' the large amplitude change between adjacent sample dots that characterizes high frequencies, so it automatically keeps its ringing, long duration pattern, to employ as crucially required tools, needed to provide the one and only correct reconstruction of the original signal waveform in the time domain.

For technical and didactic reasons, we designed a simple shaped pulse, for our digital filter time domain transient response test, by using 3 sample dots. Our shaped pulse test signal keeps the single sample dot as its peak impulse value, but then slightly widens the base of the test pulse by adding 1 sample dot on each of the side flanks, about halfway down.

Our test actually performs filter convolution, between the sinc coefficient function and our 3 dot shaped test pulse. This 3 dot pulse fully activates both the multiplication and addition components of the convolution process (the pulse above, with just 2 equal height dots, activated only the addition component). Thus, our 3 dot shaped test pulse fully obeys Goldilocks' mandate, to make the DUT actually perform its design task. The criterion for this test is the same as for the single sample 'impulse response test', or indeed for virtually any type of signal test: the DUT output should replicate the input as closely as possible.

The sinc coefficient function (i.e. the sinc bickwall filter) nails the correct time domain transient response, perfectly replicating our 3 dot shaped test pulse.

First, the sinc function, actually performing convolution, perfectly nails the first commandment of reconstruction. Its reconstructed path between and among the original sample dots actually passes directly through all these sample dots, thereby preserving all these sample dots intact. These sample dots are sacred and inviolable, since they represent correct, accurate samplings of the original signal waveform at these precise instants (and indeed constitute the only accurate information we have from and about the original pre-sampled signal waveform).

The first commandment for physicians is to first do no harm. In this case, this first commandment also applies to any and every digital reconstruction filter design: first, do no harm, by keeping the original sample dots intact, since they are the sacred inviolable guideposts (indeed only guideposts) for our reconstruction of the original signal waveform. The job of the reconstruction filter is to plot a curved path (and the one and only correct path) between the original signal's sample dots, not to change the sample dots themselves.

Second, the sinc function perfectly reconstructs the entire path of the main pulse, with correct sinusoidal curvature as befits the bandwidth limited nature of the whole digital system. This is simply and completely the perfect time domain transient response that the entire digital engineering world has been seeking as its ideal nirvana. And, amazingly, the sinc brickwall filter achieves and displays this perfect time domain transient response performance (totally and automatically erasing its ugly duckling coefficient function appearance, its awful-looking so-called 'impulse response'), merely by being actually convolved with just 3 simple sample dots.

Why oh why, oh why has no digital engineer ever thought of conducting such a simple time domain test of actual convolution performance before, either on the sinc brickwall filter design, or (especially) on the new revisionist filter designs that themselves pretend to be optimized for time domain reconstruction accuracy?

Third, the sinc function even correctly includes the tiny negative ears, at the base of this reconstructed 3 dot shaped pulse, that must be present in any and every bandwidth limited system (since no bandwidth limited system can execute the infinitely sharp time domain corner, from flat zero amplitude to suddenly rising pulse).

How do the 'time-domain-optimized' filter designs of the modern revisionist digital engineering movement fare, in this simplest possible pulse response test, when actually performing convolution to reconstruct this simple, easy 3 dot pulse, hopefully with superior time domain accuracy? Awful. They literally butcher the time domain accuracy of even this simple pulse, with gross distortion, of and in the very time domain they pretend such expertise in and devotion to.

First, all of these modern revisionist 'time-domain-optimized' filter designs, from Craven et al, and from all the thousands of such digital products sold to the consumer, flagrantly violate the very first commandment to first do no harm. They necessarily a priori, completely and utterly, fail to draw the correct reconstructed path between and among the original signal's sample dots, because they do not even draw a path between and among the sample dots at all. They do not even pretend to try passing through all the sacred sample dots that anchor and represent the accurate original signal waveform path. Instead, they plot a freewheeling path of a whole new signal waveform path that willy-nilly runs amok, wandering afar and astray, without any concern about honoring and obeying the sacred, inviolable anchor points of the correct time domain path to be reconstructed. This is supposed to be time domain reconstruction accuracy???

Second, an educated glance at their coefficient function immediately sees that it is much too wide at its base in the time domain. Thus, when actually performing convolution, their filter design has too much overlap in its 'impulse response' to adjacent sample dots. And this in turn produces a grossly bloated amplitude error in its time domain performance, 'reconstructing' our simple 3 dot pulse, as well as any real music signal waveform.

Third, this bloated amplitude distortion, caused by a too wide base in the filter design's coefficient function, also causes a lingering time domain bloat, a lingering energy shadow in time that lingers long after the original transient has stopped (and long after the original transient's sample dots, disobeyed by this filter design, have returned back down to zero).

Fourth, the very short duration of this filter's coefficient function (thus also of its 'impulse response'), which is also fixed and non-self-adjusting (as the sinc filter automatically is), means that this filter cannot come even close to correctly reconstructing the full accurate amplitude of all high frequency transient peaks that are inevitably unluckily sampled. Thus, these modern revisionist filter designs, which pretend such allegiance to time domain transient response accuracy, ironically actually do the complete opposite when attempting to reconstruct time domain transients, precisely what they purport to be optimized for !!!

When their coefficient function is actually made to perform its design task of convolution, it 'reconstructs' time domain transient peaks as too low in amplitude, as too rounded and dull in shape, and even occurring at the wrong instant in the time domain (that's 3 strikes of time domain butchering). The sonic result of this butchering is that high frequency transients are severely degraded, to sound very rounded and dull, and also too fat and lingering in the time domain. We hear all these time domain distortions from all the modern revisionist digital reconstruction filter designs we have ever heard.

What about MQA's time domain performance, while actually performing its design task of convolution, to reconstruct the original pre-sampled signal waveform? MQA's own AES technical paper describes its impulse response as being so very brief that its output approaches the common engineering ideal of an infinitesimally brief impulse as an input test signal - indeed so brief that it is even shorter than a sampling interval (in fact, the MQA AES paper even describes a dual-ramp technique for determining where such a brief spike belongs within the span of a sampling interval).

Thus, MQA falls within the aegis and spirit of the modern revisionist digital engineering movement, but takes its ideal goals to a new pinnacle of engineering achievement. Good, right? Worthy of hearty congratulations for such a design and design effort, right?

Not so fast. MQA's brilliant PhD designers clearly forgot one tiny detail. Goldilocks. The lesson that they (and all of us) learned at age 4 from Goldilocks. To actually test a filter's actual time domain transient response performance, you need to make it actually perform its design task, namely convolution. If you merely conduct a so-called 'impulse response test' using the single sample Kronecker delta as a test signal, then you are merely multiplying your own coefficient function design by 1, thus merely photocopying your own napkin sketch of your own filter design coefficient function.

(Continued on page 178)