Background Analysis:

The Importance of Digital Filtering


     All CD players and D-A processors manipulate the digital signal from the CD while it is still in the digital domain. First, at the most elementary level, the raw digital signal from the CD must be unpacked and reformatted and error checked, so that it becomes a string of bits that relate only to the music signal, and are in the proper sequence to represent the music signal. Second, the signal from the CD should be filtered, and nowadays this is almost universally accomplished by a digital filter (while the signal is still in the digital domain) rather than an analog filter (after the signal has been converted to analog).
     What is the role of a digital filter? It's a common misconception that the (digital) filter in a CD player or D-A processor chiefly exists merely to filter out ultrasonic spuriae and thereby also reduce aliasing distortion. If this were true, then it would be optional to even have the filter at all. Why bother filtering out ultrasonic spuriae if we can't even hear them (they're ultrasonic, above 20 kc), and if most amplifiers and speakers can easily handle this ultrasonic energy (even contributing themselves toward filtering it)? As to aliasing distortion, that's merely the presence of aliases (cousins) of ultrasonic spuriae. These aliases are usually not a serious practical issue on most music (aside from the occasional cymbal crash), thanks to music's naturally declining spectral energy content at higher frequencies, and also because they exist only at the very top edge of the band (around 19-20 kc) where they would be scarcely audible to most people.
     But, in point of fact, a digital filter has another function, and this other function is crucial. This other function does not relate to the mere elimination of ultrasonic energy that is scarcely audible and scarcely problematic anyway. Rather, this other function involves the literal creation and generation of all the music you hear above 2 kc. The sound of all music in that huge span from 2 kc to 20 kc depends entirely on the digital filter, and more importantly depends on exactly how this filter is implemented.
     You see, the common misconception also believes that the bits coming off a CD furnish enough dots (sample points) to reasonably outline the music waveform, and basically all a CD player or D-A processor needs to do is connect the dots. This common misconception is true only at low frequencies. In the lower midrange it begins to be only approximately or crudely true. And above 2 kc it is not true at all. In truth, above 2 kc there are not enough sample dots or points coming off the CD frequently enough to outline the correct, original music waveform, and indeed at higher frequencies the pattern of these dots or sample points coming off the CD doesn't even resemble the music waveform. Instead, the CD player or D-A processor must literally create or generate the music waveform almost out of thin air, with the sample dots off the CD being mere sketchy clues (as in a detective story) that need to be correctly interpreted to guide the music waveform creation process.
     The music waveform you will hear is actually created by a complex algorithm in the CD player or D-A processor, which is supposed to look like a boxcar filter in the frequency domain and a sinx/x function in the time domain. This complex algorithm generates a music waveform using as occasional guideposts the sketchy data sample points coming off the CD. The music waveform thus created is often far more complex than the waveform you would imagine or get if you merely connected the dots of the data sample points coming off the CD. For example, three consecutive sample dots could at most define a music waveform no more complex than the shape of a V, if you merely connected the dots or provided some sort of similar intermediary interpolative playback filter. But a correct boxcar playback filter algorithm can actually generate more complex music waveforms, for example resembling the shape of a W, from just three sample dots off the CD. In IAR issue 58 we showed this contrast mathematically, and then we actually measured two different D-A processors: one generated the correctly complex W shaped music waveform from just three sample dots off a music CD, while the other, having a simpler interpolating filter on board, generated an erroneous, simpler, cruder V shaped music waveform, from the same three sample dots, thereby robbing you of some true musical detail.
     As you can well imagine, the sound of various CD players and D-A processors varies dramatically, depending on the design of their digital filter, since this filter is actually creating the music waveform you hear, especially above 2 kc. CD players and D-A processors that use simple interpolating (connect-the-dots) filters tend to sound soft and smooth, since their overly simplistic re-creation of the music waveform misses a lot of musical details (especially at higher frequencies) and thereby presents you with a smoothed down averaged version of the music. And this sounds very different from the more revealing detail (especially at higher frequencies) heard from CD players and D-A processors that use boxcar digital filters for re-creating the music waveform.
     But, you might well ask, shouldn't all these CD players and D-A processors that use the correct boxcar filter algorithm (for re-creating the music) sound the same? After all, a boxcar filter is a boxcar filter, and a correct boxcar digital filter calculation should crank out the same numbers/bits, regardless of whose chip or computer is doing these calculations. This question, expectation, and line of thinking might have merit - if it were indeed possible to make a correct boxcar filter. But it isn't.
     A perfectly correct boxcar filter has a perfectly flat response to the band edge (say 22.05 kc), a perfectly sharp corner at that edge frequency, and then an infinitely steep slope down to nothing at that same frequency (its shape looks like a boxcar, hence the name). This ideal target filter response cannot be even crudely approximated with analog circuitry, and with a digital filter it is still physically impossible, since it would require an infinitely complex digital computer to calculate infinitely long numbers for an infinitely long time (to produce the correct boxcar filter numbers, even at the coarse 16 bit resolution level and even at the slow 44.1 kc sample rate). This means that various CD players and D-A processors can at best only approximate the correct complex algorithm for generating the music waveform above 2 kc.
     Naturally, different CD player and D-A processor designs will make different engineering compromises in approximating the correct ideal, compromises dictated by cost (the more correct complex algorithms require more complex on board computing that costs you more), by available chip designs, and by engineering preference. Where compromises are necessary, different engineers will choose different design compromises, each approximating the correct ideal music-waveform-generating algorithm in different ways and to different degrees.
     Thus, each CD player or D-A processor with a different digital filter design, which approximates the complex boxcar algorithm in different ways and to different degrees, will generate a slightly different music waveform above 2kc, and so of course it will sound different than other CD players and D-A processors. This sonic difference among CD players and D-A processors is especially noticeable and important because the human ear/brain is most sensitive to sounds and sonic differences in the spectral region above 2 kc.
     Since the digital filter plays such a crucial role in literally generating the music waveform above 2 kc, it is sometimes called a reconstruction filter, which hints at the fact that this filter's key role is far different than a filter's usual role in merely filtering out some frequencies. You now have seen that the digital filter's role is far more important than merely filtering out ultrasonic spuriae and aliases way up around 20 kc. If the digital filter's role were merely ultrasonic filtering, then this would mean that the correct in-band music waveform would already be traced out by the sample dots coming off the CD, such that a simple connect-the-dots interpolator could suffice to furnish the full music waveform. But that is not the case. We know from theory and from measurements (see IAR 58) that the sample dots coming off the CD fail to trace out all the complexities of the original music waveform for a huge 2 kc to 20 kc segment of the spectrum, well in band. We know from theory and measurements that CD players and D-A processors, whose digital filters merely connect the dots, fail to accurately re-create the original music waveform, missing (smoothing out) details that are well in band. We know from theory and measurements that digital filters closely approximating the ideal boxcar filter do re-create these musical details well in band. Obviously, then, these theoretical and measurement observations prove that a digital filter is responsible for the music waveform well in band, not just responsible for handling spuriae out of band or at the top edge of the band.
     The lesson is that money spent on a more complex, more expensive, more accurate digital filter in your CD player or D-A processor is money well spent. It will reward you with more musical detail and more accurate re-creation of the original musical detail. No digital filter can be a perfect boxcar, so none can re-create and give you perfectly accurate music above 2 kc (contrary to that early Philips claim of perfect sound forever). But the best, most expensive digital filters can get pretty close. And, given the ear/brain's incredible resolution, you want to spend as much as you can afford on a digital filter which will get as close as possible to re-creating the original musical waveform the way it should be.
     Note incidentally that a digital filter's approximations to the correct original waveform can be affected by a variety of engineering compromises in the design of this filter. For example, if the corner of the filter is too rounded, or is not at the optimum frequency, or if the skirt of the filter is not steep enough, then the filter calculation and its literal re-creation of the music waveform (in digital) will be somewhat wrong above 2 kc (the error becoming progressively worse at progressively higher frequencies within the band). Interestingly, the compromises in boxcar filter design are located way up near the band edge (around 20 kc), yet they affect re-creation of the music waveform all the way down to about 2 kc, since they affect the exact trajectory that the filter plots between the sample dots (which, again, become too scarce above 2 kc to accurately outline the correct waveform in a connect-the-dot mode). Note too that this error above 2 kc would be far worse than the filter's own native resolution, and might even be worse than 1 LSB of resolution of the DAC chip following the digital filter. CD players and D-A processors often brag about the number of bits of resolution in their digital filters and DAC chips, but this fine resolution does little good if their filter design produces erroneous music waveform numbers for the DAC chip to convert to analog. High resolution conversion of digital errors still produces erroneous music (the garbage in, garbage out principle).
     The Swiss Anagram module in the Capitole is surely a very high quality, expensive digital filter, because it re-creates details in the music waveform that other digital filters simply miss, and it must be more correct than other digital filters because the extra details it reveals sound so musically right.
     We also suspect that this Anagram module does something more than high quality digital filtering. We asked Anagram about this, but they regard the processing in their sealed module as proprietary, so they would not discuss it. But we can hazard some educated inferences.

Internal Oversampling

     An on-board computer programmed to be a high quality digital filter can also be programmed to do more, if there's extra computing horsepower available. What more is there to do? Let's briefly look at the output of a digital filter.
     A standard high quality digital filter, even if it could be perfectly executed to perform perfect boxcar filter calculations, is still only obligated to output data samples at a 44.1 kc rate (and at 16 bit resolution). In other words, the difficulties we discussed above relate to the impossibility of designing a correct boxcar filter to output merely the correct 16 bit digital values at a 44.1 kc rate, to correctly re-create the original music waveform. We weren't yet talking about trying to get any higher resolution or any higher sampling rates from a digital filter.
     Now, in order to perform its internal calculations needed to approximate a boxcar filter, the digital filter computer or chip internally expands the resolution of the 16 bit data coming off the CD (usually to 24 bits or more), and it internally expands the sampling rate by oversampling (it does this by simply repeatedly re-looking at each 44.1 kc sample (say) 4 times, essentially the same as you now reading the same word word word word repeatedly 4 times). After the digital filter has finished its calculations to generate the music waveform above 2 kc as best it can, then it can output the numerical digital results to the DAC chip. The DAC chip really only needs data at 16 bit resolution and at a 44.1 kc sample rate. So the digital filter's internal expansion of bit resolution and sampling rate to (let's say) 24 bits / 176.4 kc can be trimmed back at its output, trimmed back all the way back down to the original 16/44.1 input.
     However, there are some technical advantages to leaving the data stream in its expanded form as the digital filter internally expanded it, or at least expanded to some extent. Assuming that the DAC chip is designed to accept and process a higher sampling rate than 44.1 kc (most DAC chips nowadays can handle 88/96 kc or even 176/192 kc), then feeding data from the digital filter output through the DAC chip at this higher sampling rate allows the use of a much simpler analog filter after the DAC chip. This simpler analog filter can be simpler because it now only has to filter out very high ultrasonic frequencies (say above 100 kc). And the simplicity of this analog filter has a number of sonic advantages: it can be gentler in corner transition and slope, thus less mathematically destructive of the music signal in the time domain; it can be made out of fewer fidelity-robbing parts put in the analog music signal path; and its parameters (corner frequency, slope, etc.) are non-critical, so normal tolerance variations do not have adverse sonic effects.
     These benefits can be achieved by merely having the digital filter output its results at its higher internal clock rate, into and through the DAC chip, instead of trimming its higher internal clock rate back down to the original 44.1 kc sample rate before outputting its calculated results generating the music waveform. This is called oversampling, which is somewhat of a misnomer because, interestingly, the extra samples output at the higher clock rate can be meaningless dummy data (say set to zero value, or alternatively a 3 times repetition of the previous music waveform value), rather than meaningful music waveform sample data, so there are no extra true samples. Thus, for example, with 4 times oversampling, there is 1 true meaningful music waveform sample (with a value generated by the digital filter calculation), followed by 3 dummy data samples with value simply set at say zero (and therefore obviously not conveying any further information about the music signal's possible changes during these 3 extra sample periods).
     This standard oversampling, seen in most CD players and D-A processors nowadays, does its intended job just fine, and secures the extra benefits intended. But what a shame and waste! Here we have a fast computer used as a digital filter, and a fast DAC chip capable of accepting this 4 times faster sample rate. But then we waste of their time by having them process meaningless zeroes for of their samples!
     Could we put this empty dummy time to better use, and thereby perhaps secure even greater sonic benefits? Yes we can!


High Power Averaging


     Nowadays computational power is relatively cheap. So let's invest in a computer with some extra computational power, beyond that which we need to make high quality digital filter calculations. And then let's put this extra computational power to work. How? By filling those empty time slots with real music waveform values instead of dummy zeroes. The computer could calculate 3 new values for each set of 3 dummy time slots that the digital filter has placed intermediately between each pair of two music waveform sample points (at the 44.1 kc sampling rate), sample points that this digital filter has just finished calculating by applying its boxcar approximation to the raw data stream coming off the CD. What numerical values should we plug into these empty time slots? Let's program the computer to calculate appropriate new music waveform values. These new values could be interpolated among the older music waveform data points that the digital filter has just finished calculating, using a sophisticated averaging algorithm.
     How might such an averaging algorithm work? Let's start with an oversimple example. Suppose that the numerical values (representing waveform amplitude) of two adjacent data points were 1 and 5, as just calculated by the digital filter using its boxcar approximation. The oversampling mechanism has expanded the sampling rate so that there are now three extra dummy samples between the two genuine samples with numerical values of 1 and then 5. A simple linear interpolation for the three extra dummy samples would set their values at 2, 3, and 4, respectively. The music waveform would now have meaningful calculated values at all five sample points (the two end points and the three extra oversampled points added in between, originally with meaningless dummy values). These five meaningful values, 1, 2, 3, 4, 5, respectively, define a straight line between the two original sample points. The three extra sample points added by oversampling now contain meaningful values, which define the music waveform to a finer degree. So the music waveform has now been calculated to a finer degree of refinement, before it encounters the DAC chip for conversion to analog.
     Linear interpolation is the simplest kind of averaging. It draws only straight line segments between previously calculated music waveform sample points. Unfortunately, this is too simple. For music is not a straight line. The music waveform is constantly changing, so it is composed of curves, and indeed curves of many sorts whose shape is constantly changing. So simple linear averaging, which can create only straight line segments, is inadequate. We have just finished putting the data from the CD through an expensive digital filter, which generates a complex curved function in time (the sinx/x function) among incoming sample dots. So it would be inappropriate to now put this

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