What was measured?
The tests compare a calculated upsampled waveform with an ideal, delay-aligned reference. The input sample rate is 44,100 Hz and the output rate is 705,600 Hz, a factor of 16. We use linear-phase sinc-based FIR filters. FIR means a filter with a finite number of coefficients. These filter weights are derived from the sinc function; we call them sinc coefficients (sinc coeffs). The window adjusts the coefficients, not the sinc function itself.
For each input frequency, we calculate the steady-state response to a unit-amplitude sinusoid. We include all 16 positions of the new samples between successive input samples and find the largest error over the input tone's phase. We then search for the largest error anywhere in each reported frequency band.
This measures the wanted signal and residual upsampling images together. It is not just passband ripple, a root-sum-square of image levels, or an average error across frequency.
The two bands are DC–20 kHz and 20–22 kHz. DC means zero frequency. The upper band stops 50 Hz below the 22.05 kHz input Nyquist frequency. None of these tables claims the same accuracy all the way to Nyquist.
The figures label these Audible band: DC–20 kHz and Extended band: 20–22 kHz. This article uses 44.1 kHz input throughout. The principles apply to other sample rates, but the corresponding boundaries, required lengths and accuracy values must be evaluated for those rates. Near Nyquist, maintaining very high reconstruction accuracy requires progressively longer filters; a uniformly tiny error arbitrarily close to the boundary is not a practical finite-filter specification.
How is the error expressed in bits?
Let E be the largest absolute reconstruction error relative to the unit-amplitude reference. The equivalent accuracy is:
Accuracy in bits = −log₂(E).
Halving E adds one bit. An improvement of one bit corresponds to approximately 6.02 dB less error. Higher bit values therefore mean smaller errors.
These values describe mathematical filter-design error. They are not output bit depths, DAC resolution, measured acoustic noise floors or evidence that an error is audible. The numerical calculation is performed at substantially higher precision than the errors being reported.
How are the interpolation phases included?
For an input tone at angular frequency ω, the ideal output at each of the 16 interpolation phases is known. We compute the filter response for that phase, remove the filter's known delay, and compare it with unity. The magnitude of this complex difference gives the worst error over the real input tone's phase at that output position.
Taking the largest difference over all output positions includes the combined effect of residual images. Taking the largest value over the frequency band gives the number in the tables. A sinusoid does not need to be generated and played for many cycles to make this steady-state frequency-response calculation.
These measurements do not bound every possible peak-normalized multitone or music waveform, or include startup and shutdown transients.
Avoiding misleading ripple samples.
A frequency grid can land in ripple dips and overstate accuracy. We first sweep the response at high precision, identify competing local peaks, then refine their frequencies with direct evaluations. We also test the exact band endpoints and the partial grid intervals adjacent to them.
The comparisons check competing interior peaks, exact band endpoints and adjacent intervals, with direct higher-precision evaluations of the worst candidates.
These are numerical peak searches, not analytic proofs of the global maximum over a continuous band. The tables present accuracy rounded to the nearest bit; the charts retain measured differences.
Charts use the measured values, without whole-bit rounding of the plotted bars or curves. In bar charts labelled “reconstruction accuracy exceeds,” labels are rounded down to two decimal places. Relative-gain chart labels use decimal summaries. Table accuracy values round to the nearest bit. Compact-filter lengths are approximate, not exact minimum-length claims. This display change does not change any underlying measurement. The spectral-image and window-shape illustrations are explanatory diagrams, not additional PGGB performance tests. The PGGB shape panel is a generic illustrative example: it is neither the actual PGGB window nor the window used to calculate the results. These illustrations disclose no actual PGGB-window or HF-filter design.
Equal-length window comparison.
All four filters have about 260k output-rate taps. Rectangular applies no taper. The Kaiser comparison uses β = 10.35. Tukey uses α = 0.5. The PGGB and Tukey filters have the same total length and the same number of unchanged central sinc coeffs.
The PGGB window's construction and coefficient-generation method are not included. The measurements evaluate it as the supplied filter; they do not require publishing its design.
| Window | DC–20 kHz accuracy | 20–22 kHz accuracy |
|---|---|---|
| Rectangular | 12 bits | 7 bits |
| Kaiser | 24 bits | 19 bits |
| Tukey | 28 bits | 13 bits |
| PGGB | 207 bits | 8 bits |
The comparison is for these selected windows and parameters. It does not show that one family dominates every other design at every frequency or transition width.
Adding unchanged sinc coeffs while keeping the taper fixed.
This experiment holds the PGGB taper shape and its absolute length fixed. Only the number of unchanged central sinc coeffs is increased. It is different from increasing flat coverage while keeping total filter length fixed.
| Output-rate taps | DC–20 kHz accuracy | 20–22 kHz accuracy |
|---|---|---|
| ~1 million | 403 bits | 28 bits |
| ~2 million | 404 bits | 29 bits |
| ~4 million | 405 bits | 30 bits |
From the shortest to the longest filter, the gains are about 2 bits / 13 dB in the lower band and about 2 bits / 11 dB in the upper band. This supports approximately one extra bit per doubling for this experiment. It is not a general length law for all windows.
PGGB V8 did not use this particular construction. The article separately reports the author's observation that V8 also continued to benefit from increased lengths.
Compact filters sized for the lower band.
These filters were sized using PGGB's design method. The table gives approximate lengths; the design method itself is proprietary and not disclosed.
| DC–20 kHz target | Approximate output-rate taps | Measured DC–20 kHz | Measured 20–22 kHz |
|---|---|---|---|
| 107 bits | ~50k | 109 bits | 4 bits |
| 128 bits | ~60k | 130 bits | 4 bits |
| 256 bits | ~200k | 262 bits | 9 bits |
All three meet their lower-band targets. They also illustrate why a requirement through 20 kHz is very different from a requirement through 22 kHz. These are verified design examples, not an exhaustive search proving the absolute shortest possible filter. Their tap counts do not include every stage in the new engine.
Precision and independent checks.
The calculations use MATLAB with Advanpix multiprecision arithmetic. The equal-length comparison and compact filters were swept and refined at 386-bit precision (116 decimal digits), then the coefficients were independently regenerated and the selected worst-band points directly evaluated at 512-bit precision. The fixed-taper length comparison used 512-bit sweeps and refinement, followed by independently regenerated 1024-bit checks. The binary precision settings were checked directly in the runtime.
Every reported band result agrees at the precision shown in the article. This checks that the tiny residuals are not simply the rounding limit of the numerical evaluation. It does not imply that the commercial PGGB engine uses these verification precisions.
For context on arithmetic error during processing, see Do FFTs destroy transients? No., FFT results by filter length and the arithmetic validation. Arithmetic error depends on the operations, signals and implementation. A selected working precision is not automatically a guaranteed flat noise floor at exactly that many bits of accuracy.
Listening observations and the HF stage.
Listening tests by the author and testers found that, after the upper frequency region was sufficiently reduced, changing filter length or window parameters no longer changed the sound when reconstruction error in the retained band was below the processing's numerical error level.
That is a reported listening observation. The tables above are numerical reconstruction measurements; they are not listening scores or a calculation of audibility. The suggestion that the upper frequency region explains the earlier length-dependent listening changes remains an interpretation of the observations rather than an established scientific fact.
The actual HF-filter design, coefficients and construction are not provided. These measurements do not evaluate the new engine’s complete processing chain.
Data and scope.
The tables above summarize the measurements used in the figures. Full-precision numerical records are retained privately for validation.
References.
For the public foundations of windowed-sinc filtering, spectral images, the standard window definitions and floating-point rounding, see the article’s references and further reading. These sources provide background; the tables on this page are our own results.