The droop a zero-order hold imposes at 48 kHz
Drawn above at its default parameters, which is almost never how an essay calls it. A placement states the numbers that essay is arguing about, so the figure a reader meets is about that argument rather than about the generator — 98% of the placements on this site pass one, and the phase that raised that number from 12% found eight captions describing a figure the page was not showing.
At those defaults the edge it states is -2.64 dB at 20 kHz, -3.92 dB at half — the right-hand slot
of the caption strip, which on this site is never used for anything else, and which is read
back out of the drawing above rather than out of the code that wrote it.
It belongs to Where a signal becomes a number, which is to say a change to it is a
change to lib/figures/digital.js.
It takes a slider on sample rate (kHz) with 6 settings,
and every one of them has passed the same assertions as the frame above — a figure whose
circuit stops doing what its caption says at any setting stops the build.
Called by 8 essays
which is the blast radius of changing it
An exact answer to a different question
The reconstruction that turns samples back into a signal is normally introduced as the thing that recovers what was there. Measured on both sides of half the sample rate it does something more interesting than failing: above the boundary it returns the alias to 1.6 parts in a thousand, which is the same accuracy it returns the input with below the boundary, and it is wrong about the input by twice the amplitude. Its error is not a degradation. It is exactness about something else.
The staircase on the way out
A converter does not emit impulses. It holds each sample for a whole clock period, which is a convolution with a rectangle and therefore a multiplication by a sinc — 0.14 dB down at a tenth of the sample rate, 0.91 at a quarter, and 3.92 at half, which is exactly 20 log(2/π). Nobody chose that droop and it is nearly eight times the half-decibel ripple of a Chebyshev passband. Measured on the transform of the staircase itself, it agrees with the closed form to 0.008%.
The nulls are where nothing is
A zero-order hold multiplies the whole repeated spectrum by one sinc, so it attenuates every image at the image's own frequency and its nulls land exactly on the multiples of the clock. Nothing is ever at a null: the two first-order images straddle it, and the closer the signal comes to half the clock the closer they come to each other. The hold gives 25.6 dB of image rejection to a tone at a twentieth of the clock, 1.74 dB at 0.45 of it, and nothing at all at half — which is the frequency the band most needs it at.
Flatness, and the two currencies it is bought in
The hold's droop takes the signal and everything arriving with it down together, so it costs no signal-to-noise ratio at all — a fact that is never stated and settles what correcting it can possibly be worth. Corrected digitally the price is headroom and is exactly the disease: 2.42 dB of flatness for 2.42 dB of output level. Corrected by an analogue shelf the price is image rejection and is nearly a constant: between 3.4 and 4.7 decibels whatever the band, so it is eight times the droop at a fifth of the clock and nine tenths of it at forty-nine hundredths.
One knob, and the two exponents it turns
Oversampling is quoted as buying one thing and buys two that improve at different rates. The hold's droop at the band edge falls with the SQUARE of the ratio — fitted exponent −2.011 over six doublings, from 2.640 dB at the Nyquist rate to 0.0097 at sixteen times it — while the nearest image moves out with the first power, exponent 1.022. The third quantity, the poles a reconstruction filter needs for sixty decibels, collapses from 20.5 to 3.21 by a ratio of four alone, because it is a logarithm of the second.
The pole a straight line is worth
A converter that joins its samples with straight lines instead of holding each one is convolving with a triangle rather than a rectangle, and its transform is the hold's sinc squared. Every decibel doubles: 5.28 dB of droop at a 20 kHz band edge on a 48 kHz clock instead of 2.64, and 5.85 dB of image rejection instead of 2.92. Both exponents of the oversampling ratio are unchanged. The second sinc is worth exactly one pole of reconstruction filter at every ratio, because a sinc's rejection of the nearest image is twenty times the log of that image's distance from the band edge, identically. A single analogue pole allowed the same droop gives a fraction of that rejection. The price is a whole clock of delay rather than half.
Four rectangles and no filter
A converter's output pulse can be any number of one-clock rectangles convolved together: one is the hold, two the straight line, four the cubic B-spline. Measured on its own waveform, n rectangles give the hold's sinc to the n-th power to a few parts in ten million, so each one rejects the nearest image by exactly what one analogue pole could at best — and each saves exactly one pole of reconstruction filter until none is left. At the Nyquist rate that is three poles of nineteen and a half, bought with 10.56 dB of droop. From about 2.8 times oversampling the cubic pulse needs no analogue filter at all for 60 dB, for 0.62 dB of droop at four times and two clocks of delay. The cost is that from three rectangles on the output no longer passes through its samples.
Half a clock against a pole
Inside a digitally controlled loop a converter's delay is phase at the crossover, and a straight-line output pays one more half clock than a hold: 180 degrees times the crossover over the clock. The analogue pole that rejects the nearest image as much as the straight line's second sinc does turns out to sit at the crossover itself — 0.990 of it when the loop crosses over at a tenth of its clock, sinc(r) of it in general — so it costs forty-five degrees however slow the loop is. By the phase estimate the half clock is the cheaper filter below 26% of the clock. Marched as a sampled loop the answer is 17.6%, and at a tenth of the clock a step overshoots 13.7% through the straight line against 32.8% through the hold and its pole, for the same image rejection measured on both waveforms.