Quantisation — where it appears
Named by 10 essays across one field — each of them below, with the objects they name alongside it.
The floor a converter sets
A converter's resolution is quoted as a number of bits, which is a property of the converter. What it can actually resolve is a property of the circuit in front of it, and the two cross: measured against the Johnson noise of a 1 kΩ source in 100 kHz of bandwidth, the quantiser is the limit up to 18.80 bits and the resistor is the limit above it. Past that crossing every further bit buys a more precise measurement of thermal noise.
Six decibels a bit, and the half step blamed on it
Every converter data sheet quotes 6.02N + 1.76 dB and every bench measurement comes up short of it, which is usually explained by calling the formula an approximation. Measured one step inside full scale it agrees to 0.15 dB at every bit count from eight to sixteen. Driven to exactly full scale the same converter loses 0.83 dB at eight bits — because 327 of 8192 samples hit the top code, and a mid-tread converter's largest code is one step below full scale.
When a floor stops being a floor
Quantisation error is treated as noise and behaves like noise while the input crosses many levels. As the amplitude falls it stops: measured at the bins its harmonics occupy, the share of the error's power sitting in harmonics of the input rises from 0.29% at 230 levels to 76% at 1.3. The floor has not moved. It has become a distortion product locked to the signal, and the fix is to add noise on purpose — one whole least significant bit, and no more.
One bit, and where the noise went
Sampling faster spreads a fixed quantity of quantisation noise over a wider band, so the part inside the band of interest falls by 3.01 dB for every doubling — half a bit. Putting the quantiser inside a loop with an integrator does something different in kind: measured on a modulator marched forward one sample at a time, with its test tone inside the band the ratio is quoted over, the in-band noise falls by 8.99 dB per doubling. At an oversampling ratio of 128, one bit is worth 9.18 — and the octaves scatter by a decibel each, which is the loop telling the truth about what its error is made of.
The loop that is worse at full scale
A second integrator in a one-bit loop takes the shaping law from nine decibels an octave to fourteen, and a third to nineteen. What it charges is not in decibels at all: past six tenths of full scale the ratio starts falling, and a fourth integrator's states run away at seven tenths. The best input to a second-order modulator is 0.6 of the reference it is measured against — an amplitude boundary of exactly the kind this collection is built for, on the one object in the field that has no continuous output to draw.
Two loops, and the mismatch between them
The rung below marched single loops of second, third and fourth order and found the amplitude at which each stops working, falling with the order — which is why nobody builds a fourth-order single loop. The standard answer is two first-order loops with the first one's error fed to the second and differentiated back out, giving second-order shaping out of parts that cannot be unstable. The cancellation is between an analogue path and a digital one, and it is worth 25 decibels until the two differ by five per cent.
Zero in, and not zero out
The rung below rounds a filter's coefficients and marches it in double precision, and its own list of what it did not do names the other half: the products are rounded too. Put that in and a twelve-bit eighth-order cascade fed nothing at all settles two per cent of full scale away from zero and stays there — and the offset does not shrink with the word length, it grows as the square of how far the corner sits below the sample rate, reaching eighteen per cent at a hundred and twenty-eight times.
Which section goes first
A cascade of four biquads can be assembled in twenty-four orders, all of which realise exactly the same transfer function. They do not cost the same: the round-off floor runs from 24 to 100 least significant bits and the largest value any section reaches runs over eighteen decibels — and the two go opposite ways, so eleven of the twenty-four are genuine choices and thirteen are beaten on both counts by another arrangement.
A floor, or five tones
The field's sharpest statement about a one-bit loop is that three quarters of its in-band error sits in five lines, against the 1.953 per cent a white error would put in any five — a factor of thirty-eight. The comparison is to a white error, and a shaping loop exists to make its error anything but white. Measured against the loop's own transfer function, which the loop reproduces line by line to a part in five hundred when it is dithered, the same error is 17.4 times tonal at a fiftieth of full scale and 8.9 times at nine tenths. And 99.5 per cent of it at the quiet end is two harmonics of the input.
The dither that is a decision
One whole least significant bit is quoted everywhere as the dither, which makes a decision look like a constant. Swept, the axis is a corner and a comb. An eighth of a step leaves 64.5 per cent of the error locked to the signal and one whole step leaves 0.34; above that the sweep mean is 0.283, 0.297, 0.297 and 0.301 per cent at one, two, three and four steps and 0.526 at a step and a half, which fails at exactly the small amplitudes dither exists for. The price is 10·log₁₀(1 + L²) to 0.118 of a decibel, and four steps cost 12.41 for nothing.
Named alongside it
The objects these essays reach for when they reach for this one.
Effective bitsOversamplingDesign tradeoffDynamic rangeNoise shapingTotal harmonic distortionBiquad cascadeDitherIdle tonesModel rangeNoise floorNumerical error