Circuit Response
every model has an edge, and the figure carries it
Concept

Idle tones — where it appears

Discrete lines a quantiser inside a loop produces at frequencies unrelated to its input, from patterns the loop settles into rather than from distortion. They are what makes a shaped error a set of tones instead of a floor, and they are worst where the input is small and the loop has least to follow.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

A second integrator is 5.8 more decibels an octave, and a third 5.4 more. computed by solving, not by drawing. Modulators of order 1, 2, 3 marched one sample at a time with a one-bit quantiser, their in-band noise read from the transform of the error with the test tone a quarter of the way up the band. The measured slopes are 8.17, 13.93, 19.33 decibels per doubling of the oversampling ratio, against the 9, 15, 21 the white-noise argument predicts and the 3.01 plain oversampling gives. Each order is short of its own prediction by about a decibel, in the same direction, which is the error in the band not being white. The straight lines are each order's prediction drawn through its own first point, so what is compared is a slope against a slope.

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.

digital · Noise-shaping Against its own shaping the error is 17.4× tonal, not 51×. computed by solving, not by drawing. The share of an order-1 loop's in-band error sitting in its five largest lines, from 0.02 to 0.9 of full scale, with both nulls drawn. The lower level is 1.953 per cent — five lines of a FLAT error over 256 — and it is what this measurement has always been quoted against. The upper level is 5.74 per cent, which is what five lines of the loop's OWN shaping hold with no tone anywhere in them. Measured against the first the error is 51 times tonal at 0.02 of full scale and 26 at 0.9; against the second, 17.4 and 8.9. The direction survives the correction and the size does not.

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.

digital · Noise-shaping A second-order four-bit loop whose integrators are built round amplifiers of finite gain: the ratio in the band against oversampling. computed by solving, not by drawing: each integrator's pole at 1 − 1/A, half-scale input, half a step of shaped dither, three seeds averaged, squared Hann window. With perfect integrators the ratio rises 15.1 dB an octave, to 138.9 dB at 512. With A = 1000: 138.5 dB at 512, 0.4 below; with A = 100: 122.3 dB at 512, 16.6 below; with A = 30: 102.9 dB at 512, 35.9 below; with A = 10: 85.0 dB at 512, 53.9 below. Each curve leaves the ideal one near a ratio of about πA; for A = 10 the last octave gains 2.7 dB against the ideal's 15.1 — toward the three an octave a floor that is not shaped improves at.

The integrator that leaks the zero away

A noise-shaping loop puts its error's zero at zero frequency because its integrators never forget. An integrator built round an amplifier of gain A forgets a part in A every sample, and the zero moves off zero frequency to a floor. The cost is a function of one number, the oversampling ratio over π times the gain: for two leaky integrators it is 10 log(1 + (10/3)x² + 5x⁴), which the marched loops follow to within two decibels from gains of 10 to 300, and keeping it under a decibel needs a gain of 1.2 times the oversampling ratio. A first-order loop pays a second way: inputs within exactly 1/(2A − 1) of zero leave its output where it was.

digital · Noise-shaping

Named alongside it

The objects these essays reach for when they reach for this one.

Noise-shapingOversamplingQuantisationSigma deltaSpectral densityVerificationDesign tradeoffDynamic rangeEffective bitsIntegratorMeasurement conditionModel range

All concepts