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

Aperture jitter — the series

6 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. What 10 ps of aperture jitter is worth, in bits. computed by solving, not by drawing. Samples are taken at instants displaced by a seeded Gaussian of 10 ps and the error is measured against the same sinusoid sampled exactly. The line is −20 log(2π f × jitter), which the measurement matches to 0.12 dB across three decades. The penalty is exactly twenty decibels a decade of input frequency, because the error is the signal's slope times the timing error and nothing else — so a converter holds 16 bits only up to 199 kHz and 12 bits up to 3.18 MHz. An aperture figure quoted without an input frequency states no resolution at all.

    A picosecond, read as bits

    Every other boundary in this collection has a frequency, an amplitude or a size on its axis. This one has a duration. A converter that samples at t + δ instead of t gets a value wrong by the slope times δ, so the damage is proportional to input frequency and to nothing else about the part: ten picoseconds holds sixteen bits up to 199 kHz and twelve bits up to 3.18 MHz, falling at exactly twenty decibels a decade. An aperture figure quoted without an input frequency states no resolution at all.

    part 1 · digital
  2. Same 2 ps of jitter: a floor at -121 dB, or a line at -91. computed by solving, not by drawing. A 3.337 MHz sinusoid sampled at 10 MHz with two clocks of identical root-mean-square jitter. One clock's timing error is independent gaussians and the other's is a sinusoid at 130 kHz. The total error power is the same — -87.43 and -87.55 decibels below the signal, against the closed form's -87.55 — and the pictures are not. The random clock spreads it over 2048 bins, -120.5 dB each; the modulated one puts it into two lines at 3.337 ± 0.130 MHz, -91.5 dB down. A specification in picoseconds does not choose between them.

    A floor, or a line

    Two clocks with identical two-picosecond jitter sample the same sinusoid, and the total error power is the same to a tenth of a decibel — the closed form an earlier essay computed, right for both. One of them puts that error across two thousand bins at −120 dB each; the other puts it into two lines at −91.5. The gap is the processing gain, it grows with the record length because a density falls and a line does not, and nothing in a specification quoted in picoseconds root-mean-square distinguishes the two cases.

    part 2 · digital
  3. A band from zero to its top frequency takes 4.77 dB less from 10 ps of jitter than a tone at the top: it is the rms frequency that counts, not the highest. computed by solving, not by drawing. Forty-eight equal tones at random phases filling a band from 0 to 1 of its top frequency, sampled 16,384 times at instants displaced by a seeded 10 ps Gaussian, at seven top frequencies from 100 kHz to 100 MHz. The measured signal-to-error ratio (dots) is 88.64 dB at 1 MHz, against −20 log(2π × rms frequency × jitter) = 88.76 dB from the band's power-weighted rms frequency, 580 kHz — and against 84.04 dB for a tone at 1 MHz itself. The gap is 4.77 dB at every frequency, 10 log(3/(1 + r + r²)) for a flat band from r to 1 of the top: 4.77 dB (a factor of √3 in frequency) for a band from zero, nothing for a narrow one. At 1 MHz the band holds 14.43 bits where the tone holds 13.67.

    The frequency a band moves at

    The jitter rule is written for a sinusoid: ten picoseconds holds a tone at 1 MHz to 84.04 dB. A signal that fills the band up to 1 MHz holds 88.64 dB against the same clock, because each sample's error is its own slope times its own timing error, and a band's slope is set by its rms frequency — the top frequency over √3 for a flat band, 4.77 dB better. Pink content up to 1 MHz is 11 dB better than the rule and content falling as 1/f² is 29 dB better. So the same ten picoseconds hold sixteen bits to 199 kHz for a tone, 344 kHz for a flat band and 734 kHz for a pink one: an aperture figure needs a signal as well as a frequency.

    part 3 · digital
  4. The same 0.1 ps of period jitter: 107.0 dB at any record length if each edge is independent, 80.3 dB over 4096 samples if the edges walk. computed by solving, not by drawing, averaged over 8 seeded clocks at each of 5 record lengths. A 10.01 MHz tone sampled at 160 MHz by a clock whose every period has a spread of 0.1 ps, in two ways: each edge independently displaced (white), and each edge displaced from where the last one landed (a free-running oscillator's walk). The sampled tone is fitted by least squares for amplitude, delay and a small frequency offset, and what is left is the error. White jitter gives 107.2, 107.0, 107.0, 107.1, 107.0 dB — the closed form 107.0 dB, at every length. The walk gives 92.6, 86.4, 80.3, 74.5, 68.2 dB for 256, 1024, 4096, 16384, 65536 samples, falling 10.1 dB a decade, as the period spread times √(n/15): a jitter with no value until the length of the look is named.

    A jitter that grows with the look

    Two clocks with the same 0.1 ps spread in every period sample the same 10 MHz tone. If each edge is displaced independently, the error is 107.0 dB down at any record length. If each edge is displaced from where the last one landed — a free-running oscillator — the edges walk, and the error after fitting the best delay and frequency is 92.6 dB down over 256 samples, 80.3 over 4,096 and 68.2 over 65,536: ten decibels a decade of record, as the period spread times √(n/15). The one oscillator honestly has jitter figures from 0.10 to 4.7 ps depending on the length named. A locking loop is what gives the jitter a value: with a time constant of a hundred periods it stops growing at 87.0 dB.

    part 4 · digital
  5. With a 2 ps reference and a 0.1 ps-a-period oscillator the best loop has a time constant of 20.0 periods, where the clock's error is 0.453 ps — 90.9 dB on a 10 MHz tone. computed by solving, not by drawing, with seeded locked clocks as dots. A free-running oscillator whose period has a spread of 0.1 ps, locked by a first-order loop to a reference whose timing error, as the phase detector sees it each period, is 2 ps rms and white. The loop passes the reference's error below its corner and the oscillator's walk above it, so the stationary error is σₒ²/(1 − a²) + σᵣ²(1 − a)/(1 + a) with a = exp(−1/τ). The oscillator's part rises as √(τ/2)·σₒ and the reference's falls as σᵣ/√(2τ); they are equal, and their sum least, at τ = 20.00 periods, the ratio σᵣ/σₒ, where the clock's error is 0.4528 ps — √(σᵣ·σₒ). Seeded clocks of 65,536 periods: τ = 2, 1.000 ps; τ = 5, 0.656 ps; τ = 20, 0.452 ps; τ = 80, 0.649 ps; τ = 300, 1.187 ps; τ = 1000, 2.105 ps.

    The loop a reference allows

    A sampling clock is a free-running oscillator held by a loop to a reference, and the reference has an error of its own. With 0.1 ps of period spread in the oscillator and 2 ps of white error in the reference, a first-order loop of 20 periods leaves the clock 0.453 ps from ideal — 90.9 dB on a 10 MHz tone — and that is the best any first-order loop can do: the oscillator's share rises as the square root of the loop's time constant, the reference's falls as one over it, and they are equal at σᵣ/σₒ. The clock's error there is the geometric mean of the two, so a reference a hundred times worse costs ten. On a spectrum the rule reads as one instruction: put the loop's corner where the free oscillator's skirt meets the reference's floor. And it holds only for long captures. Below about seven loop time constants the free oscillator beats every loop.

    part 5 · digital
  6. With a 100 kHz flicker corner the 0.1 ps oscillator's ratio stops falling ten decibels a decade of capture and starts falling twenty, past 2,277 samples — 1.42 periods of the corner. computed by solving, not by drawing, from the exact expected residual of a summed process, with seeded clocks as dots (24 at each length to 16,384). A free-running oscillator with a 0.1 ps period spread and a flicker of its frequency whose timing spectrum meets the walk's at 100 kHz, sampling a 10.01 MHz tone at 160 MHz. Signal-to-timing-error after removing each capture's best delay and frequency: the walk alone falls ten decibels a decade, 85.7 dB at 1024, 67.6 dB at 65536; with the flicker, 84.1 and 52.9 dB. The flicker's share equals the walk's at 2,277 samples, 14.2 µs, which is 1.42 periods of the corner frequency; beyond it the ratio falls 19.5 dB a decade. Removing a quadratic as well buys 3.80 dB at 65,536 samples and changes no slope.

    A frequency that wanders

    Below its flicker corner an oscillator's phase noise falls thirty decibels a decade instead of twenty, because the oscillator's frequency itself wanders. A clock with 0.1 ps of period spread and a 100 kHz corner samples a 10 MHz tone at 84.1 dB over 1,024 samples and 52.9 over 65,536 — 14.7 dB worse than the walk alone — because past 2,277 samples its error grows as the capture rather than its square root. That crossing is 1.42 periods of the corner frequency for every corner. Fitting a parabola as well as a line buys 3.88 dB at every length and changes no slope; each further term buys a fixed fraction. And a first-order locking loop cannot stop it: the locked clock's variance grows with the logarithm of the record, 1.92 dB lost between 4,096 and 262,144 samples, until a loop with an integrator holds it at 87.5.

    part 6 · digital

All series