Unified Notation Table

β: This English translation is in beta — the Traditional-Chinese original is the authoritative version.

Different papers write the same thing with different symbols. This page unifies them; every later chapter follows this table. If you encounter a different convention in one of the papers, come back here to cross-reference.

How to use this page: skim it once to get acquainted; when actually reading the derivations, come back and look up any symbol you do not recognize. Every quantity is labeled with its unit — doing a dimension check is the fastest way to catch mistakes.

Main symbols

Symbol	Meaning (intuition)	Unit	Used in	Notes
$t$	time	s	all	—
$\tau$	injection instant of the noise/impulse	s	[P1]	the ISF's argument is the injection phase $\omega_0\tau$
$T$	oscillation period $T=1/f_0$	s	all	—
$\omega_0$	oscillation angular frequency $=2\pi f_0$	rad/s	all	—
$f_0$	oscillation (carrier) frequency	Hz	all	e.g. 5 GHz
$\phi(t)$	excess phase (the deviation beyond the ideal phase)	rad	[P1][P2]	phase noise / jitter lives here
$\Delta\phi$	phase step / phase error	rad	all	the jump caused by one impulse
$A(t)$	instantaneous amplitude	V or normalized	[P1][P4]	perturbations get pulled back (see [P4] APF)
$\Gamma(\omega_0\tau)$	ISF, the oscillator's "phase sensitivity" to noise; dimensionless, $2\pi$-periodic	—	[P1]	not the noise itself, but a weighting function
$q_{max}$	maximum node charge swing $=C\cdot V_{max}$	C	[P1]	used for normalization; the larger it is, the lower the phase noise
$\Delta q$	injected charge $=\int i\,dt$	C	[P1]	e.g. 1 fC
$i_n(t)$	noise current	A	[P1][P2]	the noise source injected into the node
$\overline{i_n^2}/\Delta f$	current-noise power spectral density (single-sided)	A²/Hz	[P1]	white: independent of frequency
$S_i(f)$	current-noise PSD	A²/Hz	all	another way of writing the same thing
$S_\phi(f)$	phase PSD (single-sided)	rad²/Hz	all	integrating over $f$ gives $\sigma_\phi^2$
$\mathcal{L}(\Delta f)$	SSB phase noise (single-sideband phase noise)	dBc/Hz	all	$\approx\frac12 S_\phi$
$\Delta f,\ \Delta\omega$	offset frequency (how far from the carrier)	Hz, rad/s	all	$\Delta\omega=2\pi\Delta f$
$c_0$	DC Fourier coefficient of the ISF (DC value $=c_0/2$)	—	[P1]	the key to 1/f upconversion
$c_n,\ \theta_n$	amplitude / phase of the ISF's $n$-th harmonic	—	[P1]	moves noise near $n\omega_0$ onto the carrier
$\Gamma_{rms}$	rms value of the ISF	—	[P1][P2]	sets the magnitude of the 1/f² phase noise
$\Gamma_{eff}$	effective ISF (including cyclostationarity)	—	[P1]	$\Gamma_{eff}=\Gamma\cdot\alpha$
$\alpha(\omega_0 t)$	noise-modulating function (NMF): when the device is "leaking noise"	—	[P1]	$0\le\alpha\le1$, periodic
$\sigma_t$	rms timing jitter	s	[P2]	what SerDes cares about most
$\sigma_\phi$	rms phase	rad	all	$\sigma_t=\sigma_\phi/(2\pi f_0)$
$\kappa$	proportionality constant of ring accumulated jitter	$\sqrt{\mathrm{s}}$	[P2]	$\sigma_{\Delta t}=\kappa\sqrt{\Delta t}$
$\omega_{1/f}$	the device's 1/f-noise corner	rad/s	[P1]	note: ≠ the phase-noise 1/f³ corner
$N$	number of ring-oscillator stages	—	[P2]	$\Gamma_{rms}\propto N^{-3/2}$
$Q$	tank quality factor	—	[P1]	appears in the Leeson comparison
$\eta$	proportionality constant in the ring frequency / FOM	—	[P2]	$f_0=1/(2N\tau_D)$

The four "dialects" of jitter

Many people lump all jitter together, when in fact the measured quantities differ:

Name	Definition	Intuition
period jitter	$T_k-T$ (a single period vs. nominal)	how long/short this beat is
cycle-to-cycle jitter	$T_{k+1}-T_k$ (difference between two adjacent beats)	how fast the beat changes from one to the next
accumulated / long-term jitter	timing error between two edges separated by $\Delta t$, $\sigma_{\Delta t}=\kappa\sqrt{\Delta t}$	an open-loop oscillator drifts further the longer it runs
random jitter (RJ)	Gaussian, unbounded; described by $\sigma$	what SerDes BER uses to estimate eye closure

See psd_phase_noise_jitter and serdes_clocking_connection for details.

Symbol correspondence across the papers (where unification is needed)

Concept	This site's symbol	Papers' notation / remarks
ISF	$\Gamma(\omega_0\tau)$	[P1][P2] use $\Gamma$; some later literature uses $h$ or "ISF"
maximum charge	$q_{max}$	[P1] $q_{max}=C_{node}V_{max}$; in rings it corresponds to the per-stage node charge
offset frequency	$\Delta\omega$ or $\Delta f$	[P1] mostly uses $\Delta\omega$; datasheets use $\Delta f$ (Hz)
amplitude counterpart of phase sensitivity	(see [P4]) APF $\tilde\Lambda$	[P4] amplitude perturbation function; ideal LC fundamental $\tilde\Lambda_1=\frac{\tau_0}{q_{max}}\angle0°$, in quadrature with the ISF; $\tau_0=2Q/\omega_0$
dimensioned ISF	$\tilde\Gamma=\Gamma/q_{max}$	[P3] Eq.(26): Hong uses the dimensioned version (rad/C); this site's core uses the dimensionless $\Gamma$
phase equation (injection)	generalized Adler	[P3] Eq.(30),(33): $\frac{d\theta}{dt}=(\omega_0-\omega_{inj})+\Omega(\theta)$, $\Omega=\langle\tilde\Gamma\,i_{inj}\rangle$
PPV / adjoint / Floquet	—	not in these 5 PDFs; external literature (Demir et al.), see effective_isf

Notation trap: $c_0$ is the Fourier coefficient, while the DC value of the ISF is $c_0/2$ (see Eq.(12)). This factor is an easy source of error when computing the 1/f³ corner (Eq.(24)); later chapters will keep reminding you.