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

Cheat Sheet

The most frequently used formulas, numbers and knobs of the whole site, condensed onto one page. Every entry links back to its full derivation page.

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Core formulas (all verified against the original PDFs)

Topic	Formula	Source
ISF operational definition	$\Delta\phi=\dfrac{\Gamma(\omega_0\tau)}{q_{max}}\Delta q$	[P1] Eq.(10)(11) → impulse_to_phase_shift
LTV phase response	$\phi(t)=\dfrac{1}{q_{max}}\displaystyle\int_{-\infty}^{t}\Gamma(\omega_0\tau)\,i_n(\tau)\,d\tau$	[P1] Eq.(11) → convolution
ISF Fourier	$\Gamma=\dfrac{c_0}{2}+\sum_n c_n\cos(n\omega_0\tau+\theta_n)$	[P1] Eq.(12) → fourier
Parseval / rms	$\sum_{n=0}^\infty c_n^2=2\Gamma_{rms}^2$	[P1] Eq.(20) → rms_isf
White noise 1/f²	$\mathcal{L}=10\log_{10}\!\Big(\dfrac{\Gamma_{rms}^2}{q_{max}^2}\dfrac{\overline{i_n^2}/\Delta f}{4\,\Delta\omega^2}\Big)$	[P1] Eq.(21) → white_noise
Flicker 1/f³	$\mathcal{L}=10\log_{10}\!\Big(\dfrac{c_0^2}{q_{max}^2}\dfrac{\overline{i_n^2}/\Delta f}{8\,\Delta\omega^2}\dfrac{\omega_{1/f}}{\Delta\omega}\Big)$	[P1] Eq.(23) → flicker
1/f³ corner	$\Delta\omega_{1/f^3}=\omega_{1/f}\dfrac{c_0^2}{2\Gamma_{rms}^2}$	[P1] Eq.(24)
SSB↔PSD	$\mathcal{L}(\Delta f)\approx\tfrac12 S_\phi(\Delta f)$	psd
phase→time	$\Delta t=\dfrac{\Delta\phi}{2\pi f_0}$	standard
rms jitter	$\sigma_t=\dfrac{1}{2\pi f_0}\sqrt{\displaystyle\int_{f_1}^{f_2}S_\phi\,df}$	serdes
Accumulated jitter	$\sigma_{\Delta t}=\kappa\sqrt{\Delta t}$, $\kappa=\dfrac{\Gamma_{rms}}{q_{max}}\sqrt{\tfrac12\tfrac{\overline{i_n^2}}{\Delta f}}$	[P2] Eq.(8)(12)
Ring frequency	$f_0=\dfrac{1}{2N\tau_D}$	[P2] Eq.(15)
Ring $\Gamma_{rms}$	$\Gamma_{rms}=\sqrt{\dfrac{2\pi^2}{3\eta^3}}\;\dfrac{1}{N^{1.5}}\Rightarrow\Gamma_{rms}\propto N^{-3/2}$ (at $\eta=0.75$, $\approx4/N^{1.5}$, the solid line in [P2] Fig.8; the radical covers only the constant)	[P2] Eq.(16)
Ring FOM	$\mathcal{L}=\dfrac{8}{3\eta}\dfrac{kT}{P}\dfrac{V_{DD}}{V_{char}}\Big(\dfrac{f_0}{\Delta f}\Big)^2$ (no $N$!)	[P2] Eq.(23)
Generalized Adler	$\dfrac{d\theta}{dt}=(\omega_0-\omega_{inj})+\Omega(\theta)$, $\Omega=\langle\tilde\Gamma\,i_{inj}\rangle$	[P3] Eq.(30)(33)
APF / amplitude decay	$\tau_0=\dfrac{2Q}{\omega_{osc}}$, $\tilde\Lambda_1=\dfrac{\tau_0}{q_{max}}\angle0°$ (in quadrature with the ISF)	[P4] Eq.(25)(26)

Canonical numbers (consistent site-wide)

Example	Setup	Result
A: impulse→time	$q_{max}=1$ pC, $\Delta q=1$ fC, $\Gamma=0.5$, $f_0=5$ GHz	$\Delta\phi=5\times10^{-4}$ rad, $\Delta t=15.9$ fs
B: white-noise $\mathcal{L}$	$\Gamma_{rms}=0.5$, $q_{max}=1$ pC, $S_i=10^{-24}$ A²/Hz, $\Delta f=1$ MHz	$\mathcal{L}=-148$ dBc/Hz
C: jitter integral	$\mathcal{L}(1\text{MHz})=-100$ dBc/Hz, 1/f², 1→100 MHz, 5 GHz	$\sigma_t=447.9$ fs
Ring FOM	$\gamma=2/3$, $V_{DD}/V_{char}=3$, $P=1$ mW, others as above	$\mathcal{L}\approx-91$ dBc/Hz

Want to sweep the parameters yourself? Use the interactive calculator.

Conversion anchors at 5 GHz

• $1$ mrad $\approx 32$ fs; $1$ rad $\approx 31.8$ ps; period $=200$ ps.
• dBc/Hz → linear: $10^{\mathcal{L}/10}$; $S_\phi=2\times$ linear.
• $2\times q_{max}$ or $\tfrac12\Gamma_{rms}$ → $\mathcal{L}$ drops by 6 dB.

Design knobs (to lower phase noise)

To lower	Knob	Why
1/f² (white noise)	↑ $q_{max}$ (swing/energy), ↓ $\Gamma_{rms}$	$\mathcal{L}\propto\Gamma_{rms}^2/q_{max}^2$
1/f³ (close-in)	make the waveform symmetric → ↓ $c_0$	corner $\propto c_0^2/\Gamma_{rms}^2$
Jitter (time domain)	same as above (same $\Gamma_{rms}^2/q_{max}^2$); or lock in a PLL/CDR	$\kappa\propto\Gamma_{rms}/q_{max}$
Where noise is injected	avoid phases where $\lvert\Gamma\rvert$ is large (small slope)	cyclostationary $\Gamma_{eff}=\Gamma\alpha$

v5–v8 quick reference (jitter kernels / κ↔D↔linewidth / App.B / M:N locking / PLL / FOM / SerDes)

Rule for this block: every entry is copied verbatim from its source page, convention flags included. The link is the single source of truth for the derivation.

Jitter kernels (three kernels + white-FM closed form) — jitter_kernels

Kernel	Formula (one-sided $S_\phi$, $\int_0^\infty$ convention)	White-FM closed form
TIE (0th order)	$\sigma_{TIE}^2=\dfrac{1}{\omega_0^2}\displaystyle\int_{f_1}^{f_2}S_\phi\,df$	—
N-period (1st-order difference)	$\sigma_P^2(N)=\dfrac{1}{\omega_0^2}\displaystyle\int_0^\infty S_\phi\,4\sin^2(\pi fNT)\,df$	$=\kappa^2NT$ (exact match to [P2] Eq.(8))
Cycle-to-cycle (2nd-order difference)	$\sigma_{c2c}^2=\dfrac{1}{\omega_0^2}\displaystyle\int_0^\infty S_\phi\,16\sin^4(\pi fT)\,df$	$=2\kappa^2T\Rightarrow\sigma_{c2c}=\sqrt2\,\sigma_P(1)$

The prefactor is $1/\omega_0^2$, not $2/\omega_0^2$ (that 2 belongs to the double-sided-spectrum or $\mathcal{L}=\tfrac12S_\phi$ bookkeeping).

κ↔D↔linewidth↔ADEV↔$S_\phi$ dictionary (with the 19.9/39.8 mHz canonical numbers) — diffusion_dictionary

Lead quantity: $\kappa^2=\dfrac{\Gamma_{rms}^2}{2q_{max}^2}\dfrac{\overline{i_n^2}}{\Delta f}$ ([P2] Eq.(11)/(12)); canonical ($\Gamma_{rms}=0.5$) gives $\kappa^2=0.125$ rad²/s, true LC ($\Gamma_{rms}=1/\sqrt2$) gives $0.25$.

"Outfit"	Formula	Canonical value
$\kappa$ (phase) / $\kappa_t$ (time)	$\sigma_{\Delta\phi}=\kappa\sqrt{\Delta t}$; $\kappa_t=\kappa/\omega_0$	$\kappa=0.354$ rad/$\sqrt{\text{s}}$
$D$ convention A / B	$\mathrm{Var}=D_{A}\vert t\vert=2D_{B}\vert t\vert\Rightarrow D_{A}=\kappa^2,\ D_{B}=\kappa^2/2$	$0.125$ / $0.0625$
Lorentzian 3-dB linewidth	$\Delta f_{3\mathrm{dB}}=\kappa^2/(2\pi)$	19.9 mHz (true LC 39.8 mHz)
$S_\phi$ coefficient (one-sided)	$S_\phi=2\kappa^2/(2\pi f)^2$	$b_{-2}=6.33\times10^{-3}$ rad²·Hz
White-FM ADEV	$\sigma_y(\tau)=\kappa/(2\pi f_0\sqrt\tau)$	$1.13\times10^{-11}$@1s (5 GHz)

Three factor-of-2 families to settle before you convert: one-sided vs double-sided PSD, $\mathrm{Var}=D\vert t\vert$ vs $2D\vert t\vert$, and SSB $/2$ vs $/4$.

App.B closed forms: $\Gamma_{rms}(A,N)$, $c_0$, corner — asymmetric_isf_closed_form

$$\Gamma_{rms}^2=\frac{2\pi^2}{3\eta^3}\frac{1}{N^3}\left[4\frac{1+A^3}{(1+A)^3}\right],\quad \Gamma_{dc}=\frac{2\pi}{\eta^2}\frac{1}{N^2}\left(\frac{1-A}{1+A}\right),\quad c_0=2\Gamma_{dc}$$ $$f_{1/f^3}=f_{1/f}\cdot\frac{3}{2\eta N}\cdot\frac{(1-A)^2}{1-A+A^2}\qquad([\text{P2}]\ \text{Eq.}(52)\text{–}(57),\ \text{p.803})$$

$A\equiv f'_{rise}/f'_{fall}$; $A=1$ degenerates exactly to [P2] Eq.(16) ($\Gamma_{rms}\propto N^{-1.5}$); the corner is quadratically flat at $A=1$, symmetric under $A\to1/A$, and $\propto1/N$. Convention flag: this corner is the [P2] Eq.(7)/(57) value; [P1] Eq.(24) (substituting $c_0=2\Gamma_{dc}$) $=2\times$ this value.

[P1] appendix: $\Gamma=f'/(f'^2+f''^2)$ — isf_from_waveform

$$\Gamma(x)=\frac{f'}{f'^{\,2}+f''^{\,2}}\qquad([\text{P1}]\ \text{Eq.}(37),\ \text{p.193})$$

Substituting $f=\cos x$: the denominator $\sin^2+\cos^2=1$, giving $\Gamma=-\sin x$ exactly, and it stays bounded at the waveform peak (this is what resolves the $1/\text{slope}$ heuristic's divergence). Ranking of the three methods: A direct impulse injection (most accurate) → B this closed form (one period of waveform suffices) → C slope approximation $\Gamma=f'/f_{max}'^2$ (cheapest, ring-specific, Eq.(38)).

Injection-locked noise shaping corner $=\omega_L\cos\theta_{ss}$ — injection_locking_noise Part A

$$\omega_c\equiv\omega_L\cos\theta_{ss}=\sqrt{\omega_L^2-\Delta\omega^2}\qquad([\text{P3}]\ \text{Eq.}(40)\text{'s pull-in frequency})$$

At lock center ($\Delta\omega=0$) the corner is widest; at the lock edge ($\Delta\omega\to\omega_L$), $\cos\theta_{ss}\to0$ and the high-pass suppression of the oscillator's own noise disappears entirely. (First-order-PLL picture: own noise is high-passed, reference noise is low-passed.)

Optimal injection waveform $\omega_L^*=I_{rms}\tilde\Gamma_{rms}$ — injection_locking_noise, final section

For a fixed $I_{rms}\equiv\sqrt{\langle i_{inj}^2\rangle}$, Cauchy–Schwarz gives the lock-range upper bound ([P3] Eq.(43)–(45), pp.2119–2120):

$$\omega_L^*=I_{rms}\,\tilde\Gamma_{rms},\qquad i_{inj,0}^*(x)=\pm\frac{I_{rms}}{\tilde\Gamma_{rms}}\tilde\Gamma(x)$$

Equality holds iff the injection waveform matches the ISF's shape (matched filter). Pure-sinusoid ISF: sinusoidal injection is already optimal (gain 1). Ring narrow-pulse ISF: gain $\approx\sqrt{\eta N/3}$ ($\approx2.06$ at $N=17$, matching [P3] Fig.19's "almost doubled").

M:N subharmonic locking / ILFD: $\lvert\tilde\Gamma_N\rvert$ — paper_004

When locked at $M\omega_{inj}=N\omega_{osc}$ ($M=1$ is the ÷$N$ ILFD case), the half lock range rides on only the $N$-th ISF harmonic ([P4] Eq.(28)–(30), p.2129, verified):

$$\Omega(\theta)=\frac12 I_{inj}\lvert\tilde\Gamma_N\rvert\cos(N\theta+\angle\tilde\Gamma_N)\ \Rightarrow\ \omega_L=\frac12 I_{inj}\lvert\tilde\Gamma_N\rvert=\frac{I_{inj}\,c_N}{2\,q_{max}}$$

A half-wave-symmetric ISF ($c_2=c_4=\cdots=0$) means ÷2 cannot lock at first order — the same differential-node symmetry that's good news for phase noise is bad news for ILFD (fix: switch to the tail node to pick up $c_2$).

Subharmonic multiplication (×N, ILCM): $\omega_L$, $\beta$, corner — subharmonic_injection

The other half of the duality: at $N=1$, locking is carried by the injection waveform's own $N$-th harmonic (not the ISF's). Canonical: $f_0=5$ GHz, $q_{max}=1$ pC, $N=20$ ($f_{ref}=250$ MHz), $q_{inj}=50$ fC.

Quantity	Formula	Canonical value
Multiplier lock range	$\omega_L=\tfrac12\vert I_N\vert\vert\tilde\Gamma_1\vert$ (impulse train: $\Delta\omega_L=\dfrac{q_{inj}}{q_{max}}\cdot\dfrac{f_0}{N}$)	$f_L=1.989$ MHz
Realignment factor	$\beta\equiv-q_{inj}\tilde\Gamma'(\theta_{ss})$ (stable for $0\lt\beta\lt2$, converges in $\approx1/\beta$ injections)	$0.0498$
Noise corner	$f_c\approx\beta f_{ref}/2\pi$ (= $\Delta f_L$ at lock center)	$2.09$ MHz

ADEV floor: $\sqrt{2\ln2\cdot h_{-1}}$ — allan_variance

$$\sigma_{y,\text{floor}}=\sqrt{2\ln2\cdot h_{-1}}\approx1.1774\sqrt{h_{-1}}\qquad(\text{flicker FM, the }\tau\text{-independent floor})$$

Canonical (with a $1/f^3$ corner of $3.2$ kHz): floor $\approx1.06\times10^{-9}$ (1.1 ppb); knee $\tau_{knee}=0.3607/f_c\approx113\ \mu$s (where the white-FM $\tau^{-1/2}$ segment meets the floor).

PLL peaking: $\zeta=0.707\to2.09$ dB — pll_noise_budget, supplementary derivation

A type-II loop with a zero always peaks: $f_{pk}=f_n\sqrt{2/(s+1)}$, $s=\sqrt{1+1/\zeta^4}$ (golden-ratio easter egg: at $\zeta=1/\sqrt2$ the peak value $=\varphi=1.618$).

$\zeta$	$f_{pk}/f_n$	Peaking (dB)	Phase margin
0.707	0.786	2.09	65.5°
1.0	0.707	1.25	76.3°

Cascading 20 SONET regenerator stages: $20\times2.09=41.8$ dB, which is why specs cap peaking at the 0.1 dB level per stage.

FOM $=173.8-10\log_{10}F_{eff}$ — fom_limit

$$\mathrm{FOM}=173.8\ \text{dB}-10\log_{10}F_{eff}\qquad(T=300\ \text{K, paired with }1\cdot kT\text{, not }2kT)$$

Ring ceiling ($\gamma=2/3,\eta=1$): $F_{eff,min}=32/9\Rightarrow\mathrm{FOM}_{max}^{ring}=168.32$ dB; the LC ceiling rises with $Q$ ($F_{eff}\propto1/Q^2$).

Aperture SNR: $-20\log_{10}(2\pi f_{in}\sigma_t)$ — adc_aperture_jitter

$$\text{SNR}_{jitter}=-20\log_{10}(2\pi f_{in}\sigma_t)\ \text{dB},\qquad \text{ENOB}=\frac{\text{SNR}-1.76}{6.02}\ \text{bit}$$

Canonical $\sigma_t=447.9$ fs: at $f_{in}=f_0=5$ GHz, $2\pi f_{in}\sigma_t=\sigma_\phi=14.07$ mrad $\Rightarrow$ SNR $=37.0$ dB (carried over directly from example C).

TJ $=\mathrm{DJ}_{\delta\delta}+2Q^{-1}(\mathrm{BER})\sigma$ — dj_dual_dirac

$$\mathrm{TJ}(\mathrm{BER})=\mathrm{DJ}_{\delta\delta}+2\,Q^{-1}(\mathrm{BER})\,\sigma$$

At BER$=10^{-12}$, $Q^{-1}=7.034$, so $\mathrm{TJ}=\mathrm{DJ}_{\delta\delta}+14.07\,\sigma$. DJ does not change with BER (bounded, paid once); the RJ term grows only slowly as BER tightens ($10^{-12}\to10^{-15}$: $14.07\sigma\to15.88\sigma$).

×N / ÷N / buffer bookkeeping — clock_chain_budget

Element	Phase relation	$\mathcal{L}(f)$ bookkeeping
Ideal ×N multiplication	$\phi_{out}=N\phi_{in}$	$\mathcal{L}+20\log_{10}N$
Ideal ÷N division	$\phi_{out}=\phi_{in}/N$	$\mathcal{L}-20\log_{10}N$
Through a PLL (×N)	in-band tracks ref, out-of-band tracks VCO	$N^2S_{ref}\lvert H_{lp}\rvert^2+S_{vco}\lvert H_{hp}\rvert^2$
Buffer/divider additive floor	$\phi_{out}=\phi_{in}+\phi_{add}$ (uncorrelated)	$\mathcal{L}_{out}=10\log_{10}(10^{\mathcal{L}_{in}/10}+10^{\mathcal{L}_{buf}/10})$ (convert to linear, add, then convert back to dB)

Conserved quantity: under ideal ×N/÷N, the time-domain $\sigma_t$ (in seconds) never changes — only the $\mathcal{L}$/rad bookkeeping does.

$K_{push}$ path: $S_\phi=K^2S_v/\Delta f^2$ — varactor_tuning_supply_pushing

$$K_{VCO}\equiv\frac{\partial f_0}{\partial V_{tune}},\quad K_{push}\equiv\frac{\partial f_0}{\partial V_{DD}},\qquad S_\phi(\Delta f)=\frac{K_{VCO}^2\,S_v(\Delta f)}{\Delta f^2}\ \ (\text{the supply version is identical, with }K_{VCO}\to K_{push},\ S_v\to S_{v,DD})$$

Side by side with the ISF white-noise result: both have a $\Delta\omega^2$/$\Delta f^2$ integrator in the denominator, differing only in what feeds it ($\Gamma_{rms}/q_{max}$ vs $2\pi K_{VCO}$). White tune/supply noise $\to 1/f^2$; $1/f$ tune/supply noise $\to 1/f^3$ (parallel to the device $c_0$ mechanism). lab_38 gives a first-principles measurement of $K_{push}$ (ratio to $\beta$ of 1.002).

The five papers in one line each

• [P1] Hajimiri–Lee 1998: the ISF theory itself (the oscillator is LTV).
• [P2] 1999: ISF applied to rings — closed-form jitter/PN, $N$-independence.
• [P3] Hong 2019 I: ISF generalizes Adler → generalized injection locking.
• [P4] Hong 2019 II: the APF (amplitude counterpart of the ISF), $\tau_0=2Q/\omega_0$, frequency division.
• [P5] a sense amplifier paper, unrelated to the ISF (honestly labeled).

Full cross-reference: paper_summary_table, equation_index.