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

Paper Summary Table

The source folder contains 5 PDFs. This page lays them out in a single table, then organizes their division of labor from a teaching viewpoint: which paper is the ISF core, which extends to rings, which covers injection, which is actually off-topic, and which equations are the same thing written differently, plus which notation differs and needs to be unified.

Honesty note (up front): of these 5, 4 are Hajimiri-series oscillator phase-noise/ injection papers, and 1 (Hajimiri_ISCS_98.pdf = [P5]) is actually a cross-coupled sense-amplifier paper, unrelated to ISF — this site treats it only as a footnote. Also, the frequently co-mentioned PPV / adjoint / PSS / PNoise / Floquet topics have no dedicated paper among these 5 PDFs — they belong to external literature and are honestly flagged below.

One-page summary table

Data comes from extracted/paper_metadata.json. Citation strings follow the site convention verbatim (see references).

Paper	Year	Main Contribution	Key Equations	Key Figures	What We Use It For
[P1] Hajimiri & Lee, A General Theory of Phase Noise in Electrical Oscillators, IEEE JSSC 33(2):179–194 (general.pdf, paper_001)	1998	Establishes the LTV / ISF theory of oscillator noise: introduces $\Gamma(\omega_0\tau)$ and the $q_{max}$ normalization, derives closed forms for $1/f^2$ and $1/f^3$ plus design rules (large $q_{max}$, small $\Gamma_{rms}$, waveform symmetry to suppress $c_0$).	Eq.(1) output decomposition; (9) $\Delta V=\Delta q/C$; (10)(11) ISF impulse response and convolution; (12) Fourier series; (19)(20) summation and Parseval; (21) the signature $1/f^2$ formula; (22)(23)(24) flicker $1/f^3$ and corner	Fig. 4 (peak vs ZC injection), Fig. 6 ($\Delta\phi$–$\Delta q$ linearity), Fig. 7 (LC/ring ISF shapes), Fig. 8 ($n\omega_0$→carrier downconversion), Fig. 11/12 ($1/f^3,1/f^2$,floor panorama and corner)	The ISF core foundation — the theory source of chapters 02/03 and most labs.
[P2] Hajimiri, Limotyrakis & Lee, Jitter and Phase Noise in Ring Oscillators, IEEE JSSC 34(6):790–804 (jitter_ring.pdf, paper_002)	1999	Applies the ISF to ring oscillators: closed forms for jitter/phase noise, the $\Gamma_{rms}\propto N^{-3/2}$ scaling, "ring phase noise nearly independent of stage count $N$ at fixed power and frequency", and how rise/fall symmetry suppresses $1/f$ upconversion.	(8) accumulated jitter $\sigma_{\Delta t}=\kappa\sqrt{\Delta t}$; (11)(12) $\kappa^2\propto\Gamma_{rms}^2/q_{max}^2\cdot\overline{i_n^2}/\Delta f$; (15) $f_0=1/(2N\tau_D)$; (16) $\Gamma_{rms}=\sqrt{2\pi^2/(3\eta^3)}\cdot1/N^{1.5}\propto N^{-3/2}$; text result $\mathcal{L}\vert _{1/f^2}\approx\frac{8}{3\eta}\,\frac{V_{DD}}{V_{char}}\,\frac{kT}{P}(\omega_0/\Delta\omega)^2$ ($\eta$ is the stage-delay proportionality constant of Eq.14, $\approx 1$; $\gamma$ enters only through $V_{char}=\Delta V/\gamma$)	Fig. 5 (ring single-stage ISF), Fig. 8 ($\Gamma_{rms}$ vs $N$; solid line is $4/N^{1.5}$@$\eta=0.75$), Fig. 17 (phase noise vs symmetry control voltage, minimum at the symmetric point)	The ring-oscillator extension — chapter 06 lc_vs_ring, symmetry, and lab_03.
[P3] B. Hong & A. Hajimiri, Injection Locking and Pulling…Part I: Time-Synchronous Modeling and Injection Waveform Design, IEEE JSSC 54(8):2109–2121 (BHongGenTheor-I_JSSC2019_Postprint.pdf, paper_003)	2019	Builds a time-synchronous injection locking/pulling model on the ISF, generalizing Adler 1946 to arbitrary oscillators and arbitrary injection waveforms: a single first-order (generalized Adler) equation predicts lock range, locked phase, and mode stability.	classic Adler $\dot\phi=-\omega_L\sin\phi+\Delta\omega_{inj}$; generalized Adler (time-averaged) $\dot\theta=\omega_0-\omega_{inj}-\frac{1}{q_{max}}\langle\Gamma(\omega_{inj}t+\theta)\,i_{inj}\rangle$ = Eq.(30) (this site's $\Gamma$ uses the sign convention opposite to [P3], hence the $-$ before the averaged term; numerically equivalent, the original PDF Eq.(30) has $+$); sinusoidal lock range $\omega_L=\tfrac{1}{2}I_{inj}\vert\tilde\Gamma_1\vert$ = Eq.(35), p.2114 (verified)	Fig. 1–3 (time-synchronous injection model and ISF formulation)	Advanced injection-locking deep dive (chapter 05, elective); demonstrates the same $\Gamma$ extending beyond phase noise.
[P4] B. Hong & A. Hajimiri, …Part II: Amplitude Modulation in LC Oscillators, Transient Behavior, and Frequency Division, IEEE JSSC 54(8):2122–2139 (BHongGenTheor-II_JSSC2019_Postprint.pdf, paper_004)	2019	Introduces the APF (amplitude perturbation function) — the amplitude analogue of the ISF (units $\mathrm{A^{-1}}$): handles amplitude modulation under injection, transient locking, and injection-locked frequency division. In an ideal LC the ISF and APF are orthogonal.	APF definition $\Delta(\phi):=\int_0^\infty D(\tau,\phi)\,d\tau$ (units $\mathrm{A^{-1}}$) = Eq.(19), p.2126 (decomposition $D=\tilde\Lambda(\phi)\,d(\tau,\phi)$ = Eq.(18); ideal-LC orthogonality = Eq.(26), p.2128, verified)	Fig. 5, p.2126 (effect of instantaneous charge injection on the oscillator: ISF / excess phase / amplitude decay / APF; ISF and APF orthogonal in an ideal LC)	APF / advanced deep dive (chapter 05, elective); chapter 02's phase_vs_amplitude_noise borrows it to explain "why amplitude noise decays".
[P5] A. Hajimiri & R. Heald, Design Issues in Cross-Coupled Inverter Sense Amplifier, Proc. IEEE ISCAS 1998 (Hajimiri_ISCS_98.pdf, paper_005)	1998	Analytic design of the cross-coupled-inverter sense amplifier (regeneration speed, mismatch offset, FOM). Completely unrelated to ISF/phase noise; included only because it was in the source folder with the same author.	None (off-topic, equations not transcribed) TODO: equations not transcribed because off-topic	None	Footnote — honestly explains the mislabeling; the only conceptual link is the cross-coupled-pair regeneration/positive-feedback mechanism shared with latches/LC oscillation.

⚠️ = manual_verification_needed = true, exact constants/forms still need comparison against the original PDFs. See claims_cross_reference and equation_index for details.

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Teaching viewpoint: who teaches what

Which paper is the ISF core foundation

[P1] is the foundation of the whole site. Every LTV/ISF concept — $\Gamma(\omega_0\tau)$, the $q_{max}$ normalization, the convolutional phase response, Fourier downconversion, the $1/f^2$/$1/f^3$ closed forms — all comes from here. Chapters 02 (foundations) and 03 (isf core theory) map almost section-by-section to [P1] Secs. III–IV, and lab_01/02/04/05/06/07 all reproduce figures and equations from [P1].

Which paper extends to rings

[P2] grounds [P1]'s general theory in the ring oscillator: the same $\Gamma_{rms}^2/q_{max}^2$ logic yields accumulated jitter (claim C6), $\Gamma_{rms}\propto N^{-3/2}$ (claim C8), and "ring phase noise nearly independent of $N$ at fixed power/frequency" (claim C7). Chapter 06's lc_vs_ring, symmetry and lab_03 use it.

Which paper covers adjoint / PPV / PSS / PNoise / simulation

Honestly: none of these 5 papers is dedicated to PPV/adjoint/PSS/PNoise.

• PPV (perturbation projection vector) / adjoint method / Floquet theory is the rigorous mathematical foundation behind the ISF, but it comes from the broader literature (e.g. Demir–Mehrotra–Roychowdhury 2000, Kaertner), not among the 5 downloaded PDFs. This site treats them as standard external literature, explicitly flagged in effective_isf (claim C13).
• PSS (periodic steady-state) / PNoise are numerical methods in commercial simulators (e.g. SpectreRF) for extracting ISF/ phase noise, and likewise are not the subject of any of these 5 papers; this site cites them only conceptually and does not claim they come from the downloaded PDFs.
• The closest thing to "simulation" is actually [P1]'s own limitation that "real circuits require transient impulse response or adjoint/PSS methods to extract the ISF" — i.e., simulation extraction is mentioned, but has no dedicated paper. All labs on this site are teaching toy models (not transistor-level), reproducing the concepts in Python, not a real PSS/PNoise flow.

Which paper corrects / critiques / extends prior work

[P1] supersedes Leeson's empirical formula. Leeson 1966's $\mathcal{L}$ expression (with $2FkT/P_s$, $(\omega_0/2Q\Delta\omega)^2$, $1+\omega_{1/f^3}/|\Delta\omega|$) is an empirical fit: it can draw the $1/f^3$, $1/f^2$, floor regions, but $F$ (noise factor) and the $1/f^3$ corner are after-the-fact fitting parameters, not computed from first principles. [P1]'s contribution is precisely to compute these three regions from first principles with the ISF, and to point out that the $1/f^3$ corner is not the device's $1/f$ corner, but $\omega_{1/f}(c_0/c_1)^2$ ([P1] Eq.(24), claim C5) — a key correction to the Leeson empirical model. The Leeson formula itself is not one of the 5 downloaded PDFs; this site uses it only for comparison (see equation_index row 19, flagged reference ⚠️).

In addition, [P3] extends/generalizes Adler 1946's injection-locking equation, and [P4] supplies the amplitude dimension (APF) missing from [P1]/[P3]. Neither Adler's nor Leeson's original paper is among these 5 PDFs.

Which equations are the same thing written differently across papers

Concept	[P1] form	Other papers' form	Unifying note
phase sensitivity	$\Gamma(\omega_0\tau)$ (ISF)	[P2] keeps $\Gamma$; [P3] writes $\Gamma(\theta+\phi)$ inside the injection inner product	the same $\Gamma$; [P3] just uses it as the injection weighting kernel
phase evolution	$\phi(t)=\frac{1}{q_{max}}\int\Gamma i_n\,d\tau$ (Eq.11, noise view)	[P3] $\dot\phi=\Delta\omega-\frac{1}{q_{max}}\langle\Gamma i_{inj}\rangle$ (injection view)	the same LTV phase equation, one driven by random noise, one by deterministic injection
$1/f^2$ phase noise	Eq.(21) $\propto\Gamma_{rms}^2/q_{max}^2$	[P2] $\kappa^2\propto\Gamma_{rms}^2/q_{max}^2$ (jitter version)	the same $\Gamma_{rms}^2/q_{max}^2$ ratio; phase noise and accumulated jitter are the same physics (claims C3=C6 share the source)
the two directions of sensitivity	phase version $\Gamma$ only	[P4] adds the amplitude version, APF $\Lambda$	$\Gamma$ (tangential/phase) and $\Lambda$ (radial/amplitude) are two orthogonal projections on the limit cycle

Which notation differs and needs unifying

Different papers use different symbols for the same quantity; this site always follows notation:

• Offset frequency: [P1] mostly uses $\Delta\omega$ (rad/s), datasheets/SerDes use $\Delta f$ (Hz); this site uses both, $\Delta\omega=2\pi\Delta f$.
• The ISF's DC: note that $c_0$ is a Fourier coefficient, while the ISF's DC value is $c_0/2$ (Eq.(12)) — very easy to get wrong when computing the $1/f^3$ corner (Eq.(24)); this site repeats the reminder.
• Amplitude sensitivity: [P4]'s APF is written $\Lambda(\phi)$ with units $\mathrm{A^{-1}}$; it has different dimensions from the dimensionless ISF $\Gamma$ and must not be mixed up.
• PPV/adjoint/Floquet: this site does not use their dedicated notation (e.g. Demir's $v_1^T(t)$) in the main track, mentioning them only as external literature in effective_isf.

For the full symbol comparison (including a "per-paper notation / remarks" column) see the "per-paper notation comparison" section of notation.

Key takeaways

• [P1] = ISF core; [P2] = ring extension; [P3]/[P4] = injection locking / APF advanced; [P5] = sense amplifier, unrelated to ISF (honestly flagged).
• None of the papers is a dedicated PPV/adjoint/PSS/PNoise paper; those belong to external literature (e.g. Demir 2000).
• [P1] supersedes the Leeson empirical formula from first principles, and corrects "$1/f^3$ corner ≠ device $1/f$ corner".
• Across papers, $\Gamma$, the phase equation, and $\Gamma_{rms}^2/q_{max}^2$ are often the same thing seen from different angles; notation is always unified per notation.

Further reading

• Every equation → derivation page → source: equation_index
• Every figure's script/formula/source: figure_index
• Teaching-claims cross-reference (C1–C13): claims_cross_reference
• Paper-by-paper deep dives: paper deep dives
• Why the sources include one off-topic PDF: build_report