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

References

See also: glossary (terminology intuition), equation_index (equation ↔ page-number index); external literature is used in derivation_leeson ([E1]) and derivation_floquet_ppv ([E2], [E3])

Every formula and conclusion on this site is tagged with its source. This page collects: (A) the 5 PDFs in the download folder (site-internal citation codes [P1]–[P5], citation strings copied verbatim from Section 1 of the author's spec), (B) external supplementary literature that comes up in teaching but is not in the download folder, and (C) citation conventions plus a TODO list of items still awaiting manual verification.

Honesty principle: the formulas in [P1]–[P4] have all been verified verbatim against the original PDF rendering; for external literature [E1]–[E4] (flagged as not among the 5 downloaded PDFs), the volume/issue/page/DOI have been verified online, but the internal formulas of those papers are background only. [P5] is unrelated to the ISF.

A. Core literature (the 5 PDFs in the download folder)

[P1] — the foundational paper of ISF theory

A. Hajimiri and T. H. Lee, "A General Theory of Phase Noise in Electrical Oscillators," IEEE J. Solid-State Circuits, vol. 33, no. 2, pp. 179–194, Feb. 1998. (file general.pdf, paper_001)

One-line contribution: establishes the view that an oscillator's response to noise is LTV (linear time-variant) rather than LTI, introduces the ISF $\Gamma(\omega_0\tau)$ and $q_{max}$ normalization, and derives closed forms and design rules for $1/f^2$ and $1/f^3$ .
Used on this site in: impulse_to_phase_shift, isf_definition, convolution_derivation, fourier_series_of_isf, white_noise_to_phase_noise, flicker_noise_upconversion.
Key equations: Eqs.(1),(9),(10),(11),(12),(13),(15)–(24), pp.181–185 (see equation_index).

[P2] — extension to the ring oscillator

A. Hajimiri, S. Limotyrakis, and T. H. Lee, "Jitter and Phase Noise in Ring Oscillators," IEEE J. Solid-State Circuits, vol. 34, no. 6, pp. 790–804, Jun. 1999. (file jitter_ring.pdf, paper_002)

One-line contribution: applies the ISF framework to the ring oscillator, giving closed forms for jitter/phase noise, the $\Gamma_{rms}\propto N^{-3/2}$ scaling, and the conclusion that "at fixed power and frequency, single-ended ring phase noise is nearly independent of stage count $N$ ."
Used on this site in: lab_03, lc_vs_ring, symmetry.
Key equations (verified): Eq.(8) $\sigma_{\Delta t}=\kappa\sqrt{\Delta t}$ p.792, Eq.(12) $\kappa=\frac{\Gamma_{rms}}{q_{max}}\sqrt{\tfrac12\tfrac{\overline{i_n^2}}{\Delta f}}$ p.793, Eq.(15) $f_0=1/(2N\tau_D)$ (Eq.(14) is the normalized stage delay $\hat t_D$ ), Eq.(16) $\Gamma_{rms}=\sqrt{\dfrac{2\pi^2}{3\eta^3}}\;\dfrac{1}{N^{1.5}}$ p.794 (re-verified in v7: the square root covers only the constant, $\Gamma_{rms}\propto N^{-3/2}$ ; triple-checked against the body-text $4/N^{1.5}$ @ $\eta=0.75$ and App.B Eq.(55), at $\eta=0.75$ this is $\approx4/N^{1.5}$ , matching the solid line in [P2] Fig.8; v3 had mis-read this as $N^{-3/4}$ ), Eq.(23) FOM $\frac{8}{3\eta}\frac{kT}{P}\frac{V_{DD}}{V_{char}}(f_0/\Delta f)^2$ p.796 ( $\eta$ is the stage-delay proportionality constant from Eq.(14), $\approx 1$ ; $\gamma$ enters only through $V_{char}=\Delta V/\gamma$ . v2 had mis-edited the leading coefficient to $8/(3\gamma)$ ; v3 corrected it against the original PDF p.796).

[P3] — injection locking (advanced)

B. Hong and A. Hajimiri, "A General Theory of Injection Locking and Pulling in Electrical Oscillators—Part I: Time-Synchronous Modeling and Injection Waveform Design," IEEE J. Solid-State Circuits, vol. 54, no. 8, pp. 2109–2121, Aug. 2019. (file BHongGenTheor-I_JSSC2019_Postprint.pdf, paper_003)

One-line contribution: builds a time-synchronous theory of injection locking/pulling using the ISF, generalizing Adler 1946 to arbitrary oscillators and arbitrary injection waveforms.
Used on this site in: paper_003_injection_locking_part1.
Key equations (verified): $\tilde\Gamma=\Gamma/q_{max}$ (Eq.26); the generalized Adler equation $\frac{d\theta}{dt}=(\omega_0-\omega_{inj})+\Omega(\theta)$ , $\Omega(\theta)=\frac{1}{T_{osc}}\int\tilde\Gamma(\omega_0 t+\theta)i_{inj}\,dt$ (Eq.33); lock range $\omega_L=\frac12 I_{inj}\lvert\tilde\Gamma_1\rvert$ (Eq.35), pp.2113–2114.

[P4] — APF / advanced

B. Hong and A. Hajimiri, "...Part II: Amplitude Modulation in LC Oscillators, Transient Behavior, and Frequency Division," IEEE J. Solid-State Circuits, vol. 54, no. 8, pp. 2122–2139, Aug. 2019. (file BHongGenTheor-II_JSSC2019_Postprint.pdf, paper_004)

One-line contribution: introduces the APF (Amplitude Perturbation Function) — the amplitude-domain counterpart of the ISF (units 1/A), covering amplitude modulation under injection, transient locking, and injection-locked frequency division in LC oscillators.
Used on this site in: paper_004_injection_locking_part2, phase_vs_amplitude_noise (uses its ISF/APF quadrature diagram to explain why amplitude perturbations decay).
Key equations (verified): APF decomposition $D(\tau,\phi)=\tilde\Lambda(\phi)\,d(\tau,\phi)$ (Eq.18); APF definition $\Delta(\phi):=\int_0^\infty D\,d\tau$ (units 1/A, Eq.19); amplitude change (Eq.20); augmented pulling equation (Eq.21, sinusoidal form Eq.22), all in Fig.5, p.2126; ideal-LC quadrature (ISF fundamental $\angle90°$ , APF fundamental $\angle0°$ ) $\tilde\Gamma_1=\frac{1}{q_{max}}\angle90°$ , $\tilde\Lambda_1=\frac{\tau_0}{q_{max}}\angle0°$ (Eq.26), p.2128; amplitude decay $d(t,\phi)=e^{-t/\tau_0}$ , $\tau_0=2Q/\omega_{osc}$ in the same ideal-LC section, p.2127–2128.

[P5] — unrelated to the ISF (honesty note)

A. Hajimiri and R. Heald, "Design Issues in Cross-Coupled Inverter Sense Amplifier," Proc. IEEE ISCAS, 1998. (file Hajimiri_ISCS_98.pdf, paper_005)

Honesty note: this is a paper on cross-coupled-inverter sense amplifiers, on the topic of regeneration speed and mismatch offset — unrelated to oscillator phase noise / the ISF. It appears in the download folder only because it shares an author (Hajimiri).
The only conceptual link: the cross-coupled-pair's regeneration / positive feedback mechanism, which is also the underlying mechanism for startup in latch-type and LC oscillators — this site uses it only as a peripheral note and does not draw any ISF formula from it (corresponds to claim C12).
Key equations: TODO: equations not transcribed because this PDF is unrelated to ISF/phase noise.

B. External supplementary literature (not among the 5 downloaded PDFs)

The following literature comes up in teaching but is not in the download folder. Volume/issue/pages/DOI have been verified online; but for these papers' internal formulas this site provides background only, without re-deriving them verbatim.

[E1] Leeson 1966 — empirical phase-noise model (for comparison)

D. B. Leeson, "A Simple Model of Feedback Oscillator Noise Spectrum," Proc. IEEE, vol. 54, no. 2, pp. 329–330, Feb. 1966. DOI: 10.1109/PROC.1966.4682.

Role: the most widely used empirical phase-noise model before ISF theory; [P1]'s introduction treats it as "the special case that ISF theory subsumes and surpasses." For the comparison and derivation see derivation_leeson:

\mathcal{L}(\Delta\omega)=10\log_{10}\!\left[\frac{2FkT}{P_s}\left(1+\left(\frac{\omega_0}{2Q\,\Delta\omega}\right)^2\right)\left(1+\frac{\omega_{1/f^3}}{|\Delta\omega|}\right)\right].

Status: volume/issue/pages/DOI verified; $F$ (noise figure) is an empirically fitted parameter (varies slightly by implementation), used here for comparison only.

[E2] Demir–Mehrotra–Roychowdhury 2000 — PPV (rigorizing the ISF)

A. Demir, A. Mehrotra, and J. Roychowdhury, "Phase Noise in Oscillators: A Unifying Theory and Numerical Methods for Characterization," IEEE Trans. Circuits Syst. I: Fundam. Theory Appl., vol. 47, no. 5, pp. 655–674, May 2000. DOI: 10.1109/81.847872.

Role: using the PPV (Perturbation Projection Vector) and a nonlinear phase equation, gives the ISF a rigorous mathematical foundation (the PPV is the first principal Floquet vector, obtainable numerically via the adjoint). Background equation: $\dot{\phi}(t)=v_1^T(t)\,B(t)\,\xi(t)$ .
Status: volume/issue/pages/DOI verified; the PPV framework is detailed in derivation_floquet_ppv.

[E3] Kärtner 1990 — perturbation analysis of oscillator noise (background)

F. X. Kärtner, "Analysis of White and $f^{-\alpha}$ Noise in Oscillators," Int. J. Circuit Theory Appl., vol. 18, no. 5, pp. 485–519, 1990. DOI: 10.1002/cta.4490180505.

Role: one of the perturbation analyses of oscillator noise predating [P1], often listed alongside Demir as a "mathematical precursor of the ISF/PPV."
Status: volume/issue/pages/DOI verified; mentioned on this site as background only.

[E4] Adler 1946 — the original injection-locking paper (the target of [P3]'s generalization)

R. Adler, "A Study of Locking Phenomena in Oscillators," Proc. IRE, vol. 34, no. 6, pp. 351–357, Jun. 1946. DOI: 10.1109/JRPROC.1946.229930. (Reprinted in Proc. IEEE, vol. 61, no. 10, pp. 1380–1385, Oct. 1973.)

Role: the original source of the classical Adler equation; [P3] generalizes it to arbitrary oscillators / arbitrary injection waveforms.
Status: volume/issue/pages verified; the classical and generalized Adler equations appear in paper_003.

[E5] Zadeh 1950 — frequency-domain analysis of linear time-variant systems (the source of the HTM)

L. A. Zadeh, "Frequency Analysis of Variable Networks," Proc. IRE, vol. 38, no. 3, pp. 291–299, Mar. 1950. DOI: 10.1109/JRPROC.1950.231083.

Role: defines the time-variant transfer function $H(f,t)$ (system function), the source of the harmonic transfer matrix and of the strict LTV view that "the ISF is the vector that maps phase output onto each harmonic."
Status: volume/issue/pages/DOI verified; the framework appears in ltv_htm (not among the 5 PDFs).

C. Citation conventions

Site-internal citation format: inline usage like [P1] Eq.(21), p.185; every definition/formula/conclusion/figure drawn from a paper is tagged with its source (see Section 1 of the author's spec).
Codes: the core 5 papers use [P1]–[P5] (corresponding to paper_001–paper_005); external supplements use [E1]–[E3], and always carry the note "not among the 5 downloaded PDFs."
LaTeX sources: the formula LaTeX in [P1] has been confirmed against the PDF rendering pages (manual_verification_needed=false); some constants/forms in [P2]–[P4] are flagged ⚠️, see the TODO below.
Units and notation: consistently follows notation site-wide.

D. TODO list awaiting manual verification

After the v3/v4 audit, all of [P1]; the [P2] ring constants (Eqs.8/12/14/16/17/21/23, where the leading coefficient of Eq.(23) was corrected in v3 from the mis-transcribed $8/(3\gamma)$ to $8/(3\eta)$ and re-verified; Eq.(16) was re-verified in v7, see the table below); the [P3] generalized Adler equations (Eqs.26/30/33/34/35); and [P4]'s APF (correctly numbered as Eqs.(18)–(22) + Fig.5, p.2126; ideal-LC quadrature Eq.(26), p.2128) have all been verified verbatim against the original PDF rendering; for external literature, [E1]–[E5]'s volume/issue/pages/DOI have been verified online, and the design-related external citations (Hegazi-Sjöland-Abidi JSSC 2001, Andreani JSSC 2002/2005, Romanò TCAS-I 2006, Behbahani JSSC 2001) have likewise had their volume/issue/pages verified. Only the following minor items remain:

Item	Content	Source	Nature
Re-verified (v7)	[P2] Eq.(16), p.794 (re-verified in v7: the square root covers only the constant, $\Gamma_{rms}\propto N^{-3/2}$ ; triple-checked against the body-text $4/N^{1.5}$ @ $\eta=0.75$ and App.B Eq.(55). v3 had mis-read this as $N^{-3/4}$ )	p.794	Corrected (square-root scope had been mis-read)
Verified	[P2] Fig.17 (phase noise vs. symmetry voltage, y-axis = $1/f^3$ corner frequency, drops sharply at the symmetry point)	p.802	Minor (figure detail, verified)
`TODO`	Stage-allocation details of [P4]'s dual-modulus prescaler	Sec. VIII, p.2135	Minor (advanced circuit)
Note	The internal formulas of [E1]–[E4] are background only, not re-derived verbatim (volume/issue/DOI verified)	External	Background
Note	[P5]'s sense-amplifier formulas are deliberately not transcribed (unrelated to the ISF)	—	Honesty note

Factor-of-2 reminder: [P1] Eq.(21) writes the denominator as $4\Delta\omega^2$ , while a clean time-domain derivation gives $2\Delta\omega^2$ ; the factor-of-2 difference is an SSB accounting convention (a well-known minor point of contention in the literature), not a citation error; see white_noise_to_phase_noise for details.

A. Core literature (the 5 PDFs in the download folder)​

[P1] — the foundational paper of ISF theory​

[P2] — extension to the ring oscillator​

[P3] — injection locking (advanced)​

[P4] — APF / advanced​

[P5] — unrelated to the ISF (honesty note)​

B. External supplementary literature (not among the 5 downloaded PDFs)​

[E1] Leeson 1966 — empirical phase-noise model (for comparison)​

[E2] Demir–Mehrotra–Roychowdhury 2000 — PPV (rigorizing the ISF)​

[E3] Kärtner 1990 — perturbation analysis of oscillator noise (background)​

[E4] Adler 1946 — the original injection-locking paper (the target of [P3]'s generalization)​

[E5] Zadeh 1950 — frequency-domain analysis of linear time-variant systems (the source of the HTM)​

C. Citation conventions​

D. TODO list awaiting manual verification​

Further reading​