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

Stochastic Noise Basics

Prerequisites: lti_vs_ltv · Notation | Next: white_noise_to_phase_noise

This page answers a prerequisite question: before we can talk about the ISF, we have to pin down "noise" as a stochastic process. Phase jitter in an oscillator originates from the random currents $i_n(t)$ of transistors and resistors. Before feeding it into the phase integral of [P1] Eq.(11), we need to know: how to quantify the "strength" of this random current (PSD), how to recover time-domain power from the PSD (Parseval), whether it is correlated across time instants (autocorrelation), whether a single measurement can represent the whole ensemble (ergodicity), and — most critical for oscillators — whether the noise intensity itself varies periodically with the oscillation phase (cyclostationary).

That last point is the core of effective_isf later on: the transistor leaks noise only during the short window when it conducts, and where that window falls in the waveform's phase determines which segment of the ISF weights it.

Physical intuition (big picture first): think of noise as "an invisible hand randomly kicking the oscillator at every instant." We need two independent pieces of information: (1) how hard the hand kicks on average (intensity, measured by the PSD); (2) when the hand kicks hardest (time modulation, described by cyclostationarity / the NMF). The ISF is the third item: "at which phase of the waveform a kick changes the phase the most." Multiply the three together and integrate — that is phase noise.

1. White noise

White noise (a stochastic process whose power spectrum is flat at all frequencies). The name comes from white light — all frequency components with equal strength. Its single-sided PSD (power spectral density) is constant:

$$S_i(f)=\frac{\overline{i_n^2}}{\Delta f}=\text{const}.$$

• Units: current-noise PSD is in $\text{A}^2/\text{Hz}$ (mean-square current per Hz of bandwidth). Typical thermal-noise expressions: $\overline{i_n^2}/\Delta f=4kT/R$ (resistor) or $4kT\gamma g_m$ (MOS channel).
• Dimension check: $\overline{i_n^2}$ is $\text{A}^2$; dividing by the bandwidth $\Delta f$ (Hz) gives $\text{A}^2/\text{Hz}$ ✓.
• Physical origin: thermal noise (random thermal motion of carriers) and shot noise (discreteness of carriers crossing a barrier) can both be treated as white over the frequency range we care about.
• Canonical value: this site's white-noise source uses $S_i=10^{-24}\ \text{A}^2/\text{Hz}$ (see canonical example B). This corresponds to a current-noise density of $\sqrt{S_i}=10^{-12}\ \text{A}/\sqrt{\text{Hz}}=1\ \text{pA}/\sqrt{\text{Hz}}$.

Why oscillators care about white noise: white noise has equal-strength components near every $n\omega_0$; each ISF harmonic $c_n$ "downconverts" the white noise of the corresponding band to near the carrier, forming the 1/f² ($-20$ dB/dec) phase-noise skirt. This is exactly the content of [P1] Eq.(21) — see white_noise_to_phase_noise.

2. Flicker (1/f) noise

Flicker noise (also called 1/f noise: low-frequency noise whose power rises inversely with frequency). In MOS devices it mainly comes from carriers being randomly captured/released by interface traps. Its PSD is not flat:

$$S_{i,1/f}(f)=\overline{i_n^2}\cdot\frac{\omega_{1/f}}{\Delta\omega}\qquad(\Delta\omega < \omega_{1/f}),$$

which is exactly the device flicker model of [P1] Eq.(22), p.185 (site spec Section 3, formula 13).

• Units: still $\text{A}^2/\text{Hz}$. $\omega_{1/f}$ is the device's 1/f corner (angular frequency, rad/s) — the frequency at which the 1/f component's strength equals the white-noise floor.
• Dimension check: $\overline{i_n^2}$ ($\text{A}^2/\text{Hz}$) $\times\ \omega_{1/f}/\Delta\omega$ (dimensionless ratio) $=\text{A}^2/\text{Hz}$ ✓.
• Key pitfall: the device 1/f corner $\omega_{1/f}$ is not equal to the phase-noise 1/f³ corner. The latter is $\Delta\omega_{1/f^3}=\omega_{1/f}\cdot c_0^2/(2\Gamma_{rms}^2)$ ([P1] Eq.(24)), upconverted only through the ISF's DC coefficient $c_0$. A symmetric waveform (small $c_0$) can push the 1/f³ corner far below the device corner — see symmetry (claim C5).

Noise type	PSD shape	Physical origin	Becomes, in phase noise
white	flat (const $\text{A}^2/\text{Hz}$)	thermal / shot	1/f² ($-20$ dB/dec)
flicker (1/f)	$\propto 1/f$	trap capture/release	1/f³ ($-30$ dB/dec), via $c_0$ only

3. PSD and Parseval: frequency-domain intensity ↔ time-domain power

The PSD (power spectral density) tells you "how much power per unit bandwidth." Integrating it over frequency gives the total time-domain power (variance). This is Parseval's theorem (time-domain energy = frequency-domain energy) in its stochastic-process form:

$$\overline{i_n^2}=\int_0^{\infty}S_i(f)\,df\qquad(\text{單邊 PSD}).$$

• Units: left side $\text{A}^2$; right side $(\text{A}^2/\text{Hz})\times\text{Hz}=\text{A}^2$ ✓.
• Intuition: the PSD is a "density of power," and integrating means "multiply density by bandwidth and add it up." For white noise this integral diverges — so any real system has a finite bandwidth, and white noise is only an in-band approximation.
• Why it matters: phase noise plays the same trick — integrate the phase PSD $S_\phi(f)$ over offset frequency to get the phase variance $\sigma_\phi^2$, then convert to rms jitter. Full derivation in psd_phase_noise_jitter.

Convention: this site uses the single-sided PSD (positive frequencies only, $0\le f<\infty$). If you switch to the double-sided ($-\infty<f<\infty$) PSD, the value at each frequency is divided by 2. This factor-of-2 convention is the same class of bookkeeping issue as the SSB phase-noise relation $\mathcal{L}\approx\frac12 S_\phi$, discussed in detail in white_noise_to_phase_noise.

4. Autocorrelation and the real definition of "white"

Autocorrelation (how correlated one stochastic process is between two time instants):

$$R_i(\tau)=\overline{i_n(t)\,i_n(t+\tau)}.$$

• Units: $\text{A}^2$ (the expectation of a product of two currents).
• Wiener–Khinchin theorem: autocorrelation and PSD are a Fourier pair. White noise has a flat PSD, corresponding to a delta-function autocorrelation: $R_i(\tau)=\dfrac{\overline{i_n^2}}{\Delta f}\,\delta(\tau)$.
• Intuition: white noise's "value now" is completely uncorrelated with its "value the next instant" (zero memory). This is exactly why, in the phase integral of [P1] Eq.(11), noise contributions at different instants can be superposed independently — it turns the phase-variance calculation into "square and add term by term" with no cross-correlation terms to handle.
• Flicker counterexample: 1/f noise has a long autocorrelation tail (long memory, strong correlation), which is also why it is hard to integrate and hard to simulate (see the aliasing caveats on the DSP page).

5. Ergodicity

Ergodicity (the property that "time average = ensemble average").

• Plain language: you have only one oscillator and measure one long waveform (time average); the theory speaks of the statistics of infinitely many identical oscillators at the same instant (ensemble average). Ergodicity guarantees the two are equal — otherwise "measuring one device" could never validate "the ensemble statistics predicted by theory."
• Applicability: the process must be stationary (statistics do not drift with time) before ergodicity is even on the table. Thermal noise qualifies.
• Failure warning: the oscillator's excess phase $\phi(t)$ itself is not stationary — it is a random walk (variance grows linearly with time; see accumulated jitter $\sigma_{\Delta t}=\kappa\sqrt{\Delta t}$, [P2] Eq.(8)). So we do statistics on "the time derivative of phase (frequency)" or on "phase differences," not on the absolute value of $\phi$. This subtlety is retold in dsp_view_of_phase_noise from the "1/(jω) integrator turns white noise into 1/f²" viewpoint.

6. Cyclostationary noise — this page's focus, straight into the effective ISF

Cyclostationary noise (a stochastic process whose statistics vary "periodically" with time). [P1] Sec. III-D, p.186 says it plainly:

"the channel noise of a MOS device in an oscillator is cyclostationary because the noise power is modulated by the gate-source overdrive which varies with time periodically." ([P1], p.186, cyclostationary paragraph)

In other words: the transistor does not leak the same amount of noise at every instant. It is noisy only during the short window when it conducts (large current, large overdrive); the rest of the time it is nearly silent. And "conducting" is locked to the oscillation waveform — repeating once per period $T$. That is "cyclostationarity."

[P1] decomposes a white cyclostationary current as ([P1] Eq.(25), p.186):

$$i_n(t)=i_{n0}(t)\cdot\alpha(\omega_0 t),$$

where $i_{n0}(t)$ is a white stationary process (fixed intensity) and $\alpha(\omega_0 t)$ is a deterministic, periodic, normalized (max = 1) function describing "when the noise amplitude gets turned up or down." This site calls it the NMF — noise-modulating function, symbol $\alpha(\omega_0 t)$, $0\le\alpha\le1$, dimensionless (see the notation page).

• Units: $\alpha$ is dimensionless (it is a normalized envelope); $i_{n0}$ and $i_n$ are both in A.
• Dimension check: $\text{A}=\text{A}\times(\text{無因次})$ ✓.
• Meaning of $\alpha^2$: because the PSD deals in power ($i^2$), the noise's instantaneous mean-square power is proportional to $\alpha^2(\omega_0 t)$. The maximum instantaneous PSD $=\overline{i_n^2}/\Delta f$; at all other times it is suppressed by $\alpha^2\le1$.

Why this leads straight to the ISF: the effective ISF

Substitute the cyclostationary decomposition into the phase integral [P1] Eq.(11): the noise contribution at instant $\tau$ is weighted by $\Gamma(\omega_0\tau)$, while its own intensity is modulated by $\alpha(\omega_0\tau)$. The product of the two periodic functions can be packaged into a single effective ISF ([P1] Eq.(27), p.186):

$$\boxed{\ \Gamma_{eff}(\omega_0\tau)=\Gamma(\omega_0\tau)\cdot\alpha(\omega_0\tau)\ }$$

In [P1]'s own words:

"the cyclostationary noise can be treated as a stationary noise applied to a system with an effective ISF." ([P1], p.186)

Meaning: just replace $\Gamma$ with $\Gamma_{eff}=\Gamma\cdot\alpha$, and every downstream formula ($c_n$, $\Gamma_{rms}$, Eq.(21), Eq.(23)) carries over unchanged. Cyclostationarity does not need a new theory; it only changes which curve you take the Fourier coefficients of.

• Units: $\Gamma_{eff}$ is still dimensionless (dimensionless $\Gamma$ × dimensionless $\alpha$) ✓.
• Teaching point: when computing $c_n$ and $\Gamma_{rms}$, use $\Gamma_{eff}$ (i.e., as [P1] puts it, "the effective ISF should be used in all subsequent calculations"), not the bare $\Gamma$.

LC vs ring: the cyclostationary impact differs enormously ([P1]'s beautiful conclusion)

[P1], p.186 gives two contrasting examples of great pedagogical value:

Oscillator	Large current (large $\alpha$) occurs at	$\Gamma$ there	Cyclostationary impact
Colpitts LC	tank-voltage minimum (trough)	small (ISF is small at peaks/troughs)	$\Gamma_{eff}$ differs a lot from $\Gamma$; cannot be ignored
ring (single-ended)	transitions (zero crossings, max slope)	large (ISF is large at transitions)	$\Gamma_{eff}\approx\Gamma$; small impact

[P1]'s reading: the Colpitts transistor "turns on only where the ISF is small" — nature effectively injects its noise at insensitive phases, so cyclostationarity helps the LC. The ring is unlucky: the maximum current occurs exactly at the most sensitive transitions, so noise and sensitivity peak simultaneously. [P1] calls it out directly:

"This unfortunate coincidence is one of the reasons why ring oscillators in general have inferior phase noise performance compared to a Colpitts LC oscillator." ([P1], p.186)

This is one of the two big reasons "LC is cleaner than ring" (the other: a ring burns all of its stored energy every period). Quantitative discussion in [P2] and lc_vs_ring.

Toy-model reminder: later pages use toy shapes for $\alpha(\omega_0 t)$ (e.g., setting the NMF to a conduction window) to demonstrate how $\Gamma_{eff}=\Gamma\cdot\alpha$ changes $c_0$ and $\Gamma_{rms}$; these are pedagogical toy models, not transistor-level. The real $\alpha$ must be extracted from the device noise model and operating point ([P1] states "$\alpha$ can be derived easily from device noise characteristics and operating point").

Connecting this page to the ISF in one sentence

The full phase-noise recipe is the product of three periodic/stochastic quantities, then integrated:

The two "×" signs in the middle are the origin of the effective ISF $\Gamma_{eff}=\Gamma\cdot\alpha$; the final integrator is what turns white noise into 1/f² (see the DSP viewpoint linked in the next section).

Applicability and failure conditions

Condition	When it holds	When it fails
stationary (thermal noise)	ergodicity holds; measuring one device represents the ensemble	drift/aging → re-measure
white approximation	flat PSD, delta autocorrelation, contributions superpose independently	inside the 1/f region, switch to the flicker model
cyclostationarity packaged as $\Gamma_{eff}=\Gamma\cdot\alpha$	small perturbation; $\alpha$ periodic and normalized	strong nonlinearity coupling $\alpha$ to the signal calls for more care
phase described as a random walk	do statistics on phase differences/frequency	time-averaging the absolute value of $\phi$ directly diverges

Key takeaways

• White noise: flat PSD ($\text{A}^2/\text{Hz}$), delta autocorrelation, memoryless; becomes 1/f² under the ISF.
• Flicker (1/f) noise: $S\propto1/f$ ([P1] Eq.(22)); upconverts to 1/f³ only via $c_0$; device corner $\neq$ 1/f³ corner.
• PSD ↔ Parseval: integrating the PSD gives time-domain power (variance); units self-consistent $\text{A}^2/\text{Hz}\times\text{Hz}=\text{A}^2$.
• Autocorrelation: white = delta (memoryless); 1/f = long tail (strongly correlated).
• Ergodicity: time average = ensemble average; but $\phi$ is a random walk, non-stationary — do statistics on phase differences/frequency.
• Cyclostationary (this page's protagonist): noise intensity varies periodically with the oscillation phase (the transistor leaks noise only while conducting); described by the NMF $\alpha(\omega_0 t)$ and packaged as $\Gamma_{eff}=\Gamma\cdot\alpha$ ([P1] Eq.(25)–(27)); LC benefits, ring suffers.
• Sources: [P1] Sec. III-D (cyclostationary) Eqs.(25)–(27), p.186; device flicker Eq.(22), p.185; claims C9.

Further reading

• How the phase PSD integrates into jitter: psd_phase_noise_jitter
• The DSP view of phase = a stochastic process weighted by the ISF and processed by an integrator: dsp_view_of_phase_noise
• Full derivation and extraction of the effective ISF: effective_isf
• How white noise becomes 1/f² phase noise: white_noise_to_phase_noise
• ISF Fourier coefficients $c_n$, $c_0$, and symmetry: fourier_series_of_isf, symmetry