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

Waveform Symmetry and Flicker Upconversion

Prerequisites: flicker_noise_upconversion (the full flicker → $1/f^3$ derivation; this page is its design-facing counterpart), fourier_series_of_isf (the ISF's Fourier coefficients $c_0,c_n$ and Parseval), device_noise_mapping (the $c_0$ of the effective ISF $\Gamma_{eff}=\Gamma\cdot\alpha$ is the real culprit) | Next: waveform_slope, serdes_clocking_connection

This page answers a question that must be settled at the layout/topology stage: why does a waveform with symmetric rise/fall have noticeably lower close-in (near-carrier) 1/f³ phase noise? The answer is hidden entirely in one Fourier coefficient of the ISF — the DC term $c_0$.

Physical intuition (conclusion first): a device's flicker noise (1/f noise, slow-varying noise in the gate/channel) is a near-DC, low-frequency noise. Low-frequency noise should not normally contaminate a high-frequency carrier — but the ISF is a periodic function, and its DC component $c_0/2$ acts like a "rectifier": it accumulates low-frequency device noise with a persistent sign into the phase, upconverting it to a close-in 1/f³ skirt near the carrier. If the rise/fall is perfectly symmetric, the ISF's positive and negative areas cancel over one period, $c_0\to0$, and this upconversion channel is shut off.

Step 1: why only $c_0$ upconverts flicker

Write the ISF as a Fourier series ([P1] Eq.(12), p.183):

$$\Gamma(\omega_0\tau)=\frac{c_0}{2}+\sum_{n=1}^{\infty}c_n\cos(n\omega_0\tau+\theta_n)$$

Substitute into the LTV phase response ([P1] Eq.(13), p.183), splitting the phase into contributions from each harmonic:

$$\phi(t)=\frac{1}{q_{max}}\!\left[\frac{c_0}{2}\!\int_{-\infty}^{t}\!i_n\,d\tau+\sum_{n=1}^{\infty}c_n\!\int_{-\infty}^{t}\!i_n\cos(n\omega_0\tau+\theta_n)\,d\tau\right]$$

• Key observation: device flicker-noise energy is concentrated near DC ($\Delta\omega\ll\omega_0$). In the expression above, every $c_n$ term ($n\ge1$) carries a $\cos(n\omega_0\tau+\theta_n)$ factor, which multiplies the low-frequency noise by a high-frequency carrier — a mixing operation that moves the noise to near $\pm n\omega_0$, away from DC.
• Only the $c_0/2$ term has no carrier: it performs a pure integration of the low-frequency noise. Low-frequency noise stays nearly the same sign over an extended interval, so the integral accumulates without cancelling, continuously driving the close-in phase → upconverted into 1/f³.
• In one line: $c_n$ acts like a mixer (it moves the noise away), while $c_0$ acts like an integrator (it retains and amplifies DC noise).

Step 2: 1/f³ phase noise and the corner formula

Feed the device flicker model ([P1] Eq.(22), p.185)

$$\overline{i_{n,1/f}^2}=\overline{i_n^2}\cdot\frac{\omega_{1/f}}{\Delta\omega}$$

into the phase noise from the $c_0$-only channel to get the 1/f³ region ([P1] Eq.(23), p.185):

$$\mathcal{L}\{\Delta\omega\}=10\log_{10}\!\left(\frac{c_0^2}{q_{max}^2}\cdot\frac{\overline{i_n^2}/\Delta f}{8\,\Delta\omega^2}\cdot\frac{\omega_{1/f}}{\Delta\omega}\right)$$

• Slope dimension check: $1/\Delta\omega^2$ (from the phase integration) times $1/\Delta\omega$ (from flicker's $\omega_{1/f}/\Delta\omega$ factor) = $1/\Delta\omega^3$ → $-30$ dB per decade, exactly 1/f³. ✓
• Key point: the numerator is $c_0^2$. A symmetric waveform drives $c_0\to0$ → the entire 1/f³ region is suppressed.

Intersecting the 1/f³ region with the 1/f² region ([P1] Eq.(21)) defines the 1/f³ corner ([P1] Eq.(24), p.185):

$$\Delta\omega_{1/f^3}=\omega_{1/f}\cdot\frac{c_0^2}{2\,\Gamma_{rms}^2}\approx\omega_{1/f}\left(\frac{c_0}{c_1}\right)^2$$

Step-by-step algebra: how the corner falls out of "the two regions are equal." The corner is defined as the offset where the 1/f³ region and the 1/f² region are exactly equal in height. Setting the two (linear, not-yet-log) power spectral densities equal:

$$\begin{aligned} \underbrace{\frac{c_0^2}{q_{max}^2}\cdot\frac{\overline{i_n^2}/\Delta f}{8\,\Delta\omega^2}\cdot\frac{\omega_{1/f}}{\Delta\omega}}_{\text{1/f}^3\text{ region (inside the brackets of Eq.23)}} &=\underbrace{\frac{\Gamma_{rms}^2}{q_{max}^2}\cdot\frac{\overline{i_n^2}/\Delta f}{4\,\Delta\omega^2}}_{\text{1/f}^2\text{ region (inside the brackets of Eq.21)}} \\[4pt] \frac{c_0^2}{8}\cdot\frac{\omega_{1/f}}{\Delta\omega}&=\frac{\Gamma_{rms}^2}{4} \qquad(\text{both sides cancel }q_{max}^2,\ \overline{i_n^2}/\Delta f,\ \Delta\omega^2) \\[4pt] \frac{\omega_{1/f}}{\Delta\omega}&=\frac{\Gamma_{rms}^2}{4}\cdot\frac{8}{c_0^2}=\frac{2\,\Gamma_{rms}^2}{c_0^2} \\[4pt] \Rightarrow\quad \Delta\omega=\Delta\omega_{1/f^3}&=\omega_{1/f}\cdot\frac{c_0^2}{2\,\Gamma_{rms}^2}. \end{aligned}$$

• What each step uses: the second line is just "dividing out the factors common to both sides" (pure algebra, no new physics); note that $\overline{i_n^2}/\Delta f$, $q_{max}^2$, and $\Delta\omega^2$ are the same value in both regions, so they cancel.

• Why $\Delta\omega^2$ also cancels: the 1/f³ region is $1/\Delta\omega^3$, the 1/f² region is $1/\Delta\omega^2$; dividing leaves a single $1/\Delta\omega$ — exactly the $\omega_{1/f}/\Delta\omega$ factor still on the left in line 2 above. Solving this linear equation gives the corner.

• Getting to $\approx(c_0/c_1)^2$: using Parseval ([P1] Eq.(20)) $2\Gamma_{rms}^2=\sum c_n^2$; if the ISF is dominated by its fundamental ($c_1\gg c_2,c_3,\dots$), then $2\Gamma_{rms}^2\approx c_1^2$, giving $\Delta\omega_{1/f^3}\approx\omega_{1/f}(c_0/c_1)^2$.

• Dimension check: on the right, $\omega_{1/f}$ is rad/s, and $c_0^2/(2\Gamma_{rms}^2)$ is dimensionless/dimensionless = dimensionless, so $\Delta\omega_{1/f^3}$ is rad/s ✓.

• The most important design implication (claim C5): the 1/f³ corner is not equal to the device's 1/f corner $\omega_{1/f}$! It is scaled by a factor $c_0^2/(2\Gamma_{rms}^2)$. The smaller $c_0$ is, the further the corner is pushed below $\omega_{1/f}$ — i.e., for the same transistor with the same device 1/f corner, simply making the waveform symmetric can push the 1/f³ skirt's knee down several decades in frequency.

• Notation trap (see notation): $c_0$ is the Fourier coefficient; the ISF's DC value is $c_0/2$. Eq.(24) uses the coefficient $c_0$ — don't drop a factor of 2 here.

Step 3: how symmetry drives $c_0\to0$

$c_0$ is (twice) the ISF's average over one period:

$$\frac{c_0}{2}=\frac{1}{2\pi}\int_0^{2\pi}\Gamma(x)\,dx\quad\Rightarrow\quad c_0=\frac{1}{\pi}\int_0^{2\pi}\Gamma(x)\,dx$$

• The ISF's shape is roughly proportional to the waveform slope (large slope near a zero crossing → large $|\Gamma|$; zero slope at the peak → $\Gamma\approx0$; see waveform_slope).
• If the rise and fall segments have symmetric shapes (rise slope = mirror image of fall slope), the ISF values over the rising and falling half-periods are equal in magnitude, opposite in sign, and cancel when integrated over one period → $c_0=0$.
• An ideal LC's $\Gamma(\theta)=-\sin\theta$ is exactly this kind of odd symmetry: $\int_0^{2\pi}(-\sin\theta)\,d\theta=0$, so an ideal LC naturally has $c_0=0$ and very weak 1/f³ upconversion.
• Asymmetry (e.g. fast rise, slow fall, or even-order harmonics that make the waveform top/bottom asymmetric) shifts the ISF's average away from 0 → $c_0\neq0$.

The figure below overlays a symmetric and an asymmetric ISF: the symmetric one has its DC average pinned at 0, while the asymmetric one is lifted by a "DC offset" of $c_0/2$ — that offset is the culprit behind flicker upconversion.

Feeding actual device flicker into a simulation makes the close-in difference very clear: the asymmetric waveform shows a steep 1/f³ skirt, while the symmetric waveform shows almost none (only 1/f² and the floor remain).

Both figures are pedagogical toy models (not transistor-level): the ISF uses an analytic shape (symmetric $\cos\theta$, asymmetric $\cos\theta+0.4$, etc.), and flicker is a synthesized 1/f sequence. They faithfully illustrate "$c_0$ determines 1/f³" causally, but are not a measurement of any real transistor. Full scripts: simulations/lab_05_fourier_isf.py, simulations/lab_07_flicker_noise.py.

Numerical example (building intuition)

The first example uses an assumed $c_0$ (illustrative); the second switches to the closed forms of [P2] Appendix B and computes $c_0$ and the corner directly from the topology parameters $(N,A)$ — full derivation in asymmetric_isf_closed_form.

Symmetric waveform ($c_0\approx0$): theoretically $\Delta\omega_{1/f^3}=\omega_{1/f}\cdot c_0^2/(2\Gamma_{rms}^2)\to0$, and the 1/f³ corner is pushed to extremely low frequency — in practice it is dominated by the small residual $c_0$ set by mismatch (see the knobs table below).

Asymmetric waveform (illustrative, $c_0$ assumed): take $c_0=0.4$, $\Gamma_{rms}=0.5$ (so $c_0^2/(2\Gamma_{rms}^2)=0.16/(2\times0.25)=0.32$). If the device $f_{1/f}=1$ MHz, then

$$f_{1/f^3}=f_{1/f}\cdot\frac{c_0^2}{2\Gamma_{rms}^2}=1\ \text{MHz}\times0.32=320\ \text{kHz}.$$

• Intuition: lowering $c_0$ from 0.4 to 0.04 (10× smaller) drops the corner by 100× ($c_0^2$) — from 320 kHz down to 3.2 kHz. In other words, each order-of-magnitude improvement in symmetry shrinks the reach of the 1/f³ skirt by two orders of magnitude — an extremely cost-effective design lever.

Asymmetric waveform (computed, directly from ring topology parameters): a ring with $N=5$, $\eta=1$, whose rise is 1.5× steeper than its fall (asymmetry ratio $A=f'_{rise}/f'_{fall}=1.5$). The [P2] App. B closed forms (Eq.(55)/(56)/(57), p.803; derivation in asymmetric_isf_closed_form) directly give

$$\Gamma_{rms}=0.2428,\qquad c_0=2\,\Gamma_{dc}=-0.1005,\qquad f_{1/f^3}=f_{1/f}\cdot\frac{3}{2\eta N}\cdot\frac{(1-A)^2}{1-A+A^2}=42.9\ \text{kHz}.$$

• Convention flag: 42.86 kHz is the [P2] Eq.(7)/(57) bookkeeping; substituting $c_0=2\Gamma_{dc}$ back into [P1] Eq.(24) above yields $2\times=85.71$ kHz (a DC-channel weighting convention difference; see the flag on the new page. Scalings and ratios are unaffected). The numbers are verified by simulations/lab_33_asymmetry_corner.py (closed forms vs numeric integration, error $\sim10^{-9}$).
• New design message ([P2]'s own sentence, p.803): at fixed $A$ the corner is $\propto1/N$ — rings with fewer stages have a higher flicker corner (at $A=1.5$: $N=3\to71.43$ kHz, $N=15\to14.29$ kHz). White-region phase noise is approximately N-independent ([P2] Eq.(23)), but the 1/f³ knee is pushed down by the stage count.
• Corresponding experimental evidence: [P2] Fig. 17, p.802 measures ring-oscillator phase noise vs. "symmetry control voltage," which shows a minimum at the symmetry point — direct support for the design rule "symmetric → low 1/f³." (Verified: [P2] Fig. 17, p.802, "Phase noise versus symmetry voltage for oscillator number 7" — the y-axis is the 1/f³ corner frequency, showing a clear dip to a minimum at the symmetry point.)

Design knobs for lowering $c_0$ (checklist)

Knob	How	Why it lowers $c_0$	Cost/notes
Symmetric rise/fall	NMOS/PMOS drive-strength or pull-up/pull-down symmetry (ring); differential topology	ISF's upper and lower half-periods cancel → average → 0	needs sizing/bias tuning; process offset leaves residual $c_0$
Differential / even-harmonic suppression	fully differential, symmetric loads, suppress even harmonics	even harmonics make the waveform top/bottom asymmetric → raise $c_0$	2× devices, area, power
Symmetric load (ring)	use a symmetric load ([P2]'s approach) instead of a single-ended inverter delay cell	matches rise/fall shape	[P2] Fig. 17's "symmetry voltage" tunes exactly this
Reduce DC-bias-point drift	control duty cycle near 50%	duty deviating from 50% means a DC-asymmetric waveform → $c_0\neq0$	needs duty-cycle correction
Directly lower device flicker	use large-area, PMOS, buried-channel devices	lowers $\omega_{1/f}$ itself (does not change $c_0$, but lowers the magnitude of 1/f³)	large area → large parasitic capacitance → lowers $f_0$

Note the two categories: the first four knobs change $c_0$ / the $f_{1/f^3}$ corner location; the last one changes the device $\omega_{1/f}$ / the overall height of 1/f³. Both can be used in design, but "making it symmetric" is usually free (no extra power) — take that shot first.

Validity and failure conditions

Condition	Holds when	Fails when
Small perturbation, ISF known and fixed	$c_0$ fully determines 1/f³	under large injection/strong nonlinearity the ISF itself changes
Device noise is pure 1/f	the $\omega_{1/f}/\Delta\omega$ model holds	with RTS/burst noise, a separate calculation is needed
Cyclostationarity already folded into the effective ISF	compute $c_0$ using $\Gamma_{eff}=\Gamma\cdot\alpha$	if $\alpha$ is also asymmetric, it can "recreate" a nonzero effective $c_0$ (see device_noise_mapping)

Important warning: what actually determines the upconversion is the $c_0$ of the effective ISF $\Gamma_{eff}=\Gamma\cdot\alpha$, not just the $c_0$ of the bare $\Gamma$. Even if $\Gamma$ is symmetric, if the device only "leaks noise" during half the period ($\alpha$ is asymmetric), the effective $c_0$ of $\Gamma_{eff}$ can still be nonzero. See device_noise_mapping and effective_isf.

Worked examples

The following two examples demonstrate the most cost-effective design lever — "improving symmetry lowers the 1/f³ corner" — reusing this site's canonical $\Gamma_{rms}=0.5$, device $f_{1/f}=1$ MHz. Both use an illustrative assumed $c_0$ (conversion practice); to compute the real $c_0$ from topology parameters $(N,A)$, see asymmetric_isf_closed_form.

Example 1 (baseline: compute the 1/f³ corner for an asymmetric waveform; $c_0$ is an illustrative assumed value) Given $c_0=0.4$, $\Gamma_{rms}=0.5$, device 1/f corner $f_{1/f}=1$ MHz, find the 1/f³ corner $f_{1/f^3}$.

Step-by-step substitution (with units), using the just-derived $f_{1/f^3}=f_{1/f}\cdot c_0^2/(2\Gamma_{rms}^2)$:

$$\begin{aligned} \frac{c_0^2}{2\Gamma_{rms}^2}&=\frac{(0.4)^2}{2\times(0.5)^2}=\frac{0.16}{0.50}=0.32\quad(\text{dimensionless}) \\[4pt] f_{1/f^3}&=f_{1/f}\times0.32=1\ \text{MHz}\times0.32=0.32\ \text{MHz}=320\ \text{kHz}. \end{aligned}$$

• Result: $f_{1/f^3}=320$ kHz — lower than the device's own 1 MHz corner, because $c_0^2/(2\Gamma_{rms}^2)=0.32 < 1$.
• Dimension check: $[\text{Hz}]\times[\text{dimensionless}]=[\text{Hz}]$ ✓ (a ratio of frequencies works equally in Hz or rad/s, since $f_{1/f^3}/f_{1/f}=\omega_{1/f^3}/\omega_{1/f}$ and the $2\pi$'s cancel).
• One-line Python check (using the canonical conversion; simulations/common/ has no dedicated corner function, so compute it directly):

c0, gamma_rms, f_1f = 0.4, 0.5, 1e6
f_1f3 = f_1f * c0**2 / (2 * gamma_rms**2)
print(f_1f3 / 1e3, "kHz")   # -> 320.0 kHz

Example 2 (improving symmetry by one order of magnitude → corner drops two orders of magnitude) Make the waveform more symmetric so $c_0$ drops from $0.4$ to $0.04$ (10× smaller), with $\Gamma_{rms}$ and $f_{1/f}$ unchanged. Find the new corner, and express in dB how much the heights of the two 1/f³ asymptotes (extrapolated skirts) differ.

Step-by-step substitution (with units):

$$\begin{aligned} f_{1/f^3}'&=f_{1/f}\cdot\frac{(c_0')^2}{2\Gamma_{rms}^2}=1\ \text{MHz}\times\frac{(0.04)^2}{2\times(0.5)^2} =1\ \text{MHz}\times\frac{0.0016}{0.5}=1\ \text{MHz}\times3.2\times10^{-3}=3.2\ \text{kHz}. \end{aligned}$$

The corner drops from $320$ kHz → $3.2$ kHz (a 100× reduction $=$ the square of the $c_0$ ratio, $10^2$). Note that after improvement the corner falls to $3.2$ kHz, so $\Delta f=10$ kHz for the improved waveform now lies in the 1/f² region (no longer 1/f³); what follows compares the heights of the two 1/f³ asymptotes (extrapolated skirts). 1/f³ phase noise ([P1] Eq.(23)) $\propto c_0^2$, so the change in height is

$$\Delta\mathcal{L}=10\log_{10}\!\left(\frac{(c_0')^2}{c_0^2}\right)=10\log_{10}\!\left(\frac{0.04^2}{0.4^2}\right)=10\log_{10}(0.01)=-20\ \text{dB}.$$

• Result: lowering $c_0$ by 10× → the 1/f³ skirt overall drops 20 dB, and the corner drops 100× (to 3.2 kHz). "Each order-of-magnitude improvement in symmetry shrinks the reach of 1/f³ by two orders of magnitude" is exactly this $c_0^2$ law.
• Dimension check: dB is a log of a power ratio (dimensionless) ✓; the corner is still in Hz ✓.
• One-line Python check:

import numpy as np
c0_old, c0_new = 0.4, 0.04
print("corner ratio:", (c0_new/c0_old)**2,            # -> 0.01  (3.2 kHz / 320 kHz)
      "; dL =", 10*np.log10((c0_new/c0_old)**2), "dB") # -> -20.0 dB

Both examples are pedagogical toys (not transistor-level): $c_0$ uses assumed values representing "residual asymmetry." A real circuit's $c_0$ must be extracted from the effective ISF (including cyclostationary $\alpha$); see device_noise_mapping.

Key takeaways

• Only the ISF's DC coefficient $c_0$ upconverts device 1/f noise into close-in 1/f³ ([P1] Eq.(23)).
• 1/f³ corner $=\omega_{1/f}\cdot c_0^2/(2\Gamma_{rms}^2)$, not equal to the device's 1/f corner ([P1] Eq.(24)).
• Symmetric rise/fall → the ISF cancels over one period → $c_0\to0$ → the 1/f³ corner is pushed to very low frequency.
• Lowering $c_0$ by 10× lowers the 1/f³ corner by 100× ($c_0^2$): illustrative $c_0=0.4$, $f_{1/f}=1$ MHz → corner 320 kHz.
• A ring's $c_0$ can be computed directly from topology: the [P2] App. B closed forms give $c_0=2\Gamma_{dc}=\frac{4\pi}{\eta^2N^2}\frac{1-A}{1+A}$ and corner $\propto\frac{(1-A)^2}{1-A+A^2}\cdot\frac{1}{N}$ ($N=5$, $A=1.5\to42.9$ kHz; [P1] Eq.(24) convention $\times2$) — see asymmetric_isf_closed_form.
• Design levers: differential, symmetric loads, 50% duty; must look at the $c_0$ of the effective ISF (including $\alpha$).
• Experiment: [P2] Fig. 17, phase noise vs. symmetry voltage, shows a minimum.

Further reading

• Full upconversion derivation: flicker_noise_upconversion
• Closed forms for $\Gamma_{rms}$, $c_0$ and the corner of the asymmetric triangular ISF ([P2] App. B): asymmetric_isf_closed_form
• Why $\Gamma$'s shape tracks the waveform slope: waveform_slope
• Effective ISF and $\alpha$: device_noise_mapping, effective_isf
• Fourier coefficients and Parseval: fourier_series_of_isf
• Why SerDes cares more about close-in: serdes_clocking_connection