Vocabulary, Physical Quantities and Units for the Measurement of Amplitude Noise and Phase Noise

This paper addresses the confusion caused by non-SI quantities and misleading terminology in phase and amplitude noise measurements, advocating for the universal adoption of the International System of Units (SI) and clear, unambiguous language to standardize the field.

The Gist

Imagine you are trying to listen to a pure, perfect musical note played on a violin. In a perfect world, that note would be a steady, unwavering tone. But in the real world, the violin string wobbles slightly (changing the pitch) and the bow pressure varies (changing the volume). These wobbles are what scientists call Phase Noise (pitch wobble) and Amplitude Noise (volume wobble).

This paper is essentially a group of international scientists (from France, Germany, Italy, and the USA) raising their hands to say: "Hey, we are using the wrong dictionary and the wrong ruler to measure these wobbles, and it's causing a massive amount of confusion."

Here is the breakdown of their argument using simple analogies:

1. The Problem: A Messy Kitchen

The authors argue that the scientific community has been cooking in a kitchen where everyone uses different measuring cups.

• The Confusion: Some people measure noise in "decibels relative to the carrier" (dBc/Hz), while others use standard physics units.
• The Result: It's like one baker saying a cake needs "2 cups of flour" and another saying it needs "2 'magic-spoonfuls' of flour." Without a standard, you can't compare recipes or build things reliably.

2. The "Weird" Unit: The Phantom Ruler

The paper focuses heavily on a specific quantity called $L(f)$, which is currently the industry standard. The authors say this unit is "weird" and "improper."

• The Analogy: Imagine you are measuring the height of a building. The standard way is to say "100 meters." But the current industry standard is to say "100 'Building-Units'."
  • The problem is, a "Building-Unit" isn't a real unit of length. It's a ratio that changes depending on how big the building is.
  • The authors point out that the current unit ($dBc/Hz$) is mathematically inconsistent. It's like saying "This angle is 81 degrees" when it should be "1 radian." It's a unit that doesn't actually exist in the official International System of Units (SI).

3. The "Single Sideband" Misunderstanding

The paper attacks the term "SSB Noise" (Single Sideband Noise).

• The Analogy: Imagine you are listening to a radio station. The signal has a main voice (the carrier) and two echoes on either side (sidebands).
• The current method says, "Let's just measure the echo on the left side and assume that tells us everything about the noise."
• The Flaw: The authors say this is like trying to understand a person's personality by only listening to their left ear. Sometimes the left and right echoes behave differently (one might be pitch noise, the other volume noise). By only looking at one side, you lose crucial information about what is actually happening.

4. The Solution: Go Back to Basics

The authors want the scientific world to stop using these "magic units" and go back to the International System of Units (SI), which is the global standard for all science (like meters, kilograms, and seconds).

• For Pitch Wobbles (Phase Noise): Instead of the weird "dBc/Hz," they want to use radians squared per Hertz ($rad^2/Hz$).
  • Think of it this way: If you are measuring how much a clock hand wobbles, you should measure the angle of the wobble in radians, not in some made-up "wobble-unit."
• For Volume Wobbles (Amplitude Noise): They want to use decibels relative to a ratio (dB/Hz), which is a clear, standard way to measure how much the volume changes.

5. Why Does This Matter?

You might ask, "Who cares if we use weird units as long as we know what we are talking about?"

The authors say it matters because:

1. Calibration is Broken: The only lab in the world currently certified to calibrate these instruments (LNE in France) is using a "broken ruler." If you calibrate your instrument with a broken ruler, every measurement you take for the rest of your life is technically wrong.
2. Future-Proofing: Science is moving forward. As we redefine how we measure time and frequency, sticking to these old, messy units will make it impossible to compare data between different labs in the future.
3. Clarity: It stops the confusion between "pitch noise" and "volume noise."

The Bottom Line

The authors are calling for a global cleanup. They want the scientific community to agree on a new set of rules:

• Stop using the "SSB" shortcut.
• Stop using the "dBc/Hz" magic unit.
• Start using the official SI units (radians and standard ratios) so that a measurement taken in Paris means exactly the same thing as a measurement taken in Boulder, Colorado.

They are essentially saying: "Let's all speak the same language and use the same ruler, so we can finally stop arguing about the noise and start fixing it."

Technical Summary

Based on the provided paper, here is a detailed technical summary covering the problem, methodology, key contributions, results, and significance.

1. Problem Statement

The paper identifies a critical lack of standardization and clarity in the measurement and terminology of phase noise and amplitude noise. The current domain suffers from:

• Non-SI Units: The widespread use of the non-SI quantity $L(f)$ and its unit dBc/Hz (decibels relative to the carrier per Hertz).
• Misleading Terminology: Terms like "SSB noise" (Single Sideband noise) and "offset frequency" are used in ways that obscure the physical nature of the noise (AM vs. PM) and imply incorrect physical models.
• Conceptual Confusion: The definition of $L(f)$ as a ratio of sideband power to carrier power in a 1 Hz bandwidth is physically flawed for large phase deviations because it violates energy conservation principles (sideband power cannot exceed carrier power without the carrier diminishing).
• Metrological Inconsistency: Major metrological institutions (like LNE) currently maintain Calibration and Measurement Capabilities (CMCs) based on these non-SI units, creating a disconnect between international standards (SI) and practical oscillator metrology.

2. Methodology

The authors employ a theoretical and metrological analysis to deconstruct current practices and propose SI-consistent alternatives:

• Mathematical Derivation: They analyze the fundamental signal equation $v(t) = V_0 [1 + \alpha(t)] \cos[2\pi ft + \phi(t)]$ to distinguish between fractional amplitude noise $\alpha(t)$ and phase noise $\phi(t)$.
• Spectral Analysis Review: They examine the standard measurement process (using the Welch algorithm) to define the Power Spectral Density (PSD) of phase, $S_\phi(f)$, and demonstrate its physical dimensions.
• Unit Consistency Check: They compare the dimensions of the current standard ($L(f)$) against the International System of Units (SI). They mathematically demonstrate that $L(f)$ implies a "square angle" unit ($\sqrt{c}$) that is physically nonsensical ($\approx 81^\circ$), whereas the PSD $S_\phi(f)$ has the dimension of time ($rad^2/Hz$).
• Limitation Analysis: They critique the "SSB noise" model, showing it fails when phase noise is large (common in real oscillators) because the sideband power is not independent of the carrier power due to energy conservation.

3. Key Contributions

The paper proposes a comprehensive revision of vocabulary, quantities, and units to align with the SI:

• Adoption of SI-Consistent Quantities:

  • Phase Noise: The authors advocate replacing $L(f)$ with the Power Spectral Density of phase, $S_\phi(f)$.
    • Unit: $rad^2/Hz$ (or $dB \cdot rad^2/Hz$).
    • Rationale: This matches the physical dimension of the measurement process (phase is an angle in radians) and is consistent with SI.
  • Amplitude Noise: The authors propose using $S_\alpha(f)$.
    • Unit: $dB/Hz$ (or $dB \cdot (V/V)^2/Hz$).
    • Rationale: Since fractional amplitude is dimensionless, the unit should reflect the ratio directly without the "carrier" reference.

• Terminology Corrections:

  • Abolish "SSB Noise": The term is misleading because a single sideband does not define the relationship between Upper and Lower Sidebands required to distinguish AM from PM.
  • Clarify "Offset Frequency": The term is deemed unsuitable for describing modulation; "Fourier frequency" is preferred.
  • Refine "Added Noise": For two-port devices, the term "added noise" is redundant; it should simply be "noise of the device."

• Mathematical Correction:

  • The paper highlights that the common relation $L(f) = \frac{1}{2}S_\phi(f)$ is only valid for small phase deviations. For large deviations, the linear approximation breaks down, and the "carrier power" denominator in the $L(f)$ definition becomes invalid.

4. Results and Observations

• Dimensional Inconsistency: The authors prove that the unit $dBc/Hz$ implies a unit of angle $\sqrt{c} \approx 81^\circ$, which is physically arbitrary and inconsistent with the radian.
• Measurement Reality: Real-world oscillators (e.g., the 100 MHz OCXO example provided) often exhibit phase noise levels where $\phi(t)$ diverges over time. In these cases, the "SSB noise" definition (ratio of sideband to carrier) fails because the sideband power is not negligible compared to the carrier.
• Current Metrological Gap: The LNE is the only lab with CMCs in the BIPM Key Comparison Data Base (KCDB), but these are based on the non-SI $L(f)$ and $dBc/Hz$. The authors note that while no objection was raised during the review, these definitions are fundamentally flawed and require rewriting.

5. Significance

• Standardization: The paper aims to unify the global metrology community (NIST, PTB, INRiM, LNE) under a single, SI-compliant framework, eliminating confusion between laboratories.
• Scientific Rigor: By enforcing SI units ($rad^2/Hz$), the field moves away from empirical, carrier-referenced ratios toward physically meaningful quantities that scale correctly regardless of signal power or modulation depth.
• Future Impact: The authors call for a revision of the BIPM Key Comparison Data Base and encourage other institutes to adopt these SI-consistent definitions. This is particularly urgent as the SI second is being redefined, necessitating a parallel update in how frequency stability and noise are quantified.
• Educational Value: The paper serves as a critical review for researchers and engineers, correcting decades of "improper and misleading" terminology that has hindered clear communication in oscillator design and analysis.