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Abstract:

This IEEE IFCS 2021 tutorial, presented by Enrico Rubiola, offers a comprehensive overview of phase noise, frequency stability, and related measurement challenges. It begins by defining fundamental concepts, including AM and PM noise, power spectral density, the commonly used script L quantity, and its relation to phase fluctuations (φ), time fluctuations (x), and fractional frequency fluctuations (y). The tutorial introduces the power-law model for describing different noise types (white phase, flicker phase, white frequency, flicker frequency, random walk frequency) and the Allan variance as a time-domain stability measure, providing methods for conversion between frequency-domain spectra (S_y(f)) and time-domain variances (σ_y(τ)).

The presentation then delves into phase noise characteristics in electronic components, distinguishing between additive and parametric noise, and phase-like vs. time-like noise behaviors, particularly in amplifiers, dividers, and synthesizers, including the impact of aliasing in digital circuits. A significant portion is dedicated to phase noise in oscillators, thoroughly explaining the Leeson effect, which models how amplifier/component noise is converted into oscillator phase noise, introducing the critical Leeson frequency (ν₀/2Q). Measurement techniques are detailed, covering traditional methods using phase detectors (mixers) within phase-locked loops (PLLs), modern digital techniques employing ADCs and FPGAs, and the crucial dual-channel cross-correlation method for rejecting instrument background noise. Finally, the tutorial addresses practical measurement pathologies, including power line interference, instrument artifacts, spectral leakage, and systematic errors introduced by components like power splitters. Resources for further learning are also provided.

Understanding Phase Noise: Key Concepts, Sources, Measurement, and Pitfalls

• 0:15 Introduction & Overview: The tutorial will cover phase noise definitions, noise in electronic devices, oscillator noise (Leeson effect), measurement instruments/methods, and common measurement problems (pathologies).
• 1:19 Clock Signal Definition: Introduces the mathematical representation of a clock signal including amplitude (α) and phase (φ) fluctuations.
• 2:22 AM vs. PM Noise Visualization: Explains how pure Amplitude Modulation (AM) noise affects signal peaks while zero-crossings remain stable, whereas pure Phase Modulation (PM) noise causes timing jitter with stable amplitude. This is also shown using vector representation.
• 5:36 Power Spectral Density (PSD): Defines PSD as power distribution over frequency (Watts/Hz or V²/Hz) and introduces S_φ(f) (radians²/Hz) for phase fluctuations.
• 7:30 Script L(f) Definition: Defines the standard engineering quantity for phase noise, L(f) = ½ S_φ(f), expressed in dBc/Hz. Critiques the obsolete SSB power definition.
• 10:13 Other Fluctuation Quantities: Introduces time fluctuation x(t) = φ(t) / (2πν₀) (seconds) and fractional frequency fluctuation y(t) = Δν(t) / ν₀ (dimensionless), highlighting their independence from carrier frequency (ν₀) in ideal synthesis.
• 13:37 Power-Law Noise Model: Describes the common model for phase noise spectra, S_φ(f), as a sum of terms proportional to f^α, covering white phase (f⁰), flicker phase (f⁻¹), white frequency / random walk phase (f⁻²), flicker frequency (f⁻³), and random walk frequency (f⁻⁴). Notes that 2-port devices typically only exhibit f⁰ and f⁻¹ noise.
• 18:08 Allan Variance (σ²_y(τ)): Introduces the Allan variance, a two-sample variance measuring time-domain frequency stability based on averaged fractional frequency differences over an averaging time τ. Explains its relationship to different power-law noise types in the frequency domain.
• 21:59 Converting S_φ(f) to σ_y(τ): Outlines the process: L(f) → S_φ(f) → S_y(f) using S_y(f) = (f/ν₀)² S_φ(f), then using conversion formulas (provided in tables/resources) to get σ²_y(τ) for each power-law term.
• 23:32 Additive vs. Parametric Noise: Explains additive noise (like thermal noise added to the signal, noise floor depends on signal power) and parametric noise (modulation process, noise floor independent of signal power, often flicker noise).
• 27:43 Additive Noise & Phase/Amplitude: Shows how additive noise near the carrier splits equally into AM and PM noise components. S_φ(f) and S_α(f) are proportional to Noise_Floor / P_carrier.
• 30:38 Combined Flicker & White Noise: Demonstrates how the combination results in a power-dependent corner frequency between the 1/f (parametric, power-independent) and white (additive, power-dependent) noise regions. Experimental verification shown.
• 34:21 Cascaded Amplifiers: Explains that flicker noise (parametric) adds linearly (in power) when cascading identical amplifiers, while white noise follows the Friis formula (dominated by the first stage, but affected by attenuation needed).
• 36:25 Phase-Like vs. Time-Like Noise: Defines phase-like noise (φ fluctuation independent of ν₀, e.g., buffer threshold noise) and time-like noise (x fluctuation independent of ν₀, e.g., delay line perturbations). Time-like noise translates to φ ~ ν₀.
• 42:49 Noise in Frequency Synthesizers (DDS Example): Shows how Direct Digital Synthesizers (DDS) exhibit time-like noise (constant x, S_φ ~ ν₀²) until limited by the output stage's noise floor.
• 45:32 Aliasing in Digital Circuits: Explains that phase noise is sampled at clock edges in digital circuits, leading to aliasing where noise from higher frequencies folds back into the baseband (up to f_carrier). The PSD increases as bandwidth decreases (e.g., 3 dB increase per halving of frequency due to aliasing).
• 49:06 Leeson Effect (Oscillator Noise): Introduces the Leeson model: an amplifier with noise feeds a resonator (defined by ν₀ and Q). Amplifier phase noise S_φ,amp(f) is transferred directly above the Leeson frequency (f_L = ν₀ / 2Q) and integrated (multiplied by (f_L/f)²) below f_L, converting flicker/white phase noise of the amplifier into flicker frequency (f⁻³) / white frequency (f⁻²) noise of the oscillator.
• 57:07 Interpreting Oscillator Spectra: Provides examples of deducing amplifier noise figure, power, resonator Q, and amplifier flicker corner frequency from an oscillator's phase noise plot using the Leeson model. Highlights cases where resonator flicker can dominate.
• 1:04:40 Noise Filtering in Oscillators: Discusses oscillators using a narrow-band filter (e.g., quartz) at the output to achieve extremely low far-from-carrier noise, potentially below the thermal noise floor expected from the buffer impedance matching, raising questions about measurement validity.
• 1:07:29 Measurement Instruments Overview: Modern instruments typically use two parallel channels and cross-correlation to measure the Device Under Test (DUT) while rejecting the noise of internal references and measurement hardware.
• 1:09:47 Mixer as Phase Detector: Explains using a Double-Balanced Mixer (DBM) as a phase detector. When inputs are strong, equal frequency, and in quadrature, the output is a DC voltage proportional to phase fluctuations (V_out = K_φ * φ). DBMs offer low intrinsic noise floors.
• 1:14:57 PLL-Based Measurement: Mixers require quadrature, often maintained by a Phase-Locked Loop (PLL). The PLL acts as a high-pass filter for noise measurement. Discusses trade-offs between slow (simple analysis, high dynamic range needed) and fast (complex analysis, lower dynamic range, suppresses injection locking) PLL bandwidths.
• 1:19:01 Dual-Channel Cross-Correlation: Explains the principle: measure the DUT with two independent measurement channels (A and B). The cross-spectrum <XY> converges to the DUT signal spectrum S_DUT(f) while the uncorrelated instrument noise terms (<AB>, <A*DUT_noise>, etc.) average towards zero proportionally to 1/√M (M = number of averages).
• 1:21:04 Digital Measurement Systems: Describes systems using fast ADCs, FPGAs for down-conversion (digital mixing), filtering, decimation, and CORDIC algorithms to extract phase. These allow direct phase measurement over multiple cycles but often have higher noise floors than mixer-based systems. Requires cross-correlation (4 ADCs for full rejection).
• 1:26:54 Advanced Instruments (e.g., R&S FSWP): Mentions high-frequency analyzers using IQ mixers for down-conversion to an IF suitable for digitization and cross-correlation processing.
• 1:32:34 Cross-Spectrum Estimators: Compares using the magnitude |<XY>| (biased, includes full noise variance) vs. the real part Re{<XY>} (unbiased, includes half the noise variance) as the estimator for S_DUT(f). Re{} is preferred.
• 1:35:07 Logarithmic Resolution Averaging: Explains that taking FFTs with constant fractional resolution (Δf/f = const) allows for significantly more averaging (larger M) at higher Fourier frequencies, improving noise rejection (extra 5 dB/decade slope to the noise floor).
• 1:38:49 Measurement Pathologies:
  • 1:38:55 Power Line Spurs: 50/60 Hz and odd harmonics often visible due to transformer saturation.
  • 1:40:24 Instrument Artifacts: Shows examples like discontinuities (sampling rate changes), bumps (spectral leakage, correlation), holes (anti-correlation, e.g., Keysight 5052B), notches (spurs interacting with anti-correlation).
  • 1:43:06 Power Splitter Issues: Explains how resistor noise in common power splitters (Wilkinson, resistive) introduces anti-correlated noise at the outputs, causing a systematic error (underestimation) in phase noise measurements, especially near the thermal floor (kT/P).
  • 1:46:11 Estimator Bias & Artifacts: Absolute value estimators can hide anti-correlated artifacts, while the real part estimator reveals them (can go negative, causing plotting issues but indicating a problem). Shows an extreme example with a mismatched DUT causing nonsensical results due to splitter noise and crosstalk.
• 1:49:20 Resources: Provides pointers to reference books (Leeson effect, synthesizers), a comprehensive reference chart, the Femto-ST measurement lab, open access lectures, and the presenter's website (rubiola.org) for further information.
• 1:53:01 Q&A Session: Includes discussions clarifying noise types in 2-port devices, Allan variance weights, trade-offs in MEMS oscillator design (high-freq/low-Q vs. low-freq/high-Q, Leeson model applicability), PLL noise scaling, carrier collapse, phase vs. time type noise in PLLs, fractional-N synthesis, and non-linear effects in resonators.