Standard Phase Noise Model

• Standard Phase Noise Model is defined by mathematical assumptions describing white and flicker noise contributions to phase fluctuations in practical devices.
• It illustrates how different architectures, such as cascaded, parallel, and regenerative amplifiers, scale noise parameters through topology-dependent rules.
• Experimental validations confirm the model’s predictive accuracy, informing oscillator design and system performance in communications and sensing platforms.

Phase noise models are fundamental for quantifying and mitigating the deleterious effects of random phase fluctuations in oscillators, amplifiers, and communication links across RF, microwave, photonic, and quantum systems. The “Standard Phase Noise Model” is defined by a set of mathematical and physical assumptions, parameterizations, and scaling rules that describe the spectral and statistical properties of these phase fluctuations in practical components and systems. Its structure and predictive power are widely validated across electronics, optoelectronics, and quantum communication platforms, serving as the essential interface between oscillator/device characterization and system-level performance analyses in communication, timing, and sensing architectures.

1. Core Types and Mathematical Description of Phase Noise

The standard phase noise model distinguishes two principal spectral components in the phase noise power spectral density (PSD) surrounding a carrier:

• White Phase Noise: Arises from broadband thermal noise, appearing as a flat (frequency-independent) contribution to the PSD. For an amplifier with noise figure $F$, carrier power $P_0$, at reference temperature $T_0$ (typically 290 K), and Boltzmann’s constant $k$, its coefficient is $b_0 = (F k T_0)/P_0$. This component scales inversely with carrier power and sets the ultimate noise floor far from the carrier.
• Flicker (1/f) Phase Noise: Exhibits a $1/f$ or $1/f^{\gamma}$ spectral slope, typically with $\gamma = 1$ (sometimes higher due to physical mechanisms such as flicker upconversion or adsorption-desorption; see below). Its coefficient $b_{-1}$, capturing the $1/f$ part, is a device-intrinsic property and, crucially, is almost independent of carrier power over the normal operating range:

$S_{\phi}(f) = b_0 + \frac{b_{-1}}{f}$

As established experimentally and theoretically, the $b_{-1}$ term arises primarily from up-conversion of near-dc 1/f noise by device nonlinearities and parametric modulation effects.

A general phase noise spectrum for oscillators and amplifiers thus takes the form:

$$S_{\phi}(f) = b_0 + \frac{b_{-1}}{f} + \frac{b_{-2}}{f^2} + \frac{b_{-3}}{f^3} + ...$$

In most practical RF and microwave amplifiers, $b_{-2}$ and $b_{-3}$ are negligible, but in oscillators, the integration of noise processes can lead to observable $1/f^2$ and $1/f^3$ dependencies, especially close to the carrier.

2. Topology Dependence and Scaling Laws

The standard model makes specific, experimentally validated predictions regarding how phase noise behaves in different amplifier and oscillator topologies:

Topology	White Noise $b_0$	Flicker Noise $b_{-1}$
Single Amplifier	$(F k T_0) / P_0$	Device constant
m Cascaded Amplifiers	$(F_1 + \frac{F_2-1}{A_1^2}+\dots)\frac{kT_0}{P_0}$	$m b_{-1}$
m Parallel Amplifiers	$\approx$ single cell (if splitters/comb combiners are ideal)	$(1/m) b_{-1}$
Recirculating Amplifier	---	$m^2 b_{-1}$

• Cascaded Amplifiers: Flicker component scales linearly with the number of stages ($b_{-1,\mathrm{chain}} = m b_{-1}$); white noise follows Friis’ formula.
• Parallel Amplifiers: Flicker noise is reduced by averaging ($b_{-1,\mathrm{parallel}} = b_{-1} / m$). White noise remains unchanged if power division/combination is ideal.
• Regenerative Amplifiers: Flicker noise increases quadratically ($b_{-1,\mathrm{RA}} = m^2 b_{-1}$), exceeding that from mere cascading due to repeated phase shifting from the same low-frequency process.

These scaling rules are analytically derived by modeling how independent or common noise sources combine through device nonlinearity and gain structures, and are confirmed by direct measurements across amplifier families.

3. Physical Origin and Parametric Upconversion of 1/f Noise

The origin of flicker ($1/f$) phase noise is attributed to the parametric conversion of near-dc fluctuators in the semiconductor or device material, such as carrier-number fluctuations, trapping/detrapping, and mobility fluctuations:

• In amplifiers: The near-dc 1/f noise is “up-converted” to the carrier through second-order device nonlinearity. Mathematically, with input $u(t) = U_0 e^{j2\pi\nu_0 t} + n(t)$ (where $n(t)$ is near-dc noise), the carrier output at the fundamental frequency can be written as:

$$v(t) = a_1 U_0 e^{j2\pi\nu_0 t} + 2a_2 [n'(t) + j n''(t)] U_0 e^{j2\pi\nu_0 t}$$

The phase noise is thus proportional to the quadrature component $n''(t)$, with:

$\phi(t) = \frac{2a_2}{a_1} n''(t)$

and $S_\phi(f) = 4(a_2^2/a_1^2) S_{n''}(f)$.

• Device and material dependence: $b_{-1}$ is set by device geometry, material properties, and internal noise mechanisms, largely decoupled from external signal levels until compression or saturation. Both experimental data and theoretical calculations (e.g., for bipolar, FET, and optoelectronic amplifiers, including photonic variants) confirm this independence across practical device power ranges.

4. Experimental Validation and Parameter Extraction

A systematic suite of measurements across commercial and legacy amplifiers, spanning from HF to microwave (10 MHz to 10 GHz), confirms the standard phase noise model’s predictions:

• White noise $b_0$ follows the $1/P_0$ law to high accuracy.
• Flicker ($b_{-1}$) is measurably invariant across substantial carrier power ranges.
• Cascaded and parallel architectures deliver additive and suppressive flicker noise scaling, respectively, with measured changes closely matching theoretical 3 dB increments (cascaded) or reductions (parallel) per stage/cell.
• Regenerative amplifiers show an m$^2$ scaling in flicker noise, as verified in oscillator loop experiments.
• The measurement setups utilize both saturated mixer schemes and carrier-suppression bridges to achieve low background and high-dynamic-range 1/f characterizations.

These results are consistent across housing technologies (GaAs, SiGe, bipolar, microwave transistor), bandwidths, and even device ages, highlighting the universality of the model.

5. Implications for System Design and Applications

Understanding the standard phase noise model is central for optimizing oscillators, frequency synthesizers, radar front-ends, telecommunication links, and sensitive measurement systems:

• Oscillator design: The phase noise of the sustaining amplifier translates to frequency noise via the Leeson effect, emphasizing the need to minimize $b_{-1}$ for achieving ultra-low oscillator noise.
• Multi-stage and distributed systems: The topological dependence (from cascading, paralleling, or regeneration) allows designers to tailor architectures for minimum flicker, given application constraints.
• Microwave photonics and radio astronomy: Flicker noise and spectral purity are governing factors for system resolution and dynamic range; parallel designs (or feedforward architectures) provide noise floors essential for cutting-edge applications.
• CAD/simulation integration: The standard phase noise model’s analytic expressions can be incorporated directly into circuit simulators (e.g., replacing empirical “corner frequency” models with physical $f_c = {b_{-1} P_0}/{F k T_0}$).

6. Extensions and Physical Limitations

While the standard model robustly predicts flicker and white contributions in linear or mildly nonlinear regimes, further refinements are warranted for:

• Strongly nonlinear/compressed regimes where device physics or high-field effects invalidate the baseline parameterizations.
• Higher-order noise mechanisms: For oscillators and optomechanical systems, physical sources such as adsorption–desorption (AD) noise can produce $1/f^3$ and even steeper ($1/f^4$) slopes, especially under ultralow temperature and pressure conditions (Mathai et al., 2019).
• Digital electronics: For digital clocks and FPGAs, “phase-type” vs “time-type” noise and aliasing phenomena must be carefully distinguished (Calosso et al., 2016). Flicker noise scales inversely with device volume, and aliasing transforms spectral power as a function of the sampling and circuit topology.
• Quantum and optical systems: The standard model is extended using parallel frameworks (e.g., in optomechanical, photonic, or quantum oscillator platforms) but always by direct mapping to the physical origin of noise terms and their propagation through device or system structure.

7. Summary Table: Standard Phase Noise Model Scaling

Parameter/Topology	Expression	Scaling with Device/Architecture
White Noise $b_0$	$b_0 = \frac{F k T_0}{P_0}$	$\propto 1/P_0$
Flicker Noise $b_{-1}$	$b_{-1} = \mathrm{constant}$ (device-intrinsic; no $P_0$ dependence)	$\propto$ device material and geometry
Cascaded $m$ Stages	$b_{-1,\mathrm{chain}} = m b_{-1}$	Linear in $m$
Parallel $m$ Cells	$b_{-1,\mathrm{parallel}} = b_{-1}/m$	Inverse in $m$
Regenerative $m$-fold Feedback	$b_{-1,\mathrm{regen}} = m^2 b_{-1}$	Quadratic in $m$
White Corner	$f_c = (b_{-1} P_0)/(F k T_0)$	Device and power dependent
Experimental Validation	Matches predicted 3 dB increments, 2.5–3 dB reductions, and $m^2$ scaling in oscillator loops	Verified from HF to GHz, all archetypes

• Phase Noise in RF and Microwave Amplifiers (Boudot et al., 2010)
• Calculation of the Performance of Communication Systems from Measured Oscillator Phase Noise (Khanzadi et al., 2013)
• Phase Noise and Jitter in Digital Electronics (Calosso et al., 2016)
• Modeling the colors of phase noise in optomechanical oscillators (Mathai et al., 2019)