Phase Noise Injection

The process is governed by a phase noise mask, a frequency-domain specification defining the single-sideband (SSB) phase noise power relative to the carrier (in dBc/Hz) at specific offset frequencies.

• Close-in noise: Dominated by flicker noise, rolling off at 30 dB/decade
• Thermal noise floor: Broadband white noise floor at larger offsets
• PLL loop bandwidth: A characteristic knee in the mask where the VCO's free-running noise is suppressed

The mask is applied to a clean carrier using a phase noise generation algorithm that filters white Gaussian noise to match the target spectral profile.

Phase noise manifests in the time domain as random jitter—the deviation of signal zero-crossings from their ideal positions. The injection process synthesizes this by modulating the carrier's instantaneous phase with a random process.

• Cycle-to-cycle jitter: Short-term variation between adjacent clock periods
• Period jitter: Deviation of a single period from the ideal
• Accumulated jitter: The unbounded phase error over long observation intervals, critical for OFDM systems

The relationship between phase noise and jitter is defined by integrating the phase noise mask over a specific bandwidth. A Wiener process model is often used for free-running oscillators, where the phase variance grows linearly with time.

The theoretical foundation for phase noise injection is Leeson's equation, which models the single-sideband phase noise of an ideal feedback oscillator.

• L(fm): Proportional to the resonator's loaded Q factor, the amplifier's noise figure, and the flicker noise corner
• 1/f³ region: Caused by the upconversion of flicker noise in the active device
• 1/f² region: Thermal noise shaped by the resonator's bandpass transfer function
• Resonator Q: Higher Q (e.g., crystal vs. LC tank) directly suppresses close-in phase noise

Synthetic injection engines parameterize Leeson's model to emulate specific oscillator types—from low-cost MEMS oscillators to high-stability oven-controlled crystal oscillators (OCXOs).

The core algorithm for phase noise injection involves spectral shaping of a white noise source to match a target phase noise profile.

• Frequency-domain filtering: An FIR filter is designed with a magnitude response matching the square root of the desired phase noise PSD
• IIR pole-zero modeling: Recursive filters efficiently model the 1/f² and 1/f³ slopes with poles near the unit circle
• Oscillator-specific templates: Pre-defined parameter sets for Colpitts, Pierce, and ring oscillators

The shaped noise sequence modulates the phase argument of the complex baseband signal: y[n] = x[n] * exp(j * φ[n]), where φ[n] is the integrated, filtered phase noise.

In a receiver, injected phase noise on the local oscillator causes reciprocal mixing, where a strong adjacent-channel blocker is convolved with the LO's noise sidebands, raising the in-band noise floor.

• Desensitization: The receiver's effective sensitivity is degraded in the presence of strong interferers
• Spectral regrowth: In transmitters, phase noise causes energy to spread beyond the intended channel mask, violating ACLR specifications
• EVM floor: Uncorrected phase noise sets a fundamental limit on achievable modulation accuracy

Synthetic phase noise injection must replicate these system-level effects to train robust fingerprinting models that can distinguish oscillator impairments from channel effects.

Real transceivers exhibit both correlated and uncorrelated phase noise contributions across the TX and RX chains.

• Shared LO architecture: In TDD systems, the same oscillator is used for upconversion and downconversion, creating a correlated phase noise signature that partially cancels
• Independent LOs: FDD systems use separate oscillators, producing uncorrelated noise that adds in quadrature
• MIMO arrays: Each RF chain may have independent or synchronized LOs, creating a spatial phase noise fingerprint

Synthetic injection must model these architectural nuances to create realistic, device-specific impairments that a deep learning classifier can exploit.

A phase noise mask is a frequency-domain specification that defines the maximum allowable single-sideband phase noise power relative to the carrier (in dBc/Hz) at various offset frequencies. It serves as the target profile for a phase noise injection algorithm. A typical mask specifies limits at offsets like 100 Hz, 1 kHz, 10 kHz, and 100 kHz, creating a template that the synthesized noise spectrum must not exceed. This mask is the primary input for generating realistic, oscillator-specific impairments.

While a phase noise mask is a frequency-domain limit, the actual impairment is often generated in the time domain as a jitter spectrum. This involves creating a sequence of timing errors that perturb the ideal zero-crossings of a clock signal. The process typically involves:

• Generating white Gaussian noise.
• Filtering it with a transfer function derived from the phase noise mask.
• Integrating the result to produce a phase error sequence. This time-domain jitter is then used to modulate the carrier, directly emulating the short-term instability of a real oscillator.

Leeson's equation is a foundational heuristic model that predicts the single-sideband phase noise of an oscillator. It models the noise as a function of offset frequency, oscillator loaded Q-factor, noise figure of the active device, and flicker noise corner. The equation reveals key regions: a 1/f³ region close to the carrier, a 1/f² region, and a flat noise floor. Understanding Leeson's model is critical for parameterizing synthetic phase noise to match the physical behavior of real oscillator circuits.

Synthetic phase noise is often modeled as a sum of independent power law noise processes, each dominating a different frequency range. These are defined by their spectral density's dependence on frequency (f):

• White PM (f⁰): Flat phase noise floor.
• Flicker PM (f⁻¹): Noise increasing at 10 dB/decade.
• White FM (f⁻²): Noise increasing at 20 dB/decade.
• Flicker FM (f⁻³): Noise increasing at 30 dB/decade.
• Random Walk FM (f⁻⁴): Noise increasing at 40 dB/decade. Combining these processes allows for the precise synthesis of a complete, realistic phase noise profile.

Allan variance (or its square root, Allan deviation) is a time-domain stability metric used to characterize and validate synthetic phase noise. Unlike a phase noise mask, it quantifies frequency stability as a function of averaging time. For a simulated oscillator, the Allan deviation plot should match the target device's behavior, revealing the contributions of different noise processes (e.g., white FM has a τ⁻¹/² slope). It is an essential tool for verifying that the injected noise produces the correct long-term drift characteristics.

The direct consequence of phase noise injection is the degradation of the carrier-to-noise ratio (CNR). In a communication system, phase noise spreads the carrier's power into adjacent frequencies, raising the noise floor and causing reciprocal mixing in the receiver. For a synthetic fingerprint, the specific pattern of this degradation—how the CNR varies across the signal bandwidth—becomes a unique, identifying feature. This is distinct from additive white Gaussian noise (AWGN) as it is a multiplicative, signal-dependent impairment.