Phase-Tracking Reference Signals (PTRS)

• Phase-Tracking Reference Signals (PTRS) are specially structured reference symbols embedded in OFDM and DFT-s-OFDM systems to directly sample and compensate phase noise.
• PTRS design includes sparse distributed patterns and dedicated block configurations, balancing overhead, estimation error, and support for high-order modulations.
• Advanced estimation methods like LMMSE IF and ML/LS algorithms improve phase noise compensation, reducing RMS error and enhancing EVM and BLER performance.

Phase-Tracking Reference Signals (PTRS) are specially structured reference symbols embedded within transmission frames of OFDM and DFT-spread-OFDM (DFT-s-OFDM) systems. PTRS enable robust estimation and compensation of phase noise (PN)—the dominant impairment in high-frequency, wideband wireless and distributed RF environments—by directly sampling the instantaneous PN process. Rigorous PTRS design and associated estimation methodologies are essential to achieving ultra-reliable high-throughput communications at mmWave/sub-THz bands and to maintaining exceptional phase coherence in precision RF synchronization systems.

1. Phase Noise and System Model

Phase noise arises from oscillator imperfections, exhibiting a correlated, typically Wiener-type stochastic process. For discrete-time samples at period $T_s$, the aggregate PN at sample $n$ is $\phi[n]$, with covariance

$$[\mathcal{R}_\Phi]_{m, n} = \mathbb{E}\{e^{j(\phi[m] - \phi[n])}\}$$

and a power spectral density (PSD) comprising $1/f^2$ and $1/f^3$ regions as per the 3GPP PN model TS 38.803 (Bello et al., 2022).

In DFT-s-OFDM, post-equalization, the received frequency-domain sample on subcarrier $k$ (neglecting CFO) is

$r_k = s_k e^{j \phi'_k} + \beta_k + w_k$

where $s_k$ is the QAM symbol, $\phi'_k \approx \phi[k]$ the sampled PN, $n$0 the PN-induced inter-carrier interference (ICI), and $n$1 additive white Gaussian noise (Bello et al., 2022). Parallel models describe OFDM, while in time-domain transmission (e.g., pre-DFT PTRS insertion), PTRS allow direct sampling of $n$2 within the DFT-s-OFDM block (Ibrahim et al., 20 Jan 2025).

2. Formal PTRS Structures in 5G and Beyond

PTRS patterning is dictated by both system needs and PN dynamics:

• Distributed (sparse) PTRS: Pilots at selected frequency bins and/or time-symbols within each OFDM/DFT-s-OFDM block. For uplink DFT-s-OFDM in 3GPP NR, a typical contiguous pattern divides each symbol into $n$3 groups of $n$4 consecutive subcarriers, yielding $n$5 PTRS per block (Bello et al., 2022, Qi et al., 2018).
• Block PTRS: A contiguous subband (block) dedicated to PTRS, fully or partly occupying one DFT-s-OFDM/OFDM symbol, designed such that the PTRS pulse bandwidth equals its sampling rate ($n$6) and achieves alias-free PN acquisition (Ibrahim et al., 20 Jan 2025, Levanen et al., 2019).
• Time-frequency tiling: Multiresolution tiling based on sampling theory, e.g., grouping PTRS in time to minimize aliasing and maximize tracking performance; this includes pre-DFT insertion for DFT-s-OFDM and frequency-domain pilot grouping for OFDM (Ibrahim et al., 20 Jan 2025, Levanen et al., 2019).
• RF synchronization lines: In distributed systems (e.g., LCLS-II), PTRS are continuous-wave phase references distributed via cables, with phase-locked averaging of forward and reverse signals for absolute phase definition and drift compensation (Murthy et al., 2022).

3. Phase-Noise Estimation and Compensation Algorithms

PTRS-based PN estimation exploits known pilot positions and the coupled/uncoupled statistics of PN and ICI:

• DFT-s-OFDM LMMSE IF Algorithm: The LMMSE Interpolation-Filter (IF) forms the optimal estimate (in the MSE sense) of the per-subcarrier PN rotation via

$n$7

where $n$8 are PTRS observations, $n$9 and $\phi[n]$0 encapsulate PTRS and PN covariances, and $\phi[n]$1 is a precomputable filter. The final estimate for all active subcarriers is

$\phi[n]$2

This structure minimizes residual PN and ICI, outperforming linear/spline interpolation and DCT estimation, particularly in low-density PTRS regimes (Bello et al., 2022).

• Block PTRS ML/LS Estimation: Block PTRS enables direct sampling of the PN process with bandwidth matched to its repetition rate. Estimation is performed via angle-difference or maximum-likelihood on block samples, followed by MMSE/interpolation to yield per-symbol corrections. The scheme reduces aliasing of unsampled PN components and improves overall RMS phase error and EVM relative to sparse PTRS allocation (Ibrahim et al., 20 Jan 2025, Levanen et al., 2019).
• Common Phase Error Removal in OFDM: For conventional OFDM, CPE is estimated by aggregating distributed pilots within an OFDM symbol: $\phi[n]$3 where $\phi[n]$4 and $\phi[n]$5 are the received and known PT-RS symbols. Residual ICI is left largely unaddressed unless block PTRS or advanced MMSE techniques are used (Qi et al., 2018, Levanen et al., 2019).
• Bidirectional Phase Reference Loops: In synchronization systems (e.g., LCLS-II PRL), digital phase-averaging tracking loops lock the mean of forward- and reverse-propagating PTRS signals to a master oscillator, nulling cable drift and suppressing environmental 1/f phase noise down to the DSP/system noise floor (Murthy et al., 2022).

4. Practical Architectures and Implementation

PTRS schemes differ in complexity, integration point, and runtime costs:

Method	PTRS Placement	Runtime Complexity	Integration Point
3GPP DFT-s-OFDM IF	Sparse/contiguous	$\phi[n]$6 matrix–vector	Post-equalization
Block PTRS (DFT/OFDM)	Dedicated block	One FFT/IFFT, interpolation	Symbol demodulation
OFDM PT-RS (CPE)	Sparse, per-symbol	Symbol-wise rotation	Equalization
PRL tracking loop	Continuous-wave	DSP, FPGA accumulator	Synchronization

• The LMMSE IF filter can be precomputed for a worst-case PN profile or updated in real time using pilot observations (Bello et al., 2022).
• Implementation in fielded systems (e.g., LCLS-II) achieves sub-millidegree phase stability and sub-microsecond latency; in wireless PHYs, integration is typically after channel equalization and before QAM demapping (Murthy et al., 2022, Bello et al., 2022).

5. Performance Trade-Offs and Design Guidelines

PTRS density and allocation strategy trade off overhead, estimation error, and support for high-order modulation:

• Overhead: Block PTRS or enhanced DFT-s-OFDM PTRS achieves full 256-QAM support at $\phi[n]$7 overhead; standard 3GPP sparse PTRS is insufficient beyond 64-QAM or at SCS $\phi[n]$8 kHz (Levanen et al., 2019, Ibrahim et al., 20 Jan 2025).
• Estimation error: Enhanced PTRS patterns reduce PN interpolation error $\phi[n]$9 by up to $$[\mathcal{R}_\Phi]_{m, n} = \mathbb{E}\{e^{j(\phi[m] - \phi[n])}\}$$0 compared to standard allocation (Levanen et al., 2019).
• EVM and BLER: Block PTRS provides $$[\mathcal{R}_\Phi]_{m, n} = \mathbb{E}\{e^{j(\phi[m] - \phi[n])}\}$$1 dB SNR advantage in high-order modulations at mmWave/sub-THz frequencies, directly improving error vector magnitude (EVM) and block error rate (BLER) (Levanen et al., 2019, Ibrahim et al., 20 Jan 2025).
• PAPR penalty: Block PTRS increases PAPR by $$[\mathcal{R}_\Phi]_{m, n} = \mathbb{E}\{e^{j(\phi[m] - \phi[n])}\}$$2 dB, an acceptable cost for robust phase tracking (Ibrahim et al., 20 Jan 2025).
• Frequency and time tiling: Time-density M and frequency-density L in the PT-RS grid should be tailored to modulation order and bandwidth; M=1 is mandated for 256-QAM, with frequency density L decreasing as bandwidth increases (Qi et al., 2018).

6. Advanced Applications: Distributed RF and Synchronization Lines

Distributed high-precision RF and synchronization systems also use PTRS, albeit as analog or mixed-signal phase reference lines:

• LCLS-II PRL: Implements bidirectional averaging at 1300 MHz across hundreds of meters, achieving jitter $$[\mathcal{R}_\Phi]_{m, n} = \mathbb{E}\{e^{j(\phi[m] - \phi[n])}\}$$3 RMS (integration over 60 Hz–1 kHz), and floor phase noise $$[\mathcal{R}_\Phi]_{m, n} = \mathbb{E}\{e^{j(\phi[m] - \phi[n])}\}$$4 dBrad$$[\mathcal{R}_\Phi]_{m, n} = \mathbb{E}\{e^{j(\phi[m] - \phi[n])}\}$$5/Hz. Phase-averaging tracking loops compensate cable drift and secure cavity-to-beam stability (Murthy et al., 2022).
• Tracking-loop design: Digital feedback, phase accumulator, and real-time closed-loop update eliminate drift and environmental noise without burdening front-end cavity measurements (Murthy et al., 2022).

7. Current Trends, Limitations, and Research Directions

• Block PTRS and multiresolution allocation: Block PTRS, combined with time-frequency tiling, represents the state-of-the-art for alias-free, high-fidelity PN estimation, with demonstrated performance gains at $$[\mathcal{R}_\Phi]_{m, n} = \mathbb{E}\{e^{j(\phi[m] - \phi[n])}\}$$6 GHz and for sub-THz PHYs (Ibrahim et al., 20 Jan 2025, Levanen et al., 2019).
• PTRS randomization and multi-TRP: Frequency randomization and orthogonal PTRS assignment to different transmission points (TRPs) avoid pilot collision and support MU-MIMO/CoMP (Qi et al., 2018).
• PTRS in future NR releases: PTRS frameworks continue to evolve, with recent work advocating dynamic PTRS block placement, adaptive density based on channel and PN profiles, and exploiting DFT-s-OFDM for downlink user data at extreme carrier frequencies (Levanen et al., 2019, Ibrahim et al., 20 Jan 2025).
• PTRS beyond wireless: The conceptual framework for PTRS in distributed RF synchronization is directly translatable to advanced photonic, quantum, and large-scale accelerator control systems.

PTRS remain foundational to the viability of high-SNR, high-throughput mmWave and sub-THz communication links, as well as to precision RF synchrony in large-scale distributed systems. Advances in allocation and estimation methodologies continue to push the boundaries for modulation scheme support, coverage, and spectral efficiency (Bello et al., 2022, Ibrahim et al., 20 Jan 2025, Levanen et al., 2019, Murthy et al., 2022, Qi et al., 2018).