Joint Sampling Frequency Offset Estimation and Compensation Based on the Farrow Structure

Conventional receivers use the Farrow structure for Sampling Frequency Offset (SFO) compensation and a separate stage for estimation. This paper eliminates that separation: the same Farrow output computed for compensation is directly used to estimate $\Delta$ (SFO) and $\varepsilon$ (Sampling Time Offset, STO) via Newton's method — joint estimation and compensation in a single structure, with no dedicated estimator needed.

Joint Sampling Frequency Offset Estimation and Compensation Based on the Farrow Structure
Deijany Rodriguez Linares, Oksana Moryakova, and Håkan Johansson
2025 25th International Conference on Digital Signal Processing (DSP)
doi: https://doi.org/10.1109/DSP65409.2025.11074995 arXiv: https://arxiv.org/abs/2503.07577

Related work

An extended version of this work is available as a preprint:

Joint Sampling Frequency Offset Estimation and Compensation Algorithms Based on the Farrow Structure
Deijany Rodriguez Linares, Oksana Moryakova, and Håkan Johansson
arXiv: https://arxiv.org/abs/2603.00627

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Key Results

Bandpass-filtered white noise, SNR = 60 dB, $\Delta = -200$ ppm, $\varepsilon = 300$ ppm:

Metric	Value
MSE before compensation	0.0623
MSE with true $\Delta$, $\varepsilon$	$6.21 \times 10^{-6}$
MSE with estimated $\hat{\Delta}$, $\hat{\varepsilon}$	$1.53 \times 10^{-5}$
NMSE (estimated parameters)	$1.35 \times 10^{-4}$

Estimated parameters: $\hat{\Delta} = -202.09$ ppm (true: $-200$ ppm), $\hat{\varepsilon} = 313.43$ ppm (true: $300$ ppm).

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How It Works

• The Farrow structure implements a polynomial fractional-delay filter with output $y_c(n, \Delta, \varepsilon) = \sum_{k=0}^{L} d^k(n), u_k(n)$, where $d(n) = n\Delta + \varepsilon$ and $u_k(n)$ are fixed subfilter outputs.
• A squared-error cost function $F(\Delta, \varepsilon) = \frac{1}{2}\sum_n \bigl(y_c(n,\Delta,\varepsilon) - x_0(n)\bigr)^2$ is minimized via Newton's method to jointly estimate $\hat{\Delta}$ and $\hat{\varepsilon}$.
• The gradient and Hessian are computed via cascaded accumulators, reusing the subfilter outputs $u_k(n)$ already computed for compensation — no additional filtering required.
• The estimator operates on the real part of the signal only, reducing implementation complexity.

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Run the notebook in Colab

Click below to launch the notebook in Google Colab:

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Contents

File	Role
SFO_filtered_white_noise_minimal.ipynb	Main demo: joint SFO+STO estimation & compensation on bandpass-filtered white noise
myclasses/farrow_optimizers_minimal.py	Newton–Raphson optimizer implementing the proposed algorithm
myclasses/filter_design.py	FIR filter design utilities
myclasses/generator_minimal.py	Signal generator (bandpass noise + controlled SFO/STO injection)
myclasses/plotting.py	Frequency-domain visualization helpers
functions/farrow_interpolator_minimal.*.so	Precompiled Cython: Farrow interpolator (x86-64 Linux + Apple Silicon)
farrow_structure/farrow_filter.npy	Precomputed Farrow filter coefficients

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How to cite

If you use this code in your research, please cite:

@INPROCEEDINGS{11074995,
  author={Rodriguez-Linares, Deijany and Moryakova, Oksana and Johansson, Håkan},
  booktitle={2025 25th International Conference on Digital Signal Processing (DSP)}, 
  title={Joint Sampling Frequency Offset Estimation and Compensation Based on the Farrow Structure}, 
  year={2025},
  pages={1-5},
  doi={10.1109/DSP65409.2025.11074995}
}

Extended preprint:

@misc{rodriguez-linares2026joint,
  author={Rodriguez-Linares, Deijany and Moryakova, Oksana and Johansson, Håkan},
  title={Joint Sampling Frequency Offset Estimation and Compensation Algorithms Based on the Farrow Structure},
  year={2026},
  eprint={2603.00627},
  archivePrefix={arXiv},
  primaryClass={eess.SP},
  url={https://arxiv.org/abs/2603.00627}
}