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Suppressing Multiple GNSS Interferers with Local-Gradient Adaptive Notch Filtering

Burak Soner, Felix Ott

arXiv:2610.13443v1eess.SP

TL;DR

Cascaded NLMS-ANFs can fail when comparable-power chirps produce competing gradient contributions. The paper proposes a locally conditioned gradient for the ANF update. Simulations report improved suppression in pathological cases, and hardware processing extends position-lock survival.

  • Problem

    Comparable-power chirp interferers can make full-band gradient contributions cancel, while multiple simultaneous chirps are a practically relevant GNSS threat.

    Source: §1 · p. 4, §3 · p. 5

  • Method

    The LG-ANF locally bandlimits the gradient input to attenuate competing chirps before adaptation, while retaining the conventional update at maximum local bandwidth.

    Source: §I · p. 2, §VI · p. 9

  • Results

    24 s: the LG-ANF extended position-lock survival beyond the standard ANF after jammer onset in recorded hardware data processed through GNSS-SDR.

    Source: §2 · p. 9, §VI · p. 10

  • Takeaways & Limitations

    On recorded hardware data, the simulated suppression gain translated into a measurable positioning benefit.

    Source: §VI · p. 10

    The practical-parameter improvement is limited to up to 4 dB, and downstream GNSS signal observables remain to be studied.

    Source: §VI · p. 10

Abstract

from arXiv · show

Adaptive notch filters (ANFs) based on the normalized least mean squares (NLMS) algorithm are an established technique for suppressing narrowband interference in GNSS receivers. The traditional solution for multiple interferers is simply cascading the NLMS-ANF into multiple stages but this solution degrades performance when multiple chirp jammers of similar power are present. We propose a low-complexity modification to the cascaded NLMS-ANF update rule to resolve this: a local gradient (LG) formulation: LG-ANF. The LG-ANF forms a locally bandlimited gradient signal by translating the input to baseband around the current frequency estimate, low-pass filtering it, and translating it back, thereby attenuating competing interferers before adaptation. The modification generalizes the traditional NLMS-ANF to handle the similar-power multiple interferer problem, using just one additional parameter. We characterize the gradient-averaging failure analytically and visually, demonstrate independent chirp tracking in the worst-case two-chirp scenarios, and evaluate statistical suppression performance by sweeping amplitude ratio against a standard NLMS-ANF, a frequency-locked-loop (FLL)-based ANF and a per-sample oracle upper bound. We further characterize the LG advantage over various ANF parameters and chirp conditions. Simulations show that the LG-ANF reduces to the standard NLMS-ANF outside the pathological multi-chirp regime, while recovering up to 4dB of in-band suppression around pathological amplitude ratios under per-condition optimization of the local bandwidth, and up to 13dB for selected edge-case parameter combinations. These optimized results represent an upper bound on the achievable LG gain. In a hardware experiment using a fixed alpha_lp=0.5, the LG-ANF extends position-lock survival by 24s relative to the standard ANF.

BIOGRAPHY

Multiple chirp jammers are a relevant GNSS threat, but cascaded NLMS-ANFs can fail when their gradient contributions compete. The paper introduces a local-gradient modification and evaluates it through simulations and GNSS-SDR experiments.

I. INTRODUCTION

  • Multiple simultaneous chirps can come from nearby independent PPDs or from one jammer transmitting several components.Their time-varying interference is practically relevant because GNSS satellite signals arrive with very low power.Source: §I · p. 1
  • Comparable-power chirps can make full-band gradient contributions cancel, leaving a cascaded notch between chirps rather than tracking either.The proposed LG-ANF aims to avoid this localized failure while retaining conventional NLMS behavior when local conditioning is unnecessary.Source: §I · p. 2
  • The LG-ANF computes its gradient from a locally bandlimited signal while leaving the notch-filter data path on the full input.It adds one first-order low-pass filter and two complex mixings per stage, controlled by one bandwidth parameter, and collapses to standard NLMS at maximum bandwidth.Source: §I · p. 2
  • The paper analyzes gradient averaging, proposes the local-gradient update, evaluates suppression in simulations and experiments, and releases recordings and metadata.The released materials support reproducibility and further research.Source: §I · p. 2

II. RELATED WORK

Prior ANF work improves filter structures and addresses chirp tracking, but does not directly correct the comparable-power multi-chirp gradient-averaging failure. Learning-based work spans classification, mitigation selection, tuning, and optimizer changes; LG-ANF instead modifies gradient computation.

1. Adaptive Notch Filters (ANFs)

  • Earlier ANF research improved filter realizations and studied GNSS notch filtering, while prior work also observed cascaded-filter degradation without offering a minimal NLMS-gradient correction.FLL-based ANFs address swept jammers, while MPANF adds multiple notch parameters and a pulse blanker; the paper positions LG-ANF as retaining a lightweight ANF structure.Source: §1 · pp. 2–3

2. Machine Learning in Interference Mitigation

  • Learning-based approaches address classification, mitigation selection, parameter tuning, or optimizer changes, whereas LG-ANF directly modifies the NLMS gradient input.Some approaches require array observations or add optimization complexity, unlike the single-channel pre-correlation ANF setting described here.Source: §2 · p. 3

III. PROPOSED METHOD

The proposed method addresses cascaded ANFs' multi-chirp failure by computing the adaptation gradient from a locally bandlimited version of the input, attenuating competing chirps before the update.

III. PROPOSED METHOD

  • The full-band gradient in cascaded ANFs contains competing contributions from all interferers, which can cancel and steer a notch toward a frequency occupied by none.The method overview contrasts that failure with a local-gradient correction.Source: §III · p. 3, Figure 1 · p. 4
  • LG-ANF translates the input around the current frequency estimate, low-pass filters it, and translates it back before forming the adaptation gradient.This locally bandlimited gradient signal attenuates competing chirps before the update.Source: Figure 1 · p. 4, §III · p. 4
  • The overview presents the local-gradient approach as a cascaded ANF that targets competing gradients while retaining the full-band filter data path.The adaptation gradient, rather than the notch-filter data path, is computed from the locally conditioned signal.Source: Figure 1 · p. 4

1. Problem Definition

The paper considers simultaneous sawtooth chirps as a practical GNSS interference scenario. It reviews the first-order complex NLMS-ANF, whose cascaded full-band gradient can fail when competing chirps have comparable power.

1. Problem Definition

  • Multiple chirp interferers can arise from independent PPDs or from one device transmitting several components, with each chirp having its own sweep rate, frequency range, and amplitude.The paper frames this simultaneous presence as a practically relevant and understudied threat scenario.Source: §1 · p. 4

2. The Classical NLMS Adaptive Notch Filter

  • The first-order complex NLMS-ANF tracks a single chirp's instantaneous frequency with a scalar state and updates its estimate by gradient descent on output power.The update uses a running power estimate for step-size normalization, and cascading stages is the standard approach for multiple interferers.Source: §2 · p. 4
  • In the double-chirp case, oppositely signed gradient components practically cancel, leaving the total gradient without useful directional information.The figure contrasts this with the single-chirp case.Source: Figure 2 · p. 5

3. The Gradient Averaging Problem

The section analyzes why cascaded ANFs can fail when comparable chirps compete in the full-band gradient. In the pathological case, the gradient terms cancel or oscillate, steering the notch away from both interferers.

3. The Gradient Averaging Problem

  • The analysis uses a batch objective with θ fixed or slowly varying, accounting for its propagation through the full IIR chain.It defines a sensitivity state s[n] and derives its recursion to obtain the exact output gradient.Source: §3 · p. 5
  • For two input chirps, linearity makes the output and its derivative additive, yielding an exact three-part gradient decomposition.The parts include each chirp's self-term and the cross-term.Source: §3 · p. 5
  • The exact batch-gradient formulation generalizes the stochastic gradient, which is recovered by freezing the previous sensitivity state.The paper gives the resulting stochastic derivative as (ka − 1) ż0xr[n − 1].Source: §3 · p. 5
  • When the estimate lies between two chirps, their self-gradient terms cancel and the cross-term oscillates, so the notch can track neither interferer.The paper reports a simulation in which the ANF tracks the mean frequency until approximately 1.5 ms.Source: §3 · p. 5

4. Local-Gradient Cascaded ANFs

The LG-ANF computes its adaptation gradient from a locally bandlimited input around the current frequency estimate. This attenuates competing chirps before adaptation while retaining the full input in the interference-mitigation data path.

4. Local-Gradient Cascaded ANFs

  • The LG-ANF translates the input to baseband around θ[n], low-pass filters it, and translates it back to form a local signal for gradient computation.A separate filter driven by this local signal supplies the gradient and power normalization.Source: §4 · p. 6
  • Because xloc retains spectral content near θ[n], the competing chirp is attenuated and the gradient is dominated by the nearest chirp.The notch adaptation uses the local signal, but the mitigation data path continues to operate on the full input x[n].Source: §4 · p. 6
  • The modification adds one scalar parameter, αlp, and uses K cascaded LG-ANF stages for K simultaneous chirps.The power-estimator time constant αP is already present in the canonical NLMS-ANF.Source: §4 · p. 6

1. Equal-Amplitude Chirp Tracking

In a two-stage, equal-amplitude two-chirp example, the LG-ANF tracks the chirps independently, whereas the standard ANF can oscillate between them. The example uses sawtooth chirps and is described as an exaggerated failure scenario for the standard method.

1. Equal-Amplitude Chirp Tracking

  • The LG-ANF converges to and tracks each equal-amplitude chirp independently after a short snapping phase.The stages do not maintain an exclusive stage-to-chirp identity match.Source: §1 · p. 6, Figure 3 · p. 6
  • The standard ANF's two stages can fail to lock onto either chirp, oscillating between interferer frequencies as the sweeps move.The caption identifies the dashed lines as true instantaneous frequencies and describes the standard ANF oscillation in both stages.Source: §1 · p. 6, Figure 3 · p. 6
  • The illustrative simulation uses two equal-amplitude sawtooth chirps sampled at 20 MHz over 60 µs with 30 dB SNR.The chirps sweep 1–5 MHz and 2–5 MHz, with periods of 20 µs and 40 µs; the standard method's low αP exaggerates the failure scenario.Source: §1 · p. 6

2. Amplitude-Ratio Sweep

The amplitude-ratio sweep finds that LG-ANF’s advantage concentrates in a pathological band near equal chirp amplitudes, while outside that band it reduces to the standard NLMS update. Parameter maps show that the largest gains are selected edge cases, with chirp speed, notch width, and power smoothing affecting the advantage.

2. Amplitude-Ratio Sweep

  • The sweep compares standard NLMS-ANF, LG-ANF, FLL-ANF, and a per-sample oracle across amplitude ratios, with the single-chirp point included for reference.At each ratio, the best αlp and FLL loop bandwidth are selected; the oracle is an upper bound for this notch structure.Source: §2 · p. 7
  • LG-ANF recovers up to 4 dB of in-band suppression over standard NLMS-ANF at two pathological amplitude ratios near equal amplitude; outside them, the methods coincide.The optimized low-pass coefficient narrows to about 0.1 only across the pathological band and otherwise saturates at 1.0.Source: Figure 4 · p. 7, §2 · p. 7

3. LG Advantage Over Notch Width and Chirp Speed

  • At pathological amplitude ratio 0.6624, selected notch-width and chirp-speed combinations yield up to 13 dB LG advantage, an optimized upper bound rather than a nominal gain.The advantage is largest with realistic power smoothing settings αP = 0.1 and 0.5.Source: §3 · p. 8, Figure 5 · p. 8
  • LG advantage decreases as chirps get faster, while its dependence on notch width ka is non-monotonic and inconclusive; ka is constrained by GNSS group-delay and timing-bias effects.The paper uses ka = 0.9 as a representative operating point rather than the global maximum.Source: §3 · p. 8
  • Power smoothing strongly affects the advantage: αP = 1.0 is infeasible in practice, while αP = 0.1 lets EMA power estimation and local-gradient band-limiting work together.The local low-pass excludes out-of-band noise and the competing chirp from the update, improving power-estimate and step-size conditioning.Source: §3 · p. 8

V. REAL-WORLD EXPERIMENTAL EVALUATION

In a recorded two-chirp GPS L1 scenario, the fixed-parameter LG-ANF preserves a usable position fix longer than the standard ANF. The experiment uses a verified position reference and evaluates the downstream positioning response to jammer exposure.

V. REAL-WORLD EXPERIMENTAL EVALUATION

  • The recorded scenario uses two overlapping chirp jammers around GPS L1, switched on at 100 s and off at 200 s, and processes the IQ through a GNSS-SDR.The test uses a USRP B200 receiver, an active GPS L1 patch antenna, and ADALM Pluto SDRs for the jammers.Source: §V · p. 8, Figure 6 · p. 9

1. Ground-Truth Reference

  • The static reference location was verified against Google Maps imagery and is accurate to approximately 10 cm, well below the positioning errors of interest.The reference is therefore treated as ground truth.Source: §1 · p. 8

2. Position-Lock Persistence

  • With αlp fixed at 0.5, LG-ANF extends position-lock survival by 24 s over standard ANF, maintaining a valid fix until about 174 s rather than losing lock near 150 s.The primary benefit is improved fix availability and delayed loss of lock, not lower average horizontal error during all valid epochs.Source: §2 · p. 9

3. Dataset Release

On recorded hardware, the LG-ANF extended position-lock survival, while the paper also releases the evaluation dataset. The conclusion identifies remaining validation and deployment questions.

3. Dataset Release

  • The released dataset includes raw IQ recordings, jammer and run metadata, hardware logs, and verified reference logs for reproducing the reported processing and GNSS-SDR results.The authors identify the release as support for reproducibility and further research.Source: §3 · p. 9

VI. CONCLUSION

  • Simulations report up to 4 dB of in-band suppression in typical pathological scenarios and up to 13 dB in edge cases.The largest gains occur with realistic power smoothing, where the power-estimation EMA and local-gradient filters work synergistically.Source: §VI · pp. 9–10
  • The local-gradient update addresses gradient averaging among comparable chirps and reduces to the standard NLMS update at αlp = 1.The paper characterizes the failure analytically and empirically; the method is a one-parameter generalization of the standard filter.Source: §VI · p. 9
  • 24 s: On recorded hardware, LG-ANF extended position-lock survival beyond standard ANF after jammer onset.Figure 7 shows standard ANF losing lock near 150 s and LG-ANF retaining it to approximately 174 s.Source: Figure 7 · p. 10, §VI · p. 10
  • Further work should examine signal-observable impacts and broader operating conditions; the practical parameter advantage is limited to up to 4 dB.The conclusion also identifies lightweight switching between standard and local-gradient modes as an open direction.Source: §VI · p. 10
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