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Beyond the Doppler Dilemma: Improved Fast Weather Radar Unambiguous Doppler Velocity Spectrum Reconstruction from Sparse Aperiodic Sweeps

Tworit Dash, S. A. K. Syed Mohamed, Oleg Krasnov, Alexander Yarovoy

arXiv:2610.09172v1eess.SP

TL;DR

Short-dwell weather radar faces Doppler ambiguity under periodic slow-time sampling, alongside long illumination requirements for range. The paper changes only the waiting times between FMCW chirps and reconstructs spectra with CGP likelihood methods. Real precipitation measurements show extended-domain branch recovery, while broader spectra and dwell constraints limit the ambiguity benefit.

  • Problem

    Long FMCW illumination needed for range and processing gain reduces the slow-time update rate under periodic triggering, constraining the unambiguous Doppler interval.

    Source: §I · p. 1

  • Method

    The paper changes FMCW chirp waiting times to create aperiodic samples and uses a moment-space ambiguity analysis with CGP spectrum reconstruction.

    Source: §XII · p. 21

  • Results

    Across 979 precipitation-supported range cells, log-periodic CGP stayed on the central branch; for normalized Doppler shifts from 0.5 to 6.5, it recovered every branch in all 979 cells and preserved fitted width.

    Source: §XII · p. 22

  • Takeaways & Limitations

    Aperiodic slow-time triggering offers a practical route to weakening the coupling among illumination time, operational range, Doppler resolution, and unambiguous velocity in continuously transmitting radars.

    Source: §XII · p. 22

    Timing design alone cannot restore long-lag coherence for very broad spectra, and the preferred design depends on expected spectral width and available dwell.

    Source: §XI · p. 21

Abstract

from arXiv · show

The problem of fast Doppler counter-aliasing for short-dwell weather-radar measurements is addressed. The proposed approach realizes a log-periodic slow-time design by changing only the waiting time between otherwise unchanged frequency-modulated continuous-wave (FMCW) chirps. The fast-time waveform, bandwidth, and range-processing chain are preserved while the acquisition changes the Doppler ambiguity structure. We formulate a novel moment-space ambiguity function for distributed weather spectra. It measures statistical ambiguity jointly in mean Doppler velocity and spectral width and expresses the competing sidelobes through their expected relative likelihood. A complex Gaussian process (CGP) maximum-likelihood estimator reconstructs the spectrum from the measured aperiodic samples. A deterministic basin-selection and optimization strategy accelerates the same CGP likelihood without defining a different statistical estimator. Real precipitation measurements establish central-branch stability over an extended search domain. Using 64 slow-time sweeps, we compare three acquisitions and their associated estimators. Periodic data are processed by the discrete Fourier transform (DFT) and parametric spectral estimation (PSE), log-periodic data by the nonuniform discrete Fourier transform (NUDFT) and CGP, and staggered data by velocity-difference and CGP estimators. The differences become more pronounced when the dwell is reduced to 32 sweeps. The results show that aperiodic time sampling enables extended-domain branch identification without degrading spectral-width estimation.

I. INTRODUCTION

Periodic slow-time sampling creates Doppler branches that often require post-acquisition context to resolve, while long FMCW illumination further limits the unambiguous Doppler interval. This work tests aperiodic timing as a measurement-design alternative and introduces moment-space ambiguity analysis and CGP inversion for weather spectra.

II. RELATED WORK AND RATIONALE

  • Conventional de-aliasing commonly selects among periodic branches using spatial or temporal continuity, external constraints, or information combined across scans.These approaches address ambiguity after acquisition rather than testing whether one short aperiodic dwell can support extended-domain spectral inversion.Source: §I · pp. 1–2, §II · pp. 2–3

I. INTRODUCTION

  • The log-periodic realization advances prior simulations to measured FMCW precipitation echoes, while the acquisition principle can extend to other targets with a suitable covariance model.The earlier work proposed the counter-aliasing principle and analytical timing rule.Source: §I · p. 2
  • Aperiodic slow-time timing changes the ambiguity structure while preserving the FMCW chirp, bandwidth, and range-processing chain.The work realizes the log-periodic waiting-time design on an FMCW weather radar and evaluates it using measured precipitation data and controlled phase modulation.Source: §I · p. 2
  • The moment-space ambiguity function compares candidate mean Doppler and spectral-width pairs through their sampled-covariance similarity.The deterministic basin-selection strategy accelerates the existing CGP maximum-likelihood estimator without introducing a different statistical estimator.Source: §I · p. 2

II. RELATED WORK AND RATIONALE

  • Staggered timing can enlarge the theoretical unambiguous interval, but its structured ambiguity sidelobes can remain sensitive to noise and spectral width.Nonuniform Fourier methods handle arbitrary sample times, while finite-window sidelobes may bias moments for sparse short dwells; PSE models the finite window on uniform data, and CGP directly reconstructs spectra from short-dwell samples.Source: §II · p. 3

III. SIGNAL MODEL AND NOISE CALIBRATION

The signal model represents each range cell as a complex Gaussian process with a Gaussian-shaped Doppler spectrum, and uses its covariance for likelihood-based inversion. It defines normalized Doppler coordinates and relates the formal ambiguity interval of aperiodic sampling to the recorded clock grid.

III. SIGNAL MODEL AND NOISE CALIBRATION

  • The precipitation signal is modeled as a zero-mean proper complex Gaussian process whose covariance depends on signal power, normalized mean Doppler frequency, and spectral width.Aperiodic sampling produces a general Hermitian covariance rather than the Toeplitz covariance of periodic sampling, because equal sample-index separations need not have equal time lags.Source: §III · pp. 3–4
  • The likelihood optimizes normalized Doppler mean and width while fixing signal power and noise variance from preprocessing and calibration.Signal power is set from average range-FFT power; noise variance is estimated from separate noise-only measurements and held fixed across estimators.Source: §III · p. 4
  • The physical interpretation of retrieved moments is restricted to precipitation-supported range cells above the calibrated noise background.In weak-echo or clutter-contaminated cells, the extended-interval likelihood may be broad or multimodal.Source: §III · p. 4

A. Normalized Doppler Frequency

  • Normalized Doppler frequency is defined relative to the first slow-time interval, and recorded aperiodic timestamps determine the finest repeated clock-grid unit and formal fold.The timestamps use a 120 MHz clock with an approximately 8.33 ns tick; the practical search interval is chosen as a controlled interval larger than the periodic Nyquist interval.Source: A. Normalized Doppler Frequency · p. 4

B. Sampling Sequences

The paper compares periodic, staggered, and log-periodic slow-time grids while keeping the FMCW chirp and range processing fixed. The log-periodic design uses exact recorded timestamps, with its curvature controlling the lag pattern and a trade-off between sidelobe spreading and dwell.

B. Sampling Sequences

  • The log-periodic sequence uses exact recorded timestamps, and its curvature parameter d2 controls departure from uniform sampling and the pairwise-lag distribution.The limiting case d2 → 0 is periodic sampling.Source: B. Sampling Sequences · pp. 4–5
  • Increasing d2 spreads ambiguity sidelobes at the cost of a longer dwell when the minimum interval and sample count are fixed.The implemented design is selected within this trade-off.Source: B. Sampling Sequences · p. 5
  • The three sampling sequences alter the slow-time grid while retaining the same FMCW chirp, bandwidth, and range processing.Periodic samples lie on a uniform grid, staggered samples alternate short and long separations, and log-periodic samples follow progressively perturbed placements.Source: B. Sampling Sequences · p. 5
  • The timing-strategy comparison uses the same minimum sampling interval, with no measured sample on a finer slow-time grid.The staggered intervals are represented by measured integer clock-cycle timestamps, and their first short-to-long interval ratio is close to, but not exactly, 2/3.Source: B. Sampling Sequences · p. 5

V. MOMENT-SPACE AMBIGUITY FUNCTION FOR EXTENDED DOPPLER SPECTRA

The moment-space ambiguity function assesses whether candidate weather spectra with different mean Doppler and width produce distinguishable sampled covariances. Log-periodic sampling separates remote branches best for narrow and moderate spectra, but increasing width weakens that advantage.

Moment-space ambiguity function

  • The ambiguity function compares covariance distributions over candidate mean Doppler and spectral width; its remote sidelobes indicate plausible competing solutions.The central-lobe extent represents local joint resolution, while sidelobe levels reflect relative plausibility of separated mean–width pairs.Source: §V · pp. 5–6

Interpretation

  • The same ambiguity surface links local moment resolution to estimator variance through its Fisher-information curvature, while remote sidelobes mark global false basins.A candidate matching the true moments has ambiguity level 0 dB; positive KL divergence produces a negative ambiguity level.Source: §V · pp. 6–7

Sampling strategy comparison

  • For narrow spectra, periodic sampling repeats branches exactly, staggered sampling retains strong structured sidelobes, and log-periodic sampling isolates the true branch.As spectral width increases, long-lag covariance weakens, reducing sidelobe separation; in the broad-spectrum limit, covariance becomes independent of mean frequency, so timing cannot recover the branch.Source: §V · p. 7

Timing-law tuning

  • For σfn,0 = 0.01, the strongest remote sidelobe falls from approximately 0 dB in the periodic limit to −11.1 dB at d2 = 0.007423; at σfn,0 = 0.05, it falls to −6.6 dB.Increasing d2 also lengthens the dwell, so the timing law can be tuned to expected width and permissible dwell.Source: §V · p. 7

False detection metric

  • The false-detection rate counts estimates outside a neighborhood of the true spectrum, testing correct extended-domain selection rather than wrapped velocity error.The grid search uses 200 realizations per true parameter pair over Θ = [−8, 8] × [0.008, 0.35].Source: §V · p. 7

False detection results

  • At most 1% false detections occurred for log-periodic sampling at widths σfn,0 ≤ 0.15 across the tested means; the rate rose to 7.1% at 0.175 and 27.7% at 0.20.Periodic sampling has a 93.75% analytical lower bound for false branch selection with 16 indistinguishable branches; two detections in 200 trials imply an approximately 3% one-sided 95% upper confidence limit.Source: §V · p. 7

A. Extended DFT and NUDFT

The extended DFT and NUDFT evaluate periodic and log-periodic slow-time samples on a shared extended frequency grid. Their spectral moments are computed from the resulting periodogram after noise-floor subtraction, with widths interpreted in light of the finite observation window.

Extended frequency evaluation

  • Periodic samples use the DFT and log-periodic samples use the NUDFT to evaluate the direct Fourier periodogram on an extended frequency grid.For periodic sampling, this is a dense extended-axis DFT; for log-periodic sampling, it is the Schuster periodogram for nonuniform samples.Source: A. Extended DFT and NUDFT · p. 8

Moment calculation

  • The strongest extended-grid peak sets the estimated mean location, and spectral moments are computed over one periodic Nyquist interval.After subtracting the known noise floor, the estimates are formed from the periodogram bins.Source: A. Extended DFT and NUDFT · pp. 8–9

Finite-window interpretation

  • Fourier estimates retain finite-window convolution, so their widths are interpreted as finite-window periodogram moments.Negative residual bins are not clipped; only a negative final second central moment is clipped to zero before taking its square root.Source: A. Extended DFT and NUDFT · p. 9

B. Staggered PRT Velocity-Difference Estimator

The staggered-PRT estimator resolves ambiguity by comparing two lag-one pulse-pair estimates across candidate ambiguity branches. The selected branch yields the dealiased velocity estimate.

B. Staggered PRT Velocity-Difference Estimator

  • The two alternating intervals provide separate lag-one autocorrelation estimates for the velocity-difference calculation.The corresponding pulse-pair correlations are denoted R̂1 and R̂2.Source: B. Staggered PRT Velocity-Difference… · p. 9
  • Each candidate branch predicts a difference Cℓ between the two wrapped pulse-pair estimates, using integer ambiguity corrections for the constituent PRTs.The branch sequence is generated from the two constituent unambiguous frequencies, 1/(2∆t1) and 1/(2∆t2).Source: B. Staggered PRT Velocity-Difference… · p. 9
  • The selected branch ℓ⋆ gives the staggered dealiased estimate f̂_stag,ℓ⋆ using the standard velocity-difference estimator.The branch sequence starts at (p0, q0) = (0, 0) and alternately increments qℓ and pℓ, including corresponding negative branches, until the transfer-function interval is covered.Source: B. Staggered PRT Velocity-Difference… · p. 9

C. CGP Multi-Start Maximum-Likelihood Estimation

The reference CGP maximum-likelihood estimator searches the extended mean-Doppler and spectral-width domain from multiple random starts. A deterministic basin-selection procedure retains the same likelihood while reducing initialization variability, and the fitted model supports direct Doppler-domain spectrum reconstruction.

C. CGP Multi-Start Maximum-Likelihood Estimation

  • Random multi-start CGP maximum-likelihood estimation retains the lowest negative log likelihood, but its variable computational load is not ideal for real-time use.The reference constrains normalized width to [0.005, 0.5].Source: C. CGP Multi-Start Maximum-Likelihood… · p. 10

D. Fast CGP Basin Selection

  • Fast CGP changes only initialization: it builds candidates from NUDFT peaks, integer Nyquist translations, local Doppler offsets, and trial widths, then refines the strongest basins before one constrained optimization.The final optimization uses the same likelihood and constraints as the reference; the deterministic candidate set is ranked by the raw CGP negative log likelihood.Source: D. Fast CGP Basin Selection · p. 11

VII. FREQUENCY-DOMAIN SPECTRUM RECONSTRUCTION

  • The fast-CGP procedure returns the final raw maximum-likelihood solution after scoring candidates and two progressively narrower local grids.It retains the best K1 candidates for the first grid and the best K2 basins for the second grid before starting one optimizer.Source: §VII · p. 11
  • After fitting θ̂, the posterior mean reconstructs the spectrum directly on a frequency grid without slow-time interpolation.The likelihood's Cholesky factorization can be reused, and the inverted covariance matrix size depends on the number of samples in the dwell.Source: §VII · p. 11

VIII. REAL-DATA ANALYSIS AND PREPROCESSING

  • The measurement keeps the FMCW chirp, bandwidth, fast-time sampling, and range FFT fixed while changing the waiting time between chirps.Doppler analysis uses precipitation-present cells 40–1200, approximately 0.12–3.60 km; range-overlaid signals are outside the experiment's scope. Periodic PSE estimates remain within the principal interval and are not assigned an extended-domain branch.Source: §VIII · pp. 11–12

A. Controlled Doppler-Branch Injection

Controlled branch injection translates each measured complex slow-time record by a known Doppler shift while preserving its measured amplitudes and timing. Recovery is assessed against the imposed increment and expected branch, with error metrics sensitive to both offsets and wrong-branch selections.

A. Controlled Doppler-Branch Injection

  • A known Doppler translation is imposed on each measured record, and the recovered increment is evaluated against the known shift and expected branch.The reference is the imposed shift fs, not an estimator's output; eligible range cells must pass the SNR gate and have finite estimates before and after injection.Source: A. Controlled Doppler-Branch Injection · p. 13
  • The injected phase modulation translates the full weather spectrum while retaining instantaneous amplitudes, spectral fluctuations, SNR, and the measured timing grid.The unknown original physical mean is not assumed known.Source: A. Controlled Doppler-Branch Injection · p. 13
  • The recovered-increment RMSE is sensitive to both a common offset and isolated wrong-branch selections.The conversion uses 90.5 m s−1 per normalized Doppler unit.Source: A. Controlled Doppler-Branch Injection · p. 13
  • The width change is the translated-record estimate minus the unshifted estimate, summarized by the median and the 25th and 75th percentiles across retained cells.The same shifts and precipitation-support gate are used for all three acquisitions.Source: A. Controlled Doppler-Branch Injection · p. 13
  • Phase injection tests statistical identifiability on known extended-domain branches, but does not reproduce physical range migration or FMCW range-Doppler displacement.At |fn| = 8, the target would move through about 7.6–9.6 range cells during the measured dwells, depending on timing law.Source: A. Controlled Doppler-Branch Injection · p. 13

IX. RESULTS

Across the tested widths, parametric estimators recover narrower spectra than the Fourier estimators, while log-periodic CGP retains low width-estimation spread at both dwell lengths. Reconstructions also show that log-periodic sampling removes exact periodic repetition, with CGP concentrating power near the likelihood-supported branch.

A. Known-Width Finite-Dwell Comparison

  • At N = 64 and 20 dB for σfn = 0.011, absolute width biases are 0.0184 for periodic DFT, 0.0296 for log-periodic NUDFT, below 2 × 10−4 for periodic PSE and staggered velocity-difference, and 4.07 × 10−4 for both log-periodic CGP implementations.The corresponding standard deviations are 0.0137 for periodic DFT, 0.0234 for log-periodic NUDFT, 0.0119 for periodic PSE, 0.00565 for staggered velocity-difference, and 0.000971 for both log-periodic CGP implementations.Source: A. Known-Width Finite-Dwell Comparison · pp. 13–14
  • At N = 32, width standard deviations are 0.0210 for periodic DFT, 0.0201 for log-periodic NUDFT, 0.0175 for periodic PSE, 0.00739 for staggered velocity-difference, and 0.00186 for both log-periodic CGP implementations.The shorter record increases every estimator's spread, while fast CGP follows the local CGP reference throughout the informative width range.Source: A. Known-Width Finite-Dwell Comparison · p. 14
  • Fourier width estimates are broadened mainly by the finite observation window, whereas PSE and CGP can recover narrower widths and log-periodic sampling reduces competing ambiguity branches.The tested widths include spectra narrower than the periodic Fourier resolution.Source: A. Known-Width Finite-Dwell Comparison · pp. 13–14

B. Spectral Reconstructions at Selected Range Cells

  • Periodic DFT spectra repeat at each periodic Nyquist interval, and periodic PSE narrows the fitted spectrum without identifying its branch.Log-periodic NUDFT removes exact repetition but retains visible sidelobes; log-periodic CGP concentrates power near the likelihood-supported branch, while staggered velocity-difference returns moments rather than a full spectrum.Source: B. Spectral Reconstructions at… · p. 14

C. Quantitative Moment and Branch Comparison

Controlled phase shifts show that log-periodic CGP recovers extended Doppler branches while preserving widths; in real precipitation data, log-periodic sampling concentrates branch selections centrally, unlike periodic and staggered sampling.

C. Quantitative Moment and Branch Comparison

  • For 955 periodic cells, median width is 3.96 m s−1 with DFT and 0.94 m s−1 with PSE; median principal-interval velocities are 4.27 and 4.16 m s−1, respectively.The middle-half width ranges are 1.73–6.38 m s−1 for DFT and 0.45–1.56 m s−1 for PSE.Source: C. Quantitative Moment and Branch… · p. 14, Table I · p. 15
  • In real data, log-periodic NUDFT selects the central branch for 99.1% of 979 cells and CGP for all cells, while both staggered estimators select several remote branches.Periodic sampling leaves repeated branches equivalent, so its branch percentage is not reported.Source: C. Quantitative Moment and Branch… · p. 14, Figure 9 · p. 17
  • A power enhancement near 2.2–2.5 km accompanies locally increased spectral width, particularly in the staggered acquisition, consistent with a melting-layer signature.Source: C. Quantitative Moment and Branch… · p. 14
  • For log-periodic data, the zero-Doppler bin is a median 15.4 dB below the strongest principal-interval periodogram bin and is within 1 dB of the local maximum in only 1.7% of cells.The vertically pointed radar sampled hydrometeors aloft rather than a stationary ground surface.Source: C. Quantitative Moment and Branch… · p. 14

D. Controlled Doppler-Branch Recovery

  • Log-periodic CGP recovers the correct branch in all 979 cells at every imposed shift, with increment RMSE from 3.88 × 10−5 to 6.85 × 10−5 m s−1.The log-periodic NUDFT recovers every branch at shifts 0.5 and 1.5, and 99.90% at larger shifts; one remote-branch error raises its RMSE despite a median error of zero.Source: D. Controlled Doppler-Branch Recovery · p. 15
  • For staggered data, velocity-difference branch recovery falls below 0.22% from shift 2.5 onward; periodic DFT does not identify the extended branch.Velocity-difference correct-branch rates are 98.30% and 97.77% for the first two shifts; periodic DFT’s apparent rate drops from 18.43% at shift 0.5 to 0.10% at 6.5.Source: D. Controlled Doppler-Branch Recovery · p. 15
  • Across imposed shifts, median width changes stay near zero for periodic PSE, DFT, log-periodic NUDFT, and CGP; staggered velocity-difference changes range from −0.001 to −0.013 m s−1.Periodic PSE preserves within-branch width but cannot identify the imposed extended branch because its mean is restricted to the principal interval.Source: D. Controlled Doppler-Branch Recovery · pp. 15–16
  • With every second sample and 32 sweeps, the shorter record broadens the NUDFT response, producing more remote estimates and larger widths; both CGP implementations remain close.With 64 samples, the narrow precipitation spectra often lead NUDFT to select the same branch as CGP.Source: D. Controlled Doppler-Branch Recovery · pp. 16–17

E. Range-Doppler Maps

The range-Doppler maps show that log-periodic timing avoids repeated ambiguity branches, while CGP places precipitation power on one branch. The discussion relates practical branch identification to spectral width, timing design, and the limits of the tested model.

E. Range-Doppler Maps

  • Log-periodic NUDFT removes exact range-Doppler repetition but leaves sidelobes, while full and fast CGP maps place precipitation power on one branch.Periodic DFT repeats the echo at every Nyquist interval, and periodic PSE retains repeated branches despite narrowing each copy.Source: E. Range-Doppler Maps · pp. 17–18, Figure 14 · p. 20

X. COMPUTATIONAL COMPLEXITY

  • 3.01 s versus 0.338 s: deterministic fast CGP was 8.9× faster than the 64-start reference, with both selecting the correct basin in all ten trials.The controlled benchmark used N = 64 and ten synthetic weather realizations; timings depend on implementation and hardware.Source: §X · p. 20

XI. DISCUSSION

  • For narrow and moderate spectra, log-periodic timing suppresses remote ambiguity sidelobes; for the tested design, false detections rise near σfn = 0.16 as broad spectra lose long-lag information.The tested design retains low false-detection rates through σfn = 0.15; broadening raises sidelobes toward the main lobe for every timing law.Source: §XI · p. 20
  • Increasing d2 lowers the strongest remote sidelobe for narrow and moderate spectra but increases dwell, so its setting trades ambiguity reduction against acquisition time.The preferred value depends on expected spectral width and available dwell; timing alone cannot restore long-lag coherence for very broad spectra.Source: §XI · pp. 20–21
  • 0.94 m s−1 for periodic PSE versus 3.96 m s−1 for periodic DFT: PSE’s median width is narrower, but it cannot identify a translated periodic branch.The likelihood model determines estimated width, whereas timing determines whether competing Doppler branches can be distinguished.Source: §XI · p. 21
  • With 32 samples, NUDFT sidelobes and width errors become more visible; large velocities in weak cells indicate unresolved sidelobes, not precipitation fall speeds.Measured precipitation spectra are narrow over most of the analyzed range, and the SNR gate excludes weak-cell selections from quantitative comparison.Source: §XI · p. 21
  • The staggered sequence’s large hidden-grid interval does not ensure reliable branch selection because its conventional estimator uses two pulse-pair phases, not the full sampled covariance.Fast CGP instead uses deterministic candidates and local refinements before one final constrained optimization, retaining the existing CGP likelihood.Source: §XI · p. 21
  • The extended search interval is the domain available to the sampled model, not a guarantee of arbitrary-velocity reconstruction when target motion or FMCW range displacement occurs.The phase-injection test does not introduce range migration or FMCW range-Doppler displacement.Source: §XI · p. 21

XII. CONCLUSION

The paper realizes Doppler counter-aliasing by changing only FMCW chirp waiting times and introduces a moment-space ambiguity measure for distributed spectra. Real precipitation measurements show extended-domain branch recovery, while the stated scope is Gaussian precipitation spectra.

XII. CONCLUSION

  • The acquisition changes only waiting times between FMCW chirps, preserving chirp duration, bandwidth, fast-time sampling, and range processing.This changes the Doppler ambiguity pattern while retaining the illumination and processing gain used for weather-radar range measurements.Source: §XII · p. 21
  • The moment-space ambiguity function compares expected likelihood similarity across candidate mean Doppler and spectral width, revealing ambiguity sidelobes and their dependence on spectrum width.For the tested design, low false-detection rates persist over 16 conventional periodic Nyquist intervals through σfn = 0.15, with a rapid transition near σfn = 0.16.Source: §XII · p. 21
  • Across 979 precipitation-supported range cells, log-periodic CGP stays on the central branch; for normalized Doppler shifts from 0.5 to 6.5, it recovers every branch and preserves fitted width.The result uses the measured log-periodic acquisition; the N = 32 comparison highlights covariance-based reconstruction as the nonuniform Fourier response broadens.Source: §XII · p. 22
  • 3.01 s to 0.338 s: deterministic basin selection reduces median single-cell computation while retaining the CGP likelihood and final optimizer.The controlled benchmark corresponds to the proposed procedure’s reduction from the reference computation.Source: §XII · p. 22
  • The demonstrated inversion uses a Gaussian precipitation spectrum model; other target classes require an appropriate covariance or parametric Doppler description.Within that scope, the work establishes the acquisition principle and demonstrates extended-domain inversion on real weather-radar measurements.Source: §XII · p. 22
  • The ambiguity function offers a compact tool for designing sampling frameworks for user-defined problems.The paper presents the moment-domain measure as a way to design the ambiguity response.Source: §XII · p. 22
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