Frequency-domain conditioning
Once a grouped FFT spectrum exists, a handful of conditioning steps turn a raw transform into something you can read a field off. They live in the Conditioning and Exclusions sections of the Fourier panel and apply, in order, to the averaged spectrum on its canonical frequency axis: pulse-response compensation, a robust baseline offset, and frequency-range exclusions. The display unit (MHz, Gauss, or Tesla) and the Real+Imag view are display choices layered on top. None of these change the underlying transform — they are the last mile between the FFT and the number you quote.
Field axis: reading a spectrum in Gauss
A precession line at frequency \(\nu\) reports the local field through the Larmor relation
so a transverse-field spectrum can be displayed against frequency or field interchangeably. The unit selector above the frequency plot switches between Frequency (MHz), Field (G), and Field (T); the FFT and the maximum- entropy reconstruction share one axis, so the same run peaks at the same field in either view. For a transverse field of 200 G the Larmor line sits at \(27.1\) MHz, equivalently 200 G — the field axis simply relabels it.
When to use this. Switch to Gauss or Tesla whenever the physics is a field — vortex-lattice internal-field distributions, knight shifts, applied-field calibration. Stay in MHz when the physics is a frequency — hyperfine couplings, F–μ–F entanglement, muonium pairs.
Pulse-response compensation
At a pulsed source the muons arrive spread over a finite pulse, so a signal oscillating fast compared with the pulse width is averaged across the spread of arrival times and its amplitude is suppressed: the pulse acts as a passband filter that rolls the spectrum off at high frequency. Compensation divides each bin by the pulse amplitude \(R(\nu)\) — the same parabola-times-Lorentzian lineshape the maximum-entropy forward model folds into its kernel, so the two methods correct one physical pulse, not two approximations of it.
Because \(R(\nu) \to 0\) near the first node of the pulse transform, the raw correction \(1/R(\nu)\) grows without bound. Asymmetry caps the gain (default \(25\times\)) and cuts the spectrum off at the node: beyond it the pulse has erased the signal and no multiplication can recover it. This is the deliberate difference from WiMDA, whose Gaussian \(\exp[(\pi f \tau)^2]\) factor amplifies high-frequency noise without limit. The pulse half-width defaults from the run’s instrument metadata; set it explicitly in the panel to override.
When to use this. Turn it on for ISIS (pulsed-source) transverse-field data with lines above a few MHz, where the rolloff visibly suppresses peak heights. Leave it off for continuous-source (PSI, TRIUMF) data, which has no pulse envelope, and at low frequency where \(R(\nu) \approx 1\). Pitfall: the cap and cutoff are physical limits, not bugs — if a line vanishes past the cutoff, the pulse genuinely could not record it; reduce the pulse width only if you know it.
Robust baseline offset
A power or magnitude spectrum sits on a positive pedestal from the redistributed counting noise. The baseline control estimates that pedestal and subtracts it. Robust σ-clip (the default when enabled) iteratively re- estimates the median and width of the inlier set, discarding points more than \(\kappa\) standard deviations away until the width converges; sharp peaks are excluded as outliers and survive the subtraction intact. WiMDA single-pass is the one-iteration, mean-location special case, kept for parity. The converged width doubles as a baseline-noise estimate that feeds the signal-to-noise readout.
When to use this. Enable it whenever you want peak heights or areas measured above a true zero — comparing intensities across runs, integrating a line. The robust mode is the right default; the single-pass mode reproduces a WiMDA number when you need to match a legacy analysis. Pitfall: on a spectrum that is mostly signal (a single dominant line filling the window) the inlier set is small; widen \(\kappa\) or leave the baseline off.
Frequency-range exclusions
The Exclusions table zeroes the spectrum inside up to ten symmetric \((\text{centre} \pm \text{half-width})\) windows. A PSI RF harmonics convenience sits on top of the table: it fills the table with DC and the first five harmonics of 50.63 MHz — the radio-frequency pickup that contaminates PSI continuous-source spectra.
The diamagnetic line has its own dedicated control (see Diamagnetic line removal below), not a slot in this table.
When to use this. Exclude a known artefact (RF pickup, a saturating diamagnetic line, a mains harmonic) that would otherwise dominate the plot or bias a baseline or peak search. Pitfall: exclusion blanks data — never use it to hide a feature you have not identified, and keep the half-widths tight so you do not remove signal next to the artefact.
Real + imaginary view
The Real+Imag display mode overlays the cosine (real) and sine (imaginary) quadratures of the averaged spectrum on one axis. A correctly phased absorption line is purely real with a flat imaginary part; residual structure in the imaginary channel is the visual signature of an imperfect phase correction.
When to use this. Reach for it while tuning the phase — manual, per-group, or the entropy optimiser — to see at a glance whether the real channel carries all the line. For a final quantitative spectrum, switch back to a single channel.
Signal-to-noise readout
After an averaged FFT the panel reports the mean per-bin error and the peak signal-to-noise — the largest ratio of spectrum amplitude to error across the band, with the DC bin excluded so the average-subtraction spike does not masquerade as signal. When the robust baseline is active its noise estimate anchors the denominator. Use the peak S/N to judge whether a marginal line is real before committing to a frequency-domain fit.
Burg all-poles pole scan (diagnostic)
The Resolution (Burg) display mode is a diagnostic, not a measurement. Where the FFT models the spectrum as a sum of sinusoids (an all-zeroes transform), the Burg method models it as an all-poles autoregressive process,
with \(m\) poles placed exactly where the data demand sharp lines. Because \(m\) can be far smaller than the number of time points, the method resolves close lines from short windows that the FFT cannot. The pole count is chosen automatically by minimising the Final Prediction Error (FPE) over the scan range (default 2–40); if the optimum lands on an endpoint the log warns that the range is too narrow.
What it is good for:
Qualitative super-resolution. On a window of 48 bins at 50 ns (FFT resolution \(0.42\) MHz), a \(0.25\) MHz doublet collapses to a single FFT peak but the Burg estimate shows two — a quick way to ask “is that one line or two?” before committing to a fit.
A line-count hint. The FPE-optimal pole count rises with the number of resolvable lines, so it suggests how many components a time-domain fit should include.
Its pathologies — why it is never the quantitative answer:
Spurious splitting and baseline peaks. Pushing the pole count well above the FPE optimum splits strong lines and invents peaks in the baseline that are not in the data. The peak heights are not amplitudes and the areas are not populations.
Noise-dependent bias and small position offsets. The estimate shifts with the noise realisation and nudges line positions; it carries no uncertainties.
Treat a Burg spectrum as a magnifying glass for spotting structure. Once you know how many lines are present and roughly where, measure them with a frequency-domain fit or with maximum entropy — those are the quantitative methods. See Choosing a spectral estimator: FFT, MaxEnt, or Burg for where Burg sits among the three (Blundell, De Renzi, Lancaster & Pratt, Muon Spectroscopy: An Introduction (Oxford University Press, Oxford, 2022), §15.5; Burg 1975).
Diamagnetic line removal
In a transverse field the unshifted diamagnetic muon line is often the largest feature and can bury a weaker shifted or radical line. A single Diamagnetic line control offers three mutually exclusive choices — Leave, Fit & subtract, and Exclude band.
Set it to Fit & subtract to fit a single damped cosine, \(s(t)=A\cos(2\pi(\nu t+\phi))\,e^{-\lambda t}+c\), to the time-domain signal before the FFT — seeding its frequency from the run’s applied field so it locks onto the diamagnetic line — and subtract the oscillatory part. The fitted frequency, converted to field, is reported in the log and overlaid on the time-domain plot so you can judge the subtraction. If the run has no transverse field (below a ~5 G seed) there is no line to lock onto: the fit is skipped and the panel status says so, leaving the spectrum unsubtracted. Exclude band hard-zeroes a band of half-width Band half-width (MHz) centred on the Larmor frequency after the FFT instead.
When to use this. Turn it on to reveal a shifted line sitting close to the diamagnetic peak, or to read the applied field off the fitted frequency as an independent calibration. Pitfall: the fit assumes a single damped cosine; if several lines overlap the diamagnetic frequency the subtraction is approximate — inspect the overlay, and prefer a full multi-line time-domain fit when the removal leaves visible residual.
Fit-and-subtract or exclude the band?
Two mechanisms suppress the diamagnetic line, and they differ in what they cost the rest of the spectrum:
Fit and subtract removes the line before the transform by fitting a damped cosine and subtracting only its oscillatory part. It leaves the neighbouring frequency bins intact — nothing is blanked — and it reports the fitted field as a by-product. Prefer it whenever a shifted line sits close to the diamagnetic peak and you cannot afford to blank the region around it: muoniated-radical and \(A_\mu\)-coupling work (Muoniated-radical correlation spectrum) depends on the satellite structure that a band exclusion would erase, and the fitted field is itself a useful calibration.
Exclude the band (the diamagnetic band in the exclusions table; see Exclusions: a glossary of six mechanisms) hard-zeroes a window of the displayed spectrum centred on the Larmor frequency. It makes no assumption about the line shape, so it is the robust fallback when the diamagnetic line is too strong or too distorted to fit cleanly — and the fit-and-subtract path itself falls back to nothing below a ~5 G seed field, where it silently has no line to lock onto. The cost is the blanked window: any signal inside it is gone too, so keep the half-width tight.
In short: fit-and-subtract preserves data and yields the field, and is the default for correlation and shifted-line work; band exclusion is the assumption-free fallback for lines that will not fit.
References
S. J. Blundell, R. De Renzi, T. Lancaster, and F. L. Pratt, Muon Spectroscopy: An Introduction (Oxford University Press, Oxford, 2022), §15.5 (frequency domain) and §14.2 (pulsed sources).
J. P. Burg, Maximum Entropy Spectral Analysis, Ph.D. thesis (Stanford University, 1975).
Á. Sánchez-Monge, P. Schilke, A. Ginsburg, R. Cesaroni, and A. Schmiedeke, Astron. Astrophys. 609, A101 (2018) — STATCONT robust continuum estimation by iterative σ-clipping.