Data Conditioning and Whitening

Conditioning maps calibrated strain into the numerical representation used by an analysis. Resampling, high-pass filtering, line treatment, gating, and whitening can improve numerical behavior and make departures from the noise model visible, but every operation has a transfer function and an edge response. A valid workflow records that transformation and applies an equivalent signal model; otherwise preprocessing can create or erase apparent physics.

Required background. Probability and Inference for Gravity supplies covariance, stationary processes, and likelihood weighting.

Helpful background. Gravitational Radiation and Waveforms fixes waveform and Fourier conventions. Computational Relativity supplies discretization, interpolation, and convergence testing.

Conditioning as an analysis operator

Write the selected calibrated strain as $d(t)=h(t)+n(t)$. A linear conditioning chain has frequency response $K(f)$,

$$\widetilde d_c(f)=K(f)\widetilde d(f), \qquad \widetilde h_c(f;\theta)=K(f)\widetilde h(f;\theta).$$

The same $K$ must act on data and signal model, or its effect must be included analytically in the likelihood. This scalar frequency response describes a linear, time-invariant operation. A fixed gate is linear but time dependent and mixes Fourier bins; selecting the gate from the data, clipping, and fitting a subtraction from the same data can make the complete algorithm nonlinear. Such procedures require injections passed through the complete algorithm.

For an invertible linear filter, the noise also transforms as $S_{n,c}=\lvert K\rvert^2S_n$. Consequently,

$$\frac{\lvert\widetilde d_c-\widetilde h_c\rvert^2}{S_{n,c}} =\frac{\lvert\widetilde d-\widetilde h\rvert^2}{S_n}$$

where $K\ne0$. Thus consistent filtering changes the representation without increasing the information. A notch, gate, or discarded band loses information and cannot be undone by dividing by its response.

Operations need not commute. A loud impulse passed through a long whitening filter can ring across many samples, whereas taper-gating the impulse before whitening can localize its influence. Conversely, gating after a narrow notch can reintroduce broadband spectral leakage. The pipeline order is part of the scientific method, not an implementation detail.

A practical chain often follows this logic:

1. select a calibrated, analysis-ready interval and remove invalid samples;
2. pad beyond the retained interval and remove offsets or trends as declared;
3. taper or gate identified large transients;
4. anti-alias filter before resampling;
5. apply broad band limits and any declared line treatment;
6. estimate or load the PSD from an appropriate interval;
7. whiten, then discard filter transients and retain only the prespecified core.

Different analyses can choose a different order, but must test the consequences.

Resampling and band limitation

Sampling at rate $f_s$ represents frequencies only up to the Nyquist frequency $f_s/2$. Before downsampling to $f_s'$, suppress power above $f_s'/2$ with an anti-alias filter. Decimation without that filter maps a component at $f$ to an alias

$$f_{\rm alias}=\left\lvert f-kf_s'\right\rvert$$

for the integer $k$ that places the result in $[0,f_s'/2]$. Instrumental lines above the new Nyquist frequency can therefore become false in-band features.

An anti-alias filter is not an ideal brick wall. Its passband ripple, transition width, stopband rejection, and group delay must meet the analysis tolerance. A causal filter delays signals; a forward–backward filter can remove phase delay but is acausal, has squared magnitude response $|H(f)|^2$, and doubles the effective filter order. Either is acceptable for offline analysis when its action and edge handling are reproduced in the signal model.

High-pass and band-pass filters suppress regions where the noise model or instrument response is unsuitable. Their cutoff labels alone are insufficient: filter family, order, phase response, and transition band determine signal distortion. Never interpret a transient inside the settling time of a filter without an explicit end-to-end test.

Lines, notches, and regression

A narrow line can be handled by excluding its frequency bins, applying a notch, fitting a deterministic or wandering-line model, or regressing a witnessed contribution. These choices answer different questions.

• A notch removes both noise and any astrophysical signal in its stopband.
• Bin exclusion changes the likelihood domain and is clean when leakage is controlled.
• A line model retains surrounding information but introduces nuisance parameters.
• Witness regression can subtract a coupling only if the witness is safe, sufficiently informative, and measured with a transfer function valid for the interval.

A zero-phase notch can have symmetric pre- and post-ringing; a causal notch has phase delay and post-ringing. Both can turn a short disturbance into a longer feature. Tapering the finite analysis segment and resolving off-bin lines reduce leakage but do not justify erasing unexplained structure.

Gating without inventing an edge

Gating multiplies the time series by a window $g(t)$ that smoothly approaches zero around a contaminated interval:

$$d_g(t)=g(t)d(t), \qquad \widetilde d_g(f)=\widetilde g*\widetilde d(f).$$

The convolution shows why a hard rectangular gate is dangerous: its slowly decaying sinc sidelobes distribute power across the spectrum. A smooth taper reduces leakage but removes more neighboring data. Gate center, zero-width, taper duration, threshold, and detection algorithm must be fixed or reported.

If a plausible signal overlaps the proposed gate, simply removing it can bias inference. Alternatives include an explicit glitch model, joint signal–glitch inference, or a sensitivity study over gate choices. Hardware or software injections placed near gate boundaries test whether the retained signal amplitude and phase remain within tolerance.

The practical roles of high-pass filtering, notching, resampling, and transient mitigation in interferometer analyses are surveyed in LIGO Scientific Collaboration and Virgo Collaboration 2020, §§4–6, PDF.

Whitening and the noise-weighted geometry

For one-sided PSD $S_n(f)$, define the standard noise-weighted inner product

$$(a\mid b) =4\operatorname{Re}\int_{f_{\rm low}}^{f_{\rm high}} \frac{\widetilde a(f)\widetilde b^*(f)}{S_n(f)}\,df.$$

Whitening applies an inverse square-root noise filter. For positive-frequency coefficients, one convenient convention is

$$\widetilde d_w(f) =\sqrt{\frac{2}{S_n(f)}}\,\widetilde d(f).$$

With this convention, the whitened one-sided PSD is $S_w=2$ within the retained band. For real data, Parseval’s identity gives $(a\mid b)=\int a_w(t)b_w(t)\,dt$ when the same two-sided whitening filter and band are used. A band of width $B=f_{\rm high}-f_{\rm low}$ has time-domain noise variance $2B$ in this continuous normalization; unit-variance sampled noise requires a further sampling-dependent scale. Whitening makes equal-width independent Fourier degrees of freedom equally weighted, but a software implementation must still verify its FFT normalization.

Whitening is not merely dividing an ASD curve point by point. A finite impulse-response realization needs padding and truncation; an infinite impulse-response realization needs stable poles and initial conditions; frequency-domain whitening needs a taper and treatment of DC and Nyquist bins. A floor or smooth PSD model may be required near very deep notches to keep the inverse filter from amplifying numerical noise.

The PSD used to whiten data should describe the noise relevant to the inference. If a glitch or strong signal contaminates the estimate, it downweights the same frequencies where the feature lies. If the spectrum drifts, a distant off-source PSD can misweight the event. Repeat the analysis with reasonable PSD intervals and estimators rather than assuming one curve is exact.

End-to-end evidence and convergence

A conditioning benchmark should contain colored Gaussian noise, an off-bin line, a known chirp, a large impulse, and quiet padding. Record the random seed and analytic injections. Measure:

• alias suppression after downsampling;
• passband amplitude and phase error for the chirp;
• leakage and ringing from the line and gate;
• residual covariance and spectral flatness after whitening;
• mismatch between conditioned injection and conditioned template;
• dependence on padding, sample rate, PSD segment length, taper, gate, and calibration-response sample.

Then replay the same versioned workflow on a declared public strain interval. Pin the dataset, release, detector, GPS interval, sample rate, quality mask, file hash, PSD recipe, line list, filter coefficients, gate definition, retained core, and software environment through the GWOSC API documentation available on 2026-08-18. Public tutorials illustrate access patterns but are not evidence that a production collaboration pipeline has been reproduced.

Convergence means that a scientific output—matched-filter signal-to-noise ratio, waveform mismatch, residual distribution, or recovered injection parameters—stabilizes under justified refinements. A visually flat spectrum alone is not enough. Report edge failures, nonstationarity, calibration interpolation, and veto or gate sensitivity explicitly. No public-data execution is claimed on this page.

Common pitfalls

Downsampling before anti-alias filtering. Once out-of-band power has folded into the retained band, a later filter cannot identify or remove it reliably.

Whitening data but not templates. This changes the signal relative to the likelihood and biases amplitude, phase, and timing.

Cropping until ringing disappears. The retained interval should be determined from the filter impulse response and a declared tolerance, not from whether a candidate looks cleaner.

Exercises

1. Find an alias. A $3000\,\mathrm{Hz}$ line is present when data are downsampled to $4096\,\mathrm{Hz}$ without adequate anti-alias filtering. At what in-band frequency does it appear?

Solution

The new Nyquist frequency is $2048\,\mathrm{Hz}$. Taking $k=1$ gives

$$f_{\rm alias}=\left\lvert3000-4096\right\rvert\,\mathrm{Hz} =1096\,\mathrm{Hz}.$$

A filter applied after decimation cannot distinguish this alias from a genuine $1096\,\mathrm{Hz}$ component.

2. Check whitening weights. At two independent frequency bins, a template has equal magnitude $\lvert\widetilde h\rvert=H$, while the noise PSDs are $S$ and $4S$. What fraction of $(h\mid h)$ comes from each bin, assuming equal bin widths?

Solution

The contributions are proportional to $H^2/S$ and $H^2/(4S)$. Their ratio is $4:1$, so the quieter bin contributes $4/5$ of the total and the noisier bin $1/5$. Whitening therefore encodes statistical weighting; it is not intended to preserve the unweighted spectral shape.

References

• Gravitational Wave Open Science Center. “API Documentation.” Version available 2026-08-18. Official documentation.
• LIGO Scientific Collaboration and Virgo Collaboration. “A Guide to LIGO–Virgo Detector Noise and Extraction of Transient Gravitational-Wave Signals.” Classical and Quantum Gravity 37 (2020): 055002. DOI. Open PDF.