Likelihood Ratios and Matched Filtering

A matched filter is the likelihood-ratio statistic for a known waveform in additive Gaussian noise with known covariance. That sentence contains both its power and its boundary: the filter is optimal only for the declared signal family, noise model, frequency band, and nuisance treatment. This page derives the statistic, checks its null distribution and normalization, and then explains why an operational gravitational-wave search must add template coverage, data-quality controls, background calibration, and signal-consistency information before assigning significance.

Required background. Probability and Statistical Inference supplies probability densities, likelihood ratios, Gaussian covariance, and hypothesis tests.

Helpful background. Gravitational Radiation and Waveforms supplies the strain models used as templates, while Computational Relativity supplies numerical waveform products and validation principles.

The Gaussian likelihood ratio

Let $d(t)$ be a calibrated dimensionless strain channel over a selected analysis interval. Compare

$$H_0:d=n, \qquad H_1:d=A h+n,$$

where $h$ is a fixed template shape, $A$ is a real amplitude, and $n$ is zero-mean stationary Gaussian noise. For a one-sided power spectral density $S_n(f)$ and positive frequencies, define

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

Strain is dimensionless, $\widetilde a$ has units of seconds, and $S_n$ has units of seconds, so this inner product is dimensionless. The Gaussian log likelihood is, up to a term independent of the hypothesis,

$$\ln p(d\mid A,H_1) =-\frac12(d-Ah\mid d-Ah).$$

Subtracting the $H_0$ likelihood gives

$$\ln\Lambda(A) =A(d\mid h)-\frac{A^2}{2}(h\mid h).$$

This is the Neyman–Pearson likelihood ratio for the two stated simple hypotheses before $A$ is optimized. Its derivation and the corresponding colored-Gaussian geometry are given in Jaranowski and Królak 2012, §§3.3–4.1. It is not yet a false-alarm probability, a Bayes factor, or a posterior probability for a signal.

Maximization and signal-to-noise ratio

Maximizing over the unconstrained amplitude yields

$$\widehat A=\frac{(d\mid h)}{(h\mid h)}, \qquad 2\ln\Lambda_{\max} =\frac{(d\mid h)^2}{(h\mid h)} \equiv\rho^2.$$

Thus the normalized matched-filter output is

$$\rho=\frac{(d\mid h)}{\sqrt{(h\mid h)}}.$$

If the physical amplitude is restricted to $A\ge0$ at a fixed phase, the maximum becomes $\widehat A=\max[0,(d\mid h)/(h\mid h)]$. The likelihood-ratio statistic is then $[\max(0,\rho)]^2$, rather than $\rho^2$: a large negative correlation is evidence against that fixed-sign signal.

For a waveform whose unknown phase acts as a rotation in a two-dimensional template space, introduce orthonormal quadratures $h_c$ and $h_s$. This describes the usual dominant-harmonic, nonprecessing compact-binary approximation; general higher-mode or precessing waveforms need a more complete treatment of phase. The complex filter

$$z=(d\mid h_c)+i(d\mid h_s)$$

has amplitude- and phase-maximized statistic $|z|$. To see this, a unit template at phase $\phi$ is $h_\phi=h_c\cos\phi+h_s\sin\phi$, so maximizing $(d\mid h_\phi)$ gives the length $\sqrt{(d\mid h_c)^2+(d\mid h_s)^2}$. With the transform $\widetilde x(f)=\int x(t)e^{+2\pi i f t}\,dt$, shifting the template to coalescence time $t_c$ as $h(t-t_c)$ multiplies it by $e^{+2\pi i f t_c}$. An inverse Fourier transform then evaluates all sampled coalescence times efficiently. FINDCHIRP develops this discrete implementation, including PSD weighting, normalization, inverse-spectrum truncation, and the bank loop, in Allen et al. 2012, §§II–VIII, PDF.

There are two separate maximizations here. Phase and time are nuisance transformations of one waveform. Searching over intrinsic masses and spins selects among different waveform shapes and introduces a trials and template-coverage problem treated in Template Banks and Signal Manifolds.

An independent null-distribution check

Normalize $u=h/\sqrt{(h\mid h)}$. Under $H_0$, $(n\mid u)$ is a standard normal variable because its variance is

$$\mathbb E[(n\mid u)^2]=(u\mid u)=1.$$

Therefore a one-quadrature statistic has $\rho\sim\mathcal N(0,1)$, while two independent quadratures give $|z|^2\sim\chi^2_2$. The unconstrained one-amplitude likelihood ratio has $\rho^2\sim\chi^2_1$; restricting $A\ge0$ instead produces probability $1/2$ at zero and probability $1/2$ distributed as $\chi^2_1$. Under a perfectly matched signal $A h$, the mean signed output is $A\sqrt{(h\mid h)}$. These checks expose how nuisance parameters and physical bounds change a statistic’s null distribution.

The check also exposes a common mistake. Maximizing over amplitude changes the statistic’s sampling distribution; one cannot insert $\rho^2$ into the fixed-amplitude likelihood-ratio distribution. Likewise, maximizing across time, templates, detectors, and categories changes the relevant tail. The single-template Gaussian distribution is a control calculation, not the final search background.

From an optimum statistic to a search ranking

Real detector data are neither exactly stationary nor exactly Gaussian. The PSD is estimated from neighboring data, calibration has frequency-dependent amplitude and phase uncertainty, spectral lines break the smooth covariance model, and transient glitches can overlap the signal band. Windowing, gating, line handling, data-quality segments, and signal-consistency tests therefore become part of the statistic’s operational definition. The detector-noise guide traces those departures from calibrated strain through searches and inference in LIGO Scientific Collaboration and Virgo Collaboration 2020, §§6–8, PDF.

A compact-binary pipeline may combine matched-filter amplitude with a frequency-binned consistency statistic, coincidence or coherent network information, signal duration, template-dependent rates, and data-quality categories. Such a reweighted ranking statistic need not have a closed-form Gaussian null distribution. Its tail is measured from off-source or time-shifted data, with the analysis choices fixed before the candidate is interpreted.

The current pipeline example is bounded by an evidence cutoff of 2026-08-18. The GWTC-5.0 methods paper records the O4b searches, waveform inputs, sensitivity measurements, candidate data-quality checks, and characterization settings in LVK Collaboration 2026, §§2–5, PDF. Those documented rankings are versioned pipeline products. They do not alter the stable Gaussian derivation above, and no threshold or catalog membership is treated here as timeless.

Data, calibration, selection, and reproduction boundary

A reproducible matched-filter calculation names the GWOSC release or controlled synthetic data, detector, GPS interval, sample rate, calibration version, quality mask, window, frequency band, PSD estimator, template normalization, waveform version, time/phase grid, and ranking threshold. For a public-data check, file hashes should be retained before conditioning. Calibration uncertainty may be propagated by filtering response-perturbed strain or by carrying response nuisance samples downstream; treating the nominal response as exact is an approximation.

The smallest independent control uses generated Gaussian frequency bins with known $S_n$, adds a template at a declared amplitude, and checks three statements: noise-only $\rho$ has unit variance, the injected mean equals the optimal signal-to-noise ratio, and two quadratures produce the predicted $\chi^2_2$ distribution. A public-data extension may use a versioned GWOSC event segment, but reproducing one filter peak does not reproduce a collaboration search: bank generation, vetoes, category definitions, background livetime, and pipeline software remain additional inputs.

Check your understanding

A unit-normalized, fixed-phase template gives $(d\mid h)=5$ and $(h\mid h)=1$. Find the maximum-likelihood amplitude and $2\ln\Lambda_{\max}$. Does either number by itself give the search false-alarm probability?

Solution

The amplitude estimate and maximized statistic are

$$\widehat A=5, \qquad 2\ln\Lambda_{\max}=\frac{(d\mid h)^2}{(h\mid h)}=25.$$

Neither is a search false-alarm probability. Under the ideal fixed-template null, an unconstrained amplitude tests $\lvert\rho\rvert\ge5$ and therefore uses a two-sided normal tail; a prespecified positive-amplitude signal uses the one-sided tail $\rho\ge5$. An actual search also maximizes over time, phase, templates, and other categories and may use a reweighted statistic in non-Gaussian data. Its false-alarm probability requires the corresponding calibrated background and trials accounting.

References

• Allen, Bruce, et al. “FINDCHIRP: An Algorithm for Detection of Gravitational Waves from Inspiraling Compact Binaries.” Physical Review D 85 (2012): 122006. DOI. Open PDF.
• Jaranowski, Piotr, and Andrzej Królak. “Gravitational-Wave Data Analysis: Formalism and Sample Applications: The Gaussian Case.” Living Reviews in Relativity 15 (2012): 4. DOI.
• 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.
• LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration. “GWTC-5.0: Methods for Identifying and Characterizing Gravitational-Wave Transients.” arXiv:2605.27224 [gr-qc] (2026). Open PDF.