Noise-Weighted Waveform Inner Products

A waveform difference becomes a task-dependent distance only after a frequency band and positive weighting are chosen. For detector applications the conventional weight is an assumed one-sided noise power spectral density, but the geometry can be studied without adopting any current instrument curve. This page defines that geometry, checks its normalization, and explains which waveform components it deliberately ignores. Event-specific inference and current detector noise curves are excluded.

Required background. Probability and Statistical Inference supplies covariance weighting, likelihood geometry, and the distinction between a norm and a posterior conclusion.

Helpful background. Einstein Equations supplies the radiative field. Waves, Green Functions, and Geometric Optics supplies Fourier transforms and positive-frequency representations. Gravitational Radiation and Waveforms supplies strain and frame conventions. Computational Relativity supplies numerical waveform products and error estimates.

A positive-frequency inner product

Use

$$h(t)=\int_{-\infty}^{\infty} \widetilde h(f)e^{-2\pi ift}\,df, \qquad \widetilde h(-f)=\widetilde h(f)^*.$$

Let $S_n(f)\gt0$ be a declared one-sided weighting PSD on $f\in[f_{\min},f_{\max}]$. Define the real inner product

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

The factor four combines the negative-frequency contribution of real time series with the convention that $S_n$ is one sided. If a two-sided PSD is used instead, its positive-frequency value is half as large and the prefactor changes. The noise-weighted construction is introduced in Cutler and Flanagan 1994, §§II–III, PDF.

For stationary real noise, the convention at positive frequencies is

$$\left\langle\widetilde n(f)\widetilde n(f')^*\right\rangle =\frac12 S_n(f)\delta(f-f').$$

Equivalently, its two-sided PSD is $P_n(f)=S_n(\lvert f\rvert)/2$. Starting from $\int_{-\infty}^{\infty}\widetilde a\widetilde b^*/P_n\,df$, pair the $f$ and $-f$ terms of the real signals. They are complex conjugates; their sum supplies $2\operatorname{Re}$, and $1/P_n=2/S_n$ supplies the other factor of two. Stationarity fixes the spectral covariance; Gaussianity is needed only when this quadratic geometry is used as an exact Gaussian log likelihood.

Norm, units, and positivity

The squared norm is

$$\lVert a\rVert^2=(a\mid a) =4\int_{f_{\min}}^{f_{\max}} \frac{\lvert\widetilde a(f)\rvert^2}{S_n(f)}\,df\ge0.$$

Strain is dimensionless, so $[\widetilde a]=\mathrm{s}$ and $[S_n]=\mathrm{s}=\mathrm{Hz}^{-1}$. Multiplying the ratio by $df$ leaves a dimensionless norm. It vanishes only when the waveform is zero almost everywhere in the weighted band. Components outside the band, or inside a deliberately zero-weight region, are identified with zero in this geometry even if the spacetime waveforms differ there.

Symmetry follows from the real part. Linearity over real scalars and positivity make this a real inner-product space after quotienting unweighted components. Cauchy–Schwarz gives

$$\lvert(a\mid b)\rvert \le\lVert a\rVert\,\lVert b\rVert.$$

This inequality is the reason normalized overlaps lie between $-1$ and $1$ before phase maximization.

Whitening and Euclidean geometry

Define a positive-frequency whitened waveform

$$\widetilde a_w(f)= \frac{2\widetilde a(f)}{\sqrt{S_n(f)}}.$$

Then

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

The PSD therefore stretches waveform space: a fixed physical error counts more where the declared noise is small and less where it is large. Changing $S_n$, the band, or the response changes the geometry without changing the underlying waveform. A claim that one model is “closer” than another is incomplete until these choices are stated.

Field, response, and frame

The weight applies to detector-frame channel waveforms. A source-frame asymptotic mode must first be redshifted, rotated, summed over harmonics, scaled by distance, and contracted with a declared response. Weighting a source-frame $H=r(h_+-ih_\times)$ directly by a detector PSD mixes incompatible units and frames.

For correlated channels, define the one-sided Hermitian spectral matrix by $\langle\widetilde n_I(f)\widetilde n_J(f')^*\rangle=\tfrac12 C_{IJ}(f)\delta(f-f')$ at positive frequencies. The natural extension is

$$(a\mid b)=4\operatorname{Re}\int \widetilde a_I(f)^* [C^{-1}(f)]^{IJ} \widetilde b_J(f)\,df.$$

Positive definiteness of $C$ replaces positivity of scalar $S_n$. Singular channel combinations must be projected out or regularized explicitly; they cannot be inverted as though independently measured.

Difference norms and accuracy use

Using the chapter’s truth-minus-model convention, $\delta h=h_*-h_m$,

$$\lVert\delta h\rVert^2 =4\int \frac{\lvert\delta\widetilde h\rvert^2}{S_n}\,df.$$

This is a useful summary of weighted discrepancy, not a universal scientific verdict. It does not say whether the error is parallel to a parameter degeneracy, whether a prior absorbs it, or whether a calibration uncertainty is correlated with it. Norm-based sufficient accuracy conditions and their assumptions are derived in Lindblom, Owen, and Brown 2008, §§II–IV, PDF.

Common pitfalls

Mixing one- and two-sided spectra. Re-derive the variance normalization before importing a factor of two or four.

Calling an out-of-band difference zero physics. It is zero distance only in the chosen quotient geometry. Another band or instrument can expose it.

Using a mutable curve silently. This chapter uses declared reference weights. Current instrument curves and event-specific conclusions belong to Gravitational-Wave Astronomy.

Exercises

1. Verify the factor and units. Suppose $\widetilde a=A$ and $S_n=S_0$ are constant over a positive-frequency band of width $\Delta f$. Find $\lVert a\rVert^2$.

Solution

Direct integration gives

$$\lVert a\rVert^2 =4\frac{\lvert A\rvert^2}{S_0}\Delta f.$$

Since $A$ has units seconds, $S_0$ has units seconds, and $\Delta f$ has units inverse seconds, the result is dimensionless. A two-sided PSD convention would redistribute the prefactor while preserving the same physical norm.

2. Quotient an unweighted component. Let $a=b+c$, where $c$ has Fourier support entirely above $f_{\max}$. Compare $\lVert a-b\rVert$ in this band with the unweighted time-domain difference.

Solution

Inside the declared integral $\widetilde c=0$, so

$$\lVert a-b\rVert=\lVert c\rVert=0.$$

The time-domain waveforms can nevertheless differ because $c(t)$ is nonzero. The weighted space has quotiented out that high-frequency component. Extending the band or choosing a different task makes the difference visible again.

References

• Cutler, Curt, and Éanna É. Flanagan. “Gravitational Waves from Merging Compact Binaries: How Accurately Can One Extract the Binary’s Parameters from the Inspiral Waveform?” Physical Review D 49 (1994): 2658–2697. DOI. Open PDF.
• Lindblom, Lee, Benjamin J. Owen, and Duncan A. Brown. “Model Waveform Accuracy Standards for Gravitational Wave Data Analysis.” Physical Review D 78 (2008): 124020. DOI. Open PDF.