= Condition for preserving a spectral signal under linear compression
{title2=$J_B=J\iff C^{-1}\mathbf s\in\operatorname{range}(B^T)$}
The \b[condition for preserving a spectral signal under linear compression] specifies when compression retains the <covariance>-weighted signal measure of a known shape. Let the mean be $\theta\mathbf s$, with fixed positive-definite noise <covariance> $C$, and compress by a full-row-rank matrix $B$. The measures in <noninvertible spectral compression versus a coordinate change> obey
$$
J_B=J\quad\Longleftrightarrow\quad C^{-1}\mathbf s\in\operatorname{range}(B^T).
$$
Thus the rows of the compression must retain the noise-weighted signal direction, not necessarily the unweighted plotted shape.
For a short proof, whiten the shape as $\mathbf t=C^{-1/2}\mathbf s$ and put $M=BC^{1/2}$. The parent's orthogonal projection formula gives
$$
J-J_B=\|(I-P)\mathbf t\|^2,\qquad P=M^T(MM^T)^{-1}M.
$$
Equality holds exactly when $\mathbf t$ lies in the row space of $M$. Multiplying that subspace relation by $C^{-1/2}$ yields the stated condition. Linear <covariance> propagation gives the compressed <covariance> $BCB^T$.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[Appendix A, equation A-3]{ref}
One useful construction is the scalar <matched filter> statistic $z=\mathbf s^TC^{-1}\mathbf y$. Its mean is $\theta J$ and <variance> $J$, so $z/J$ has <variance> $1/J$, the original optimal linear amplitude precision. In the Gaussian known-<covariance> model, this statistic also retains the likelihood's dependence on that single amplitude.
For an original example, choose $C=\operatorname{diag}(1,4)$ and $\mathbf s=(1,2)^T$. Then $J=2$ and the retaining row is $B=(1,1/2)$. Its signal response is two and noise <variance> two, giving $J_B=4/2=2$. An unweighted sum $B=(1,1)$ instead has response three and <variance> five, giving $J_B=9/5=1.8$: ten percent of this signal measure is lost.
<Signal-subspace preservation for several spectral amplitudes> requires retaining the corresponding family of weighted mean directions when all those amplitudes are to be preserved. Unknown nuisance shapes can also change the relevant estimation problem. A compression designed for one known shape does not automatically preserve every spectral question or uncertainty component.
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