Condition for preserving a spectral signal under linear compression ($J_B=J⟺C^{-1}\mathbf{s}\in \mathrm{range}(B^T)$)

The 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 obeyThus 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 $M=B{C}^{1/2}$. The parent's orthogonal projection formula givesEquality 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 $BC{B}^T$.^[ref]

One useful construction is the scalar matched filter statistic $z={\mathbf{s}}^T{C}^{-1}\mathbf{y}$. 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 $C=\mathrm{diag}(1,4)$ 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.

Signal-subspace preservation for several spectral amplitudes requires a compression to retain every noise-weighted mean direction relevant to those amplitudes. Let $\mathbf{y}$ have mean $A\mathbf{\theta }$, fixed positive-definite noise covariance $C$, and a full-column-rank shape matrix $A$. Compress it as $B\mathbf{y}$, with $B$ full row rank. The original and compressed amplitude-information matrices areFor Gaussian noise these are Fisher information matrices. Without that distributional assumption, they still describe the corresponding known-covariance linear estimation problem.

Whiten the shapes as $T=C^{-1/2}A$ and put $M=B{C}^{1/2}$. The orthogonal projection onto the row space of $M$ is $P=M^T(M{M}^T{)}^{-1}M$, givingThis difference is positive semidefinite. It vanishes exactly when every column of $T$ lies in that row space, equivalentlyThus preserving one selected shape does not establish preservation of the whole signal subspace. Linear covariance propagation supplies $BC{B}^T$ for the compressed observations.^[ref]

For an original example, take $C=\mathrm{diag}(1,4,9)$ and shape columns $(1,2,0{)}^T$ and $(0,0,3{)}^T$. Then $J=\mathrm{diag}(2,1)$. Choosinggives $BA=\mathrm{diag}(2,1)$ and $BC{B}^T=\mathrm{diag}(2,1)$, hence $J_B=J$. Keeping only the first compressed row retains information2 for the first amplitude but destroys sensitivity to the second. An unweighted first row $(1,1,0)$ would instead reduce the first information entry to1.8.

At least $p$ independent compressed directions are needed to preserve $p$ independent shape amplitudes. One construction is $B=A^T{C}^{-1}$, whose compressed signal has mean $J\mathbf{\theta }$ and covariance $J$. When $J$ is invertible, $J^{-1}B\mathbf{y}$ has mean $\mathbf{\theta }$ and covariance $J^{-1}$. In the fixed-covariance Gaussian model, expanding the likelihood function shows that its parameter dependence is retained through $A^T{C}^{-1}\mathbf{y}$.

Unknown nuisance shapes, changing covariance or additional spectral parameters can alter the subspace that needs preservation. The proof concerns the specified family of mean shapes and fixed noise model, rather than guaranteeing that one compression preserves every future spectral question.