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Orthogonal coordinates diagonalize a finite-dimensional Gaussian noise law

Statement

Assume AC. Let $K \in R^{d \times d}$ be symmetric positive definite and let $Z \sim N_{d} (0, K)$ be the additive noise in Finite-dimensional additive Gaussian channel with average power. Then there are an orthogonal matrix $U$ and positive numbers $λ_{1}, \dots, λ_{d}$ such that $K = U^{T} diag (λ_{1}, \dots, λ_{d}) U$ and the coordinates of $U Z$ are independent with laws $N (0, λ_{j})$ . For every input $X$ , $∥ U X ∥_{2} = ∥ X ∥_{2}$ and $U (X + Z) = U X + U Z$ , so this change of coordinates preserves the power constraint and the channel law.

Facts & Assumptions

Given: AC, a positive-definite covariance matrix $K$ , and a centered Gaussian vector $Z$ with covariance $K$ .

[A1]

The Axiom of Choice: AC is assumed by the multivariate-normal construction and by the orthogonal-normal invariance supplier.

[F1]

Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis: a real symmetric endomorphism has an orthonormal eigenbasis; positive definiteness makes each eigenvalue positive.

[F2]

A non-negative operator has a unique non-negative square root: a positive-semidefinite symmetric matrix has a unique positive-semidefinite square root.

[F3]

Multivariate normal law, including singular covariance and Standard normal and normal laws: $N_{d} (0, K)$ is realized as $K^{1 / 2} G$ with independent standard-normal coordinates in $G$ .

[F4]

Orthogonal transformations preserve the standard normal product law: under AC, if $G \sim N_{d} (0, I_{d})$ has independent standard-normal coordinates and $U^{T} U = I_{d}$ , then $U G$ again has independent standard-normal coordinates.

Proof

1.1F1F2givenchoose

By [F1], choose an orthogonal eigenbasis $U$ for $K$ , so $K = U^{T} Λ U$ with $Λ = diag (λ_{1}, \dots, λ_{d})$ and $λ_{j} > 0$ . The positive square root is $K^{1 / 2} = U^{T} Λ^{1 / 2} U$ : this matrix is positive semidefinite, squares to $K$ , and is therefore the unique square root in [F2].

2.1A1F3F4step 1.1algebra

By [F3], write $Z$ in law as $K^{1 / 2} G$ . Since $U U^{T} = I_{d}$ , $U Z$ therefore has the law of $Λ^{1 / 2} U G$ . By [A1] and [F4], the coordinates of $U G$ are independent standard normals; multiplication by $\sqrt{λ_{j}}$ gives independent $N (0, λ_{j})$ coordinates.

3.1step 1.1step 2.1algebra∎

Orthogonality gives $∥ U X ∥_{2}^{2} = X^{T} U^{T} U X = ∥ X ∥_{2}^{2}$ , and linearity gives $U (X + Z) = U X + U Z$ . Thus the transformed channel has parallel independent Gaussian noise and exactly the same input power and code/error correspondence.

Depends on

Used by

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Sources

Yury Polyanskiy and Yihong Wu, Information Theory: From Coding to Learning, Chapter 20 §20.4 (standard reference, not scraped)
Dmitry Panchenko, MIT 18.443 Statistics for Applications, Fall 2006, Lecture 4 (standard reference, not scraped)