LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generated

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Gaussian feedback estimation-error covariance update

Statement

Let $(X_{1}, X_{2})$ be centered jointly Gaussian with variances $P_{1}, P_{2} > 0$ , correlation $ρ \in [0, 1)$ , and let $Z \sim N (0, N)$ be independent, $N > 0$ . Put $X = (X_{1}, X_{2})^{T}$ , $a = (1, 1)^{T}$ , $K = (\begin{array}{cc} P_{1} & ρ \sqrt{P_{1} P_{2}} \\ ρ \sqrt{P_{1} P_{2}} & P_{2} \end{array})$ , $Y = a^{T} X + Z$ , and $D = N + a^{T} K a$ . The residual $X - E [X ∣ Y]$ is independent of $Y$ and has covariance $K - K a a^{T} K / D$ . In particular, $Var (X_{1} ∣ Y) = \frac{P_{1} [N + P_{2} (1 - ρ^{2})]}{D}, Var (X_{2} ∣ Y) = \frac{P_{2} [N + P_{1} (1 - ρ^{2})]}{D},$ $Cov (X_{1} - E [X_{1} ∣ Y], X_{2} - E [X_{2} ∣ Y]) = \frac{ρ \sqrt{P_{1} P_{2}} N - P_{1} P_{2} (1 - ρ^{2})}{D} .$ After scaling the two errors to variances $P_{1}, P_{2}$ and reversing the sign of the second, their correlation is $ρ_{n e w} = \frac{\sqrt{P_{1} P_{2}} (1 - ρ^{2}) - ρ N}{\sqrt{[N + P_{1} (1 - ρ^{2})] [N + P_{2} (1 - ρ^{2})]}} .$

Facts & Assumptions

Given: A centered jointly Gaussian pair with positive variances, a correlation $0 \leq ρ < 1$ , and independent Gaussian observation noise of variance $N > 0$ .

[F1]

Two-user real AWGN MAC with perfect causal output feedback: the output is the sum of the two real inputs and independent Gaussian noise; feedback encoders may normalize causal estimation errors.

Proof

1.1givenF1algebra

Write $K = (\begin{array}{cc} P_{1} & c \\ c & P_{2} \end{array})$ with $c = ρ \sqrt{P_{1} P_{2}}$ , $a = (1, 1)^{T}$ , and $D = N + a^{T} K a$ . Set $R = X - (K a / D) Y$ . Then $Cov (R, Y) = K a - (K a / D) D = 0$ . Since $(R, Y)$ is jointly Gaussian, this block-diagonal covariance makes its joint characteristic function factor, so $R$ and $Y$ are independent. Thus $E [X ∣ Y] = K a Y / D$ and the residual covariance is $Cov (R) = K - K a a^{T} K / D$ .

2.1step 1.1algebra

Since $K a = (P_{1} + c, P_{2} + c)^{T}$ and $N + a^{T} K a = D$ , subtracting the rank-one matrix gives diagonal entries $P_{1} [N + P_{2} (1 - ρ^{2})] / D$ , $P_{2} [N + P_{1} (1 - ρ^{2})] / D$ , and off-diagonal entry $[c N + c^{2} - P_{1} P_{2}] / D$ .

3.1step 2.1algebra∎

Dividing the off-diagonal covariance by the square root of the two diagonal variances and changing its sign yields the displayed $ρ_{n e w}$ ; all denominators are positive because $N > 0$ .

Depends on

Two-user real AWGN MAC with perfect causal output feedback

Used by

Dependency tree · two levels

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Sources

Abbas El Gamal and Young-Han Kim, Network Information Theory, Lecture Notes 18, AWGN MAC with feedback (standard reference, not scraped)
Amos Lapidoth and Michèle Wigger, On the AWGN MAC with Imperfect Feedback, Appendix C, perfect-feedback recursion (standard reference, not scraped)