Entropy estimates for complex Gaussian interference converses

If $(T,S)$ are jointly Gaussian-smoothed finite-second-moment scalar variables with nonsingular joint Gaussian noise, then $h(T\mathrm{\mid }S)=h(T-kS\mathrm{\mid }S)\le h(T-kS)\le {\log }_2(\pi e\mathrm{Var}(T-kS))$ for every complex $k$. The first identity is conditional translation and the first inequality is conditional mutual-information nonnegativity; the second is [F2]. A finite-dimensional convolution with a nonsingular Gaussian has a bounded density, finite second moment, and positive-definite real covariance; [F3], applied after identifying complex vectors with real vectors, therefore makes every relevant joint, marginal, and residual differential entropy finite and verifies the hypotheses of [F2]. In particular the pairs $(Y_1^n,S_1^n)$ and $(Y_2^n,S_2^n)$ have nonsingular independent receiver-noise coordinates, and the degraded pair below has nonsingular joint Gaussian noise when $s>t$. Their conditional entropies are differences of these finite joint and marginal entropies. For conditioning additionally on $X_1^n$, which need not have a density, first subtract its known translated contribution; independence removes that conditioning and leaves the nonsingular smoothed pair $(h_{22}{X}_2^n+Z_2^n,S_2^n)$. For a channel with conditional density $w(t\mathrm{\mid }x)$ and positive output density $f_T(t)$, the joint-to-product density ratio is $w(t\mathrm{\mid }x)\mathrm{/}{f}_T(t)$. Thus [F5] gives $I(X;T)=\int \ln (w(t\mathrm{\mid }x)\mathrm{/}{f}_T(t))d{P}_{X,T}$. When $T=BX+Z$ with independent nonsingular Gaussian $Z$, the two log integrals are finite by [F3] and the quadratic Gaussian log density, and this integral equals $(\ln 2)[h(T)-h(Z)]$. The same calculation with conditional densities justifies the conditional entropy identities used below, with every base-two entropy difference equal to the corresponding mutual information divided by $\ln 2$, since their joint and marginal entropies are finite as established above. At zero signal power the same Gaussian density argument applies without alteration. Coordinate entropy chain rules give $h(T^n\mathrm{\mid }{S}^n)\le {\sum }_{j}h(T_j\mathrm{\mid }{S}_j)$.

Suppose $s>t$ and set $U^n=\sqrt{s}{X}^n+Z^n$, $V^n=\sqrt{t\mathrm{/}s}{U}^n+\sqrt{1-t\mathrm{/}s}{W}^n$. The marginal law of $V^n$ is the lower-SNR channel and $X^n-U^n-V^n$ is Markov. The chain rule and step 1.1 give $h(U^n)-h(V^n)=[I(X^n;U^n)-I(X^n;V^n)]\mathrm{/}\ln 2=I(X^n;U^n\mathrm{\mid }{V}^n)\mathrm{/}\ln 2=h(U^n\mathrm{\mid }{V}^n)-n{\log }_2(\pi e[1-t\mathrm{/}s])$: the conditional noise is $Z^n$ given $\sqrt{t\mathrm{/}s}{Z}^n+\sqrt{1-t\mathrm{/}s}{W}^n$. For a coordinate variance $p_j$, minimizing the quadratic residual variance of $U_j$ given $V_j$ gives $(1-t\mathrm{/}s)(1+s{p}_j)\mathrm{/}(1+t{p}_j)$. Step 1.1 therefore bounds the entropy difference by ${\sum }_j{\log }_2((1+s{p}_j)\mathrm{/}(1+t{p}_j))$. This logarithmic ratio has derivative $(s-t)\mathrm{/}((1+sp)(1+tp)\ln 2)$, which is nonnegative and decreasing. Jensen and the average variance bound yield the first claimed inequality. If $s=t$ the two marginal entropies agree; if $t=0<s$ the same construction and computation apply. If $s=t=0$ both outputs are pure Gaussian noise.

Let $p_{i,j}=\mathrm{Var}(X_{i,j})$. The optimal scalar linear estimate of $Y_{1,j}$ from $S_{1,j}$ has residual variance $1+\mathrm{\mid }{h}_{21}{\mathrm{\mid }}^2{p}_{2,j}+\mathrm{\mid }{h}_{11}{\mathrm{\mid }}^2{p}_{1,j}\mathrm{/}(1+\mathrm{\mid }{h}_{12}{\mathrm{\mid }}^2{p}_{1,j})$: independence of the two user vectors and the noises eliminates all other covariances. Step 1.1 bounds the conditional entropy by the logarithm of this variance times $\pi e$. The map $x\mapsto x\mathrm{/}(1+kx)$ is increasing and concave on $[0,\mathrm{\infty })$ for $k\ge 0$, with second derivative $-2k\mathrm{/}(1+kx{)}^3$; hence the residual variance is increasing and concave in the two powers, and its logarithm is also concave and increasing. Jensen and $n^{-1}{\sum }_j{p}_{i,j}\le P_i$ give the first conditional-output bound. Reflection gives the second. This calculation has no division by a channel gain and remains valid when any gain vanishes.

The output variances are $1+\mathrm{\mid }{h}_{11}{\mathrm{\mid }}^2{p}_{1,j}+\mathrm{\mid }{h}_{21}{\mathrm{\mid }}^2{p}_{2,j}$ and $1+\mathrm{\mid }{h}_{12}{\mathrm{\mid }}^2{p}_{1,j}+\mathrm{\mid }{h}_{22}{\mathrm{\mid }}^2{p}_{2,j}$. Applying the coordinate chain rule, [F2], and logarithmic Jensen gives the two unconditioned-output bounds. Given $X_1^n$, subtract its contribution from $Y_2^n$; independence leaves the linear-estimation problem for $h_{22}{X}_2^n+Z_2^n$ from $h_{21}{X}_2^n+Z_1^n$. Its coordinate residual variance is $1+\mathrm{\mid }{h}_{22}{\mathrm{\mid }}^2{p}_{2,j}\mathrm{/}(1+\mathrm{\mid }{h}_{21}{\mathrm{\mid }}^2{p}_{2,j})$. The same concavity calculation proves the final conditional bound. Reflection proves its counterpart.