On the Fragile Rates of Linear Feedback Coding Schemes of Gaussian Channels with Memory

TL;DR

This paper analyzes linear feedback coding schemes for Gaussian channels with memory, revealing their fragility due to unbounded growth of coding coefficients, which makes them impractical under model mismatch.

Contribution

It demonstrates that linear feedback codes with positive feedback capacity necessarily have unbounded growth in their coefficients, leading to practical limitations and fragility.

Findings

Linear feedback codes grow unboundedly if they achieve positive feedback capacity.

Such codes are highly sensitive to model mismatch and perturbations.

Simulations confirm the theoretical unbounded growth of coding coefficients.

Abstract

In \cite{butman1976} the linear coding scheme is applied, $X_t =g_t\Big(\Theta - {\bf E}\Big\{\Theta\Big|Y^{t-1}, V_0=v_0\Big\}\Big)$, $t=2,\ldots,n$, $X_1=g_1\Theta$, with $\Theta: \Omega \to {\mathbb R}$, a Gaussian random variable, to derive a lower bound on the feedback rate, for additive Gaussian noise (AGN) channels, $Y_t=X_t+V_t, t=1, \ldots, n$, where $V_t$ is a Gaussian autoregressive (AR) noise, and $\kappa \in [0,\infty)$ is the total transmitter power. For the unit memory AR noise, with parameters $(c, K_W)$, where $c\in [-1,1]$ is the pole and $K_W$ is the variance of the Gaussian noise, the lower bound is $C^{L,B} =\frac{1}{2} \log \chi^2$, where $\chi =\lim_{n\longrightarrow \infty} \chi_n$ is the positive root of $\chi^2=1+\Big(1+ \frac{|c|}{\chi}\Big)^2 \frac{\kappa}{K_W}$, and the sequence $\chi_n \triangleq \Big|\frac{g_n}{g_{n-1}}\Big|, n=2, 3, \ldots,$ satisfies a certain recursion, and conjectured that $C^{L,B}$ is the feedback capacity. In this correspondence, it is observed that the nontrivial lower bound $C^{L,B}=\frac{1}{2} \log \chi^2$ such that $\chi >1$, necessarily implies the scaling coefficients of the feedback code, $g_n$, $n=1,2, \ldots$, grow unbounded, in the sense that, $\lim_{n\longrightarrow\infty}|g_n| =+\infty$. The unbounded behaviour of $g_n$ follows from the ratio limit theorem of a sequence of real numbers, and it is verified by simulations. It is then concluded that such linear codes are not practical, and fragile with respect to a mismatch between the statistics of the mathematical model of the channel and the real statistics of the channel. In particular, if the error is perturbed by $\epsilon_n>0$ no matter how small, then $X_n =g_t\Big(\Theta - {\bf E}\Big\{\Theta\Big|Y^{t-1}, V_0=v_0\Big\}\Big)+g_n \epsilon_n$, and $|g_n|\epsilon_n \longrightarrow \infty$, as $n \longrightarrow \infty$.