On the Fragile Rates of Linear Feedback Coding Schemes of Gaussian Channels with Memory
In \cite{butman1976} the linear coding scheme is applied, $X_t =g_t\Big(Θ- {\bf E}\Big\{Θ\Big|Y^{t-1}, V_0=v_0\Big\}\Big)$, $t=2,\ldots,n$, $X_1=g_1Θ$, with $Θ: Ω\to {\mathbb R}$, 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 $κ\in [0,\infty)$ 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 $C^{L,B} =\frac{1}{2} \log χ^2$, where $χ=\lim_{n\longrightarrow \infty} χ_n$ is the positive root of $χ^2=1+\Big(1+ \frac{|c|}χ\Big)^2 \fracκ{K_W}$, and the sequence $χ_n \triangleq \Big|\frac{g_n}{g_{n-1}}\Big|, n=2, 3, \ldots,$ 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 $C^{L,B}=\frac{1}{2} \log χ^2$ such that $χ>1$, 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 $ε_n>0$ no matter how small, then $X_n =g_t\Big(Θ- {\bf E}\Big\{Θ\Big|Y^{t-1}, V_0=v_0\Big\}\Big)+g_n ε_n$, and $|g_n|ε_n \longrightarrow \infty$, as $n \longrightarrow \infty$.