Gaussian channels with memory and with noiseless feedback have been widely studied in the information theory literature. However, a coding scheme to achieve the feedback capacity is not available. In this paper, a coding scheme is proposed to achieve the feedback capacity for Gaussian channels. The coding scheme essentially implements the celebrated Kalman filter algorithm, and is equivalent to an estimation system over the same channel without feedback. It reveals that the achievable information rate of the feedback communication system can be alternatively given by the decay rate of the Cramer-Rao bound of the associated estimation system. Thus, combined with the control theoretic characterizations of feedback communication (proposed by Elia), this implies that the fundamental limitations in feedback communication, estimation, and control coincide. This leads to a unifying perspective that integrates information, estimation, and control. We also establish the optimality of the Kalman filtering in the sense of information transmission, a supplement to the optimality of Kalman filtering in the sense of information processing proposed by Mitter and Newton. In addition, the proposed coding scheme generalizes the Schalkwijk-Kailath codes and reduces the coding complexity and coding delay. The construction of the coding scheme amounts to solving a finite-dimensional optimization problem. A simplification to the optimal stationary input distribution developed by Yang, Kavcic, and Tatikonda is also obtained. The results are verified in a numerical example.
Spatial interference avoidance is a simple and effective way of mitigating interference in multi-antenna wireless networks. The deployment of this technique requires channel-state information (CSI) feedback from each receiver to all interferers, resulting in substantial network overhead. To address this issue, this paper proposes the method of distributive control that intelligently allocates CSI bits over multiple feedback links and adapts feedback to channel dynamics. For symmetric channel distributions, it is optimal for each receiver to equally allocate the average sum-feedback rate for different feedback links, thereby decoupling their control. Using the criterion of minimum sum-interference power, the optimal feedback-control policy is shown using stochastic-optimization theory to exhibit opportunism. Specifically, a specific feedback link is turned on only when the corresponding transmit-CSI error is significant or interference-channel gain large, and the optimal number of feedback bits increases with this gain. For high mobility and considering the sphere-cap-quantized-CSI model, the optimal feedback-control policy is shown to perform water-filling in time, where the number of feedback bits increases logarithmically with the corresponding interference-channel gain. Furthermore, we consider asymmetric channel distributions with heterogeneous path losses and high mobility, and prove the existence of a unique optimal policy for jointly controlling multiple feedback links. Given the sphere-cap-quantized-CSI model, this policy is shown to perform water-filling over feedback links. Finally, simulation demonstrates that feedback-control yields significant throughput gains compared with the conventional differential-feedback method.
In cite{butman1976} the linear coding scheme is applied, $X_t =g_tBig(Theta - {bf E}Big{ThetaBig|Y^{t-1}, V_0=v_0Big}Big)$, $t=2,ldots,n$, $X_1=g_1Theta$, 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 $cin [-1,1]$ 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 $chi =lim_{nlongrightarrow infty} chi_n$ 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, $lim_{nlongrightarrowinfty}|g_n| =+infty$. 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 $X_n =g_tBig(Theta - {bf E}Big{ThetaBig|Y^{t-1}, V_0=v_0Big}Big)+g_n epsilon_n$, and $|g_n|epsilon_n longrightarrow infty$, as $n longrightarrow infty$.