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تسجيل مستخدم جديدGaussian Channels with Feedback: Optimality, Fundamental Limitations, and Connections of Communication, Estimation, and Control
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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.
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In this paper, we establish the connections of the fundamental limitations in feedback communication, estimation, and feedback control over Gaussian channels, from a unifying perspective for information, estimation, and control. The optimal feedback
communication system over a Gaussian necessarily employs the Kalman filter (KF) algorithm, and hence can be transformed into an estimation system and a feedback control system over the same channel. This follows that the information rate of the communication system is alternatively given by the decay rate of the Cramer-Rao bound (CRB) of the estimation system and by the Bode integral (BI) of the control system. Furthermore, the optimal tradeoff between the channel input power and information rate in feedback communication is alternatively characterized by the optimal tradeoff between the (causal) one-step prediction mean-square error (MSE) and (anti-causal) smoothing MSE (of an appropriate form) in estimation, and by the optimal tradeoff between the regulated output variance with causal feedback and the disturbance rejection measure (BI or degree of anti-causality) in feedback control. All these optimal tradeoffs have an interpretation as the tradeoff between causality and anti-causality. Utilizing and motivated by these relations, we provide several new results regarding the feedback codes and information theoretic characterization of KF. Finally, the extension of the finite-horizon results to infinite horizon is briefly discussed under specific dimension assumptions (the asymptotic feedback capacity problem is left open in this paper).
The feedback sum-rate capacity is established for the symmetric $J$-user Gaussian multiple-access channel (GMAC). The main contribution is a converse bound that combines the dependence-balance argument of Hekstra and Willems (1989) with a variant of
the factorization of a convex envelope of Geng and Nair (2014). The converse bound matches the achievable sum-rate of the Fourier-Modulated Estimate Correction strategy of Kramer (2002).
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 f
eedback 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 $n longrightarrow infty$.
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tting, an active cooperation between two trusted users is enabled through using channel feedback in order to improve the communication efficiency. Based on rate-splitting and decode-and-forward strategies, achievable secrecy rate regions are derived for both discrete memoryless and Gaussian channels. Results show that channel feedback improves the achievable secrecy rates.
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