Control-theoretic Approach to Communication with Feedback: Fundamental Limits and Code Design

Feedback communication is studied from a control-theoretic perspective, mapping the communication problem to a control problem in which the control signal is received through the same noisy channel as in the communication problem, and the (nonlinear and time-varying) dynamics of the system determine a subclass of encoders available at the transmitter. The MMSE capacity is defined to be the supremum exponential decay rate of the mean square decoding error. This is upper bounded by the information-theoretic feedback capacity, which is the supremum of the achievable rates. A sufficient condition is provided under which the upper bound holds with equality. For the special class of stationary Gaussian channels, a simple application of Bode's integral formula shows that the feedback capacity, recently characterized by Kim, is equal to the maximum instability that can be tolerated by the controller under a given power constraint. Finally, the control mapping is generalized to the N-sender AWGN multiple access channel. It is shown that Kramer's code for this channel, which is known to be sum rate optimal in the class of generalized linear feedback codes, can be obtained by solving a linear quadratic Gaussian control problem.Comment: Submitted to IEEE Transactions on Automatic Contro

LQG Control Approach to Gaussian Broadcast Channels with Feedback

A code for communication over the k-receiver additive white Gaussian noise broadcast channel with feedback is presented and analyzed using tools from the theory of linear quadratic Gaussian optimal control. It is shown that the performance of this code depends on the noise correlation at the receivers and it is related to the solution of a discrete algebraic Riccati equation. For the case of independent noises, the sum rate achieved by the proposed code, satisfying average power constraint P, is characterized as 1/2 log (1+P*phi), where the coefficient "phi" in the interval [1,k] quantifies the power gain due to the presence of feedback. When specialized to the case of two receivers, this includes a previous result by Elia and strictly improves upon the code of Ozarow and Leung. When the noises are correlated, the pre-log of the sum-capacity of the broadcast channel with feedback can be strictly greater than one. It is established that for all noise covariance matrices of rank r the pre-log of the sum capacity is at most k-r+1 and, conversely, there exists a noise covariance matrix of rank r for which the proposed code achieves this upper bound. This generalizes a previous result by Gastpar and Wigger for the two-receiver broadcast channel.Comment: Submitted to IEEE Transactions on Information Theor

Feedback communication systems : fundamental limits and control-theoretic approach

Feedback links from the receivers to the transmitters are natural resources in many real-world communication networks which are utilized to send back information about the decoding process as well as the channel dynamics. However, the theoretical understanding of the role of feedback in communication systems is yet far from being complete. In this thesis, we apply techniques from information theory, estimation and control, and optimization theory to investigate the benefits of feedback in improving fundamental limits on information flow in communication networks. We focus on three network models: Gaussian multiple access channels, Gaussian broadcast channels, and wiretap channels. First, combining the Lagrange duality technique and tools from information theory we derive an upper bound on the sum rate achievable by linear codes for the Gaussian multiple access channel with feedback. This upper bound is further shown to coincide with the known lower bound, hence establishing the linear sum capacity. Next, we study the application of tools from the theory of linear quadratic Gaussian (LQG) control in designing codes for feedback communications. For the Gaussian broadcast channel with feedback, we construct a linear code based on the LQG optimal control and establish the best known lower bound on the sum rate. In addition, depending on the spatial correlation of the noise across different receivers, it is shown that in the high signal-to-noise ratio regime, the sum rate achieved by this code can increase linearly with the number of receivers. Third, we consider the wiretap channel with an eavesdropper and study the benefits of a rate-limited feedback link. We propose a new technique based on which we derive an upper bound on the maximum rate of reliable and secure communication. For the special case in which the eavesdropper's signal is a degraded version of the legitimate receiver's signal, this upper bound matches the known lower bound establishing the secrecy capacity. Finally, we present results for the binary multiplying channel, one of the simplest two-way channels for which the capacity region is not known. We apply tools from stochastic control to establish sufficient conditions for optimality, and use the concept of directed information to analyze the performance of coding scheme

Wiretap channel with secure rate-limited feedback

Abstract—This paper studies the problem of secure communication over a wiretap channel p(y; z j x) with a secure feedback link of rate Rf, where X is the channel input, and Y and Z are channel outputs observed by the legitimate receiver and the eavesdropper, respectively. It is shown that the secrecy capacity, the maximum data rate of reliable communication while the intended message is not revealed to the eavesdropper, is upper bounded as By relaxing the secrecy requirement mildly while exploiting the better quality of the Alice-Bob channel than that of the Alice-Eve channel, Wyner showed that information can be transmitted securely at a positive rate, and characterized the secrecy capacity, the supremum of all achievable rates of secure communication, as Cs(R f) max minfI(X; Y); I(X; Y j Z) +Rf g: p(x) (1) The proof of the bound crucially depends on a recursive argument which is used to obtain the single-letter characterization. This upper bound is shown to be tight for the class of physically degraded wiretap channels. A capacity-achieving coding scheme is presented for this case, in which the receiver securely feeds back fresh randomness with rate Rf, generated independent of the received channel output symbols. The transmitter then uses this shared randomness as a secret key on top of Wyner’s coding scheme for wiretap channels without feedback. Hence, when a feedback link is available, the receiver should allocate all resources to convey a new key rather than sending back the channel output. Index Terms—Common randomness, rate-limited feedback, secrecy capacity, wiretap channel