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Register a new userOn MMSE Properties of Codes for the Gaussian Broadcast and Wiretap Channels
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Added by
Ronit Bustin
Publication date
2015
fields
Informatics Engineering
and research's language is
English
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This work concerns the behavior of good (capacity achieving) codes in several multi-user settings in the Gaussian regime, in terms of their minimum mean-square error (MMSE) behavior. The settings investigated in this context include the Gaussian wiretap channel, the Gaussian broadcast channel (BC) and the Gaussian BC with confidential messages (BCC). In particular this work addresses the effects of transmitting such codes on unintended receivers, that is, receivers that neither require reliable decoding of the transmitted messages nor are they eavesdroppers that must be kept ignorant, to some extent, of the transmitted message. This work also examines the effect on the capacity region that occurs when we limit the allowed disturbance in terms of MMSE on some unintended receiver. This trade-off between the capacity region and the disturbance constraint is given explicitly for the Gaussian BC and the secrecy capacity region of the Gaussian BCC.
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This paper extends the single crossing point property of the scalar MMSE function, derived by Guo, Shamai and Verdu (first presented in ISIT 2008), to the parallel degraded MIMO scenario. It is shown that the matrix Q(t), which is the difference between the MMSE assuming a Gaussian input and the MMSE assuming an arbitrary input, has, at most, a single crossing point for each of its eigenvalues. Together with the I-MMSE relationship, a fundamental connection between Information Theory and Estimation Theory, this new property is employed to derive results in Information Theory. As a simple application of this property we provide an alternative converse proof for the broadcast channel (BC) capacity region under covariance constraint in this specific setting.
The scalar additive Gaussian noise channel has the single crossing point property between the minimum-mean square error (MMSE) in the estimation of the input given the channel output, assuming a Gaussian input to the channel, and the MMSE assuming an arbitrary input. This paper extends the result to the parallel MIMO additive Gaussian channel in three phases: i) The channel matrix is the identity matrix, and we limit the Gaussian input to a vector of Gaussian i.i.d. elements. The single crossing point property is with respect to the snr (as in the scalar case). ii) The channel matrix is arbitrary, the Gaussian input is limited to an independent Gaussian input. A single crossing point property is derived for each diagonal element of the MMSE matrix. iii) The Gaussian input is allowed to be an arbitrary Gaussian random vector. A single crossing point property is derived for each eigenvalue of the MMSE matrix. These three extensions are then translated to new information theoretic properties on the mutual information, using the fundamental relationship between estimation theory and information theory. The results of the last phase are also translated to a new property of Fishers information. Finally, the applicability of all three extensions on information theoretic problems is demonstrated through: a proof of a special case of Shannons vector EPI, a converse proof of the capacity region of the parallel degraded MIMO broadcast channel (BC) under per-antenna power constrains and under covariance constraints, and a converse proof of the capacity region of the compound parallel degraded MIMO BC under covariance constraint.
Polar codes are introduced for discrete memoryless broadcast channels. For $m$-user deterministic broadcast channels, polarization is applied to map uniformly random message bits from $m$ independent messages to one codeword while satisfying broadcast constraints. The polarization-based codes achieve rates on the boundary of the private-message capacity region. For two-user noisy broadcast channels, polar implementations are presented for two information-theoretic schemes: i) Covers superposition codes; ii) Martons codes. Due to the structure of polarization, constraints on the auxiliary and channel-input distributions are identified to ensure proper alignment of polarization indices in the multi-user setting. The codes achieve rates on the capacity boundary of a few classes of broadcast channels (e.g., binary-input stochastically degraded). The complexity of encoding and decoding is $O(n*log n)$ where $n$ is the block length. In addition, polar code sequences obtain a stretched-exponential decay of $O(2^{-n^{beta}})$ of the average block error probability where $0 < beta < 0.5$.
We propose a new scheme of wiretap lattice coding that achieves semantic security and strong secrecy over the Gaussian wiretap channel. The key tool in our security proof is the flatness factor which characterizes the convergence of the conditional output distributions corresponding to different messages and leads to an upper bound on the information leakage. We not only introduce the notion of secrecy-good lattices, but also propose the {flatness factor} as a design criterion of such lattices. Both the modulo-lattice Gaussian channel and the genuine Gaussian channel are considered. In the latter case, we propose a novel secrecy coding scheme based on the discrete Gaussian distribution over a lattice, which achieves the secrecy capacity to within a half nat under mild conditions. No textit{a priori} distribution of the message is assumed, and no dither is used in our proposed schemes.
In this paper, we focus on the two-user Gaussian interference channel (GIC), and study the Han-Kobayashi (HK) coding/decoding strategy with the objective of designing low-density parity-check (LDPC) codes. A code optimization algorithm is proposed which adopts a random perturbation technique via tracking the average mutual information. The degree distribution optimization and convergence threshold computation are carried out for strong and weak interference channels, employing binary phase-shift keying (BPSK). Under strong interference, it is observed that optimized codes operate close to the capacity boundary. For the case of weak interference, it is shown that via the newly designed codes, a nontrivial rate pair is achievable, which is not attainable by single user codes with time-sharing. Performance of the designed LDPC codes are also studied for finite block lengths through simulations of specific codes picked from the optimized degree distributions.
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