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Register a new userGeometric Relationships Between Gaussian and Modulo-Lattice Error Exponents
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Added by
Uri Erez
Publication date
2013
fields
Informatics Engineering
and research's language is
English
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Lattice coding and decoding have been shown to achieve the capacity of the additive white Gaussian noise (AWGN) channel. This was accomplished using a minimum mean-square error scaling and randomization to transform the AWGN channel into a modulo-lattice additive noise channel of the same capacity. It has been further shown that when operating at rates below capacity but above the critical rate of the channel, there exists a rate-dependent scaling such that the associated modulo-lattice channel attains the error exponent of the AWGN channel. A geometric explanation for this result is developed. In particular, it is shown how the geometry of typical error events for the modulo-lattice channel coincides with that of a spherical code for the AWGN channel.
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The combination of source coding with decoder side-information (Wyner-Ziv problem) and channel coding with encoder side-information (Gelfand-Pinsker problem) can be optimally solved using the separation principle. In this work we show an alternative scheme for the quadratic-Gaussian case, which merges source and channel coding. This scheme achieves the optimal performance by a applying modulo-lattice modulation to the analog source. Thus it saves the complexity of quantization and channel decoding, and remains with the task of shaping only. Furthermore, for high signal-to-noise ratio (SNR), the scheme approaches the optimal performance using an SNR-independent encoder, thus it is robust to unknown SNR at the encoder.
The problem of estimating an arbitrary random vector from its observation corrupted by additive white Gaussian noise, where the cost function is taken to be the Minimum Mean $p$-th Error (MMPE), is considered. The classical Minimum Mean Square Error (MMSE) is a special case of the MMPE. Several bounds, properties and applications of the MMPE are derived and discussed. The optimal MMPE estimator is found for Gaussian and binary input distributions. Properties of the MMPE as a function of the input distribution, SNR and order $p$ are derived. In particular, it is shown that the MMPE is a continuous function of $p$ and SNR. These results are possible in view of interpolation and change of measure bounds on the MMPE. The `Single-Crossing-Point Property (SCPP) that bounds the MMSE for all SNR values {it above} a certain value, at which the MMSE is known, together with the I-MMSE relationship is a powerful tool in deriving converse proofs in information theory. By studying the notion of conditional MMPE, a unifying proof (i.e., for any $p$) of the SCPP is shown. A complementary bound to the SCPP is then shown, which bounds the MMPE for all SNR values {it below} a certain value, at which the MMPE is known. As a first application of the MMPE, a bound on the conditional differential entropy in terms of the MMPE is provided, which then yields a generalization of the Ozarow-Wyner lower bound on the mutual information achieved by a discrete input on a Gaussian noise channel. As a second application, the MMPE is shown to improve on previous characterizations of the phase transition phenomenon that manifests, in the limit as the length of the capacity achieving code goes to infinity, as a discontinuity of the MMSE as a function of SNR. As a final application, the MMPE is used to show bounds on the second derivative of mutual information, that tighten previously known bounds.
We propose a new coding scheme using only one lattice that achieves the $frac{1}{2}log(1+SNR)$ capacity of the additive white Gaussian noise (AWGN) channel with lattice decoding, when the signal-to-noise ratio $SNR>e-1$. The scheme applies a discrete Gaussian distribution over an AWGN-good lattice, but otherwise does not require a shaping lattice or dither. Thus, it significantly simplifies the default lattice coding scheme of Erez and Zamir which involves a quantization-good lattice as well as an AWGN-good lattice. Using the flatness factor, we show that the error probability of the proposed scheme under minimum mean-square error (MMSE) lattice decoding is almost the same as that of Erez and Zamir, for any rate up to the AWGN channel capacity. We introduce the notion of good constellations, which carry almost the same mutual information as that of continuous Gaussian inputs. We also address the implementation of Gaussian shaping for the proposed lattice Gaussian coding scheme.
We respond to [1] which claimed that Modulo-SK scheme outperforms Deepcode [2]. We demonstrate that this statement is not true: the two schemes are designed and evaluated for entirely different settings. DeepCode is designed and evaluated for the AWGN channel with (potentially delayed) uncoded output feedback. Modulo-SK is evaluated on the AWGN channel with coded feedback and unit delay. [1] also claimed an implementation of Schalkwijk and Kailath (SK) [3] which was numerically stable for any number of information bits and iterations. However, we observe that while their implementation does marginally improve over ours, it also suffers from a fundamental issue with precision. Finally, we show that Deepcode dominates the optimized performance of SK, over a natural choice of parameterizations when the feedback is noisy.
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.
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