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Register a new userA Simple Converse Proof and a Unified Capacity Formula for Channels with Input Constraints
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Added by
Youjian (Eugene) Liu
Publication date
2008
fields
Informatics Engineering
and research's language is
English
Authors
Youjian Liu
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Given the single-letter capacity formula and the converse proof of a channel without constraints, we provide a simple approach to extend the results for the same channel but with constraints. The resulting capacity formula is the minimum of a Lagrange dual function. It gives an unified formula in the sense that it works regardless whether the problem is convex. If the problem is non-convex, we show that the capacity can be larger than the formula obtained by the naive approach of imposing constraints on the maximization in the capacity formula of the case without the constraints. The extension on the converse proof is simply by adding a term involving the Lagrange multiplier and the constraints. The rest of the proof does not need to be changed. We name the proof method the Lagrangian Converse Proof. In contrast, traditional approaches need to construct a better input distribution for convex problems or need to introduce a time sharing variable for non-convex problems. We illustrate the Lagrangian Converse Proof for three channels, the classic discrete time memoryless channel, the channel with non-causal channel-state information at the transmitter, the channel with limited channel-state feedback. The extension to the rate distortion theory is also provided.
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In non-coherent wideband fading channels where energy rather than spectrum is the limiting resource, peaky and non-peaky signaling schemes have long been considered species apart, as the first approaches asymptotically the capacity of a wideband AWGN channel with the same average SNR, whereas the second reaches a peak rate at some finite critical bandwidth and then falls to zero as bandwidth grows to infinity. In this paper it is shown that this distinction is in fact an artifact of the limited attention paid in the past to the product between the bandwidth and the fraction of time it is in use. This fundamental quantity, called bandwidth occupancy, measures average bandwidth usage over time. For all signaling schemes with the same bandwidth occupancy, achievable rates approach to the wideband AWGN capacity within the same gap as the bandwidth occupancy approaches its critical value, and decrease to zero as the occupancy goes to infinity. This unified analysis produces quantitative closed-form expressions for the ideal bandwidth occupancy, recovers the existing capacity results for (non-)peaky signaling schemes, and unveils a trade-off between the accuracy of approximating capacity with a generalized Taylor polynomial and the accuracy with which the optimal bandwidth occupancy can be bounded.
Flat-fading channels that are correlated in time are considered under peak and average power constraints. For discrete-time channels, a new upper bound on the capacity per unit time is derived. A low SNR analysis of a full-scattering vector channel is used to derive a complimentary lower bound. Together, these bounds allow us to identify the exact scaling of channel capacity for a fixed peak to average ratio, as the average power converges to zero. The upper bound is also asymptotically tight as the average power converges to zero for a fixed peak power. For a continuous time infinite bandwidth channel, Viterbi identified the capacity for M-FSK modulation. Recently, Zhang and Laneman showed that the capacity can be achieved with non-bursty signaling (QPSK). An additional contribution of this paper is to obtain similar results under peak and average power constraints.
It is known that for a discrete channel with correlated additive noise, the ordinary capacity with or without feedback both equal $ log q-mathcal{H} (Z) $, where $ mathcal{H}(Z) $ is the entropy rate of the noise process $ Z $ and $ q $ is the alphabet size. In this paper, a class of finite-state additive noise channels is introduced. It is shown that the zero-error feedback capacity of such channels is either zero or $C_{0f} =log q -h (Z) $, where $ h (Z) $ is the {em topological entropy} of the noise process. A topological condition is given when the zero-error capacity is zero, with or without feedback. Moreover, the zero-error capacity without feedback is lower-bounded by $ log q-2 h (Z) $. We explicitly compute the zero-error feedback capacity for several examples, including channels with isolated errors and a Gilbert-Elliot channel.
We propose two modifi
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