Consider a Gaussian multiple-input multiple-output (MIMO) multiple-access channel (MAC) with channel matrix @math and a Gaussian MIMO broadcast channel (BC) with channel matrix @math . For the MIMO MAC, the integer-forcing architecture consists of first decoding integer-linear combinations of the transmitted codewords, which are then solved for the original messages. For the MIMO BC, the integer-forcing architecture consists of pre-inverting the integer-linear combinations at the transmitter so that each receiver can obtain its desired codeword by decoding an integer-linear combination. In both cases, integer-forcing offers higher achievable rates than zero-forcing while maintaining a similar implementation complexity. This paper establishes an uplink-downlink duality relationship for integer-forcing, i.e., any sum rate that is achievable via integer-forcing on the MIMO MAC can be achieved via integer-forcing on the MIMO BC with the same sum power and vice versa. Using this duality relationship, it is shown that integer-forcing can operate within a constant gap of the MIMO BC sum capacity. Finally, the paper proposes a duality-based iterative algorithm for the non-convex problem of selecting optimal beamforming and equalization vectors, and establishes that it converges to a local optimum.
For the sake of notational simplicity, we will state all of our results for real-valued channels. Analogous results can be obtained for complex-valued channels via real-valued decompositions. Recent efforts have shown that compute-and-forward can also be realized for more general algebraic structures @cite_35 . For instance, building lattices from the Eisenstein integers yields better approximations for complex numbers on average, and can increase the average performance of compute-and-forward @cite_16 .
@cite_35 @cite_16
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In a distributed communications scenario, a relay applying the compute-and-forward strategy for the real-valued channel model aims to decode an integer linear combination of transmitted messages, a task of very complex nature for which general efficient algorithms for dimension n > 1 have not yet been developed. Nonetheless, the maximum-likelihood decoding metric related to solving for the desired lattice point exhibits interesting properties which lead to partial design criteria for lattice codes in compute-and-forward. This article generalizes maximum-likelihood decoding at the relay to allow for arbitrary real lattice codes at the transmitters, and studies the behavior of the resulting decoding metric using an approximation of the theta series of a lattice, which is itself derived in this article. For the first time, the resulting random sums of lattices over whose points the relay needs to perform a sum are analyzed and furthermore, previous related work is extended to the case of K > 2 transmitters. The specific cases K = 2 and K = 3 are studied empirically. We develop a systematic approach, based on convex programming and real analysis, for obtaining upper bounds on the capacity of the binary deletion channel and, more generally, channels with i.i.d. insertions and deletions. Other than the classical deletion channel, we give a special attention to the Poisson-repeat channel introduced by Mitzenmacher and Drinea (IEEE Transactions on Information Theory, 2006). Our framework can be applied to obtain capacity upper bounds for any repetition distribution (the deletion and Poisson-repeat channels corresponding to the special cases of Bernoulli and Poisson distributions). Our techniques essentially reduce the task of proving capacity upper bounds to maximizing a univariate, real-valued, and often concave function over a bounded interval. The corresponding univariate function is carefully designed according to the underlying distribution of repetitions and the choices vary depending on the desired strength of the upper bounds as well as the desired simplicity of the function (e.g., being only efficiently computable versus having an explicit closed-form expression in terms of elementary, or common special, functions). Among our results, we show that the capacity of the binary deletion channel with deletion probability d is at most (1−d) logϕ for d ≥ 1 2, and, assuming the capacity function is convex, is at most 1−d log(4 ϕ) for d Furthermore, we derive the first set of capacity upper bounds for the Poisson-repeat channel. Our results uncover further striking connections between this channel and the deletion channel, and suggest, somewhat counter-intuitively, that the Poisson-repeat channel is actually analytically simpler than the deletion channel and may be of key importance to a complete understanding of the deletion channel. Finally, we derive several novel upper bounds on the capacity of the deletion channel. All upper bounds are maximums of efficiently computable, and concave, univariate real functions over a bounded domain. In turn, we upper bound these functions in terms of explicit elementary and standard special functions, whose maximums can be found even more efficiently (and sometimes, analytically, for example for d=1 2). Along the way, we develop several new techniques of potentially independent interest. For example, we develop systematic techniques to study channels with mean constraints over the reals. Furthermore, we motivate the study of novel probability distributions over non-negative integers, as well as novel special functions which could be of interest to mathematical analysis. We address an open question, regarding whether a lattice code with lattice decoding (as opposed to maximum-likelihood (ML) decoding) can achieve the additive white Gaussian noise (AWGN) channel capacity. We first demonstrate how minimum mean-square error (MMSE) scaling along with dithering (lattice randomization) techniques can transform the power-constrained AWGN channel into a modulo-lattice additive noise channel, whose effective noise is reduced by a factor of spl radic (1+SNR SNR). For the resulting channel, a uniform input maximizes mutual information, which in the limit of large lattice dimension becomes 1 2 log (1+SNR), i.e., the full capacity of the original power constrained AWGN channel. We then show that capacity may also be achieved using nested lattice codes, the coarse lattice serving for shaping via the modulo-lattice transformation, the fine lattice for channel coding. We show that such pairs exist for any desired nesting ratio, i.e., for any signal-to-noise ratio (SNR). Furthermore, for the modulo-lattice additive noise channel lattice decoding is optimal. Finally, we show that the error exponent of the proposed scheme is lower bounded by the Poltyrev exponent. In this work, we introduce @math -expansion, a notion borrowed from number theory, as a theoretical framework to study fast construction of polar codes based on a recursive structure of universal partial order (UPO) and polarization weight (PW) algorithm. We show that polar codes can be recursively constructed from UPO by continuously solving several polynomial equations at each recursive step. From these polynomial equations, we can extract an interval for @math , such that ranking the synthetic channels through a closed-form @math -expansion preserves the property of nested frozen sets, which is a desired feature for low-complex construction. In an example of AWGN channels, we show that this interval for @math converges to a constant close to @math when the code block-length trends to infinity. Both asymptotic analysis and simulation results validate our theoretical claims.