Through refined asymptotic analysis based on the normal approximation, we study how higher-order coding performance depends on the mean power as well as on finer statistics of the input power. We introduce a multifaceted power model in which the expectation of an arbitrary (but finite) number of arbitrary functions of the normalized average power is constrained. The framework generalizes existing models, recovering the standard maximal and expected power constraints and the recent mean and variance constraint as special cases. Under certain growth and continuity assumptions on the functions, our main theorem gives an exact characterization of the minimum average error probability for Gaussian channels as a function of the first- and second-order coding rates. The converse proof reduces the code design problem to minimization over a compact (under the Prokhorov metric) set of probability distributions, characterizes the extreme points of this set and invokes the Bauer's maximization principle. Our results for the multifaceted power model serve as more precise benchmarks for practical modulation schemes with multiple amplitude levels, probabilistic shaping and nonuniform constellation geometries.

Through refined asymptotic analysis based on the normal approximation, we study how higher-order coding performance depends on the mean power as well as on finer statistics of the input power. We introduce a multifaceted power model in which the expectation of an arbitrary (but finite) number of arbitrary functions of the normalized average power is constrained. The framework generalizes existing models, recovering the standard maximal and expected power constraints and the recent mean and variance constraint as special cases. Under certain growth and continuity assumptions on the functions, our main theorem gives an exact characterization of the minimum average error probability for Gaussian channels as a function of the first- and second-order coding rates. The converse proof reduces the code design problem to minimization over a compact (under the Prokhorov metric) set of probability distributions, characterizes the extreme points of this set and invokes the Bauer’s maximization principle. Our results for the multifaceted power model serve as more precise benchmarks for practical modulation schemes with multiple amplitude levels, probabilistic shaping and nonuniform constellation geometries.

We consider channel coding for Gaussian channels with the recently introduced mean and variance cost constraint. Through matching converse and achievability bounds, we charac-terize the optimal first- and second-order performance. The main technical contribution of this paper is an achievability scheme which uses random codewords drawn from a mixture of three uniform distributions on$(n-1)$-spheres of radii$R_{1}, R_{2}$and$R_{3}$, where$R_{i}=O(\sqrt{n})$and$\vert R_{i}-R_{j}\vert =O(1)$. To analyze such a mixture distribution, we prove a lemma giving a uniform$O(\log n)$bound, which holds with high probability, on the$\log$ratio of the output distributions$Q_{i}^{cc}$and$Q_{j}^{cc}$, where$Q_{i}^{cc}$is induced by a random channel input uniformly distributed on an$(n - 1)$-sphere of radius$R_{i}$. To facilitate the application of the usual central limit theorem, we also give a uniform$O(\log n)$bound, which holds with high probability, on the log ratio of the output distributions$Q_{i}^{cc}$and$Q_{i}^{\ast}$, where$Q_{i}^{\ast}$is induced by a random channel input with i.i.d. components.

Channel coding for discrete memoryless channels (DMCs) with mean and variance cost constraints has been recently introduced. We show that there is an improvement in coding performance due to cost variability, both with and without feedback. We demonstrate this improvement over the traditional almost-sure (per-codeword) cost constraint that prohibits any cost variation above a fixed threshold. Our result simultaneously shows that feedback does not improve the second-order coding rate of simple-dispersion DMCs under the almost-sure cost constraint. This finding parallels similar results for unconstrained simple-dispersion DMCs, additive white Gaussian noise (AWGN) channels and parallel Gaussian channels.

We consider channel coding for discrete memoryless channels (DMCs) with a novel cost constraint that constrains both the mean and the variance of the cost of the codewords. We show that the maximum (asymptotically) achievable rate under the new cost formulation is equal to the capacity-cost function; in particular, the strong converse holds. We further characterize the optimal second-order coding rate of these cost-constrained codes; in particular, the optimal second-order coding rate is finite. We then show that the second-order coding performance is strictly improved with feedback using a new variation of timid/bold coding, significantly broadening the applicability of timid/bold coding schemes from unconstrained compound-dispersion channels to all cost-constrained channels. Equivalent results on the minimum average probability of error are also given.

Presents corrections to the paper, (Errata to “Channel Coding With Mean and Variance Cost Constraints”).

We consider channel coding for Gaussian channels with the recently introduced mean and variance cost constraints. Through matching converse and achievability bounds, we characterize the optimal first- and second-order performance. The main technical contribution of this paper is an achievability scheme which uses random codewords drawn from a mixture of three uniform distributions on (n−1)-spheres of radiiR1,R2andR3, whereRi=O( √n) and |Ri−Rj| = O(1). To analyze such a mixture distribution, we prove a lemma giving a uniformO(logn) bound, which holds with high probability, on the log ratio of the output distributionsQcciandQccj, whereQcciis induced by a random channel input uniformly distributed on an (n− 1)-sphere of radiusRi. To facilitate the application of the usual central limit theorem, we also give a uniformO(logn) bound, which holds with high probability, on the log ratio of the output distributionsQcciandQ∗i, whereQ∗iis induced by a random channel input with i.i.d. components.

We consider channel coding for discrete memoryless channels (DMCs) with a novel cost constraint that constrains both the mean and the variance of the cost of the codewords. We show that the maximum (asymptotically) achievable rate under the new cost formulation is equal to the capacity-cost function; in particular, the strong converse holds. We further characterize the optimal second-order coding rate of these cost-constrained codes; in particular, the optimal second-order coding rate is finite. We then show that the second -order coding performance is strictly improved with feedback using a new variation of timid/bold coding, significantly broadening the applicability of timid/bold coding schemes from unconstrained compound-dispersion chan-nels to all cost-constrained channels. Equivalent results on the minimum average probability of error are also given.

We show that a variation of the timid/bold coding scheme for feedback communication can be used to improve the second-order coding rate for most discrete memoryless channels with a cost constraint, even in the simple-dispersion case for which the optimal input distribution is unique.

We consider a novel variant of$d$-semifaithful lossy coding in which the distortion measure is revealed only to the encoder and only at run-time, as well as an extension of it in which the distortion constraint$d$is also revealed at run-time. Two forms of rate redundancy are used to analyze the performance, and achievability results of both a pointwise and minimax nature are demonstrated. The first coding scheme uses ideas from VC dimension and growth functions, the second uses appropriate quantization of the space of distortion measures, and the third relies on a random coding argument.

We show the existence of variable-rate rate-distortion codes that meet the distortion constraint almost surely and are minimax, i.e., strongly, universal with respect to an unknown source distribution and a distortion measure that is revealed only to the encoder and only at runtime. If we only require minimax universality with respect to the source distribution and not the distortion measure, then we provide an achievable$\tilde {O}(1/\sqrt {n})$redundancy rate, which we show is optimal. This is in contrast to prior work on universal lossy compression, which provides$O(\log n/n)$redundancy guarantees for weakly universal codes under various regularity conditions. We show that either eliminating the regularity conditions or upgrading to strong universality while keeping these regularity conditions entails an inevitable increase in the redundancy to$\tilde {O}(1/\sqrt {n})$. Our construction involves random coding with non-i.i.d. codewords and a zero-rate uncoded transmission scheme. The proof uses exact asymptotics from large deviations, acceptance-rejection sampling, and the VC dimension of distortion measures.

We consider a novel variant of lossy coding in which the distortion measure is revealed only to the encoder and only at run-time, as well as an extension of it in which the distortion constraint is also revealed at run-time. Two forms of rate redundancy are used to analyze the performance, and achievability results of both a pointwise and minimax nature are demonstrated. One proof uses appropriate quantization of the space of distortion measures while another uses ideas from VC dimension and growth functions.

We show the existence of universal, variable-rate rate-distortion codes that meet the distortion constraint almost surely and approach the rate-distortion function uniformly with respect to an unknown source distribution and a distortion measure that is only revealed to the encoder and only at runtime. If the convergence only needs to be uniform with respect to the source distribution and not the distortion measure, then we provide an explicit bound on the minimax rate of convergence. Our construction combines conventional random coding with a zero-rate uncoded transmission scheme. The proof uses exact asymptotics from large deviations, acceptance-rejection sampling, the VC dimension of distortion measures, and the identification of an explicit, code-independent, finite-blocklength quantity, which converges to the rate-distortion function, that controls the performance of the best codes.