Bit-stuffing algorithms and analysis for run-length constrained channels in two and three dimensions

A rigorous derivation is given of the coding rate of a variable-to-variable length bit-stuffing coder for a two-dimensional (1,/spl infin/)-constrained channel. The coder studied is "nearly" a fixed-to-fixed length algorithm. Then an analogous variable-to-variable length bit-stuffing algorithm for the three-dimensional (1,/spl infin/)-constrained channel is presented, and its coding rate is analyzed using the two-dimensional method. The three-dimensional coding rate is demonstrated to be at least 0.502, which is proven to be within 4% of the capacity.     

Asymptotic capacity of two-dimensional channels with checkerboard constraints

A checkerboard constraint is a bounded measurable set S/spl sub/R/sup 2/, containing the origin. A binary labeling of the Z/sup 2/ lattice satisfies the checkerboard constraint S if whenever t/spl isin/Z/sup 2/ is labeled 1, all of the other Z/sup 2/-lattice points in the translate t+S are labeled 0. Two-dimensional channels that only allow labelings of Z/sup 2/ satisfying checkerboard constraints are studied. Let A(S) be the area of S, and let A(S)/spl rarr//spl infin/ mean that S retains its shape but is inflated in size in the form /spl alpha/S, as /spl alpha//spl rarr//spl infin/. It is shown that for any open checkerboard constraint S, there exist positive reals K/sub 1/ and K/sub 2/ such that as A(S)/spl rarr//spl infin/, the channel capacity C/sub S/ decays to zero at least as fast as (K/sub 1/log/sub 2/A(S))/A(S) and at most as fast as (K/sub 2/log/sub 2/A(S))/A(S). It is also shown that if S is an open convex and symmetric checkerboard constraint, then as A(S)/spl rarr//spl infin/, the capacity decays exactly at the rate 4/spl delta/(S)(log/sub 2/A(S))/A(S), where /spl delta/(S) is the packing density of the set S. An implication is that the capacity of such checkerboard constrained channels is asymptotically determined only by the areas of the constraint and the smallest (possibly degenerate) hexagon that can be circumscribed about the constraint. In particular, this establishes that channels with square, diamond, or hexagonal checkerboard constraints all asymptotically have the same capacity, since /spl delta/(S)=1 for such constraints.     

On the capacity of log-normal fading channels

In this letter we provide an analytical expression for the moments of the capacity for the log-normal fading channel. Since the developed expression involves infinite series, we show that the error that results from the truncation of these series is insignificant. We also analyze in more details the ergodic capacity by giving a simpler expression for the remainder of the truncated series. Relying on the fact that the sum of log-normal Random Variables (RV) is well approximated by another lognormal RV, we further utilize the obtained results to approximate the capacity of diversity combining techniques in correlated lognormal fading channels. The results that we provide in this letter are an important tool for measuring the performance of communication links in a log-normal environment.     

On the capacity of MIMO relay channels

We study the capacity of multiple-input multiple- output (MIMO) relay channels. We first consider the Gaussian MIMO relay channel with fixed channel conditions, and derive upper bounds and lower bounds that can be obtained numerically by convex programming. We present algorithms to compute the bounds. Next, we generalize the study to the Rayleigh fading case. We find an upper bound and a lower bound on the ergodic capacity. It is somewhat surprising that the upper bound can meet the lower bound under certain regularity conditions (not necessarily degradedness), and therefore the capacity can be characterized exactly; previously this has been proven only for the degraded Gaussian relay channel. We investigate sufficient conditions for achieving the ergodic capacity; and in particular, for the case where all nodes have the same number of antennas, the capacity can be achieved under certain signal-to-noise ratio (SNR) conditions. Numerical results are also provided to illustrate the bounds on the ergodic capacity of the MIMO relay channel over Rayleigh fading. Finally, we present a potential application of the MIMO relay channel for cooperative communications in ad hoc networks.     

On the capacity of a cascade of channels

A class of information-theoretic problems is introduced that deals with systems in which information is transmitted through a series of noisy channels with limited or no processing between channels. Some basic questions that arise in such systems are examined in detail. The effects of ordering of the channels on overall capacity and the characteristics that cause channels to have high capacity when cascaded with themselves are dealt with     

On the capacity and normalization of ISI channels

We investigate the capacity of various intersymbol interference (ISI) channels with additive white Gaussian noise (AWGN). Previous papers showed a minimum E/sub b//N/sub 0/ of -4.6 dB, 3 dB below the capacity of a flat channel, is obtained using water-pouring capacity formulas for the 1+D channel. However, these papers did not take into account that the channel power gain can be greater than one when water-pouring is used. We present a generic power normalization method of the channel frequency response, namely, peak bandwidth normalization (PBN), to facilitate the fair capacity comparison of various ISI channels. Three types of ISI channel, i.e., adder channels, RC channels, and magnetic recording channels, are examined. By using our channel power gain normalization, the capacity curves of these ISI channels are shown.     

On the capacity of UWB over multipath channels

In this work we compute the information theoretic capacity C of binary orthogonal pulse position modulated (PPM) signals for ultra wideband (UWB) communications over multipath channels. We consider binary PPM signals with random energy and correlation values. Numerical examples are given to illustrate the capacity results.     

On the capacity of channels with unknown interference

The process of communicating in the presence of interference that is unknown or hostile is modeled as a two-person zero-sum game with the communicator and the jammer as the players. The objective function considered is the rate of reliable communication. The communicator's strategies are encoders and distributions on a set of quantizers. The jammer's strategies are distributions on the noise power subject to certain constraints. Various conditions are considered on the jammer's strategy set and on the communicator's knowledge. For the case where the decoder is uninformed of the actual quantizer chosen, it is shown that, from the communicator's perspective, the worst-case jamming strategy is a distribution concentrated on a finite number of points, thereby converting a functional optimization problem into a nonlinear programming problem. Moreover, the worst-case distributions can be characterized by means of necessary and sufficient conditions which are easy to verify. For the case where the decoder is informed of the actual quantizer chosen, the existence of saddle-point strategies is demonstrated. The analysis is also seen to be valid for a number of situations where the jammer is adaptive     

On the capacity of channels with block memory

The capacity of channels with block memory is investigated. It is shown that, when the problem is modeled as a game-theoretic problem, the optimum coding and noise distributions when block memory is permitted are independent from symbol to symbol within a block. Optimal jamming strategies are also independent from symbol to symbol within a block     

On the capacity of certain additive non-Gaussian channels

A model of an additive non-Gaussian noise channel with generalized average input energy constraint is considered. The asymptotic channel capacityC_{zeta}(S), for large signal-to-noise ratioS, is found under certain conditions on the entropyH_{ tilde{ zeta}}( zeta)of the measure induced in function space by the noise processzeta, relative to the measure induced bytilde{zeta}, where is a Gaussian process with the same covariance as that ofzeta. IfH_{ tilde{zeta}}( zeta) < inftyand the channel input signal is of dimensionM< infty, thenC_{ zeta}(S)= frac{1}{2}M ln(1 + S/M) + Q_{zeta}( M ) + {o}(1), where0 leq Q_{ zeta}( M ) leq H_{ tilde{ zeta}}( zeta). If the channel input signal is of infinite dimension andH_{ tilde{ zeta}}( zeta) rightarrow 0forS rightarrow infty, thenC_{ zeta}(S) = frac{1}{2}S+{o}(1).     

On the capacity of AM and PM channels

By representing amplitude-modulated and phase-modulated continuous signals (if the channel noise is white Gaussian) as a set of independent samples, the capacities of the corresponding channels are calculated. These are compared with that of a Gaussian channel, and with each other. The results are discussed.     

RMS bandwidth constrained signature waveforms that maximize thetotal capacity of PAM-synchronous CDMA channels

Optimum time-limited signal sets of equal and unequal energies are obtained under root mean square (RMS) bandwidth constraints. The total capacity and the total asymptotic efficiency of the PAM synchronous Gaussian CDMA (PSG-CDMA) channel are considered as the optimality criteria. The latter measure is monotonic with the determinant of the correlation matrix, R, and the former is monotonic with det(I+σ ^-2R), where σ² represents the noise level. Average as well as maximum RMS bandwidth constraints are considered in the equal-energy case, and the energy-weighted RMS bandwidth constraint is considered for unequal energy signals. For the equal-energy problem, signal sets are found that simultaneously optimize the total asymptotic efficiency under both average and maximum RMS bandwidth constraints. For the total capacity measure, such simultaneously optimal signal sets are also obtained, albeit under the restriction that the number of signals n be a Hadamard matrix dimension. When the Hadamard dimension is in particular a power of two, we obtain optimum signal sets that are shown to yield equal optimum multiuser detector asymptotic efficiencies for all users of an uncoded PSG-CDMA channel. Unequal energy signal sets are also found under an energy-weighted RMS bandwidth constraint for both optimality criteria     

On the symmetric information rate of two-dimensional finite-state ISI channels

We derive a pair of bounds (upper and lower) on the symmetric information rate of a two-dimensional finite-state intersymbol interference (ISI) channel model. For channels with small impulse response support, they can be estimated via a modified forward recursion of the Bahl-Cocke-Jelinek-Raviv (BCJR) algorithm. The convergence of the bounds is also analyzed. To relax the constraint on the size of the impulse response, a new upper bound is proposed which allows the tradeoff of the computational complexity and the tightness of the bound. These bounds are further extended to d-dimensional (d>2) ISI channels.     

On the capacity of spatially correlated MIMO Rayleigh-fading channels

In this paper, we investigate the capacity distribution of spatially correlated, multiple-input-multiple-output (MIMO) channels. In particular, we derive a concise closed-form expression for the characteristic function (c.f.) of MIMO system capacity with arbitrary correlation among the transmitting antennas or among the receiving antennas in frequency-flat Rayleigh-fading environments. Using the exact expression of the c.f., the probability density function (pdf) and the cumulative distribution function (CDF) can be easily obtained, thus enabling the exact evaluation of the outage and mean capacity of spatially correlated MIMO channels. Our results are valid for scenarios with the number of transmitting antennas greater than or equal to that of receiving antennas with arbitrary correlation among them. Moreover, the results are valid for an arbitrary number of transmitting and receiving antennas in uncorrelated MIMO channels. It is shown that the capacity loss is negligible even with a correlation coefficient between two adjacent antennas as large as 0.5 for exponential correlation model. Finally, we derive an exact expression for the mean value of the capacity for arbitrary correlation matrices.     

On the asymptotic capacity of stationary Gaussian fading channels

We consider a peak-power-limited single-antenna flat complex-Gaussian fading channel where the receiver and transmitter, while fully cognizant of the distribution of the fading process, have no knowledge of its realization. Upper and lower bounds on channel capacity are derived, with special emphasis on tightness in the high signal-to-noise ratio (SNR) regime. Necessary and sufficient conditions (in terms of the autocorrelation of the fading process) are derived for capacity to grow double-logarithmically in the SNR. For cases in which capacity increases logarithmically in the SNR, we provide an expression for the "pre-log", i.e., for the asymptotic ratio between channel capacity and the logarithm of the SNR. This ratio is given by the Lebesgue measure of the set of harmonics where the spectral density of the fading process is zero. We finally demonstrate that the asymptotic dependence of channel capacity on the SNR need not be limited to logarithmic or double-logarithmic behaviors. We exhibit power spectra for which capacity grows as a fractional power of the logarithm of the SNR     

On the capacity of multiuser wireless channels with multiple antennas

The advantages of multiuser communication, where many users are allowed to simultaneously transmit or receive in a common bandwidth, are considered for multiple-antenna systems in a high signal-to-noise ratio (SNR) regime. Assuming channel state information at receiver (CSIR) to be available, the ergodic capacity is characterized for both unbiased and biased channels, and the quantitative capacity gain of a multiple-antenna multiuser system is analyzed for multiple-access channels. For highly biased (correlated) channels, a multiuser system is shown to be inherently superior to a single-user system (a time- or frequency-division multiple-access (TDMA or FDMA) based system) due to the underlying multiuser diversity, and the sum capacity is shown to scale linearly with the number of antennas. For unbiased channels, the characteristics of ergodic capacity are shown to transfer to outage capacity when a large degree of space diversity exists, and to deterministic capacity when the number of receive antennas is large. Also, a brief discussion on the multiuser multiple-antenna communication in broadcast channel is provided.     

On the capacity of binary and Gaussian channels withrun-length-limited inputs

Bounds on the capacity of binary symmetric channels and additive Gaussian channels with run-length-limited two-level (binary, bipolar) inputs are presented, and their tightness is demonstrated for some cases. Stationary input sequences, which do not degrade capacity, are considered when deriving the bounds. Lower bounds on the magnetic recording density for a simple continuous-time recording model incorporating a minimal intertransition constraint are evaluated for soft and hard decisions. A superiority of about 1.5 dB in signal-to-noise ratio is observed for the soft-decision scheme     

On the capacity of some channels with channel state information

We study the capacity of some channels whose conditional output probability distribution depends on a state process independent of the channel input and where channel state information (CSI) signals are available both at the transmitter (CSIT) and at the receiver (CSIR). When the channel state and the CSI signals are jointly independent and identically distributed (i.i.d.), the channel reduces to a case studied by Shannon (1958). In this case, we show that when the CSIT is a deterministic function of the CSIR, optimal coding is particularly simple. When the state process has memory, we provide a general capacity formula and we give some more restrictive conditions under which the capacity has still a simple single-letter characterization, allowing simple optimal coding. Finally, we turn to the additive white Gaussian noise (AWGN) channel with fading and we provide a generalization of some results about capacity with CSI for this channel. In particular, we show that variable-rate coding (or multiplexing of several codebooks) is not needed to achieve capacity and, even when the CSIT is not perfect, the capacity achieving power allocation is of the waferfilling type     

On the capacity of cognitive broadcast channels with opportunistic scheduling

In this paper, we investigate the fundamental capacity limit of the cognitive broadcast channels with opportunistic scheduling, where cognitive users (CUs) can share the same spectrum with the primary user (PU) as long as the interference introduced to the PU is kept at an acceptable level. In this context, we analyze the capacity gains offered by this opportunistic spectrum sharing in Rayleigh fading environment. Specifically, we analyze the outage and effective capacity of the selected CU, and derive closed‐form expressions for these capacity metrics. We also obtain closed‐form expressions for the asymptotic performance as the bandwidth approaches infinity. Numerical results are provided to corroborate our theoretical analysis and quantify the effects of the system parameters. Copyright © 2011 John Wiley & Sons, Ltd.