Resource Allocation for Wireless Fading Relay Channels: Max-Min Solution

Resource allocation is investigated for fading relay channels under separate power constraints at the source and relay nodes. As a basic information-theoretic model for fading relay channels, the parallel relay channel is first studied, which consists of multiple independent three-terminal relay channels as subchannels. Lower and upper bounds on the capacity are derived, and are shown to match, and thus establish the capacity for the parallel relay channel with degraded subchannels. This capacity theorem is further demonstrated via the Gaussian parallel relay channel with degraded subchannels, for which the synchronized and asynchronized capacities are obtained. The capacity-achieving power allocation at the source and relay nodes among the subchannels is partially characterized for the synchronized case and fully characterized for the asynchronized case. The fading relay channel is then studied, which is based on the three-terminal relay channel with each communication link being corrupted by a multiplicative fading gain coefficient as well as an additive Gaussian noise term. For each link, the fading state information is assumed to be known at both the transmitter and the receiver. The source and relay nodes are allowed to allocate their power adaptively according to the instantaneous channel state information. The source and relay nodes are assumed to be subject to separate power constraints. For both the full-duplex and half-duplex cases, power allocations that maximize the achievable rates are obtained. In the half-duplex case, the power allocation needs to be jointly optimized with the channel resource (time and bandwidth) allocation between the two orthogonal channels over which the relay node transmits and receives. Capacities are established for fading relay channels that satisfy certain conditions.     

On the capacity-achieving distribution of the discrete-time noncoherent and partially coherent AWGN channels

We investigate the communication limits over rapid phase-varying channels and consider the capacity of a discrete- time noncoherent additive white Gaussian noise (NCAWGN) channel under the average power constraint. We obtain necessary and sufficient conditions for the capacity-achieving input distribution and show that this distribution is discrete and possesses an infinite number of mass points. Using this characterization of the capacity-achieving distribution we compute a tight lower bound on the capacity of the channel based on examining suboptimal input distributions. In addition, we provide some easily computable lower and upper bounds on the channel capacity. Finally, we extend some of these results to the partially coherent channel, where it is assumed that a phase-locked loop (PLL) is used to track the carrier phase at the receiver, and that an ideal interleaver and de-interleaver are employed-rendering the Tikhonov distributed residual phase errors statistically independent from one symbol interval to another.     

Capacity-achieving probability measure for a reduced number of signaling points

Putting bounding constraints on the input of a channel leads in many cases to a discrete capacity-achieving distribution with a finite support. Given a finite number of signaling points, we determine reduced subsets and the corresponding optimal probability measures to simplify the receiver design. The objective for the subset selection is to keep the channel quality high by maximizing mutual information and cutoff rate. Two approaches are introduced to obtain a capacity-achieving probability measure for the reduced subset. The first one is based on a preceded signaling point selection while the second one chooses the signaling points and corresponding probabilities simultaneously. Numerical results for both approaches show that using only a small number of signaling points achieves a very high mutual information compared to channels utilizing the full set of signaling points.     

The Capacity Region of Frequency-Selective Gaussian Interference Channels Under Strong Interference

This paper presents the capacity region of frequency-selective Gaussian interference channels under the condition of strong interference, assuming an average power constraint per user. First, a frequency-selective Gaussian interference channel is modeled as a set of independent parallel memoryless Gaussian interference channels. Using nonfrequency selective results, the capacity region of frequency-selective Gaussian interference channels under strong interference is expressed mathematically. Exploiting structures inherent in the problem, a dual problem is constructed for each independent memoryless channel, in which both mathematical and numerical analysis are performed. Furthermore, three suboptimal methods are compared to the capacity-achieving coding and power allocation scheme. Iterative waterfilling, a suboptimal scheme, provides close-to-optimum performance and has a distributed coding and power allocation scheme, which are attractive in practice.     

Capacity Analysis for Continuous Alphabet Channels With Side Information, Part II: MIMO Channels

In this second part of our two-part paper, we consider the capacity analysis for wireless mobile systems with multiple-antenna architectures. We apply the results of the first part to a commonly known baseband, discrete-time multiple-antenna system where both the transmitter and receiver know the channel's statistical law. We analyze the capacity for additive white Gaussian noise (AWGN) channels, fading channels with full channel state information (CSI) at the receiver, fading channels with no CSI, and fading channels with partial CSI at the receiver. For each type of channels, we study the capacity value as well as issues such as the existence, uniqueness, and characterization of the capacity-achieving measures for different types of moment constraints. The results are applicable to both Rayleigh and Rician fading channels in the presence of arbitrary line-of-sight and correlation profiles.     

Capacity Regions and Bounds for a Class of Z-Interference Channels

We define a class of Z-interference channels for which we obtain a new upper bound on the capacity region. The bound exploits a technique first introduced by Korner and Marton. A channel in this class has the property that, for the transmitter-receiver pair that suffers from interference, the conditional output entropy at the receiver is invariant with respect to the transmitted codewords. We compare the new capacity region upper bound with the Han/Kobayashi achievable rate region for interference channels. This comparison shows that our bound is tight in some cases, thereby yielding specific points on the capacity region as well as sum capacity for certain Z-interference channels. In particular, this result can be used as an alternate method to obtain sum capacity of Gaussian Z-interference channels. We then apply an additional restriction on our channel class: the transmitter-receiver pair that suffers from interference achieves its maximum output entropy with a single input distribution irrespective of the interference distribution. For these channels, we show that our new capacity region upper bound coincides with the Han/Kobayashi achievable rate region, which is therefore capacity-achieving. In particular, for these channels superposition encoding with partial decoding is shown to be optimal and a single-letter characterization for the capacity region is obtained.     

Characterization and computation of optimal distributions for channel coding

This paper concerns the structure of capacity-achieving input distributions for stochastic channel models, and a renewed look at their computational aspects. The following conclusions are obtained under general assumptions on the channel statistics. i) The capacity-achieving input distribution is binary for low signal-to-noise ratio (SNR). The proof is obtained on comparing the optimization equations that determine channel capacity with a linear program over the space of probability measures. ii) Simple discrete approximations can nearly reach capacity even in cases where the optimal distribution is known to be absolutely continuous with respect to Lebesgue measure. iii) A new class of algorithms is introduced based on the cutting-plane method to iteratively construct discrete distributions, along with upper and lower bounds on channel capacity. It is shown that the bounds converge to the true channel capacity, and that the distributions converge weakly to a capacity-achieving distribution.     

Channel capacity and non-uniform signalling for free-space optical intensity channels

This work considers the design of capacityapproaching, non-uniform optical intensity signalling in the presence of average and peak amplitude constraints. Although it is known that the capacity-achieving input distribution is discrete with a finite number of mass points, finding it requires complex non-linear optimization at every SNR. In this work, a simple expression for a capacity-approaching distribution is derived via source entropy maximization. The resulting mutual information using the derived discrete non-uniform input distribution is negligibly far away from the channel capacity. The computation of this distribution is substantially less complex than previous optimization approaches and can be easily computed at different SNRs. A practical algorithm for non-uniform optical intensity signalling is presented using multi-level coding followed by a mapper and multi-stage decoding at the receiver. The proposed signalling is simulated on free-space optical channels and outage capacity is analyzed. A significant gain in both rate and probability of outage is achieved compared to uniform signalling, especially in the case of channels corrupted by fog.     

The construction of M-ary (d,∞) codes that achieve capacityand have the fewest number of encoder states

The existence of 100% efficient (i.e., capacity-achieving) fixed-rate codes for input-constrained, noiseless channels is guaranteed provided the channel has rational capacity. A class of M-ary runlength-limited (M,d,∞) constraints was shown in previous work to have rational capacity. In this correspondence we present a code construction procedure for obtaining 100% efficient codes with the fewest number of encoder states for all (M,d,∞) constraints with rational capacity. The decoders are sliding-block decoders with sliding window size d+1     

Gaussian arbitrarily varying channels

The {em arbitrarily varying channel} (AVC) can be interpreted as a model of a channel jammed by an intelligent and unpredictable adversary. We investigate the asymptotic reliability of optimal random block codes on Gaussian arbitrarily varying channels (GAVC's). A GAVC is a discrete-time memoryless Gaussian channel with input power constraintP_{T}and noise powerN_{e}, which is further corrupted by an additive "jamming signal." The statistics of this signal are unknown and may be arbitrary, except that they are subject to a power constraintP_{J}. We distinguish between two types of power constraints: {em peak} and {em average.} For peak constraints on the input power and the jamming power we show that the GAVC has a random coding capacity. For the remaining cases in which either the transmitter or the jammer or both are subject to average power constraints, no capacities exist and onlylambda-capacities are found. The asymptotic error probability suffered by optimal random codes in these cases is determined. Our results suggest that if the jammer is subject only to an average power constraint, reliable communication is impossible at any positive code rate.     

The capacity of discrete-time memoryless Rayleigh-fading channels

We consider transmission over a discrete-time Rayleigh fading channel, in which successive symbols face independent fading, and where neither the transmitter nor the receiver has channel state information. Subject to an average power constraint, we study the capacity-achieving distribution of this channel and prove it to be discrete with a finite number of mass points, one of them located at the origin. We numerically compute the capacity and the corresponding optimal distribution as a function of the signal-to-noise ratio (SNR). The behavior of the channel at low SNR is studied and finally a comparison is drawn with the ideal additive white Gaussian noise channel     

On the Capacity of Mid-latitude High Frequency Ionospheric Channel

In this paper we consider the theoretical characterization of the ionospheric transmission. More accurately, we derive a closed form expression of the average capacity for Mid-latitude High Frequency (HF) ionospheric channels. Heretofore, this problem has been studied for Rayleigh channels when each tap of the impulse response has a Rayleigh distribution without characterizing the variance of this distribution. In this paper, we extend these works to HF ionospheric channels by evaluating the variance of the amplitude attenuation versus the Doppler spread and then the channel capacity. For a multipath HF ionospheric channel, we model the Doppler phenomenon as a Gaussian profile which is suggested for HF environments. Finally, we derive a closed form expression of the average channel capacity using the probability density function (pdf) of the instantaneous impulse response. Numerical results on both simulated and real measured data are derived at the end of the paper.     

Power-Efficient Resource Allocation for Time-Division Multiple Access Over Fading Channels

We investigate resource allocation policies for time-division multiple access (TDMA) over fading channels in the power-limited regime. For frequency-flat block-fading channels and transmitters having full channel state information (CSI), we first minimize power under a weighted sum average rate constraint and show that the optimal rate and time allocation policies can be obtained by a greedy water-filling approach with linear complexity in the number of users. Subsequently, we pursue power minimization under individual average rate constraints and establish that the optimal resource allocation also amounts to a greedy water-filling solution. Our approaches not only provide fundamental power limits when each user can support an infinite-size capacity-achieving codebook (continuous rates), but also yield guidelines for practical designs where users can only support a finite set of adaptive modulation and coding modes (discrete rates).      

Capacity Achieving LDPC Codes Through Puncturing

The performance of punctured low-definition parity-check (LDPC) codes under maximum-likelihood (ML) decoding is studied in this correspondence via deriving and analyzing their average weight distributions (AWDs) and the corresponding asymptotic growth rate of the AWDs. In particular, it is proved that capacity-achieving codes of any rate and for any memoryless binary-input output-symmetric (MBIOS) channel under ML decoding can be constructed by puncturing some original LDPC code with small enough rate. Moreover, it is shown that the gap to capacity of all the punctured codes can be the same as the original code with a small enough rate. Conditions under which puncturing results in no rate loss with asymptotically high probability are also given in the process. These results show high potential for puncturing to be used in designing capacity-achieving codes, and in rate-compatible coding under any MBIOS channel.      

Limiting performance of block-fading channels with multipleantennas

We derive the performance limits of a radio system consisting of a transmitter with t antennas and a receiver with r antennas, a block-fading channel with additive white Gaussian noise (AWGN), delay and transmit-power constraints, and perfect channel-state information available at both the transmitter and the receiver. Because of a delay constraint, the transmission of a codeword is assumed to span a finite (and typically small) number M of independent channel realizations; therefore, the relevant performance limits are the information outage probability and the “delay-limited” (or “nonergodic”) capacity. We derive the coding scheme that minimizes the information outage probability. This scheme can be interpreted as the concatenation of an optimal code for the AWGN channel without fading to an optimal beamformer. For this optimal scheme, we evaluate minimum-outage probability and delay-limited capacity. Among other results, we prove that, for the fairly general class of regular fading channels, the asymptotic delay-limited capacity slope, expressed in bits per second per hertz (b/s/Hz) per decibel of transmit signal-to-noise ratio (SNR), is proportional to min (t,r) and independent of the number of fading blocks M. Since M is a measure of the time diversity (induced by interleaving) or of the frequency diversity of the system, this result shows that, if channel-state information is available also to the transmitter, very high rates with asymptotically small error probabilities are achievable without the need of deep interleaving or high-frequency diversity. Moreover, for a large number of antennas, delay-limited capacity approaches ergodic capacity     

A simple derivation of the coding theorem and some applications

Upper bounds are derived on the probability of error that can be achieved by using block codes on general time-discrete memoryless channels. Both amplitude-discrete and amplitude-continuous channels are treated, both with and without input constraints. The major advantages of the present approach are the simplicity of the derivations and the relative simplicity of the results; on the other hand, the exponential behavior of the bounds with block length is the best known for all transmission rates between0and capacity. The results are applied to a number of special channels, including the binary symmetric channel and the additive Gaussian noise channel.     

Capacity Analysis for Continuous-Alphabet Channels With Side Information, Part I: A General Framework

Capacity analysis for channels with side information at the receiver has been an active area of interest. This problem is well investigated for the case of finite-alphabet channels. However, the results are not easily generalizable to the case of continuous-alphabet channels due to analytical difficulties inherent with continuous alphabets. In the first part of this two-part paper, we address an analytical framework for capacity analysis of continuous alphabet channels with side information at the receiver. For this purpose, we establish novel necessary and sufficient conditions for weak* continuity and strict concavity of the mutual information. These conditions are used in investigating the existence and uniqueness of the capacity-achieving measures. Furthermore, we derive necessary and sufficient conditions that characterize the capacity value and the capacity-achieving measure for continuous-alphabet channels with side information at the receiver.     

Capacity of RAKE reception with energy randomization

We consider a RAKE receiver for coherent binary orthogonal signaling over a frequency selective multipath Rayleigh fading channel. The receiver uses maximal-ratio combining such that the weight estimation errors are not independent of the additive noise. We find the capacity-achieving energy randomization scheme with two energy levels for the cases of imperfect and perfect channel estimates. We observe that the capacity-achieving probability gets closer to 1/2 with increase in the number of paths. We also show that the capacity is higher if channel estimates are perfect and that the channel estimation errors have more pronounced effect on the capacity at low signal-to-noise ratios.     

On approaching wideband capacity using multitone FSK

In the wideband limit, certain types of "flash" signaling, such as flash frequency-shift keying (FSK), achieve the capacity of multipath fading channels. It is not clear, however, whether these asymptotic results translate into insights for practical fading channels in the finite-bandwidth, power-limited regime. It is known that, for flash FSK, the size of the input alphabet grows slowly with increasing bandwidth, leading to very high-peak power per tone. Thus, for flash FSK, the codeword probability of error decays very slowly with bandwidth and feasible rates approach the wideband capacity limit extremely slowly. Without contradicting the above results, our results in this paper point to a more optimistic outlook, from the point of view of error exponents and achievable rates, for the applicability of flash techniques in practical scenarios. We consider multitone FSK (MFSK), which has the same asymptotic capacity-achieving property as flash FSK in the wideband limit, but allows a larger input alphabet size with the same bandwidth. First, we show, using an error exponent approach, that multitone FSK allows lower peak power per tone than flash FSK. Next, we present the capacity of single-tone and two-tone FSK schemes with hard-decision detection at finite bandwidths. For typical channel parameters, the capacities are close to the wideband capacity limit.     

Capacity-achieving input distributions for some amplitude-limited channels with additive noise (Corresp.)

An additive noise channel wherein the noise is described by a piecewise constant probability density is shown to reduce to a discrete channel by means of an explicit construction. In addition, conditions are found which describe a class of continuous amplitude-limited channels for which the capacity-achieving input distribution is binary.