We consider the problem of neural association for a network of nonbinary neurons. Here, the task is to first memorize a set of patterns using a network of neurons whose states assume values from a finite number of integer levels. Later, the same network should be able to recall the previously memorized patterns from their noisy versions. Prior work in this area consider storing a finite number of purely random patterns, and have shown that the pattern retrieval capacities (maximum number of patterns that can be memorized) scale only linearly with the number of neurons in the network. In our formulation of the problem, we concentrate on exploiting redundancy and internal structure of the patterns to improve the pattern retrieval capacity. Our first result shows that if the given patterns have a suitable linear-algebraic structure, i.e., comprise a subspace of the set of all possible patterns, then the pattern retrieval capacity is exponential in terms of the number of neurons. The second result extends the previous finding to cases where the patterns have weak minor components, i.e., the smallest eigenvalues of the correlation matrix tend toward zero. We will use these minor components (or the basis vectors of the pattern null space) to increase both the pattern retrieval capacity and error correction capabilities. An iterative algorithm is proposed for the learning phase, and two simple algorithms are presented for the recall phase. Using analytical methods and simulations, we show that the proposed methods can tolerate a fair amount of errors in the input while being able to memorize an exponentially large number of patterns.

We consider the problem of neural association, which deals with the retrieval of a previously memorized pattern from its noisy version. The performance of various neural networks developed for this task may be judged in terms of their pattern retrieval capacities (the number of patterns that can be stored), and their error-correction (noise tolerance) capabilities. While significant progress has been made, most prior works in this area show poor performance with regard to pattern retrieval capacity and/or error correction. In this paper, we propose two new methods to significantly increase the pattern retrieval capacity of the Hopfield and Bidirectional Associative Memories (BAM). The main idea is to store patterns drawn from a family of low correlation sequences, similar to those used in Code Division Multiple Access (CDMA) communications, instead of storing purely random patterns as in prior works. These low correlation patterns can be obtained from random sequences by pre-coding the original sequences via simple operations that both real and artificial neurons are capable of accomplishing.

We consider the problem of neural association for a network of non-binary neurons. Here, the task is to recall a previously memorized pattern from its noisy version using a network of neurons whose states assume values from a finite number of non-negative integer levels. Prior works in this area consider storing a finite number of purely random patterns, and have shown that the pattern retrieval capacities (maximum number of patterns that can be memorized) scale only linearly with the number of neurons in the network. In our formulation of the problem, we consider storing patterns from a suitably chosen set of patterns, that are obtained by enforcing a set of simple constraints on the coordinates (such as those enforced in graph based codes). Such patterns may be generated from purely random information symbols by simple neural operations. Two simple neural update algorithms are presented, and it is shown that our proposed mechanisms result in a pattern retrieval capacity that is exponential in terms of the network size. Furthermore, using analytical results and simulations, we show that the suggested methods can tolerate a fair amount of errors in the input.

In order to exploit the full multiplexing gain of multi-antenna multi-user downlink schemes, accurate channel state information at the transmitter (i.e., at the base station) is required. We consider the design of a closed-loop channel state information feedback scheme, where user terminals feed back their channel state information simultaneously to a multi-antenna base station. The underlying information theoretic problem consists of lossy source-channel coding of multiple independent analog sources (i.e., the users' channel coefficients) over a Gaussian multiple-input multiple-output multi-access channel (MIMO-MAC). Unlike the classical source-channel coding setting, this application requires low latency, otherwise the channel state information would be outdated. Hence, source-channel codewords can span only a single fading state of the uplink (feedback) channel. Furthermore, the transmitters are ignorant of the realization of the uplink channel coefficients. In this scenario, the scaling of the maximum of the estimated downlink channel mean-square errors with the SNR dominates the performance of the multiuser downlink. This scaling is described by the distortion SNR exponent, previously introduced in a single-user MIMO setting. This paper analyzes the max-min distortion SNR exponent of the MIMO-MAC for both separated source-channel coding, and a particular hybrid digital-analog joint source-channel coding scheme. For the case of single-antenna users, we prove that the distortion SNR exponent of separated source-channel coding can be achieved by the concatenation of scalar quantization and uncoded quadrature-amplitude modulation (QAM) transmission, with lattice decoding at the base-station receiver. The resulting scheme has very low encoding latency (only a few symbols of the uplink slot) and generally outperforms currently proposed channel state feedback schemes based on analog unquantized transmission or vector quantization with fixed codebooks.

We consider the problem of signal design for transmission over an interference channel, along the lines of the recently proposed strategy of Interference Alignment. In the framework of interference alignment, each receiver maintains an interference subspace. The transmitters shape their transmissions such that all the interference seen at a particular receiver falls into its interference subspace, hence allowing the remaining receiver space to be used for reception of the useful signal. In contrast to previously proposed iterative interference alignment algorithms for the general interference channel, our proposed technique not only minimizes the interference power that is spilled outside the interference subspace, but also minimizes the signal power that is spilled over into the interference subspace. A weighted optimization technique is proposed to tradeoff between the relative importance associated with the leakage interference and the leakage signal powers. It is proven that the algorithm converges monotonically, and simulation results reveal that the proposed technique outperforms other known algorithms in terms of throughput. Further discussions on generalizing the algorithm are also presented.

We consider the problem of designing low latency and low complexity schemes for channel state feedback over the MIMO-MAC (multiple-input multiple-output multiple access channel). We develop a framework for analyzing this problem in terms of minimizing the distortion, and come up with separated source-channel schemes and joint source-channel schemes that perform better than traditional analog feedback. The results are validated through simulations using very simple channel codes.

Linear receivers are considered as an attractive low- complexity alternative to optimal processing for multi-antenna MIMO communications. In this paper we characterize the performance of MMSE MIMO receivers in the limit of large antenna numbers in the presence of channel correlations. Using a different approach, we generalize our results obtained in [1] to correlated channels showing that the mutual information converges in distribution to a Gaussian random variable whose mean and variance can be characterized analytically. Our results agree very well with simulations even with a moderate number of antennas. We conclude by observing that for a fixed target rate and operating SNR, using a larger number of antennas and with a simple linear MMSE receiver may be a convenient design choice compared to limiting the number of antennas and insisting on more complicated non-linear receivers.

We present constructions of space-time (ST) codes based on lattice coset coding. First, we focus on ST code constructions for the short block-length case, i.e., when the block length is equal to or slightly larger than the number of transmit antennas. We present constructions based on dense lattice packings and nested lattice (Voronoi) shaping. Our codes achieve the optimal diversity-multiplexing tradeoff (DMT) of quasi-static multiple-input multiple-output (MIMO) fading channels for any fading statistics, and perform very well also at practical, moderate values of signal-to-noise ratios (SNR). Then, we extend the construction to the case of large block lengths, by using trellis coset coding. We provide constructions of trellis coded modulation (TCM) schemes that are endowed with good packing and shaping properties. Both short-block and trellis constructions allow for a reduced complexity decoding algorithm based on minimum mean-squared error generalized decision feedback equalizer (MMSE-GDFE) lattice decoding and a combination of this with a Viterbi TCM decoder for the TCM case. Beyond the interesting algebraic structure, we exhibit codes whose performance is among the state-of-the art considering codes with similar encoding/decoding complexity.

We study the dynamic decode-and-forward (DDF) protocol for a single half-duplex relay, single-antenna channel with quasi-static fading. The DDF protocol is well known and has been analyzed in terms of the diversity-multiplexing tradeoff (DMT) in the infinite block length limit. We characterize the finite block length DMT and give new explicit code constructions. The finite block length analysis illuminates a few key aspects that have been neglected in the previous literature: 1) we show that one dominating cause of degradation with respect to the infinite block length regime is the event of decoding error at the relay; 2) we explicitly take into account the fact that the destination does not generally know a priori the relay decision time at which the relay switches from listening to transmit mode. Both of the above problems can be tackled by a careful design of the decoding algorithm. In particular, we introduce a decision rejection criterion at the relay based on Forney's decision rule (a variant of the Neyman-Pearson rule), such that the relay triggers transmission only when its decision is reliable. Also, we show that a receiver based on the generalized likelihood ratio test (GLRT) rule that jointly decodes the relay decision time and the information message achieves the optimal DMT. Our results show that no cyclic redundancy check (CRC) for error detection or additional protocol overhead to communicate the decision time are needed for DDF. Finally, we investigate the use of minimum mean-squared error generalized decision feedback equalizer (MMSE-GDFE) lattice decoding at both the relay and the destination, and show that it provides near-optimal performance at moderate complexity.

Linear receivers are an attractive low-complexity alternative to optimal processing for multiple-antenna multiple-input multiple-output (MIMO) communications. In this paper, we characterize the information-theoretic performance of MIMO linear receivers in two different asymptotic regimes. For fixed number of antennas, we investigate the limit of error probability in the high-signal-to noise-ratio (SNR) regime in terms of the diversity-multiplexing tradeoff (DMT). Following this, we characterize the error probability for fixed SNR in the regime of large (but finite) number of antennas.As far as the DMT is concerned, we report a negative result: we show that both linear zero-forcing (ZF) and linear minimum mean- square error (MMSE) receivers achieve the same DMT, which is largely suboptimal even in the case where outer coding and deAcircnot coding is performed across the antennas. We also provide an apAcircnot proximate quantitative analysis of the markedly different behavior of the MMSE and ZF receivers at finite rate and nonasymptotic SNR, and show that while the ZF receiver achieves poor diversity at any finite rate, the MMSE receiver error curve slope flattens out progressively, as the coding rate increases. When SNR is fixed and the number of antennas becomes large, we show that the mutual information at the output of an MMSE or ZF linear receiver has fluctuations that converge in distribution to a Gaussian random variable, whose mean and variance can be characterized in closed form. This analysis extends to the linear reAcircnot ceiver case a well-known result previously obtained for the optimal receiver. Simulations reveal that the asymptotic analysis captures accurately the outage behavior of systems even with a moderate number of antennas.

For the quasi-static, Rayleigh-fading multiple-input multiple-output (MIMO) channel with$n_{t}$transmit and$n_{r}$receive antennas, Zheng and Tse showed that there exists a fundamental tradeoff between diversity and spatial-multiplexing gains, referred to as the diversity–multiplexing gain (D-MG) tradeoff. Subsequently, El Gamal, Caire, and Damen considered signaling across the same channel using an$L$-round automatic retransmission request (ARQ) protocol that assumes the presence of a noiseless feedback channel capable of conveying one bit of information per use of the feedback channel. They showed that given a fixed number$L$of ARQ rounds and no power control, there is a tradeoff between diversity and multiplexing gains, termed the diversity–multiplexing–delay (DMD) tradeoff. This tradeoff indicates that the diversity gain under the ARQ scheme for a particular information rate is considerably larger than that obtainable in the absence of feedback.

We study the dynamic decode and forward (DDF) protocol for a single half-duplex relay, single-antenna channel with quasi-static fading. We analyze the diversity-multiplexing tradeoff (DMT) of the DDF protocol for the case of finite block-lengths, and provide simple constructions of explicit, DMT optimal codes. Reduced complexity lattice decoding algorithms for the proposed codes are investigated.

The diversity-multiplexing tradeoff (DMT) captures in a succinct and elegant way the tradeoff between error probability and rate achieved by space-time coding in the high- SNR, high spectral efficiency regime, of MIMO quasi-static fading channels. We show that the DMT for Rayleigh and Rician fading MIMO channels with arbitrary full-rank correlation is identical to the case of i.i.d. Rayleigh fading. Previously known results for the case of full-rank Kronecker correlations are recovered as special cases of this result. Then, we argue that while the impact of correlation on capacity and error probability may be huge, the gain achieved by optimizing the input covariance matrix with respect to standard white input is essentially negligible. This observation motivates the use of standard space-time codes irrespective of the channel correlation, with possibly a rate-adaptation mechanism that allocates the rate according to the channel correlation, in order to meet the desired error probability target.

This paper addresses the issue of the design of Space-Time (ST) codes for long block-lengths that have good coding gain and minimal decoding complexity. A general technique to encode information onto cosets and points of a sequence of nested lattices is used to construct ST trellis coded modulation (TCM) schemes with good coding and shaping gains, and is termed as structured lattice space-time trellis coded modulation (S-LaST TCM). These codes permit low complexity minimum mean-squared error generalized decision feedback equalizer (MMSE-GDFE) lattice decoding. Code construction examples are presented, and simulation results reveal excellent performances.

In wireless/mobile communications, terminals adapt their rate and transmit power or, more in general, their coding and modulation scheme, depending on the time-varying channel conditions. This paper presents, in a tutorial form, the information theoretic framework underlying such “adaptive modulation” techniques. First, we review fading channel models, channel state information assumptions, and related capacity results. Then, we treat the case of input power constraint, where the optimal input distribution is Gaussian. Finally, we address the case of discrete modulations. In order to treat the latter, we make use of the recently developed method of “mercury-waterfilling,” based on the relationship between mutual information and minimum mean-square error (MMSE) estimation of the channel input from the channel output. While the traditional design of adaptive modulation schemes based on uncoded bit-error rate (BER) involves the optimization over a discrete set of signal constellations, when powerful (i.e., capacity approaching) coding schemes are used the corresponding adaptive coded modulation design becomes surprisingly simple. The regime of very powerful coding is justified by the use of modern coding schemes, such as turbo codes and low-density parity-check codes, able to perform close to channel capacity at very small BER.

The objective of this paper is to present constructions of space-time (ST) codes that are endowed with energy efficient constellations, resulting in good performance at finite values of signal-to-noise ratios (SNR). This problem is especially relevant since codes that achieve the diversity-multiplexing gain (D-MG) tradeoff, a metric of performance at high values of SNR, have been recently proposed for arbitrary number of transmit and receive antennas. We provide a technique for constructing structured lattice space-time codes derived from dense lattice packings, where constellation shaping is achieved through a nested lattice (Voronoi) construction and sphere encoding. Applying this technique to ST codes derived from cyclic division algebras, we construct codes that achieve the D-MG tradeoff over every fading channel (i.e., are approximately universal) and have excellent performance at finite SNRs. A host of simulation results using a low complexity MMSE-GDFE lattice decoder are presented

A recent result of Zheng and Tse states that over a quasi-static channel, there exists a fundamental tradeoff, referred to as the diversity–multiplexing gain (D-MG) tradeoff, between the spatial multiplexing gain and the diversity gain that can be simultaneously achieved by a space–time (ST) code. This tradeoff is precisely known in the case of independent and identically distributed (i.i.d.) Rayleigh fading, for$T geq n_t+n_r-1$where$T$is the number of time slots over which coding takes place and$n_t,n_r$are the number of transmit and receive antennas, respectively. For$T ≪ n_t+n_r-1$, only upper and lower bounds on the D-MG tradeoff are available. In this paper, we present a complete solution to the problem of explicitly constructing D-MG optimal ST codes, i.e., codes that achieve the D-MG tradeoff for any number of receive antennas. We do this by showing that for the square minimum-delay case when$T=n_t=n$, cyclic-division-algebra (CDA)-based ST codes having the nonvanishing determinant property are D-MG optimal. While constructions of such codes were previously known for restricted values of$n$, we provide here a construction for such codes that is valid for all$n$. For the rectangular,$T ≫ n_t$case, we present two general techniques for building D-MG-optimal rectangular ST codes from their square counterparts. A byproduct of our results establishes that the D-MG tradeoff for all$Tgeq n_t$is the same as that previously known to hold for$T geq n_t + n_r -1$.

In the recent landmark paper of Zheng and Tse it is shown for the quasi-static, Rayleigh-fading MIMO channel with n/sub t/ transmit and n/sub r/ receive antennas, that there exists a fundamental tradeoff between diversity gain and multiplexing gain, referred to as the diversity-multiplexing gain (D-MG) tradeoff. This paper presents the first explicit construction of space-time (ST) codes for an arbitrary number of transmit and/or receive antennas that achieve the D-MG tradeoff. It is shown here that ST codes constructed from cyclic-division-algebras (CDA) and satisfying a certain non-vanishing determinant (NVD) property, are optimal under the D-MG tradeoff for any n/sub t/,n/sub r/. Furthermore, this optimality is achieved with minimum possible value of the delay or block-length parameter T = n/sub t/. CDA-based ST codes with NVD have previously been constructed for restricted values of n/sub t/. A unified construction of D-MG optimal CDA-based ST codes with NVD is given here, for any number n/sub t/ of transmit antennas. The CDA-based constructions are also extended to provide D-MG optimal codes for all T /spl ges/ n/sub t/, again for any number nt of transmit antennas. This extension thus presents rectangular D-MG optimal space-time codes that achieve the D-MG tradeoff. Taken together, the above constructions also extend the region of T for which the D-MG tradeoff is precisely known from T /spl ges/ n/sub t/ + n/sub r/ - 1 to T /spl ges/ n/sub t/.

For the quasi-static, Rayleigh-fading MIMO channel with nttransmit and nrreceive antennas, Zheng and Tse showed that there exists a fundamental tradeoff between diversity and multiplexing gains, referred to as the diversity-multiplexing gain (D-MG) tradeoff. Explicit constructions for D-MG optimal ST codes are now available. In a subsequent paper, El Gamal, Caire and Damen considered signaling across the quasi-static ST channel using an L-round ARQ protocol that assumes the presence of a noiseless feedback channel capable of conveying one bit of information (ACK or NACK) per use of the feedback channel. They showed that given a fixed number of ARQ rounds L, there is a tradeoff between diversity and multiplexing gains under which the optimum diversity gain of the ARQ channel transmitting R = r log(SNR) bits per channel use, is that of the quasi-static channel without feedback investigated by Zheng and Tse, transmitting at (1/L)th the information rate. This tradeoff, which now is a function of the number L of ARQ rounds, is termed the diversity-multiplexing gain-delay (DMD) tradeoff. In the current paper, a sufficient condition under which a ST code will achieve the DMD tradeoff is presented for the case nrges nt. The cyclic-division-algebra-based constructions of DMG optimal ST codes by Elia et. al. are then modified to yield codes which meet this sufficient criterion and are thereby DMD-optimal. This modification requires that either nt|L or L|nt