Finite-SNR Diversity–Multiplexing Tradeoff for Correlated Rayleigh and Rician MIMO Channels

A nonasymptotic framework is presented to analyze the diversity-multiplexing tradeoff of a multiple-input-multiple-output (MIMO) wireless system at finite signal-to-noise ratios (SNRs). The target data rate at each SNR is proportional to the capacity of an additive white Gaussian noise (AWGN) channel with an array gain. The proportionality constant, which can be interpreted as a finite-SNR spatial multiplexing gain, dictates the sensitivity of the rate adaptation policy to SNR. The diversity gain as a function of SNR for a fixed multiplexing gain is defined by the negative slope of the outage probability versus SNR curve on a log-log scale. The finite-SNR diversity gain provides an estimate of the additional power required to decrease the outage probability by a target amount. For general MIMO systems, lower bounds on the outage probabilities in correlated Rayleigh fading and Rician fading are used to estimate the diversity gain as a function of multiplexing gain and SNR. In addition, exact diversity gain expressions are determined for orthogonal space-time block codes (OSTBC). Spatial correlation significantly lowers the achievable diversity gain at finite SNR when compared to high-SNR asymptotic values. The presence of line-of-sight (LOS) components in Rician fading yields diversity gains higher than high-SNR asymptotic values at some SNRs and multiplexing gains while resulting in diversity gains near zero for multiplexing gains larger than unity. Furthermore, as the multiplexing gain approaches zero, the normalized limiting diversity gain, which can be interpreted in terms of the wideband slope and the high-SNR slope of spectral efficiency, exhibits slow convergence with SNR to the high-SNR asymptotic value. This finite-SNR framework for the diversity-multiplexing tradeoff is useful in MIMO system design for realistic SNRs and propagation environments     

Explicit Space–Time Codes Achieving the Diversity–Multiplexing Gain Tradeoff

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 Tgesn_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 Tt+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 Tgesn_t is the same as that previously known to hold for Tgesn_t+n _r-1     

Optimal Throughput-Diversity-Delay Tradeoff in MIMO ARQ Block-Fading Channels

In this paper, we consider an automatic-repeat-request (ARQ) retransmission protocol signaling over a block-fading multiple-input–multiple-output (MIMO) channel. Unlike previous work, we allow for multiple fading blocks within each transmission (ARQ round), and we constrain the transmitter to fixed rate codes constructed over complex signal constellations. In particular, we examine the general case of average input-power-constrained constellations with a fixed signaling alphabet of finite cardinality. This scenario is a suitable model for practical wireless communications systems employing orthogonal frequency division multiplexing (OFDM) techniques over a MIMO ARQ channel. Two cases of fading dynamics are considered, namely, short-term static fading where channel fading gains change randomly for each ARQ round, and long-term static fading where channel fading gains remain constant over all ARQ rounds pertaining to a given message. As our main result, we prove that for the block-fading MIMO ARQ channel with a fixed signaling alphabet satisfying a short-term power constraint, the optimal signal-to-noise ratio (SNR) exponent is given by a modified Singleton bound, relating all the system parameters. To demonstrate the practical significance of the theoretical analysis, we present numerical results showing that practical Singleton-bound-achieving maximum distance separable codes achieve the optimal SNR exponent.      

On Space–Time Trellis Codes Achieving Optimal Diversity Multiplexing Tradeoff

Multiple antennas can be used for increasing the amount of diversity (diversity gain) or increasing the data rate (the number of degrees of freedom or spatial multiplexing gain) in wireless communication. As quantified by Zheng and Tse, given a multiple-input-multiple-output (MIMO) channel, both gains can, in fact, be simultaneously obtained, but there is a fundamental tradeoff (called the Diversity-Multiplexing Gain (DM-G) tradeoff) between how much of each type of gain, any coding scheme can extract. Space-time codes (STCs) can be employed to make use of these advantages offered by multiple antennas. Space-Time Trellis Codes (STTCs) are known to have better bit error rate performance than Space-Time Block Codes (STBCs), but with a penalty in decoding complexity. Also, for STTCs, the frame length is assumed to be finite and hence zeros are forced towards the end of the frame (called the trailing zeros), inducing rate loss. In this correspondence, we derive an upper bound on the DM-G tradeoff of full-rate STTCs with nonvanishing determinant (NVD). Also, we show that the full-rate STTCs with NVD are optimal under the DM-G tradeoff for any number of transmit and receive antennas, neglecting the rate loss due to trailing zeros. Next, we give an explicit generalized full-rate STTC construction for any number of states of the trellis, which achieves the optimal DM-G tradeoff for any number of transmit and receive antennas, neglecting the rate loss due to trailing zeros     

Coding for the Feedback Gel'fand–Pinsker Channel and the Feedforward Wyner–Ziv Source

We consider both channel coding and source coding, with perfect past feedback/feedforward, in the presence of side information. It is first observed that feedback does not increase the capacity of the Gel'fand-Pinsker channel, nor does feedforward improve the achievable rate-distortion performance in the Wyner-Ziv problem. We then focus on the Gaussian case showing that, as in the absence of side information, feedback/feedforward allows to efficiently attain the respective performance limits. In particular, we derive schemes via variations on that of Schalkwijk and Kailath. These variants, which are as simple as their origin and require no binning, are shown to achieve, respectively, the capacity of Costa's channel, and the Wyner-Ziv rate distortion function. Finally, we consider the finite-alphabet setting and derive schemes for both the channel and the source coding problems that attain the fundamental limits, using variations on schemes of Ahlswede and Ooi and Wornell, and of Martinian and Wornell, respectively     

Zero-Error Source–Channel Coding With Side Information

This correspondence presents a novel application of the theta function defined by Lovasz. The problem of coding for transmission of a source through a channel without error when the receiver has side information about the source is analyzed. Using properties of the Lovasz theta function, it is shown that separate source and channel coding is asymptotically suboptimal in general. By contrast, in the case of vanishingly small probability of error, separate source and channel coding is known to be asymptotically optimal. For the zero-error case, it is further shown that the joint coding gain can in fact be unbounded. Since separate coding simplifies code design and use, conditions on sources and channels for the optimality of separate coding are also derived     

Optimal Diversity‐Multiplexing Tradeoff of MIMO Multi‐way Relay Channel

A MIMO multi‐way relay channel with full data exchange in which K users exchange messages with each other via the help of a single relay is considered. For the case in which each link is quasi‐static Rayleigh fading and the relay is full‐duplex, the fundamental diversity‐multiplexing tradeoff (DMT) is investigated, and we show that a compress‐and‐forward relay protocol can achieve the optimal DMT.     

The Throughput–Reliability Tradeoff in Block-Fading MIMO Channels

We build on Zheng and Tse's elegant formulation of diversity-multiplexing tradeoff (DMT) to provide a better understanding of the asymptotic interplay between transmission rate, error probability, and signal-to-noise ratio (SNR) in block-fading multiple-input multiple-output (MIMO) channels. In particular, we identify the limitation imposed by the notion of multiplexing gain and develop a new formulation called the throughput-reliability tradeoff (TRT), that avoids this limitation. The new characterization is then used to elucidate the asymptotic trends exhibited by the outage probability curves of block-fading MIMO channels     

Capacity of Differential Versus Nondifferential Unitary Space–Time Modulation for MIMO Channels

Differential unitary space-time modulation (DUSTM) and its earlier nondifferential counterpart, USTM, permit high-throughput multiple-input multiple-output (MIMO) communication entirely without the possession of channel state information by either the transmitter or the receiver. For an isotropically random unitary input we obtain the exact closed-form expression for the probability density of the DUSTM received signal, permitting the straightforward Monte Carlo evaluation of its mutual information. We compare the performance of DUSTM and USTM through both numerical computations of mutual information and through the analysis of low- and high-signal-to-noise ratio (SNR) asymptotic expressions. In our comparisons the symbol durations of the equivalent unitary space-time signals are equal to T. For DUSTM the number of transmit antennas is constrained by the scheme to be M=T/2, while USTM has no such constraint. If DUSTM and USTM utilize the same number of transmit antennas at high SNRs the normalized mutual information of the two schemes expressed in bits/s/Hz are asymptotically equal, with the differential scheme performing somewhat better. At low SNRs the normalized mutual information of DUSTM is asymptotically twice the normalized mutual information of USTM. If, instead, USTM utilizes the optimum number of transmit antennas then USTM can outperform DUSTM at sufficiently low SNRs     

Error Exponents for Recursive Decoding of Reed–Muller Codes on a Binary-Symmetric Channel

Error exponents are studied for recursive decoding of Reed-Muller (RM) codes and their subcodes used on a binary-symmetric channel. The decoding process is first decomposed into similar steps, with one new information bit derived in each step. Multiple recursive additions and multiplications of the randomly corrupted channel outputs plusmn1 are performed using a specific order of these two operations in each step. Recalculated random outputs are compared in terms of their exponential moments. As a result, tight analytical bounds are obtained for decoding error probability of the two recursive algorithms considered in the paper. For both algorithms, the derived error exponents almost coincide with simulation results. Comparison of these bounds with similar bounds for bounded distance decoding and majority decoding shows that recursive decoding can reduce the output error probability of the latter two algorithms by five or more orders of magnitude even on the short block length of 256. It is also proven that the error probability of recursive decoding can be exponentially reduced by eliminating one or a few information bits from the original RM code     

Time-Interleaved Multirate Sigma–Delta Modulators

A time-interleaved (TI) implementation of multirate sigma-delta modulators (SDMs) is proposed. In multirate SDMs, the first integrator is clocked at a rate that is lower than that of the rest of the integrators. In the proposed architecture, each integrator clocked at a high rate is replaced by two parallel integrators operating in interleaved mode and clocked at the same low rate as the first one. The new architecture has several nice features. First, every integrator operates at the same low rate, which simplifies the clock circuitry when compared to the original multirate modulator. Second, there are no delayed cross paths, which is typical of TI-SDMs. Third, no high-rate sample-and-hold at the input of a TI-SDM is required. Finally, as time interleaving is not applied to the first integrator, the proposed modulator is robust against circuit mismatches, unlike other TI architectures. The same strategy can be applied to continuous time (CT) modulators. To the authors' knowledge, this is the first TI-CT-SDM ever reported     

Perfect Space–Time Block Codes

In this paper, we introduce the notion of perfect space-time block codes (STBCs). These codes have full-rate, full-diversity, nonvanishing constant minimum determinant for increasing spectral efficiency, uniform average transmitted energy per antenna and good shaping. We present algebraic constructions of perfect STBCs for 2, 3, 4, and 6 antennas     

A New Planar InGaAs–InAlAs Avalanche Photodiode

We discuss a new simple InGaAs–InAlAs avalanche photodiode (APD) with a planar buried multiplication region. Some of the advantages compared to standard APDs are as follows: 1) The thickness of the avalanche and the charge control regions are accurately controlled by molecular beam epitaxy growth in contrast to the standard diffusion process; 2) InAlAs is the multiplication material (avalanching faster electrons) instead of InP (avalanching slower holes); 3) InAlAs avalanche gain has a lower noise figure than that for InP; 4) a guard ring is not required; 5) fabrication is as simple as that for a p-i-n detector; 6)the APD has high wafer uniformity, and high reproducibility; 7)the InAlAs breakdown voltage is lower than InP, and its variation with temperature is three times lower than that for InP; 8) excellent aging and reliability including Telcordia GR-468 qualification for die and modules; 9) high gain-bandwidth product as high as 150 GHz; and 10) high long-range (LR-2) bit-error-rate$10^-12$receiver sensitivity of$-$29.0 dBm at 10 Gb/s,$-$28.1 at 10.7Gb/s, and$-$27.1 dBm at 12.5 Gb/s.     

Fast Wavelength Switching Lasers Using Two-Section Slotted Fabry–PÉrot Structures

Fast wavelength switching of a two-section slotted Fabry–PÉrot laser structure is presented. The slot design enables operation at five discrete wavelength channels spaced by 10 nm by tuning one section of the device. These wavelengths operate with sidemode suppression ratio in excess of 35 dB, and switching times between these channels of approximately 1 ns are demonstrated.     

Diversity–Multiplexing Tradeoff and Outage Performance for Rician MIMO Channels

In this paper, we analyze the diversity–multiplexing tradeoff (DMT), originally introduced by Zheng and Tse, and outage performance for Rician multiple-input–multiple-output (MIMO) channels. The DMT characteristics of Rayleigh and Rician channels are shown to be identical. In a high signal-to-noise ratio (SNR) regime, the log–log plot of outage probability versus SNR curve for a Rician channel is a shifted version of that for the corresponding Rayleigh channel. The SNR gap between the outage curves of the Rayleigh and Rician channels is derived. The DMT and outage performance are also analyzed for Rician multiple-input–single-output (MISO)/single-input–multiple-output (SIMO) channels over a finite SNR regime. A closed-form expression for the outage probability is derived and the finite SNR DMT characteristic is analyzed. It is observed that the maximum diversity gain can be achieved at some finite SNR–the maximum gain tends to increase linearly with the Rician factor. The finite SNR diversity gain is shown to be a linear function of the finite SNR multiplexing gain. The consistency between the DMTs for finite and infinite SNRs is also shown.      

On the Stopping Redundancy of Reed–Muller Codes

The stopping redundancy of the code is an important parameter which arises from analyzing the performance of a linear code under iterative decoding on a binary erasure channel. In this paper, we will consider the stopping redundancy of Reed-Muller codes and related codes. Let R(lscr,m) be the Reed-Muller code of length 2^m and order lscr. Schwartz and Vardy gave a recursive construction of parity-check matrices for the Reed-Muller codes, and asked whether the number of rows in those parity-check matrices is the stopping redundancy of the codes. We prove that the stopping redundancy of R(m-2,m), which is also the extended Hamming code of length 2^m, is 2m-1 and thus show that the recursive bound is tight in this case. We prove that the stopping redundancy of the simplex code equals its redundancy. Several constructions of codes for which the stopping redundancy equals the redundancy are discussed. We prove an upper bound on the stopping redundancy of R(1,m). This bound is better than the known recursive bound and thus gives a negative answer to the question of Schwartz and Vardy     

Information-Lossless Space–Time Block Codes From Crossed-Product Algebras

It is known that the Alamouti code is the only complex orthogonal design (COD) which achieves capacity and that too for the case of two transmit and one receive antenna only. Damen proposed a design for two transmit antennas, which achieves capacity for any number of receive antennas, calling the resulting space-time block code (STBC) when used with a signal set an information-lossless STBC. In this paper, using crossed-product central simple algebras, we construct STBCs for arbitrary number of transmit antennas over an a priori specified signal set. Alamouti code and quasi-orthogonal designs are the simplest special cases of our constructions. We obtain a condition under which these STBCs from crossed-product algebras are information-lossless. We give some classes of crossed-product algebras, from which the STBCs obtained are information-lossless and also of full rank. We present some simulation results for two, three, and four transmit antennas to show that our STBCs perform better than some of the best known STBCs and also that these STBCs are approximately 1 dB away from the capacity of the channel with quadrature amplitude modulation (QAM) symbols as input     

5-W Reliable Operation Over 2000 h of 5-mm-Wide 650-nm AlGaInP–GaInP–AlGaAs Laser Bars With Asymmetric Cladding Layers

Reliable operation of 650-nm laser bars with GaInP quantum wells embedded in AlGaInP waveguide layers and n-AlInP and p-AlGaAs cladding layers is reported. The 5-mm-wide bars consisting of ten emitters with 100-$muhbox m$-wide stripe width showed reliable operation over 2000 h at 5 W.     

Low-Complexity Approaches to Slepian–Wolf Near-Lossless Distributed Data Compression

This paper discusses the Slepian-Wolf problem of distributed near-lossless compression of correlated sources. We introduce practical new tools for communicating at all rates in the achievable region. The technique employs a simple "source-splitting" strategy that does not require common sources of randomness at the encoders and decoders. This approach allows for pipelined encoding and decoding so that the system operates with the complexity of a single user encoder and decoder. Moreover, when this splitting approach is used in conjunction with iterative decoding methods, it produces a significant simplification of the decoding process. We demonstrate this approach for synthetically generated data. Finally, we consider the Slepian-Wolf problem when linear codes are used as syndrome-formers and consider a linear programming relaxation to maximum-likelihood (ML) sequence decoding. We note that the fractional vertices of the relaxed polytope compete with the optimal solution in a manner analogous to that observed when the "min-sum" iterative decoding algorithm is applied. This relaxation exhibits the ML-certificate property: if an integral solution is found, it is the ML solution. For symmetric binary joint distributions, we show that selecting easily constructable "expander"-style low-density parity check codes (LDPCs) as syndrome-formers admits a positive error exponent and therefore provably good performance     

Reduced Complexity Interpolation Architecture for Soft-Decision Reed–Solomon Decoding

Reed-Solomon (RS) codes are one of the most widely utilized block error-correcting codes in modern communication and computer systems. Compared to hard-decision decoding, soft-decision decoding offers considerably higher error-correcting capability. The Koetter-Vardy (KV) soft-decision decoding algorithm can achieve substantial coding gain, while maintaining a complexity polynomial with respect to the code word length. In the KV algorithm, the interpolation step dominates the decoding complexity. A reduced complexity interpolation architecture is proposed in this paper by eliminating the polynomial updating corresponding to zero discrepancy coefficients in this step. Using this architecture, an area reduction of 27% can be achieved over prior efforts for the interpolation step of a typical (255, 239) RS code, while the interpolation latency remains the same