Limited feedback unitary precoding for orthogonal space-time block codes

Orthogonal space-time block codes (OSTBCs) are a class of easily decoded space-time codes that achieve full diversity order in Rayleigh fading channels. OSTBCs exist only for certain numbers of transmit antennas and do not provide array gain like diversity techniques that exploit transmit channel information. When channel state information is available at the transmitter, though, precoding the space-time codeword can be used to support different numbers of transmit antennas and to improve array gain. Unfortunately, transmitters in many wireless systems have no knowledge about current channel conditions. This motivates limited feedback precoding methods such as channel quantization or antenna subset selection. This paper investigates a limited feedback approach that uses a codebook of precoding matrices known a priori to both the transmitter and receiver. The receiver chooses a matrix from the codebook based on current channel conditions and conveys the optimal codebook matrix to the transmitter over an error-free, zero-delay feedback channel. A criterion for choosing the optimal precoding matrix in the codebook is proposed that relates directly to minimizing the probability of symbol error of the precoded system. Low average distortion codebooks are derived based on the optimal codeword selection criterion. The resulting design is found to relate to the famous applied mathematics problem of subspace packing in the Grassmann manifold. Codebooks designed by this method are proven to provide full diversity order in Rayleigh fading channels. Monte Carlo simulations show that limited feedback precoding performs better than antenna subset selection.     

Systematic Codebook Designs for Quantized Beamforming in Correlated MIMO Channels

The full diversity gain provided by a multi-antenna channel can be achieved by transmit beamforming and receive combining. This requires the knowledge of channel state information (CSI) at the transmitter which is difficult to obtain in practice. Quantized beamforming where fixed codebooks known at both the transmitter and the receiver are used to quantize the CSI has been proposed to solve this problem. Most recent works focus attention on limited feedback codebook design for the uncorrelated Rayleigh fading channel. Such designs are sub-optimal when used in correlated channels. In this paper, we propose systematic codebook design for correlated channels when channel statistical information is known at the transmitter. This design is motivated by studying the performance of pure statistical beamforming in correlated channels and is implemented by maps that can rotate and scale spherical caps on the Grassmannian manifold. Based on this study, we show that even statistical beamforming is near-optimal if the transmitter covariance matrix is ill-conditioned and receiver covariance matrix is well-conditioned. This leads to a partitioning of the transmit and receive covariance spaces based on their conditioning with variable feedback requirements to achieve an operational performance level in the different partitions. When channel statistics are difficult to obtain at the transmitter, we propose a universal codebook design (also implemented by the rotation-scaling maps) that is robust to channel statistics. Numerical studies show that even few bits of feedback, when applied with our designs, lead to near perfect CSI performance in a variety of correlated channel conditions.     

Polarimetric adaptive detection of range-distributed targets

We address the problem of polarimetric adaptive detection of range-spread targets in Gaussian noise with unknown covariance matrix. At the design stage, we model the target echo from each polarimetric channel as a deterministic signal known up to a scaling factor (possibly varying from cell to cell), which accounts for the polarimetric scattering properties of the target. We first show the failure of the generalized likelihood ratio test (GLRT) procedure to deal with this kind of problem, and thus, we propose a fully adaptive detector based on the method of sieves. We also derive the analytical expression for the probability of false alarm and show that the newly introduced receiver can be made bounded constant false alarm rate (CFAR). Finally, we present simulation results highlighting the performance gain that can be achieved by resorting to polarization diversity in conjunction with high resolution.     

On the performance of random vector quantization limited feedback beamforming in a MISO system

In multiple antenna wireless systems, beamforming is a simple technique for guarding against the negative effects of fading. Unfortunately, beamforming requires the transmitter to have knowledge of the forward-link channel which is often not available a priori. One way of overcoming this problem is to design the beamforming vector using a limited number of feedback bits sent from the receiver to the transmitter. In limited feedback beamforming, the beamforming vector is restricted to lie in a codebook that is known to both the transmitter and receiver. Random vector quantization (RVQ) is a simple approach to codebook design that generates the vectors independently from a uniform distribution on the complex unit sphere. This correspondence presents performance analysis results for RVQ limited feedback beamforming     

Spectrally efficient noncoherent communication

This paper considers noncoherent communication over a frequency-nonselective channel in which the time-varying channel gain is unknown a priori, but is approximately constant over a coherence interval. Unless the coherence interval is large, coherent communication, which requires explicit channel estimation and tracking prior to detection, incurs training overhead which may be excessive, especially for multiple-antenna communication. In contrast, noncoherent detection may be viewed as a generalized likelihood ratio test (GLRT) which jointly estimates the channel and the data, and hence does not require separate training. The main results in this paper are as follows. (1) We develop a "signal space" criterion for signal and code design for noncoherent communication, in terms of the distances of signal points from the decision boundaries. (2) The noncoherent metric thus obtained is used to guide the design of signals for noncoherent communication that are based on amplitude/phase constellations. These are significantly more efficient than conventional differential phase-shift keying (PSK), especially at high signal-to-noise ratio (SNR). Also, known results on the high-SNR performance of multiple-symbol demodulation of differential PSK are easily inferred from the noncoherent metric. (3) The GLRT interpretation is used to obtain near-optimal low-complexity implementations of noncoherent block demodulation. In particular, this gives an implementation of multiple symbol demodulation of differential PSK, which is of linear complexity (in the block length) and whose degradation from the exact, exponential complexity, implementation can be made as small as desired     

Noncoherent multiuser detection for CDMA systems with nonlinearmodulation: a non-Bayesian approach

This paper considers the problem of multiuser detection for a system in which each user employs nonlinear modulation, with an emphasis on noncoherent detection techniques which do not require knowledge of the users' channel parameters at the receiver. Our goals are to gain fundamental insight into the capabilities of multiuser detection in such a setting, and to provide practical algorithms that perform better than conventional matched-filter reception. We begin by providing fundamental performance benchmarks by considering coherent maximum-likelihood (ML) detection, which requires knowledge of the users' channel parameters, as well as noncoherent detection, formulated in a non-Bayesian generalized likelihood ratio test (GLRT) framework. The asymptotic performance of each detector, as the noise level vanishes, is characterized, yielding simple geometric criteria for near-far resistance. In general, both the ML and GLRT detectors have complexity which is exponential in the number of users. We, therefore, propose the more practical sequential decision projection (SDP) detector which has complexity which is quadratic in the number of users. It is shown that the SDP detector has nonzero asymptotic efficiency if the users' powers are suitably disparate     

Noncoherent space-time coding: An algebraic perspective

The design of space-time signals for noncoherent block-fading channels where the channel state information is not known a priori at the transmitter and the receiver is considered. In particular, a new algebraic formulation for the diversity advantage design criterion is developed. The new criterion encompasses, as a special case, the well-known diversity advantage for unitary space-time signals and, more importantly, applies to arbitrary signaling schemes and arbitrary channel distributions. This criterion is used to establish the optimal diversity-versus-rate tradeoff for training based schemes in block-fading channels. Our results are then specialized to the class of affine space-time signals which allows for a low complexity decoder. Within this class, space-time constellations based on the threaded algebraic space-time (TAST) architecture are considered. These constellations achieve the optimal diversity-versus-rate tradeoff over noncoherent block-fading channels and outperform previously proposed codes in the considered scenarios as demonstrated by the numerical results. Using the analytical and numerical results developed in this paper, nonunitary space-time codes are argued to offer certain advantages in block-fading channels where the appropriate use of coherent space-time codes is shown to offer a very efficient solution to the noncoherent space-time communication paradigm.     

Limited feedback unitary precoding for spatial multiplexing systems

Multiple-input multiple-output (MIMO) wireless systems use antenna arrays at both the transmitter and receiver to provide communication links with substantial diversity and capacity. Spatial multiplexing is a common space-time modulation technique for MIMO communication systems where independent information streams are sent over different transmit antennas. Unfortunately, spatial multiplexing is sensitive to ill-conditioning of the channel matrix. Precoding can improve the resilience of spatial multiplexing at the expense of full channel knowledge at the transmitter-which is often not realistic. This correspondence proposes a quantized precoding system where the optimal precoder is chosen from a finite codebook known to both receiver and transmitter. The index of the optimal precoder is conveyed from the receiver to the transmitter over a low-delay feedback link. Criteria are presented for selecting the optimal precoding matrix based on the error rate and mutual information for different receiver designs. Codebook design criteria are proposed for each selection criterion by minimizing a bound on the average distortion assuming a Rayleigh-fading matrix channel. The design criteria are shown to be equivalent to packing subspaces in the Grassmann manifold using the projection two-norm and Fubini-Study distances. Simulation results show that the proposed system outperforms antenna subset selection and performs close to optimal unitary precoding with a minimal amount of feedback.     

相对时延对ST-FSK系统性能的影响

空时频移键控(ST-FSK)结合了空时分组码和频移键控的优点,在瑞利衰落信道条件下无需信道信息,可采用非相干的最大似然(ML)检测器。该文推导了在接收端已知相对时延条件下的非相干ML判决度量,讨论了在接收端未知相对时延条件下信噪比与二进制、四进制ST-FSK系统误比特率性能的关系。理论和仿真结果都表明:(1)低信噪比弱化了相对时延对ST-FSK系统可靠性的影响,但在高信噪比条件下相对时延会严重制约系统性能;(2)相对时延对系统性能的影响与系统进制数的关系不大。     

Signature optimization for CDMA with limited feedback

We study the performance of joint signature-receiver optimization for direct-sequence code-division multiple access (DS-CDMA) with limited feedback. The receiver for a particular user selects the signature from a signature codebook, and relays the corresponding B index bits to the transmitter over a noiseless channel. We study the performance of a random vector quantization (RVQ) scheme in which the codebook entries are independent and isotropically distributed. Assuming the interfering signatures are independent, and have independent and identically distributed (i.i.d.) elements, we evaluate the received signal-to-interference plus noise ratio (SINR) in the large system limit as the number of users, processing gain, and feedback bits B all tend to infinity with fixed ratios. This SINR is evaluated for both the matched filter and linear minimum mean-squared error (MMSE) receivers. Furthermore, we show that this large system SINR is the maximum that can be achieved over any sequence of codebooks. Numerical results show that with the MMSE receiver, one feedback bit per signature coefficient achieves close to single-user performance. We also consider a less complex and suboptimal reduced-rank signature optimization scheme in which the user's signature is constrained to lie in a lower dimensional subspace. The optimal subspace coefficients are scalar-quantized and relayed to the transmitter. The large system performance of the quantized reduced-rank scheme can be approximated, and numerical results show that it performs in the vicinity of the RVQ bound. Finally, we extend our analysis to the scenario in which a subset of users optimize their signatures in the presence of random interference.     

Optimal transmitter eigen-beamforming and space-time block coding based on channel mean feedback

Optimal transmitter designs obeying the water-filling principle are well-documented; they are widely applied when the propagation channel is deterministically known and regularly updated at the transmitter. Because channel state information is impossible to be known perfectly at the transmitter in practical wireless systems, we design, in this paper, an optimal multiantenna transmitter based on the knowledge of mean values of the underlying channels. Our optimal transmitter design turns out to be an eigen-beamformer with multiple beams pointing to orthogonal directions along the eigenvectors of the correlation matrix of the estimated channel at the transmitter and with proper power loading across beams. The optimality pertains to minimizing an upper bound on the symbol error rate, which leads to better performance than maximizing the expected signal-to-noise ratio (SNR) at the receiver. Coupled with orthogonal space-time block codes, two-directional eigen-beamforming emerges as a more attractive choice than conventional one-directional beamforming with uniformly improved performance, without rate reduction, and without essential increase in complexity. With multiple receive antennas and reasonably good feedback quality, the two-directional eigen-beamformer is also capable of achieving the best possible performance in a large range of transmit-power-to-noise ratios, without a rate penalty.     

Chernoff bounds on pairwise error probabilities of space-time codes

We derive Chernoff bounds on pairwise error probabilities of coherent and noncoherent space-time signaling schemes. First, general Chernoff bound expressions are derived for a correlated Ricean fading channel and correlated additive Gaussian noise. Then, we specialize the obtained results to the cases of space-time-separable noise, white noise, and uncorrelated fading. We derive approximate Chernoff bounds for high and low signal-to-noise ratios (SNRs) and propose optimal signaling schemes. We also compute the optimal number of transmitter antennas for noncoherent signaling with unitary mutually orthogonal space-time codes.     

Adaptive Iterative Detectors for Phase-Uncertain Channels via Variational Bounding

The problem of iterative detection/decoding of data symbols transmitted over an additive white Gaussian noise (AWGN) channel in the presence of phase uncertainty is addressed in this paper. By modelling the phase uncertainty either as an unknown deterministic variable/process or random variable/ process with a known a priori probability density function, a number of non-Bayesian and Bayesian detection algorithms with various amount of suboptimality have been proposed in the literature to solve the problem. In this paper, a new set of suboptimal iterative detection algorithms is obtained by utilizing the variational bounding technique. Especially, applying the generic variational Bayesian (VB) framework, efficient iterative joint estimation and detection/decoding schemes are derived for the constant phase model as well as for the dynamic phase model. In addition, the relation of the VB-based approach to the optimal noncoherent receiver as well as to the classical approach via the expectation-maximization (EM) algorithm is provided. Performance of the proposed detectors in the presence of a strong dynamic phase noise is compared to the performance of the existing detectors. Furthermore, an incremental scheduling of the VB (or EM) algorithm is shown to reduce the overall complexity of the receiver.     

Diversity reception of nonorthogonal multipulse signals inmultiuser Nakagami fading channels

We consider the problem of multiantenna reception of nonorthogonal multipulse signals in a multiuser system operating over a flat-flat Nakagami fading channel, wherein each user sends an M-ary information symbol by transmitting one out of M available waveforms. The receiver first-stage is a decorrelative filter, which gets rid of the MAI, and whose output, is then properly processed in order to decode the information symbols from the user of interest. In particular, we derive both the optimum noncoherent and the GLRT detectors, and show that the latter receiver not only is simpler to implement, but also achieves a performance level very similar to that of the former receiver     

Antenna selection for multiple-antenna transmission systems: performance analysis and code construction

This correspondence studies antenna selection for wireless communications systems that employ multiple transmit and receive antennas. We assume that (1) the channel is characterized by quasi-static Rayleigh flat fading, and the subchannels fade independently, (2) the channel state information (CSI) is exactly known at the receiver, (3) the selection is available only at the receiver, and it is based on the instantaneous signal-to-noise ratio (SNR) at each receive antenna, and (4) space-time codes are used at the transmitter. We analyze the performance of such systems by deriving explicit upper bounds on the pairwise error probability (PEP). This performance analysis shows that (1) by selecting the set of antennas that observe the largest instantaneous SNR, one can achieve the same diversity gain as the one obtained by using all the receive antennas, provided that the underlying space-time code has full spatial diversity, and (2) in the case of rank-deficient space-time codes, the diversity gain may be dramatically reduced when antenna selection is used. However, we emphasize that in both cases the coding gain is reduced with antenna selection compared to the full complexity system. Based on the upper bounds derived, we describe code design principles suitable for antenna selection. Specifically, for systems with two transmit antennas, we design space-time codes that perform better than the known ones when antenna selection is employed. Finally, we present numerical examples and simulation results that validate our analysis and code design principles.     

Cyclic Distributed Space–Time Codes for Wireless Relay Networks With No Channel Information

In this paper, we present a coding strategy for half duplex wireless relay networks, where we assume no channel knowledge at any of the transmitter, receiver, or relays. The coding scheme uses distributed space-time coding, that is, the relay nodes cooperate to encode the transmitted signal so that the receiver senses a space-time codeword. It is inspired by noncoherent differential techniques. The proposed strategy is available for any number of relays nodes. It is analyzed, and shown to yield a diversity linear in the number of relays. We also study the resistance of the scheme to relay node failures, and show that a network with R relay nodes and d of them down behaves, as far as diversity is concerned, as a network with R-d nodes. Finally, our construction can be easily generalized to the case where the transmitter and receiver nodes have several antennas.     

Universal decoding for frequency-selective fading channels

We address the problem of universal decoding in unknown frequency-selective fading channels, using an orthogonal frequency-division multiplexing (OFDM) signaling scheme. A block-fading model is adopted, where the bands' fading coefficients are unknown yet assumed constant throughout the block. Given a codebook, we seek a decoder independent of the channel parameters whose worst case performance relative to a maximum-likelihood (ML) decoder that knows the channel is optimal. Specifically, the decoder is selected from a family of quadratic decoders, and the optimal decoder is referred to as a quadratic minimax (QMM) decoder for that family. As the QMM decoder is generally difficult to find, a suboptimal QMM decoder is derived instead. Despite its suboptimality, the proposed decoder is shown to outperform the generalized likelihood ratio test (GLRT), which is commonly used when the channel is unknown, while maintaining a comparable complexity. The QMM decoder is also derived for the practical case where the fading coefficients are not entirely independent but rather satisfy some general constraints. Simulations verify the superiority of the proposed QMM decoder over the GLRT and over the practically used training sequence approach.     

A design framework for limited feedback MIMO systems with zero-forcing DFE

We consider the design of multiple-input multiple-output communication systems with a linear precoder at the transmitter, zero-forcing decision feedback equalization (ZFDFE) at the receiver, and a low-rate feedback channel that enables communication from the receiver to the transmitter. The channel state information (CSI) available at the receiver is assumed to be perfect, and based on this information the receiver selects a suitable precoder from a codebook and feeds back the index of this precoder to the transmitter. Our approach to the design of the components of this limited feedback scheme is based on the development, herein, of a unified framework for the joint design of the precoder and the ZF-DFE under the assumption that perfect CSI is available at both the transmitter and the receiver. The framework is general and embraces a wide range of design criteria. This framework enables us to characterize the statistical distribution of the optimal precoder in a standard Rayleigh fading environment. Using this distribution, we show that codebooks constructed from Grassmann packings minimize an upper bound on an average distortion measure, and hence are natural candidates for the codebook in limited feedback systems. Our simulation studies show that the proposed limited feedback scheme can provide significantly better performance at a lower feedback rate than existing schemes in which the detection order is fed back to the transmitter.     

Leveraging coherent space-time codes for noncoherent communication via training

Training codes are introduced for the multiple-antenna, noncoherent, multiple block-Rayleigh-fading channel in which the fading coefficients, which are constant over a fixed number of dimensions (coherence interval) for each block and then change independently to a new realization, are known neither at the transmitter nor at the receiver. Each codeword of a training code consists of a part known to the receiver-used to form a minimum mean-squared error (MMSE) estimate of the channel-and a part that contains codeword(s) of a space-time block or trellis code designed for the coherent channel (in which the receiver has perfect knowledge of the channel). The channel estimate is used as if it were error-free for decoding the information-bearing part of the training codeword. Training codes are hence easily designed to have high rate and low decoding complexity by choosing the underlying coherent code to have high rate and to be efficiently decodable. Conditions for which the estimator-detector (E-D) receiver is equivalent to the optimal noncoherent receiver are established. A key performance analysis result of this paper is that the training codes when decoded with the E-D receiver achieve a diversity order of the error probability that is equal to the diversity order of the underlying coherent code. In some cases, the performance of training codes can be measured relative to coherent reception via "training efficiency," which is then optimized over the energy allocation between the training and data phases. In the limit of increasing block lengths, training codes always achieve the performance of coherent reception. The examples of training codes provided in this work have polynomial complexity in rate but an error rate comparable to the best performing unitary designs available, even though the latter require exponential decoding complexity.     

Space-Time Codebook Design for Spread Systems

The problem of designing space-time constellations for communication in spread multiple-antenna wireless systems over Rayleigh fading channel is addressed. Using the asymptotic union bound on error probability as design criterion, codes based on general complex-valued symbols are obtained with a gradient search optimization technique. New space-time constellations are constructed and their performance is assessed. Simulation results show that the proposed technique achieves significant performance gains over existing schemes. Since the designed codebooks do not seem to possess any structure that could reduce the memory requirements for storing the corresponding codebook, the problem of designing structured space-time constellations is also addressed. To overcome this problem, a discrete Fourier transform-based approach is adopted to design structured codebooks and their performance is evaluated. The new structured codebooks also demonstrate significant improvement over state-of-art structured schemes.