Transmission Subspace Tracking for MIMO Systems With Low-Rate Feedback

This paper describes a feedback algorithm for tracking the dominant subspaces of continuously time-varying channels in multiantenna communication systems. The nature of the problem is quantization of subspaces. It is well known that subspaces can be mathematically modeled as points in a Grassmann manifold. We model the variations between the dominant subspaces of channels at adjacent time instants to be along geodesics in the Grassmann manifold. Instead of quantizing the subspaces themselves, we propose to quantize the geodesic trajectory connecting two subspaces. More specifically, we quantize a key entity that characterizes a geodesic arc: the velocity matrix, which resembles angular speed in a one-dimensional complex space. Two techniques are proposed for quantizing the velocity matrix of the geodesic. In the first, a 1-bit feedback is utilized to indicate the preferred sign of a random velocity matrix of the geodesic. In the other, the velocity matrix is quantized using a Gaussian vector quantization codebook. Numerical results show that the performance of the proposed 1-bit feedback algorithm is better than a previously proposed Grassmannian subspace packing scheme at low-to-medium Doppler frequencies and better than a gradient sign feedback scheme at all Doppler frequencies. In our simulations, the Gaussian vector quantization algorithm is always better than the 1-bit feedback algorithm.     

基于Grassmann流形几何深度网络的图像集识别方法

近年来,深度学习以其强大的非线性计算能力在目标检测和识别任务中取得了巨大的突破。现有的深度学习网络几乎都是以数据的欧氏结构为前提,而在计算机视觉中许多数据都具有严格的流形结构,如图像集可表示为Grassmann流形。基于数据的流形几何结构来设计深度学习网络,将微分几何理论与深度学习理论相结合,提出一种基于Grassmann流形的深度图像集识别网络。同时在模型训练过程中,使用基于矩阵链式法则的反向传播算法来更新模型,并将权值的优化过程转换为Grassmann流形上的黎曼优化问题。实验结果表明:该方法不仅在结果上识别准确率得到了提高,同时在训练和测试速度上也有一个数量级的提升。     

Pseudo-Gray coding

A pseudo-Gray code is an assignment of n-bit binary indexes to 2" points in a Euclidean space so that the Hamming distance between two points corresponds closely to the Euclidean distance. Pseudo-Gray coding provides a redundancy-free error protection scheme for vector quantization (VQ) of analog signals when the binary indexes are used as channel symbols on a discrete memoryless channel and the points are signal codevectors. Binary indexes are assigned to codevectors in a way that reduces the average quantization distortion introduced in the reproduced source vectors when a transmitted index is corrupted by channel noise. A globally optimal solution to this problem is generally intractable due to an inherently large computational complexity. A locally optimal solution, the binary switching algorithm, is introduced, based on the objective of minimizing a useful upper bound on the average system distortion. The algorithm yields a significant reduction in average distortion, and converges in reasonable running times. The sue of pseudo-Gray coding is motivated by the increasing need for low-bit-rate VQ-based encoding systems that operate on noisy channels, such as in mobile radio speech communications     

On the Information Rate of MIMO Systems With Finite Rate Channel State Feedback Using Beamforming and Power On/Off Strategy

It is well known that multiple-input multiple-output (MIMO) systems have high spectral efficiency, especially when channel state information at the transmitter (CSIT) is available. In many practical systems, it is reasonable to assume that the CSIT is obtained by a limited (i.e., finite rate) feedback and is therefore imperfect. We consider the design problem of how to use the limited feedback resource to maximize the achievable information rate. In particular, we develop a low complexity power on/off strategy with beamforming (or Grassmann precoding), and analytically characterize its performance. Given the eigenvalue decomposition of the covariance matrix of the transmitted signal, refer to the eigenvectors as beams, and to the corresponding eigenvalues as the beam's power. A power on/off strategy means that a beam is either turned on with a constant power, or turned off. We will first assume that the beams match the channel perfectly and show that the ratio between the optimal number of beams turned on and the number of antennas converges to a constant when the numbers of transmit and receive antennas approach infinity proportionally. This motivates our power on/off strategy where the number of beams turned on is independent of channel realizations but is a function of the signal-to-noise ratio (SNR). When the feedback rate is finite, beamforming cannot be perfect, and we characterize the effect of imperfect beamforming by quantization bounds on the Grassmann manifold. By combining the results for power on/off and beamforming, a good approximation to the achievable information rate is derived. Simulations show that the proposed strategy is near optimal and the performance approximation is accurate for all experimented SNRs.     

Asymptotic performance of vector quantizers with a perceptualdistortion measure

Gersho's (1979) bounds on the asymptotic performance of vector quantizers are valid for vector distortions which are powers of the Euclidean norm. Yamada, Tazaki, and Gray (1980) generalized the results to distortion measures that are increasing functions of the norm of their argument. In both cases, the distortion is uniquely determined by the vector quantization error, i.e., the Euclidean difference between the original vector and the codeword into which it is quantized. We generalize these asymptotic bounds to input-weighted quadratic distortion measures and measures that are approximately output-weighted-quadratic when the distortion is small, a class of distortion measures often claimed to be perceptually meaningful. An approximation of the asymptotic distortion based on Gersho's conjecture is derived as well. We also consider the problem of source mismatch, where the quantizer is designed using a probability density different from the true source density. The resulting asymptotic performance in terms of distortion increase in decibels is shown to be linear in the relative entropy between the true and estimated probability densities     

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.     

The geometry of weighted low-rank approximations

The low-rank approximation problem is to approximate optimally, with respect to some norm, a matrix by one of the same dimension but smaller rank. It is known that under the Frobenius norm, the best low-rank approximation can be found by using the singular value decomposition (SVD). Although this is no longer true under weighted norms in general, it is demonstrated here that the weighted low-rank approximation problem can be solved by finding the subspace that minimizes a particular cost function. A number of advantages of this parameterization over the traditional parameterization are elucidated. Finding the minimizing subspace is equivalent to minimizing a cost function on the Grassmann manifold. A general framework for constructing optimization algorithms on manifolds is presented and it is shown that existing algorithms in the literature are special cases of this framework. Within this framework, two novel algorithms (a steepest descent algorithm and a Newton-like algorithm) are derived for solving the weighted low-rank approximation problem. They are compared with other algorithms for low-rank approximation as well as with other algorithms for minimizing a cost function on a Grassmann manifold.     

Grassmannian流形上基于高分辨率动态聚焦码本的MIMO信道预测

针对MIMO时变信道下,有限反馈发射预编码存在反馈延迟的问题,提出将多数据流空间复用传输的MIMO信道的主右奇异矩阵的列空间建模于n维复欧氏空间上p维子空间的集合Grassmannian流形Gn,p上,利用Grassmannian流形的测地线对时变信道进行跟踪预测,以补偿反馈延迟对于系统性能的影响。在此基础上,从Grassmannian流形的几何特性入手,针对Grassmannian流形的切空间提出了一种高分辨率动态聚焦的多维切空间码本。计算机仿真表明,Grassmannian流形上基于高分辨率动态聚焦码本的有限反馈预测预编码的系统性能明显优于存在反馈延迟的无记忆有限反馈预编码和采用固定码本的有限反馈预测预编码。      

基于黎曼流形监督降维的矩阵CFAR增强检测

矩阵CFAR检测是从几何流形角度处理雷达目标检测问题的新技术。为进一步提升其在复杂杂波背景下的检测性能,本文提出一种黎曼流形监督降维的矩阵CFAR增强检测方法。首先,将检测问题视为目标与杂波的分类问题,分别构建黎曼流形上目标单元与杂波单元的类内和类间权重矩阵;其次,为增强目标与杂波的可分性,采用保持类内几何距离最小,类间几何距离最大的准则建立降维目标函数,并基于Grassmann流形求解降维优化问题获得映射矩阵;最后,提出一种矩阵CFAR增强检测方法,实现目标增强检测。采用蒙特卡罗方法对仿真数据和实测海杂波数据进行实验分析,结果表明,所提出的方法能够进一步提升检测性能。      

On distortion-compensated dither modulation data-hiding with repetition coding

An exhaustive analysis of the distortion-compensated dither modulation (DC-DM) data-hiding method with repetition coding is presented. Two decoding strategies, maximum likelihood lattice decoding and Euclidean distance decoding, are discussed and some simplifications presented. An exact performance analysis in terms of the bit error rate (BER) is given; such an exact analysis is currently not available in the literature. Two methods for computing the exact BER and several approximations and bounds, most of them in closed form, are provided. These approximations are employed to propose two novel improvements on the standard DC-DM method with repetition: the use of a weighted Euclidean distance, with optimizable weights, and a vector form of the distortion compensation parameter. Both account for significant performance improvements. DC-DM is compared with quantization methods in the projected domain, showing worse performance against additive noise attacks but higher robustness to cropping attacks. A performance analysis of DC-DM under coarse quantization that can be specialized to JPEG compression is also supplied. All our results are validated with numerical simulations with both synthetic data and real images.     

Joint Scene and Signal Modeling for Wavelet-Based Video Coding with Cellular Neural Network Architecture

This paper presents a joint scene and signal modeling for the design of an adaptive quantization scheme applied to the wavelet coefficients in subband video coding applications. The joint modeling includes two integrated components: the scene modeling characterized by the neighborhood binding with Gibbs random field and the signal modeling characterized by the matching of the wavelet coefficient distribution. With this joint modeling, the quantization becomes adaptive to not only wavelet coefficient signal distribution but also the prominent image scene structures. The proposed quantization scheme based on the joint scene and signal modeling is accomplished through adaptive clustering with spatial neighborhood constraints. Such spatial constraint allows the quantization to shift its bit allocation, if necessary, to those perceptually more important coefficients so that the preservation of scene structure can be achieved. This joint modeling enables the quantization to reach beyond the limit of the traditional statistical signal modeling-based approaches which often lack scene adaptivity. Furthermore, the dynamically enforced spatial constraints of the Gibbs random field are able to overcome the shortcomings of the artificial block division which are usually the major source of distortion when the video is coded by block-based approaches at low bit rate. In addition, we introduce a cellular neural network architecture for the hardware implementation of this proposed adaptive quantization. We prove that this cellular neural network does converge to the desired steady state with the suggested update scheme. The adaptive quantization scheme based on the joint scene and signal modeling has been successfully applied to videoconferencing application and very favorable results have been obtained. We believe that this joint modeling-based video coding will have an impact on many other applications because it is able to simultaneously perform signal adaptive and scene adaptive quantization.     

Rate control for robust video transmission over burst-errorwireless channels

We study the problem of rate control for transmission of video over burst-error wireless channels, i.e., channels such that errors tend to occur in clusters during fading periods. In particular we consider a scenario consisting of packet based transmission with automatic repeat request (ARQ) error control and a back channel. We start by showing how the delay constraints in real time video transmission can be translated into rate constraints at the encoder, where the applicable rate constraints at a given time depend on future channel rates. With the acknowledgments received through the back channel we have an estimate of the current channel state. This information, combined with an a priori model of the channel, allows us to statistically model the future channel rates. Thus the rate constraints at the encoder can be expressed in terms of the expected channel behavior. We can then formalize a rate distortion optimization problem, namely, that of assigning quantizers to each of the video blocks stored in the encoder buffer such that the quality of the received video is maximized. This requires that the rate constraints be included in the optimization, since violating a rate constraint is equivalent to violating a delay constraint and thus results in losing a video block. We formalize two possible approaches. The first one seeks to minimize the distortion for the expected rate constraints given the channel model and current observation. The second approach seeks to allocate bits so as to minimize the expected distortion for the given model. We use both dynamic programming and Lagrangian optimization approaches to solve these problems. Our simulation results demonstrate that both the video distortion at the decoder and packet loss rate can be significantly reduced when incorporating the channel information provided by the feedback channel and the a priori model into the rate control algorithm     

一种融合流形学习和深度学习的时变信道自动调制识别技术

自动调制识别在军事领域和民用领域都发挥了巨大作用。现有的大多数研究都是基于高斯白噪声信道,但是时变信道下的自动调制识别才更符合实际并且具有挑战性。该文针对时变信道提出了一种融合流形学习和深度学习的自动调制识别方法,第一次将格拉斯曼流形引入到信号的特征提取,通过将信号星座图建模到格拉斯曼流形上完成特征提取。分类网络由基于流形学习和深度学习的两部分组成,流形数据先经过流形学习网络进行降维,然后映射到平滑子空间,最后通过简单的卷积神经网络完成分类。实验结果表明,与传统的卷积神经网络相比该文所提出的方案具有良好的性能,同时为自动调制识别提供了新的解决思路。      

Analysis of Multiple-Antenna Systems With Finite-Rate Feedback Using High-Resolution Quantization Theory

This paper considers the development of a general framework for the analysis of transmit beamforming methods in multiple-antenna systems with finite-rate feedback. Inspired by the results of classical high-resolution quantization theory, the problem of finite-rate quantized communication system is formulated as a general fixed-rate vector quantization problem with side information available at the encoder (or the quantizer) but unavailable at the decoder. The framework of the quantization problem is sufficiently general to include quantization schemes with general non-mean-squared distortion functions and constrained source vectors. Asymptotic distortion analysis of the proposed general quantization problem is provided by extending the vector version of the Bennett's integral. Specifically, tight lower and upper bounds of the average asymptotic distortion are proposed. Sufficient conditions for the achievability of the distortion bounds are also provided and related to corresponding classical fixed-rate quantization problems. The proposed general methodology provides a powerful analytical tool to study a wide range of finite-rate feedback systems. To illustrate the utility of the framework, we consider the analysis of a finite-rate feedback multiple-input single-output (MISO) beamforming system over independent and identically distributed (i.i.d.) Rayleigh flat-fading channels. Numerical and simulation results are presented that further confirm the accuracy of the analytical results     

A low-complexity intracardiac electrogram compression algorithm

Implantable cardioverter defibrillators (ICD's) detect, diagnose and treat the potentially fatal heart arrhythmias known as bradycardia, ventricular tachycardia (VT), and ventricular fibrillation (VF) in cases where these arrhythmias are resistant to surgical and drug-based treatments by direct sensing and electrical stimulation of the heart muscle. Since the ICD is implanted, power consumption, reliability, and size are severe design constraints. This paper targets the problems associated with increasing the signal recording capabilities of an ICD. A data-compression algorithm is described which has been optimized for low power consumption and high reliability implementation. Reliance on a patient's morphology or that of a population of patients is avoided by adapting to the intracardiac electrogram (ICEG) amplitude and phase variations and by using adaptive scalar quantization. The algorithm is compared to alternative compression algorithms which are also patient independent using a subset of VT arrhythmias from a data base of 146 patients. At low distortion the algorithm is closest to the Shannon lower bound achieving an average of 3.5 b/sample at 5% root mean square distortion for a 250-Hz sample rate. At higher distortion vector quantization and Karhunen-Loeve Transform approaches are superior but at the cost of considerable additional computational complexity     

On Joint Source and Channel Coding Using Trellis Coded CPM: Analytical Bounds on the Channel Distortion

Joint source and channel coding (JSCC) using trellis coded quantization (TCQ) in conjunction with trellis coded continuous phase modulation (CPM) is studied. The channel is assumed to be the additive white gaussian noise (AWGN) channel. Analytical bounds on the channel distortion for the investigated systems with maximum-likelihood sequence detection (MLSD) are developed. The bounds are based on the transfer function technique, which was modified and generalized to include continuous-amplitude discrete-time signals. For a memoryless uniform source, the constructed bounds for the investigated systems are shown to be asymptotically tight for increasing channel signal-to-noise ratio (SNR) values. For a memoryless nonuniform source, the constructed bounds are not as tight as the one for the uniform source, however, it still can be used as an indication to how the system performs. It is concluded that the minimum Euclidean distance of the system alone is not enough to evaluate the performance of the considered systems. The number of error events having minimum Euclidean distance and the total distortion caused by those error events also affect the asymptotic performance. This work provides an analysis tool for the investigated systems. The analysis method is very general. It may be applied to any trellis based JSCC schemes.     

Fine-coarse vector quantization

A fast method for searching an unstructured vector quantization (VQ) codebook is introduced and analyzed. The method, fine-coarse vector quantization (FCVQ), operates in two stages: a `fine' structured VQ followed by a table lookup `coarse' unstructured VQ. Its rate, distortion, arithmetic complexity, and storage are investigated using analytical and experimental means. Optimality condition and an optimizing algorithm are presented. The results of experiments with both uniform scalar quantization and tree-structured VQ (TSVQ) as the first stage are reported. Comparisons are made with other fast approaches to vector quantization, especially TSVQ. It is found that when rate, distortion, arithmetic complexity, and storage are all taken into account, FCVQ outperforms TSVQ in a number of cases. In comparison to full search quantization, FCVQ has much lower arithmetic complexity, at the expense of a slight increase in distortion and a substantial increase in storage. The increase in mean-squared error (over full search) decays as a negative power of the available storage     

Modeling, analysis, and optimum design of quantized M-band filterbanks

This paper provides a rigorous modeling and analysis of quantization effects in M-band subband codecs, followed by optimal filter bank design and compensation. The codec is represented by a polyphase decomposition of the analysis/synthesis filter banks and an embedded nonlinear gain-plus-additive noise model for the pdf-optimized scalar quantizers. We construct an equivalent time-invariant but nonlinear structure operating at the slow clock rate that allows us to compute the exact expression for the mean square quantization error in the reconstructed output. This error is shown to consist of two components: a distortion component and a dominant random noise component uncorrelated with the input signal. We determine the optimal paraunitary and biorthogonal FIR filter coefficients, compensators, and integer bit allocation to minimize this MSE subject to the constraints of filter length, average bit rate, and perfect reconstruction (PR) in the absence of quantizers. The biorthogonal filter bank results in a smaller MSE but the filter coefficients are very sensitive to signal statistics and to average bit constraints. By comparison, the paraunitary structure is much more robust. We also show that the null-compensated design that eliminates the distortion component is more robust than the optimally-compensated case that minimizes the total MSE, but only at nominal conditions. Both modeling and optimal design are validated by simulation in the two-channel case     

Scalar versus vector quantization: worst case analysis

We study the potential merits of vector quantization and show that there can be an arbitrary discrepancy between the worst case rates required for scalar and vector quantization. Specifically, we describe a random variable and a distortion measure where quantization of a single instance to within a given distortion requires an arbitrarily large number of bits in the worst case, but quantization of multiple independent instances to within the same distortion requires an arbitrarily small number of bits per instance in the worst case. We relate this discrepancy to expander graphs, representation- and cover-numbers of set systems, and a problem studied by Slepian, Wolf, and Wyner (1973)