Performance analysis of Godard-based blind channel identification

We analyze a blind channel impulse response identification scheme based on the cross correlation of blind symbol estimates with the received signal. The symbol estimates specified are those minimizing the Godard (1980) (or constant modulus) criterion, for which mean-squared symbol estimation error bounds have been derived. We derive upper bounds for the average squared parameter estimation error (ASPE) of the blind identification scheme that depend on the mean-squared error of the Wiener equalizer, the kurtoses of the desired and interfering sources, and the channel impulse response. The effects of finite data length and stochastic gradient equalizer design on ASPE are also investigated. All results are derived in a general multiuser vector-channel context     

Two-channel constrained least squares problems: solutions using power methods and connections with canonical coordinates

The problem of two-channel constrained least squares (CLS) filtering under various sets of constraints is considered, and a general set of solutions is derived. For each set of constraints, the solution is determined by a coupled (asymmetric) generalized eigenvalue problem. This eigenvalue problem establishes a connection between two-channel CLS filtering and transform methods for resolving channel measurements into canonical or half-canonical coordinates. Based on this connection, a unified framework for reduced-rank Wiener filtering is presented. Then, various representations of reduced-rank Wiener filters in canonical and half-canonical coordinates are introduced. An alternating power method is proposed to recursively compute the canonical coordinate and half-canonical coordinate mappings. A deflation process is introduced to extract the mappings associated with the dominant coordinates. The correctness of the alternating power method is demonstrated on a synthesized data set, and conclusions are drawn.     

Wiener filters in canonical coordinates for transform coding,filtering, and quantizing

Canonical correlations are used to decompose the Wiener filter into a whitening transform coder, a canonical filter, and a coloring transform decoder. The outputs of the whitening transform coder are called canonical coordinates; these are the coordinates that are reduced in rank and quantized in our finite-precision version of the Gauss-Markov theorem. Canonical correlations are, in fact, cosines of the canonical, angles between a source vector and a measurement vector. They produce new formulas for error covariance, spectral flatness, and entropy     

Decoding real block codes: activity detection Wiener estimation

New decoding procedures for real-number block codes which are constructed by imposing constraints in the discrete Fourier transform (DFT) domain are examined. The codewords are corrupted by small levels of roundoff noise and possibly occasionally by a few large excursions of random disturbances. The error-correcting procedure is separated into two parts, large activity detection followed by error value estimation, particularly the larger errors. The first part determines if large excursions are present, roughly identifying their locations, while the second part is a Wiener minimum mean-squared error estimation technique providing a stochastic correction to the corrupted components. The activity-detecting part determines locations for large increases in the Wiener estimator's gain. A computationally intensive Bayes hypothesis testing approach is shown to be very effective at locating large activity positions, but a more efficient modified Berlekamp-Massey (1969) algorithm is developed which leads to excellent mean-squared error performance. Extensive simulations demonstrate individual codeword corrective actions and compare the average mean-squared error performance between coded and unprotected data. The error level improvement ranges from three to four orders of magnitude     

Previsualized image vector quantization with optimized pre- andpostprocessors

The optimal previsualized image vector quantization method for compressing digital images to a bit rate of 0.75 bpp or below with moderately low to very low subjective distortion is presented. The encoding method incorporates a visual model as part of the distortion measure. By modeling the quantization noise as an additive signal-dependent noise process, an optimum pre- and postprocessing system, which minimizes the mean-squared error measured inside the visual model, is derived. The analysis of the system performance and a coordinate descent design algorithm are discussed. A set of experiments was conducted using the optimum system, and the results were compared to those obtained by other methods. The study shows that the images quantized by the method presented exhibit much less sawtooth, blocking, and contouring effects and higher subjective quality. Images of surprising quality have been produced by this method at a bit rate of about 0.1 bpp with a compression ratio of 80:1 relative to a normal 8 bpp original     

FIR differentiators for quantized signals

The impact of quantization noise on a signal whose rate is to be estimated using a FIR differentiator is analyzed, concentrating on the important constant-rate case in order that the filter be optimized for systems with low-frequency rates of change. Formulae for the mean-squared error of the filter, the corresponding spectral characteristics, and general formulae governing the filter coefficients are derived. The characteristics of four specific differentiators, including a representative wideband differentiator, are examined and compared. It is shown that a differentiator that is optimum in terms of its attenuation of white noise can also be considered optimum with respect to quantization noise attenuation in certain circumstances. An elegant relationship is derived between worst-case RMS error and the fractional value of the rate at which this error occurs. Minimization of this worst-case mean-squared error is shown to be achieved with a simple differentiator. However, the corresponding average error is poor, and a simple nonlinear filter that minimizes the worst-ease error, while retaining a similar average mean-squared error to that of the “optimum” differentiator, is proposed. The equivalence between FIR differentiators and the decoders used in single-loop sigma-delta modulators is also highlighted     

一种新的融合小波变换与低秩矩阵恢复的图像去噪算法

针对小波阈值图像去噪会引入量化噪声和阈值选取不当会损失图像本身有用信息的问题,提出一种新的融合小波变换与低秩矩阵恢复(Low Rank Matrtix Recovery,LRMR)的图像去噪算法.不同于传统的单一阈值的去噪算法,所提出的算法在单一阈值上结合了低秩矩阵恢复算法,这样不仅能进一步消除噪声,同时还能修复被噪声损坏的数据,而且更能适应各种不同的噪声环境.首先,选取固定阈值对图像矩阵进行小波去噪处理.其次,采用增广拉格朗日乘子算法最小化矩阵核范数.最后,将矩阵分解为低秩逼近矩阵和稀疏误差矩阵.实验结果表明,算法获得了较高的峰值信噪比,在不同噪声环境下有较高的鲁棒性.     

Linear, Worst-Case Estimators for Denoising Quantization Noise in Transform Coded Images

Transform-coded images exhibit distortions that fall outside of the assumptions of traditional denoising techniques. In this paper, we use tools from robust signal processing to construct linear, worst-case estimators for the denoising of transform compressed images. We show that while standard denoising is fundamentally determined by statistical models for images alone, the distortions induced by transform coding are heavily dependent on the structure of the transform used. Our method, thus, uses simple models for the image and for the quantization error, with the latter capturing the transform dependency. Based on these models, we derive optimal, linear estimators of the original image that are optimal in the mean-squared error sense for the worst-case cross correlation between the original and the quantization error. Our construction is transform agnostic and is applicable to transforms from block discrete cosine transforms to wavelets. Furthermore, our approach is applicable to different types of image statistics and can also serve as an optimization tool for the design of transforms/quantizers. Through the interaction of the source and quantizer models, our work provides useful insights and is instrumental in identifying and removing quantization artifacts from general signals coded with general transforms. As we decouple the modeling and processing steps, we allow for the construction of many different types of estimators depending on the desired sophistication and available computational complexity. In the low end of this spectrum, our lookup table based estimator, which can be deployed in low complexity environments, provides competitive PSNR values with some of the best results in the literature.     

运动单站无源定位中的坐标转换矩阵的分析

运动单站测角、角变化率无源定位中，观测方程和状态方程建立在不同坐标系中。本文引入一个坐标转换矩阵把状态向量在两个坐标系中的坐标联系起来，推导了坐标转换矩阵，计算出由平台姿态误差带来的坐标转换矩阵误差，并应用到跟踪滤波方程中，最后给出计算机仿真结果。     

Transform and Hybrid Transform/DPCM Coding of Images Using Pyramid Vector Quantization

The transform and hybrid transform/DPCM methods of image coding are generalized to allow pyramid vector quantization of the transform coefficients. An asymptotic mean-squared error performance expression is derived for the pyramid vector quantizer and used to determine the optimum rate assignment for encoding the various transform coefficients. Coding simulations for two images at average rates of 0.5-1 bit/pixel demonstrate a 1-3 dB improvement in signal-to-noise ratio for the vector quantization approach in the hybrid coding, with more modest improvements in signal-to-noise ratio in the transform coding. However, this improvement is quite noticeable in image quality, particularly in reducing "blockiness" in the low bit rate encoded images.     

Analog Precoder and Equalizer Designs and their Geometry for Multichannel Communication

This is a paper on modulation theory that addresses joint analog precoder and equalizer design for multichannel data transmission over the frequency-selective additive Gaussian noise (AGN) channel. The design goal is to maximize mutual information rate, minimize the mean square error, or minimize the bit error rate subject to a transmit power constraint. We assume a continuous channel model with precoder transmissions for m subchannels that lie in an n-dimensional linear subspace of L²(R). m and n are design parameters. We first design the subspace according to the channel characteristics, and then design the precoders as functions in this subspace. After the design of the optimal precoder and equalizer, we explore the geometry of these designs. We show that all of these precoder and equalizer designs are, in fact, decompositions of a virtual twochannel problem into a system of canonical coordinates, wherein variables in the canonical message channel are correlated only pairwise with corresponding variables in the canonical measurement channel. This finding clarifies the geometry of precoder and equalizer designs and illustrates that they decompose the two-channel communication problem into what might be called the Shannon channel     

Oversampled Sigma-Delta Modulation

Oversampled sigma-delta modulation has been proposed as a practical implementation for high rate analog-to-digital conversion because of its simplicity and its robustness against circuit imperfections. To date, mathematical developments of the basic properties of such systems have been based either on simplified continuous-time approximate models or on linearized discrete-time models where the quantizer is replaced by an additive white uniform noise source. In this paper, we rigorously derive several basic properties of a simple discrete-time single integrator loop sigma-delta modulator with an accumulate-and-dump demodulator. The derivation does not require any assumptions on the correlation or distribution of the quantizer error, and hence involves no linearization of the nonlinear system, but it does show that when the input is constant, the state sequence of the integrator in the encoder loop can be modeled exactly as a linear system in an appropriate space. Two basic properties are developed: 1) the behavior of the sigma-delta quantizer when driven by a constant input and its relation to uniform quantization, and 2) the rate-distortion tradeoffs between the oversampling ratio and the average mean-squared quantization error.     

Minimum mean-squared error transform coding and subband coding

Knowledge of the power spectrum of a stationary random sequence can be used for quantizing the signal efficiently and with minimum mean-squared error. A multichannel filter is used to transform the random sequence into an intermediate set of variables that are quantized using independent scalar quantizers, and then inverse-filtered, producing a quantized version of the original sequence. Equal word-length and optimal word-length quantization at high bit rates is considered. An analytical solution for the filter that minimizes the mean-squared quantization error is obtained in terms of its singular value decomposition. The performance is characterized by a set of invariants termed second-order modes, which are derived from the eigenvalue decomposition of the matrix-valued power spectrum. A more general rank-reduced model is used for decreasing distortion by introducing bias. The results are specialized to the case when the vector-valued time series is obtained from a scalar random sequence, which gives rise to a filter bank model for quantization. The asymptotic performance of such a subband coder is derived and shown to coincide with the asymptotic bound for transform coding. Quantization employing a single scalar pre- and postfilter, traditional transform coding using a square linear transformation, and subband coding in filter banks, arise as special cases of the structure analyzed here     

The Effects of Channel Errors in DPCM Systems and Comparison with PCM Systems

In a communications system the total system error is of importance. One measure of system error is the mean-square error between the input signal and the output signal. The total mean-squared error includes sampling error, quantization error, and channel error. The work reported here considers all three errors in differential pulsecode modulation (DPCM) systems and compares the results obtained with standard pulse-code modulation (PCM) systems. The total meansquared error is determined using well-defined system parameters such as quantizer levels, signal-to-noise ratio (SNR), sampling rate, etc. Using the derived formulas the design of DPCM systems is facilitated along with various tradeoff studies. The error equations are determined for both uniform and nonuniform quantizers. The error due to channel noise is obtained in closed form for both cases. DPCM and PCM are compared for three different reconstruction filters, the zero-order hold (ZOH), the linear interpolator (LI), and the ZOH followed by a low-pass filter. The optimum prediction coefficient is shown to depend on the channel noise. The optimum prediction coefficient improves the performance of DPCM systems considerably. DPCM is shown to perform better than PCM in all cases. Simulation results are presented, which verify the theoretical results.     

Transmit beamforming in multiple-antenna systems with finite rate feedback: a VQ-based approach

This paper investigates quantization methods for feeding back the channel information through a low-rate feedback channel in the context of multiple-input single-output (MISO) systems. We propose a new quantizer design criterion for capacity maximization and develop the corresponding iterative vector quantization (VQ) design algorithm. The criterion is based on maximizing the mean-squared weighted inner product (MSwIP) between the optimum and the quantized beamforming vector. The performance of systems with quantized beamforming is analyzed for the independent fading case. This requires finding the density of the squared inner product between the optimum and the quantized beamforming vector, which is obtained by considering a simple approximation of the quantization cell. The approximate density function is used to lower-bound the capacity loss due to quantization, the outage probability, and the bit error probability. The resulting expressions provide insight into the dependence of the performance of transmit beamforming MISO systems on the number of transmit antennas and feedback rate. Computer simulations support the analytical results and indicate that the lower bounds are quite tight.     

A general relationship between two quantizer design criteria (Corresp.)

Two design criteria for the quantization of signal-detection data are considered. These criteria, one of which is based on asymptotic efficiency and the other on a mean-squared error quantity, are shown to be closely related for quantizer design in a general parametric decision framework.     

PSO Based Sum Rate Optimization for MU-MIMO Linear Precoder with Imperfect CSI

It is well known that if the perfect CSI is available at the BS, achieving the maximum sum throughput is equivalent to minimizing the product of mean square error matrix determinants (PDetMSE). Due to the presence of background noise in the estimated signal, the channel estimation errors are unavoidable. Hence, in this paper, it is assumed that the imperfect CSI is available at the BS and the channel estimation error variance is known at the transmitter. It is shown that maximizing the achievable sum rate is not exactly equal to minimizing the PDetMSE if the channel estimation error variance is included in the system design. Particle Swarm Optimization algorithm is used here to solve the sum rate maximization problem under the imperfect CSI. The simulation results compare the proposed system, which considers the channel estimation error variance as an integral part of the system design, with an existing system which assumes the perfect CSI at the transmitter side.     

一种因果维纳滤波器的推导方法

维纳滤波是一种以最小均方误差为最优准则的线性滤波器。它利用了平稳随机过程的相关特性和频谱特性对混有噪声的信号进行滤波。它是目前基本的滤波方法之一,也是当今＂现代数字信号处理＂课程的经典内容之一。为了加深对这一内容的理解,利用了单边Z变换的性质,提出一种基于单边Z变换的因果维纳滤波器的推导方法,旨在对维纳滤波器的进一步研究提供了一种新的途径。     

Channel capacity and state estimation for state-dependent Gaussian channels

We formulate a problem of state information transmission over a state-dependent channel with states known at the transmitter. In particular, we solve a problem of minimizing the mean-squared channel state estimation error E/spl par/S/sup n/ - S/spl circ//sup n//spl par/ for a state-dependent additive Gaussian channel Y/sup n/ = X/sup n/ + S/sup n/ + Z/sup n/ with an independent and identically distributed (i.i.d.) Gaussian state sequence S/sup n/ = (S/sub 1/, ..., S/sub n/) known at the transmitter and an unknown i.i.d. additive Gaussian noise Z/sup n/. We show that a simple technique of direct state amplification (i.e., X/sup n/ = /spl alpha/S/sup n/), where the transmitter uses its entire power budget to amplify the channel state, yields the minimum mean-squared state estimation error. This same channel can also be used to send additional independent information at the expense of a higher channel state estimation error. We characterize the optimal tradeoff between the rate R of the independent information that can be reliably transmitted and the mean-squared state estimation error D. We show that any optimal (R, D) tradeoff pair can be achieved via a simple power-sharing technique, whereby the transmitter power is appropriately allocated between pure information transmission and state amplification.