Weighted Mean Square Error Research Articles

Linear Transceiver Design for Downlink Multiuser MIMO Systems: Downlink-Interference Duality Approach

This paper considers linear transceiver design for downlink multiuser multiple-input multiple-output (MIMO) systems. We examine different transceiver design problems. We focus on two groups of design problems. The first group is the weighted sum mean-square-error (WSMSE) (i.e., symbol-wise or user-wise WSMSE) minimization problems and the second group is the minimization of the maximum weighted mean-square-error (WMSE) (symbol-wise or user-wise WMSE) problems. The problems are examined for the practically relevant scenario where the power constraint is a combination of per base station (BS) antenna and per symbol (user), and the noise vector of each mobile station is a zero-mean circularly symmetric complex Gaussian random variable with arbitrary covariance matrix. For each of these problems, we propose a novel downlink-interference duality based iterative solution. Each of these problems is solved as follows. First, we establish a new mean-square-error (MSE) downlink-interference duality. Second, we formulate the power allocation part of the problem in the downlink channel as a Geometric Program (GP). Third, using the duality result and the solution of GP, we utilize alternating optimization technique to solve the original downlink problem. For the first group of problems, we have established symbol-wise and user-wise WSMSE downlink-interference duality. These duality are established by formulating the noise covariance matrices of the interference channels as fixed point functions. On the other hand, for the second group of problems, we have established symbol-wise and user-wise MSE downlink-interference duality. These duality are established by formulating the noise covariance matrices of the interference channels as marginally stable (convergent) discrete-time-switched systems. The proposed duality based iterative solutions can be extended straightforwardly to solve many other linear transceiver design problems. We also show that our MSE downlink-interference duality unify all existing MSE duality. In our simulation results, we have observed that the proposed duality based iterative algorithms utilize less total BS power than that of the existing algorithms.

두 셀 다중 안테나 하향링크 간섭 채널에서 비강인한/강인한 정칙화된 제로포싱 간섭 정렬 방법

In this paper, we propose transceiver design strategies for the two-cell multiple-input multiple-output (MIMO) interfering broadcast channel where inter-cell interference (ICI) exists in addition to inter-user interference (IUI). We first formulate the generalized zero-forcing interference alignment (ZF-IA) method based on the alignment of IUI and ICI in multi-dimensional subspace. We then devise a minimum weighted-mean-square-error (WMSE) method based on regularizing the precoders and decoders of the generalized ZF-IA scheme. In contrast to the existing weighted-sum-rate-maximizing transceiver, our method does not require an iterative calculation of the optimal weights. Because of this, the proposed scheme, while not designed specially to maximize the sum-rate, is computationally efficient and achieves a faster convergence compared to the known weighed-sum-rate maximizing scheme. Through analysis and simulation, we show the effectiveness of the proposed regularized ZF-IA scheme.

Joint linear processing design for distributed two-way Amplify-and-Forward MIMO relaying networks

Linear transceiver designs are investigated for distributed two-way relaying networks, which aim at minimising the Weighted Mean Square Error (WMSE) of data detections. The forwarding matrices at relays and equalization matrices at destinations are jointly optimised. To overcome the challenging limitations introduced by individual power constraints, a Semi-Definite Relaxation (SDR) called element-wise relaxation is proposed, which can transform the original optimization problem into a standard convex optimization problem. In this research, two-way relaying is understood from a pure signal processing perspective which can potentially simplify the theoretical analysis. Finally, simulation results are used for assessing the performance advantage of the proposed algorithm.

Regularized Zero-Forcing Interference Alignment for the Two-Cell MIMO Interfering Broadcast Channel

We present linear transmit-receive filter design strategies for the two-cell multiple-input multiple-output (MIMO) interfering broadcast channel (IBC) where inter-cell interference (ICI) exists in addition to inter-user interference (IUI) within the cell. We first formulate a generalized zero-forcing interference alignment (ZF-IA) method capable of handling multiple data streams for each user. We proceed to further adjust the transmit-receive filters based on the minimum weighted-mean-square-error (WMSE) principle using the weights obtained from the initial generalized ZF-IA design, in an effort to to enhance the sum-rate performance. In contrast to the existing weighted-sum-rate-maximizing scheme, our method does not require an iterative calculation of the weights. Because of this, the proposed scheme, while not designed specifically to maximize the sum-rate, is computationally efficient and delivers favorable tradeoffs between performance and information exchange overhead.

Robust precoder-decoder design for physical layer network coding-based MIMO two-way relaying system

In this paper, we investigate a robust joint precoder-decoder design scheme for a multiple-input multiple-output physical layer network coding (PNC)-based two-way relay system. An orthogonal training sequence is used to estimate the channels. The estimate is imperfect, and a robust design is proposed to find precoders at the source nodes and decoder at the relay node to facilitate PNC operations during multiple-access stage. Both channel estimation error and antenna correlations are used to formulate the optimization problem to minimize the weighted mean square error (WMSE) under a total power constraint. The problem becomes non-convex, and we propose an algorithm to solve it optimally. During the broadcast stage, an algorithm is proposed to find a precoder at the relay node and decoders at source nodes. The system performance is evaluated with estimation error and antenna correlation parameters. The effect of weighting parameters, relay location, and number of antennas at nodes are also considered in the numerical analysis. Numerical results confirm that our joint precoder-decoder algorithms provide the optimal solution to the minimization of WMSE with the total available power.

다중반응표면 최적화를 위한 가중평균제곱오차

본 다중반응표면 최적화는 다수의 반응변수(품질특성치)를 동시에 고려하여, 입력변수의 최적 조건을 찾는 것을 목적으로 한다. 지금까지 다중반응표면 최적화를 위하여 다양한 방법이 제안되어 왔는데, 그 중 평균제곱오차 최소화법은 다수의 반응변수의 평균과 표준편차를 동시에 고려하여 최적화하는 방법이다. 이 방법은 기본적으로 평균과 표준편차가 동일한 가중치를 가지고 있다는 것을 전제로 하고 있다. 그러나 문제의 상황에 따라 평균과 표준편차에 서로 다른 가중치를 부여해야 하는 경우도 있다. 이에 본 논문에서는 기존의 평균제곱오차를 확대하여 평균과 표준편차에 서로 다른 가중치도 부여할 수 있도록 가중평균제곱오차 최소화법을 제안하고자 한다. Multiple response surface optimization (MRSO) aims at finding a setting of input variables which simultaneously optimizes multiple responses. The minimization of mean squared error (MSE), which consists of the squared bias and variance terms, is an effective way to consider the location and dispersion effects of the responses in MRSO. This approach basically assumes that both the terms have an equal weight. However, they need to be weighted differently depending on a problem situation, for example, in case that they are not of the same importance. This paper proposes to use the weighted MSE (WMSE) criterion instead of the MSE criterion in MRSO to consider an unequal weight situation.

An Efficient Neighbourhood Pixel Filtering Algorithm for Wavelet-Based Image Denoising

Image denoising using the wavelet transform has been attracting much attention. Image corrupted by a noise is a classical problem in the field of signal or image processing. À Trous algorithm is introduced to overcome the problem of translation variant mechanism, existing on using discrete wavelet transform (DWT). This algorithm up samples low-pass filter by inserting zeros between the filter coefficients at each level and accordingly the low-pass and high-pass filter coefficients are modified. The efficiency of wavelet images by using this algorithm is low because the detail preservations of images at different scales are not uniform; also random noise rapidly attenuates with increasing scales. Due to this the contrast of the resulting image is weaker. In order to improve the clarity of the image an algorithm called a neighbourhood pixel filtering algorithm (NPFA) is added along with the existing à trous algorithm. In the proposed algorithm, find neighbourhood pixel difference (NPD ) by subtracting the neighbourhood pixel values from its current noisy pixel value. Also, calculate weight of each pixel which depends on this NPD. A filtered value is assigned for each current pixel in order to approximate the original pixel value of that pixel. This filtered value is generated by minimizing NPD and weighted mean square error (WMSE) using method of least square. A reduction in noise pixel is observed on replacing the optimal weight namely NPFA filter solution with the noisy value of the current pixel. Due to this NPFA filter gain the effect of both high-pass and low-pass filter. This filter behaves like a low-pass filter in smooth region by decreasing noise variance effectively and giving similar weights to all its neighbourhood pixels. This in turn cuts off only high frequency noise signal instead of all noisy signals. The resultant image thus obtained is observed to have much less blurring effect compared to the other wavelet method.

Interactive weighting of bias and variance in dual response surface optimization

In dual response surface optimization, minimizing weighted mean squared error (WMSE) is a simple yet effective way of obtaining a satisfactory solution. To minimize WMSE, the weights of the squared bias and variance should be determined in advance. Determining the weights in accordance with the decision maker (DM)’s preference structure regarding the tradeoffs between the two responses is critical and difficult. In this study, we develop an interactive weighting method where the DM provides his/her preference information in the form of pairwise comparisons. Our method estimates the weights based on the pairwise comparisons in an interactive manner. The method obtains a satisfactory solution through several pairwise comparisons in the case examples that we tested.

Bayesian analysis for weighted mean‐squared error in dual response surface optimization

AbstractDual response surface optimization considers the mean and the variation simultaneously. The minimization of mean‐squared error (MSE) is an effective approach in dual response surface optimization. Weighted MSE (WMSE) is formed by imposing the relative weights, (λ, 1−λ), on the squared bias and variance components of MSE. To date, a few methods have been proposed for determining λ. The resulting λ from these methods is either a single value or an interval. This paper aims at developing a systematic method to choose a λ value when an interval of λ is given. Specifically, this paper proposes a Bayesian approach to construct a probability distribution of λ. Once the distribution of λ is constructed, the expected value of λ can be used to form WMSE. Copyright © 2009 John Wiley & Sons, Ltd.

Incorporating spectral dynamics into LSF vector quantization with error shaping

This paper proposes an innovative technique to incorporate the information of spectral dynamics into the line spectrum frequency (LSF) vector quantization (VQ). This technique is based on a modified weighted log-spectral distortion (MWLSD) measure. It can be used to shape the spectral quantization distortion distribution into any frequencydependent shaping curve, and simultaneously reduce the spectral dynamics distortion between quantized spectra and unquantized spectra. To reduce the high computational complexity of the MWLSD measure during the search procedure in the LSF VQ, a quadratically weighted distortion (QWD) measure used to approximate the MWLSD measure is derived based on the theoretical analysis of the MWLSD measure. A simplified quadratically weighted distortion (SQWD) measure is also proposed to further reduce the computational complexity of the QWD measure for practical applications, whose computational complexity is almost equal to that of weighted mean square error (WMSE) measure. Three experiments are performed to verify the proposed technique in the LSF VQ.

On the Size Corrected Tests in Improved Estimation

In this paper we propose shrinkage preliminary test estimator (SPTE) of the coefficient vector in the multiple linear regression model based on the size corrected Wald ( W), likelihood ratio ( LR) and Lagrangian multiplier ( LM) tests. The correction factors used are those obt,ained from degrees of freedom corrections to the estimate of the error variance and those obtained from the second­order Edgeworth approximations to the exact distributions of the test statistics. The bias and weighted mean squared error (WMSE) fun ctions of the estimators are derived. With respect to WMSE, the relative efficiencies of the SPTEs relative to the maximum likelihood estimator are calculated. This study shows that the amount of conflict can be substantial when the three t ests are based on the same asymptotic chi­square critical value. The conflict among the SPTEs is due to the asymptotic tests not having the correct significance level. The Edgeworth size corrected W, LR and LM tests reduce the conflict remarkably.

Incorporating error shaping technique into LSF vector quantization

This paper presents an error shaping technique for line spectrum frequency (LSF) vector quantization. The error shaping technique based on the weighted logarithm spectral distortion (WLSD) measure can be used for shaping the spectral distortion distribution of quantization error into any different curve depending on what kind of weighting function is used. However, the high computational complexity of the WLSD measure deters this error shaping technique from practical use. To solve this problem, we approximate the WLSD measure by the quadratically weighted measure or the weighted mean squared error (WMSE) measure and propose an optimal error shaping technique of LSF vector quantization. In this proposed error shaping technique, the optimal WMSE weights (i.e., the optimal weights of LSF parameters) are determined based on the theoretical analysis of the WLSD measure. Three experiments are performed to check the performance of the proposed error shaping technique. One experiment is set up by incorporating human perception into the LSF quantization and another is set up by emphasizing the human-sensitivity frequency band in lower frequency bandwidth 0-3 kHz. In the third experiment, we apply the proposed error shaping technique to the LSF quantization of a CELP coder to test how it affects the overall speech quality in an actual speech coding algorithm.

Conflict Between Pretest Estimators Induced by Three Large Sample Tests Under a Regression Model with Student t-error

We consider the preliminary test approach to the estimation of regression parameters, in a multiple-regression model with non-normal error. Here, we consider the multivariate t-distribution as the error distribution. The preliminary test estimators (PTEs) based on the Wald, likelihood ratio (LR) and Lagrangian multiplier (LM) tests are given. Their bias, mean-square error matrix M and weighted mean-square error (WMSE) functions are derived and compared. We show that there is conflict between the three estimators with respect to their bias, M and WMSE because of certain inequalities that exist inherently among the test statistics and because of approximating them by the same asymptotic distribution. In the neighbourhood of the null hypothesis the PTE based on the LM test has the smallest WMSE followed by the LR-based estimator, and the estimator based on the Wald test is the worst. However, the PTE based on the Wald test performs the best followed by the LR-based estimator when the parameter moves away from the subspace of the restriction and the LM-based estimator is the worst. It has been shown that in the choice of the smallest level of significance to yield the best estimator in the sense of the highest minimum guaranteed efficiency the Wald-based PTE dominates the other two estimators. Further, the optimum choice of the level of significance becomes the traditional choice by using the Wald test.

Objective picture quality scale (PQS) for image coding

A new methodology for the determination of an objective metric for still image coding is reported. This methodology is applied to obtain a picture quality scale (PQS) for the coding of achromatic images over the full range of image quality defined by the subjective mean opinion score (MOS). This PQS takes into account the properties of visual perception for both global features and localized disturbances. The PQS closely approximates the MOS, with a correlation coefficient of more than 0.92, as compared to 0.57 obtained using the conventional weighted mean-square error (WMSE). Extensions and applications of the methodology and of the resulting metric are discussed.

Theoretical analysis of the high-rate vector quantization of LPC parameters

The paper presents a theoretical analysis of high-rate vector quantization (VQ) systems that use suboptimal, mismatched distortion measures, and describes the application of the analysis to the problem of quantizing the linear predictive coding (LPC) parameters in speech coding systems. First, it is shown that in many high-rate VQ systems the quantization distortion approaches a simple quadratically weighted error measure, where the weighting matrix is a "sensitivity matrix" that is an extension of the concept of the scalar sensitivity. The approximate performance of VQ systems that train and quantize using mismatched distortion measures is derived, and is used to construct better distortion measures. Second, these results are used to determine the performance of LPC vector quantizers, as measured by the log spectral distortion (LSD) measure, which have been trained using other error measures, such as mean-squared (MSE) or weighted mean-squared error (WMSE) measures of LEPC parameters, reflection coefficients and transforms thereof, and line spectral pair (LSP) frequencies. Computationally efficient algorithms for computing the sensitivity matrices of these parameters are described. In particular, it is shown that the sensitivity matrix for the LSP frequencies is diagonal, implying that a WMSE measured LSP frequencies converges to the LSD measure in high-rate VQ systems. Experimental results to support the theoretical performance estimates are provided.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A unified and efficient least-squares design of linear-phase nonrecursive filters

Abstract A method is described which can be used to design a wide class of nonrecursive linear-phase filters including those with arbitrary magnitude specifications and with time-domain constraints. The approach is based on formulating the weighted mean-square error between the amplitude responses of the practical and ideal filters as a quadratic function. The filter coefficients are obtained by solving a set of linear equations. This method leads to a lower weighted mean-square error and is computationally more efficient than both the eigenfilter method and the method based on the Remez exchange algorithm. However, the main advantage of our method lies in its computational efficiency. Design examples of many different types of filters are provided.