Linear Minimum Mean Square Error Estimator Research Articles

Consistent Reduced-Rank LMMSE Estimation With a Limited Number of Samples per Observation Dimension

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> An improved construction of the optimal reduced- rank linear minimum mean-square error (MMSE) estimator of a signal waveform of interest is derived that is consistent under a limited number of samples per filtering degree-of-freedom. The new filter design generalizes traditional filter realizations based on directly replacing the theoretical covariance matrix by its sample estimate, and being consistent when all dimensions in the model but the number of samples remain bounded. Our solution not only generalizes the conventional estimator, but also turns out to appropriately characterize model mismatch constraints due to finite sample-size limitations of fundamental importance in practical situations. The proposed implementation results from a generalized consistent estimation of the set of MMSE filter subspace coefficients on the reduced-dimensional subspace. Results are based on the theory of the spectral analysis of large-dimensional random matrices. In particular, we build on the analytical description of the asymptotic spectrum of sample-covariance-type matrices in the limiting regime defined as both the number of samples and the observation dimension grow without bound at the same rate. As a result, the proposed MMSE signal waveform estimator is shown to present a superior mean-square error performance under a finite sample-size by avoiding the breakdown experienced as the selected rank increases. </para>

A Low-Complexity Kalman Approach for Channel Estimation in Doubly-Selective OFDM Systems

In this letter, we propose a vector state-scalar observation (VSSO) Kalman filter for channel estimation in doubly-selective orthogonal frequency division multiplexing (OFDM) systems. Vector state-vector observation (VSVO) Kalman filters have been reported before in the literature for this purpose. The proposed VSSO Kalman filter achieves the same performance as the VSVO Kalman filter and results in 92% complexity savings. The Kalman filter outperforms a recently proposed linear minimum mean square error (LMMSE) estimator and achieves a high spectral efficiency of 93% as compared to the LMMSE estimator of 68%. A key aspect of this paper is that we show how the observed pilot symbol vector can be decorrelated or decoupled into uncorrelated multipath scalars. This aspect (and the proposed Kalman filter) is similar in spirit to that of a quasi-static channel.

A Fast LMMSE Channel Estimation Method for OFDM Systems

A fast linear minimum mean square error (LMMSE) channel estimation method has been proposed for Orthogonal Frequency Division Multiplexing (OFDM) systems. In comparison with the conventional LMMSE channel estimation, the proposed channel estimation method does not require the statistic knowledge of the channel in advance and avoids the inverse operation of a large dimension matrix by using the fast Fourier transform (FFT) operation. Therefore, the computational complexity can be reduced significantly. The normalized mean square errors (NMSEs) of the proposed method and the conventional LMMSE estimation have been derived. Numerical results show that the NMSE of the proposed method is very close to that of the conventional LMMSE method, which is also verified by computer simulation. In addition, computer simulation shows that the performance of the proposed method is almost the same with that of the conventional LMMSE method in terms of bit error rate (BER).

LMMSE and MAP estimators for reduction of multiplicative noise in the nonsubsampled contourlet domain

In this paper, two algorithms for multiplicative noise reduction, using the undecimated separable wavelet transform, are extended to the nonsubsampled contourlet (NSCT) domain. Such algorithms use either a maximum a posteriori (MAP) or a linear minimum mean square error (LMMSE) filtering approach, and involve the estimation of the moments of the NSCT coefficients up to the fourth order. The MAP filter relies on the conjecture that the NSCT coefficients follow a generalized Gaussian distribution (GGD), whose parameters locally vary. The extension of the denoising algorithms to the NSCT domain is not trivial, because several issues, related to the nonseparable implementation of the NSCT and the estimation of the moments, are to be considered to obtain viable solutions. Simulation results show that the MAP filter always outperforms the LMMSE one, confirming that the nonstationary GGD model is suitable for describing NSCT coefficients. Both denoising algorithms benefit from the multidirectional domain. However, the improvements on the LMMSE filter are greater than those on the MAP filter, showing that denoising in the NSCT domain is less effective when the nonstationary MAP estimator is used.

Distance Estimation From Received Signal Strength Under Log-Normal Shadowing: Bias and Variance

In source localization, one estimates the location of a source using a variety of relative position information. Many algorithms use certain powers of distances to effect localization. In practice, exact distance measurement is not directly available and must be estimated from information such as received signal strength (RSS), time of arrival, or time difference of arrival. This letter considers bias and variance issues in estimating powers of distances from RSS affected by practical log-normal shadowing. We show that the underlying estimation problem is inefficient and that the maximum likelihood estimate yields a bias and a mean-square error (MSE) that both increase exponentially with the noise power. We then characterize the class of unbiased estimates and show that there is <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">only</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">one</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">estimator</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">this</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">class</i> , but that its MSE also grows exponentially with the noise power. Finally, we provide the linear minimum mean-square error (MMSE) estimate and show that its bias and MSE are both bounded in the noise power.

Decision-directed channel estimation based on iterative linear minimum mean square error for orthogonal frequency division multiplexing systems

A decision-directed (DD) channel estimation based on iterative linear minimum mean square error (LMMSE) is proposed for orthogonal frequency division multiplexing systems. Existing DD channel estimation is well known to have the problem of error propagation because of symbol-by-symbol detection. The proposed algorithm can estimate the correction term of current channel state information (CSI) according to the error vector of previous CSI by applying the orthogonality principle, and corrects the current CSI with this correction term. Analysis and simulation results have shown that this method has no error propagation problem. The performance of the proposed algorithm is much better than the conventional DD channel estimation, and close to the optimal LMMSE estimator, but with much less computational complexity compared with the optimal LMMSE estimator.

On Training Optimization for Estimation of Correlated MIMO Channels in the Presence of Multiuser Interference

In this paper, the problem of estimating multiple-input multiple-output (MIMO) channels in a realistic environment involving correlated channel fading and multiuser interference is considered. Four estimation schemes are studied, including the linear minimum mean squared error (LMMSE), least squares (LS), and Gauss-Markov (GM) estimators, as well as a novel scheme which is derived here as an alternative to LMMSE estimation. The MSE-optimal training sequences for each of them are provided and their requirements for side information feedback are assessed. The new scheme is shown to exhibit a performance comparable to or even better than LMMSE, at a significantly lower feedback and computational cost. The analytical comparison of the estimation schemes is supported by numerous simulation results that cover a wide range of antenna configurations, relative interference power, and channel correlation strengths. The results of this paper provide a complete picture for a palette of estimation schemes, with their relative performance and costs of training.

MIMO-OFDM Channel Estimation in the Presence of Frequency Offsets

Optimal pilot design and placement for channel estimation in multiple-input multiple-output (MIMO) orthogonal frequency-division multiplexing (OFDM) systems with frequency offsets are considered. Both the single-frequency- offset case and the multiple-frequency-offset case are treated. We show that the constant-envelope (CE) condition is sufficient but not necessary for pilot design, and that pilots with multiple envelopes can also achieve the optimal performance in terms of the mean square error (MSE) minimization, provided that an additional constraint on the pilot placement is satisfied simultaneously. New pilot designs, which take into account the multiple-frequency-offset case, are proposed to eliminate inter- pilot-interference (IPI) and to optimize the MSE performance. The least-squares (LS) and linear minimum mean square error (LMMSE) channel estimators for the multiple-frequency- offset case are designed for uncorrelated and correlated MIMO-OFDM channels, respectively. The LMMSE estimator requires the channel covariance matrix. Both optimal adaptive pilot power allocation and suboptimal uniform pilot power allocation are developed for the proposed LMMSE estimator. The adaptive allocation performs 4 dB better than the uniform allocation in the high noise region, but they both perform identically in the low noise region. Performance comparisons are made against several previous pilot designs due to [1], [2]. The proposed LMMSE estimator significantly outperforms the LS estimator.

Consensus in Ad Hoc WSNs With Noisy Links—Part II: Distributed Estimation and Smoothing of Random Signals

Distributed algorithms are developed for optimal estimation of stationary random signals and smoothing of (even nonstationary) dynamical processes based on generally correlated observations collected by ad hoc wireless sensor networks (WSNs). Maximum a posteriori (MAP) and linear minimum mean-square error (LMMSE) schemes, well appreciated for centralized estimation, are shown possible to reformulate for distributed operation through the iterative (alternating-direction) method of multipliers. Sensors communicate with single-hop neighbors their individual estimates as well as multipliers measuring how far local estimates are from consensus. When iterations reach consensus, the resultant distributed (D) MAP and LMMSE estimators converge to their centralized counterparts when inter-sensor communication links are ideal. The D-MAP estimators do not require the desired estimator to be expressible in closed form, the D-LMMSE ones are provably robust to communication or quantization noise and both are particularly simple to implement when the data model is linear-Gaussian. For decentralized tracking applications, distributed Kalman filtering and smoothing algorithms are derived for any-time MMSE optimal consensus-based state estimation using WSNs. Analysis and corroborating numerical examples demonstrate the merits of the novel distributed estimators.

Robust dimension reduction, fusion frames, and Grassmannian packings

We consider estimating a random vector from its measurements in a fusion frame, in presence of noise and subspace erasures. A fusion frame is a collection of subspaces, for which the sum of the projection operators onto the subspaces is bounded below and above by constant multiples of the identity operator. We first consider the linear minimum mean-squared error (LMMSE) estimation of the random vector of interest from its fusion frame measurements in the presence of additive white noise. Each fusion frame measurement is a vector whose elements are inner products of an orthogonal basis for a fusion frame subspace and the random vector of interest. We derive bounds on the mean-squared error (MSE) and show that the MSE will achieve its lower bound if the fusion frame is tight. We then analyze the robustness of the constructed LMMSE estimator to erasures of the fusion frame subspaces. We limit our erasure analysis to the class of tight fusion frames and assume that all erasures are equally important. Under these assumptions, we prove that tight fusion frames consisting of equi-dimensional subspaces have maximum robustness (in the MSE sense) with respect to erasures of one subspace among all tight fusion frames, and that the optimal subspace dimension depends on signal-to-noise ratio (SNR). We also prove that tight fusion frames consisting of equi-dimensional subspaces with equal pairwise chordal distances are most robust with respect to two and more subspace erasures, among the class of equi-dimensional tight fusion frames. We call such fusion frames equi-distance tight fusion frames. We prove that the squared chordal distance between the subspaces in such fusion frames meets the so-called simplex bound, and thereby establish connections between equi-distance tight fusion frames and optimal Grassmannian packings. Finally, we present several examples for the construction of equi-distance tight fusion frames.

Wavelet-based image denoising with the normal inverse Gaussian prior and linear MMSE estimator

A new spatially adaptive wavelet-based method is introduced for reducing noise in images corrupted by additive white Gaussian noise. It is shown that a symmetric normal inverse Gaussian distribution is highly suitable for modelling the wavelet coefficients. In order to estimate the parameters of the distribution, a maximum- likelihood-based technique is proposed, wherein the Gauss-Hermite quadrature approximation is exploited to perform the maximisation in a computationally efficient way. A Bayesian minimum mean-squared error (MMSE) estimator is developed utilising the proposed distribution. The variances corresponding to the noise- free coefficients are obtained from the Bayesian estimates using a local neighbourhood. A modified linear MMSE estimator that incorporates both intra-scale and inter-scale dependencies is proposed. The performance of the proposed method is studied using typical noise-free images corrupted with simulated noise and compared with that of the other state-of-the-art methods. It is shown that the proposed method gives higher values of the peak signal-to-noise ratio compared with most of the other denoising techniques and provides images of good visual quality. Also, the performance of the proposed method is quite close to that of the state-of-the-art Gaussian scale mixture (GSM) method, but with much less complexity.

LMMSE turbo equalization based on factor graphs

In this paper, a vector-form factor graph representation is derived for intersymbol interference (ISI) channels. The resultant graphs have a tree-structure that avoids the short cycle problem in existing graph approaches. Based on a joint Gaussian approximation, we establish a connection between the LLR (log-likelihood ratio) estimator for a linear system driven by binary inputs and the LMMSE (linear minimum mean-square error) estimator for a linear system driven by Gaussian inputs. This connection facilitates the application of the recently proposed Gaussian message passing technique to the cycle-free graphs for ISI channels. We also show the equivalence between the proposed approach and the Wang-Poor approach based on the LMMSE principle. An attractive advantage of the proposed approach is its intrinsic parallel structure. Simulation results are provided to demonstrate this property.

An Efficient Design of Doubly Selective Channel Estimation for OFDM Systems

We find that placing each pilot tone in an equally spaced manner according to the conventional placement scheme is not suitable for doubly selective channel estimation. In this paper, we propose an efficient pilot tone placement scheme enabling accurate channel estimation in OFDM systems regardless of time variations of a channel. Since the number of channel impulse response taps to be estimated is typically much greater than the number of pilot tones, linear minimum mean square error (LMMSE) estimation schemes for time-invariant channels cannot be straightforwardly extended to doubly selective channel estimation. To overcome this problem, we propose an accurate LMMSE channel estimator that exploits a small number of pilot tones located according to the derived pilot placement. To achieve performance close to the LMMSE estimator but with lower complexity, an approximate LMMSE (ALMMSE) channel estimator is also proposed. Finally, we propose a novel iterative ALMMSE channel estimator that achieves better performance than the LMMSE and ALMMSE estimators, while having complexity in between the two.

Color Reproduction From Noisy CFA Data of Single Sensor Digital Cameras

Single sensor digital color still/video cameras capture images using a color filter array (CFA) and require color interpolation (demosaicking) to reconstruct full color images. The color reproduction has to combat sensor noises which are channel dependent. If untreated in demosaicking, sensor noises can cause color artifacts that are hard to remove later by a separate denoising process, because the demosaicking process complicates the noise characteristics by blending noises of different color channels. This paper presents a joint demosaicking-denoising approach to overcome this difficulty. The color image is restored from noisy mosaic data in two steps. First, the difference signals of color channels are estimated by linear minimum mean square-error estimation. This process exploits both spectral and spatial correlations to simultaneously suppress sensor noise and interpolation error. With the estimated difference signals, the full resolution green channel is recovered. The second step involves in a wavelet-based denoising process to remove the CFA channel-dependent noises from the reconstructed green channel. The red and blue channels are subsequently recovered. Simulated and real CFA mosaic data are used to evaluate the performance of the proposed joint demosaicking-denoising scheme and compare it with many recently developed sophisticated demosaicking and denoising schemes.

MIMO ISI Channel Estimation Using Uncorrelated Golay Complementary Sets of Polyphase Sequences

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> In this paper, optimal training sequence design for multiple-input multiple-output (MIMO) intersymbol interference channels is addressed, and several novel low-complexity channel estimators are proposed, using uncorrelated Golay complementary sets of polyphase sequences.<footnoteref refid="fnote1"/><footnote id="fnote1"> <footnotepara>A polyphase sequence is a sequence of complex numbers, each of unit magnitude.</footnotepara> </footnote> The theoretical analysis and simulation show that, when the additive noise is Gaussian, the proposed best linear unbiased estimator achieves the minimum possible classical CramÉr–Rao lower bound (CRLB) if the channel coefficients are regarded as unknown deterministics. On the other hand, the proposed linear minimum mean-square error estimator attains the minimum possible Bayesian CRLB when the underlying channel coefficients are Gaussian and independent of the additive Gaussian noise. The proposed channel estimators not only achieve the best estimation performance but can also be implemented with low complexity via DSP or application-specified integrated circuit/field programmable gate array. This has been possible due to the special structures intrinsic to uncorrelated Golay complementary sets of polyphase sequences, which make the proposed channel estimators ready to use in the practical MIMO systems. </para>

On Time-Domain Model-Based Ultrasonic Array Imaging

This paper treats time-domain model-based Bayesian image reconstruction for ultrasonic array imaging and, in particular, two reconstruction methods are presented. These two methods are based on a linear model of the array imaging system and they perform compensation in both the spatial and temporal domains using a minimum mean squared error (MMSE) criterion and a maximum a posteriori MAP) estimation approach, respectively. The presented estimators perform compensation for both the electrical and acoustical wave propagation effects for the ultrasonic array system at hand. The estimators also take uncertainties into account, and, by the incorporation of proper prior knowledge, high-contrast superresolution reconstruction results are obtained. The novel nonlinear MAP estimator constrains the scattering amplitudes to be positive, which applies in applications where the scatterers have higher acoustic impedance than the surrounding medium. The linear MMSE and nonlinear MAP estimators are compared to the traditional delay-and-sum (DAS) beamformer with respect to both resolution and signal-to-noise ratio. The algorithms are compared using both simulated and measured data. The results show that the model-based methods can successfully compensate for both sidelobes and grating lobes, and they have a superior temporal and lateral resolution compared to DAS beamforming. The ability of the nonlinear MAP estimator to suppress noise is also superior compared to both the linear MMSE estimator and the DAS beamformer.

Improved Subspace-Based Single-Channel Speech Enhancement Using Generalized Super-Gaussian Priors

Traditional single-channel subspace-based schemes for speech enhancement rely mostly on linear minimum mean-square error estimators, which are globally optimal only if the Karhunen-Loeacuteve transform (KLT) coefficients of the noise and speech processes are Gaussian distributed. We derive in this paper subspace-based nonlinear estimators assuming that the speech KLT coefficients are distributed according to a generalized super-Gaussian distribution which has as special cases the Laplacian and the two-sided Gamma distribution. As with the traditional linear estimators, the derived estimators are functions of the a priori signal-to-noise ratio (SNR) in the subspaces spanned by the KLT transform vectors. We propose a scheme for estimating these a priori SNRs, which is in fact a generalization of the decision-directed approach which is well-known from short-time Fourier transform (STFT)-based enhancement schemes. We show that the proposed a priori SNR estimation scheme leads to a significant reduction of the residual noise level, a conclusion which is confirmed in extensive objective speech quality evaluations as well as subjective tests. We also show that the derived estimators based on the super-Gaussian KLT coefficient distribution lead to improvements for different noise sources and levels as compared to when a Gaussian assumption is imposed

A Data-Bearing Approach for Pilot-Embedding Frameworks in Space-Time Coded MIMO Systems

Space-time (ST) coded MIMO systems employing coherent detectors crucially require channel state information. This paper presents a novel pilot-embedding framework for channel estimation and data detection by exploiting the null-space property and the orthogonality property of the data-bearer and pilot matrices. The ST data matrix is firstly projected onto the data bearer matrix, which is a null-space of the pilot matrix, and the resulting matrix and the pilot matrix are combined for transmitting. The data and pilot extractions are achieved independently through linear transformations by exploiting the null-space property. The unconstrained maximum-likelihood (ML) and linear minimum mean-squared error (lmmse) estimators are explored for channel estimation. Then the ML approach for data detection is developed by exploiting the orthogonality property. The mean-squared error (mse) of channel estimation, Cramer-Rao lower bound (CRLB), and the Chernoff's bound of a pair-wise error probability for ST codes are analyzed for examining the performance of the proposed scheme. The optimum power allocation scheme for data and pilot parts is also considered. Three data-bearer and pilot structures, including time-multiplexing (TM)-based, ST-block-code (STBC)-based, and code-multiplexing (CM)-based, are proposed. Simulation results show that the CM-based structure provides superior performance for nonquasi-static flat Rayleigh fading channels, while these three structures yield similar performances for quasi-static flat Rayleigh fading channels

An edge-guided image interpolation algorithm via directional filtering and data fusion

Preserving edge structures is a challenge to image interpolation algorithms that reconstruct a high-resolution image from a low-resolution counterpart. We propose a new edge-guided nonlinear interpolation technique through directional filtering and data fusion. For a pixel to be interpolated, two observation sets are defined in two orthogonal directions, and each set produces an estimate of the pixel value. These directional estimates, modeled as different noisy measurements of the missing pixel are fused by the linear minimum mean square-error estimation (LMMSE) technique into a more robust estimate, using the statistics of the two observation sets. We also present a simplified version of the LMMSE-based interpolation algorithm to reduce computational cost without sacrificing much the interpolation performance. Experiments show that the new interpolation techniques can preserve edge sharpness and reduce ringing artifacts.

Robust Distributed Estimation Using the Embedded Subgraphs Algorithm

We propose a new iterative, distributed approach for linear minimum mean-square-error (LMMSE) estimation in graphical models with cycles. The embedded subgraphs algorithm (ESA) decomposes a loopy graphical model into a number of linked embedded subgraphs and applies the classical parallel block Jacobi iteration comprising local LMMSE estimation in each subgraph (involving inversion of a small matrix) followed by an information exchange between neighboring nodes and subgraphs. Our primary application is sensor networks, where the model encodes the correlation structure of the sensor measurements, which are assumed to be Gaussian. The resulting LMMSE estimation problem involves a large matrix inverse, which must be solved in-network with distributed computation and minimal intersensor communication. By invoking the theory of asynchronous iterations, we prove that ESA is robust to temporary communication faults such as failing links and sleeping nodes, and enjoys guaranteed convergence under relatively mild conditions. Simulation studies demonstrate that ESA compares favorably with other recently proposed algorithms for distributed estimation. Simulations also indicate that energy consumption for iterative estimation increases substantially as more links fail or nodes sleep. Thus, somewhat surprisingly, sensor network energy conservation strategies such as low-powered transmission and aggressive sleep schedules could actually prove counterproductive. Our results can be replicated using MATLAB code from www.dsp.rice.edu/software