Linear Minimum Mean-square Error Filtering Research Articles

Well-Conditioned Linear Minimum Mean Square Error Estimation

Linear minimum mean square error (LMMSE) estimation is often ill-conditioned, suggesting that unconstrained minimization of the mean square error is an inadequate approach to filter design. To address this, we first develop a unifying framework for studying constrained LMMSE estimation problems. Using this framework, we explore an important structural property of constrained LMMSE filters involving a certain prefilter. Optimality is invariant under invertible linear transformations of the prefilter. This parameterizes all optimal filters by equivalence classes of prefilters. We then clarify that merely constraining the rank of the filter does not suitably address the problem of ill-conditioning. Instead, we adopt a constraint that explicitly requires solutions to be well-conditioned in a certain specific sense. We introduce two well-conditioned filters and show that they converge to the unconstrained LMMSE filter as their truncation-power loss goes to zero, at the same rate as the low-rank Wiener filter. We also show extensions to the case of weighted trace and determinant of the error covariance as objective functions. Finally, our quantitative results with historical VIX data demonstrate that our two well-conditioned filters have stable performance while the standard LMMSE filter deteriorates with increasing condition number.

Multi-Symbol Digital Signal Processing Techniques for Discrete Eigenvalue Transmissions Based on Nonlinear Fourier Transform

Optical communications based on Nonlinear Fourier Transform (NFT) and digital coherent transceivers are proposed as a new theoretical framework for communications over the nonlinear optical fiber channel. For discrete eigenvalue transmissions (or soliton transmissions), one seeks to encode as much information as possible in each degree of freedom and shorten the distance between neighboring pulses to increase the overall bit rate. However, such attempts would result in nonlinear inter-symbol interference (ISI) across multiple symbols and significantly degrade transmission performance. In this paper, we investigated joint modulation of discrete eigenvalue λ and <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</b> -coefficents <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</b> (λ) and developed a suite of multi-symbol digital signal processing (DSP) techniques to exploit the statistical correlations between the continuous and discrete eigenvalues and <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</b> -coefficents to mitigate nonlinear distortions and improve detection performance. This include 1) jointly modulating both λ and <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</b> (λ) of pairs of 1-solitons so that the mean value of λ for solitons with odd index is α + 1 <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</b> while it is - α + 1 <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</b> for solitons with even index. This is followed by decoding superimposed received waveforms as 2-solitons with twice the INFT processing time window; 2) linear minimum mean squared error (LMMSE) estimation filters to mitigate noise in discrete eigenvalue λ using continuous eigenvalue; 3) multi-symbol (MS) LMMSE filters to mitigate noise in <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</b> (λ) using discrete eigenvalue noise and 4) approximate the received signal distributions of λ and <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</b> (λ) as Gaussians with mean and covariance matrices obtained empirically from experiments followed by Maximum Likelihood (ML) detection for each symbol or multi-symbol (MS)-joint ML detection of 2-soliton signals. We jointly modulate λ with 16-QAM and <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</b> (λ) with 16-APSK and a record single-polarization discrete eigenvalue transmission of 64 Gb/s (net 54 Gb/s) over 1200 km is experimentally demonstrated with the proposed multi-symbol DSP algorithms.

On Orthogonal AMP in Coded Linear Vector Systems

Linear minimum mean square error (LMMSE) estimation based turbo detection has been extensively studied for coded linear systems since the seminal work of Wang and Poor (WP). The WP algorithm operates iteratively between a linear detector (LD) and a nonlinear detector (NLD): the LD suppresses the interference based on LMMSE filtering, and the NLD decodes the data by treating the output of the LD as an observation from an additive white Gaussian noise (AWGN) channel. In WP, the messages exchanged between LD and NLD are required to be extrinsic . For the NLD, the extrinsic message comes from the constraint imposed on feedforward error correction (FEC) codes. Therefore, WP does not work in an un-coded linear system. Recently, we proposed an orthogonal approximate message passing (OAMP) algorithm, which only requires the input/output error terms of LD and NLD to be orthogonal . We conjectured that for un-coded linear systems that involve certain large random matrices, the dynamics of OAMP can be accurately characterized by state evolution (SE). In this paper, we consider a coded linear system and develop an extrinsic message aided OAMP (EMA-OAMP) algorithm. Similar to the un-coded case, EMA-OAMP relaxes the requirements on output messages to be orthogonal instead of extrinsic. We derive an SE procedure to characterize the performance of OAMP in coded systems. We conjecture that this SE procedure is accurate, which is verified by simulation results. Under this conjecture, we show that EMA-OAMP can outperform WP under certain standard assumptions for iterative decoding. Extensive simulations results are provided to verify the advantages of OAMP in coded MIMO systems.

Speckle Noise Suppression in SAR Images Using a Three-Step Algorithm.

Speckle noise can reduce the image quality of synthetic aperture radar (SAR) and complicate image interpretation. This study proposes a novel three-step approach based on the conventional probabilistic patch-based (PPB) algorithm to minimize the impact of bright structures on speckle suppression. The first step improves the calculation accuracy of the weight by pre-processing speckle noise with a linear minimum mean-square error filter and reassessing similarity between pixels. In the second step, an iterative method is developed to avoid interfering with bright structures and acquires a more accurate homogeneous factor by adaptively changing the size of the search window. In the final step, the spreading and blurring of bright structures is corrected using a modified bias-reduction technique. Experimental results demonstrate the proposed algorithm has improved performance for both speckle suppression and preservation of edges and textures, evaluated by indicators including the equivalent number of looks, the edge preservation index, the mean, and standard deviation of ratio images.

Multi-objective noise estimator for the applications of de-noising and segmentation of MRI data

Abstract The present study proposes the noise estimation of Magnetic Resonance Imaging (MRI) data using multi-objective particle swarm optimisation (MOPSO). This adaptive noise estimation is based on the maximisation of the multiple quality measures, which enable the algorithm to achieve de-noising along with enhancement in the image features. The paper proposes two filtering approaches to de-noise MRI data. In first, MOPSO based noise estimation is followed by non-local statistics based Kalman filter, whereas, in the second approach, MOPSO based noise estimation is followed by Linear Minimum Mean Square Error (LMMSE) filter. The impact of de-noising on segmentation of MRI data has also been studied, for this purpose enhanced fuzzy c-means algorithm has been applied on filtered MRI data. The de-noising and segmentation performance of MOPSO-non local Kalman filter and MOPSO-LMMSE filters has been evaluated and compared with Wavelet filter, Wiener filter, non-local mean filter, standard Kalman and standard LMMSE filter. The proposed noise estimation approach followed by filtering is giving better de-noising and segmentation results as compared to standard filters considered.

Limited memory optimal filter for discrete-time systems with measurement delay

This paper considers the limited memory optimal filter for discrete-time systems with measurement delay. Firstly, we deal with the measurement delay using the reorganized innovation analysis approach. According to the projection theory, an optimal linear filter is provided in the linear minimum mean square error (LMMSE) sense. With the increase of time, the size of LMMSE filter problem will grow. Then, in order to avoid this problem, we explore a limited memory optimal filter calculated by applying the recent N measurements. Finally, we give a numerical example to show that the limited memory optimal filter is effective.

LMMSE Filtering in Feedback Systems With White Random Modes: Application to Tracking in Clutter

A generalized state space representation of dynamical systems with random modes switching according to a white random process is presented. The new formulation includes a term, in the dynamics equation, that depends on the most recent linear minimum mean squared error (LMMSE) estimate of the state. This can model the behavior of a feedback control system featuring a state estimator. The measurement equation is allowed to depend on the previous LMMSE estimate of the state, which can represent the fact that measurements are obtained from a validation window centered about the predicted measurement and not from the entire surveillance region. The LMMSE filter is derived for the considered problem. The approach is demonstrated in the context of target tracking in clutter and is shown to be competitive with several popular nonlinear methods.

A recursive linear MMSE filter for dynamic systems with unknown state vector means

In this contribution we extend Kalman-filter theory by introducing a new recursive linear minimum mean squared error (MMSE) filter for dynamic systems with unknown state-vector means. The recursive filter enables the joint MMSE prediction and estimation of the random state vectors and their unknown means, respectively. We show how the new filter reduces to the Kalman-filter in case the state-vector means are known and we discuss the fundamentally different roles played by the intitialization of the two filters.

Transceiver Impairments on the Performance of the LMMSE-PIC Iterative Receiver and its Mitigation

This work investigates the impact of inherent residual transceiver impairments of multiple-input multiple-output (MIMO) systems on the performance of linear minimum mean-squared error filtering and parallel interference cancellation (LMMSE-PIC) based iterative receiver. The residual transceiver impairments are first modeled as Gaussian noise at the transmitter in this work. Then a residual-transceiver-impairments-aware technique with empirically determined transmitter noise is proposed to mitigate the performance loss associated with the residual transceiver impairments. Numerical results based on MIMO systems with typical residual transceiver impairments show that the proposed method achieves significant performance gain over the conventional one which is designed for ideal transceivers.

Color Video Denoising Based on Combined Interframe and Intercolor Prediction

An advanced color video denoising scheme which we call CIFIC based on combined interframe and intercolor prediction is proposed in this paper. CIFIC performs the denoising filtering in the RGB color space, and exploits both the interframe and intercolor correlation in color video signal directly by forming multiple predictors for each color component using all three color components in the current frame as well as the motion-compensated neighboring reference frames. The temporal correspondence is established through the joint-RGB motion estimation (ME) which acquires a single motion trajectory for the red, green, and blue components. Then the current noisy observation as well as the interframe and intercolor predictors are combined by a linear minimum mean squared error (LMMSE) filter to obtain the denoised estimate for every color component. The ill condition in the weight determination of the LMMSE filter is detected and remedied by gradually removing the “least contributing” predictor. Furthermore, our previous work on the LMMSE filter applied in the adaptive luminance-chrominance space (LAYUV for short) is revisited. By reformulating LAYUV and comparing it with CIFIC, we deduce that LAYUV is a restricted version of CIFIC, and thus CIFIC can theoretically achieve lower denoising error. Experimental results verify the improvement brought by the joint-RGB ME and the integration of the intercolor prediction, as well as the superiority of CIFIC over LAYUV. Meanwhile, when compared with other state-of-the-art algorithms, CIFIC provides competitive performance both in terms of the color peak signal-to-noise ratio and in perceptual quality.

Multi-Target Performance of LMMSE Filtering in Radar

Coded orthogonal frequency division multiplexing (OFDM), like several other coded waveforms, exhibits inherent high range sidelobes after matched filtering. Consequently special processing at the receiver is required that can serve for sidelobe suppression in order to avoid target masking. However unmasking is not the only concern. It is crucial to evaluate the filtering scheme both in terms of sidelobe suppression capability and in terms of output signal-to-noise ratio (SNR). This last criterion is essential when aiming at detecting weaker reflections also. The theoretical performance of the reiterated filtering technique based on the linear minimum mean square error (LMMSE), as implemented in the adaptive pulse compression (APC) scheme, is derived and compared with the matched filter (MF). The analysis of the performance is done for a multi-target scenario where targets are close enough for the sidelobes of a target to interfere with the mainlobe of another target. Consequently the unmasking capabilities are relevant, but also output power figures. Complex-valued filter output peaks are also evaluated and compared with the MF output peaks. Moreover the performance of the method is evaluated for OFDM communication-encoded radar waveforms and compared with linear frequency modulated waveforms aiming at simultaneous use of the radar signal for both communication and radar.

Integration of Recursive Temporal LMMSE Denoising Filter Into Video Codec

The presence of noise can dramatically affect the efficiency of video compression systems. For performance improvement, most practical video compression systems adopt a denoising filter as a pre-processing module for the video encoder, or as a post-processing module for the video decoder, but the complexity introduced by denoising can be very high. This paper first presents a recursive temporal linear minimum mean squared error (LMMSE) filter for video denoising. Based on the analysis of the hybrid video compression process, two novel schemes are presented, one for video encoding and the other for video decoding, in which the proposed recursive temporal LMMSE filter is seamlessly integrated into the encoding and the decoding processes, respectively. For both of these two schemes, the denoising is implemented with nearly no extra computation introduced. Experimental results validate the effectiveness of the proposed schemes on encoding and decoding noisy video sequences.

Pilot-Based LMMSE Channel Estimation for OFDM Systems With Power–Delay Profile Approximation

In linear-minimum-mean-square-error (LMMSE) channel estimation for multicarrier systems, one needs to know the channel correlation function. This poses a problem for systems with a small number of pilots operating in time-varying channels. We propose to approximate the channel power-delay profile (PDP) with a shape that can completely be described in two parameters, namely, the mean delay and the root-mean-square (RMS) delay spread. Furthermore, we develop a simple technique to estimate these delay parameters. The approximate PDP is then used to generate the LMMSE filter coefficients for data-subcarrier channel estimation. Mathematical expressions are derived that can be used to predict the accuracy of the various estimates, and they are verified by simulation. The proposed technique is applicable to both point-to-point communication and multiaccess communication where different users may experience different channel conditions. As a practical application, we also specialize the proposed technique to Mobile WiMAX signals and investigate the resulting performance.

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.

Iterative Group-Blind Multiuser Detection and Decoding With Signal Rank Estimation for Coded CDMA Systems

In this paper, a turbo receiver structure is proposed for the uplink of coded code-division multiple-access (CDMA) systems in the presence of unknown users. The proposed receiver consists of two stages following each other. The first stage performs soft interference cancellation and group-blind linear minimum mean square error (MMSE) filtering, and the second stage performs channel decoding. The proposed group-blind linear MMSE filter suppresses the residual multiple-access interference (MAI) from known users based on the spreading sequences and the channel characteristics of these users while suppressing the interference from other unknown users using a subspace-based blind method. The proposed receiver is suitable for suppressing intercell interference in heavily loaded CDMA systems. Since the knowledge of the number of unknown users is crucial for the proposed receiver structure, a novel estimator is also proposed to estimate the number of unknown users in the system by exploiting the statistical properties of the received signal. Simulation results demonstrate that the proposed estimator can provide the number of unknown users with high accuracy; in addition, the proposed group-blind receiver integrated with the new estimator can significantly outperform the conventional turbo multiuser detector in the presence of unknown users.

On the Equivalence of Turbo Multiuser Detector using MMSE with A-Priori Information and Soft Interference Cancellation Followed by MMSE Filtering

This letter derives a turbo multiuser detector with linear minimum mean squared error (MMSE) filtering using a-priori information in order to reduce the computational complexity. The equivalence of the turbo multiuser detector developed in this letter and the one based on soft interference cancellation followed by a linear MMSE filtering is established.

Linear minimum mean-square error filtering for evoked responses: application to fetal MEG

This paper describes a linear minimum mean-squared error (LMMSE) approach for designing spatial filters that improve the signal-to-noise ratio (SNR) of multiepoch evoked response data. This approach does not rely on availability of a forward solution and thus is applicable to problems in which a forward solution is not readily available, such as fetal magnetoencephalography (fMEG). The LMMSE criterion leads to a spatial filter that is a function of the autocorrelation matrix of the data and the autocorrelation matrix of the signal. The signal statistics are unknown, so we approximate the signal autocorrelation matrix using the average of the data across epochs. This approximation is reasonable provided the mean of the noise is zero across epochs and the signal mean is significant. An analysis of the error incurred using this approximation is presented. Calculations of SNR for the exact and approximate LMMSE filters and simple averaging for the rank-1 signal case are shown. The effectiveness of the method is demonstrated with simulated evoked response data and fetal MEG data.

Iterative detection and decoding with an improved V-BLAST for MIMO-OFDM systems

Multiple-input-multiple-output (MIMO) systems provide a very promising means to increase the spectral efficiency for wireless systems. By using orthogonal frequency-division multiplexing (OFDM), wideband transmission can be achieved over frequency-selective fading radio channels. First, in this paper, we introduce an improved vertical Bell Labs layered space-time (V-BLAST) receiver which takes the decision errors into account. Second, we propose an iterative detection and decoding (IDD) scheme for coded layered space-time architectures in MIMO-OFDM systems. For the iterative process, a low-complexity demapper is developed by making use of both nonlinear interference cancellation and linear minimum mean-square error filtering. Also, a simple cancellation method based on hard decision is presented to reduce the overall complexity. Simulation results demonstrate that the proposed IDD scheme combined with the improved V-BLAST performs almost as well as the optimal turbo-MIMO approach, while providing tremendous savings in computational complexity.

Estimating the PDF of the SIC-MMSE equalizer output and its applications in designing LDPC codes with turbo equalization

We consider the analysis and design of low-density parity check (LDPC) codes for intersymbol interference (ISI) channels when used with soft interference cancellation plus linear minimum mean-square error filtering (SIC-MMSE) turbo equalization. We discuss techniques to compute the probability density function (pdf) of the extrinsic information at the output of the SIC-MMSE equalizer as a function of pdf of the input extrinsic information, channel impulse response, and the signal-to-noise ratio. For static ISI channels, we show that the output pdf can be modeled as symmetric Gaussian, and show that the mean can be evaluated without simulating the equalizer. For channels with long memory, we propose to use the unscented transform technique to compute the mean, which significantly reduces the computation required. Finally, for fading channels, we model the pdf by a mixture of symmetric Gaussian densities. Using these techniques, we are able to fairly accurately compute the thresholds for LDPC codes and design good irregular LDPC codes. Simulation results are in good agreement with the computed thresholds and the designed irregular LDPC codes outperform regular ones significantly.

Blind Multiuser Detection in Uplink CDMA With Multipath Fading: A Sequential EM Approach

We consider joint channel estimation and data detection in uplink asynchronous code-division multiple-access systems employing aperiodic (long) spreading sequences in the presence of unknown multipath fading. Since maximum-likelihood (ML) sequence estimation is too complex to perform, multiuser receivers are proposed based on the sequential expectation-maximization (EM) algorithm. With the prior knowledge of only the signature waveforms, the delays and the second-order statistics of the fading channel, the receivers sequentially estimate the channel using the sequential EM algorithm. Moreover, the snapshot estimates of each path are tracked by linear minimum mean-squared error filters. The user data are detected by a ML sequence detector, given the channel estimates. The proposed receivers that use the exact expressions have a computational complexity O(2/sup K/) per bit, where K is the number of users. Using the EM algorithm, we derive low-complexity approximations which have a computational complexity of O(K/sup 2/) per bit. Simulation results demonstrate that the proposed receivers offer substantial performance gains over conventional pilot-symbol-assisted techniques and achieve a performance close to the known channel bounds. Furthermore, the proposed receivers even outperform the single-user RAKE receiver with Nyquist pilot-insertion rate in a single-user environment.