Mean Square Error Performance Research Articles

Steady-State Mean-Square Error Performance Analysis of the Tensor LMS Algorithm

In some system identification problems, the weight vector to be estimated can be represented by a tensor product of two low dimensional component vectors. This feature is useful for the design of adaptive filters with fast convergence rate and/or low computational complexity. The tensor LMS algorithm has been proposed under this scenario and its recent extensions find applications in several contexts. This brief analyzes the steady-state mean-square error (MSE) performance of the tensor LMS algorithm, which can provide performance prediction and design guideline for engineers. Since the updates of the weight-vectors of the component adaptive filters (CAFs) are coupled, it is challenging to perform steady-state MSE analysis. To address this problem, this brief first establishes the variance relation in steady state for the CAFs and then decouples their excess MSEs by solving a system of linear equations. Finally, the steady-state MSE of the reproduced adaptive filter is composed of the excess MSEs of the CAFs and the variance of the system noise. Simulations are performed to verify the accuracy of the theoretical findings.

Robust range estimation algorithm based on hyper‐tangent loss function

Herein, the authors present a robust estimator of range against the impulsive noise using only the received signal's magnitude. The M estimator has been widely used in robust signal processing. However, the existing M estimator requires statistical testing involving a threshold which has an optimality that varies with time, hence algorithmically challenging and computationally burdensome. The statistical testing is utilised for discerning the inlier and outlier. Further, statistical testing renders the computational burden of the algorithm high since the testing must be performed for each observation. Therefore, they propose the M estimator based on the hyper-tangent loss function, which does not demand statistical testing. Conventional M estimator employing information theoretic learning also does not call for statistical testing, but the mean square error (MSE) performance for the range estimation is inferior to that of the proposed method. Furthermore, they perform an analysis for the MSE for the proposed algorithm. Monte Carlo simulations not only validate their theoretical analysis, but also demonstrate the MSE performance of the proposed method is nearly same as the existing skipped filter although it does not require the statistical testing and optimal threshold selection.

The Extended Feature LMS Algorithm: Exploiting Hidden Sparsity for Systems with Unknown Spectrum

The feature least-mean-square (F-LMS) algorithm has already been introduced to exploit hidden sparsity in lowpass and highpass systems. In this paper, by proposing the extended F-LMS (EF-LMS) algorithm, we boosted the F-LMS algorithm to exploit hidden sparsity in more general systems, those which are neither lowpass nor highpass. To this end, by means of the so-called feature matrix, we reveal the hidden sparsity in coefficients and utilize the $$l_1$$-norm to exploit the exposed sparsity. As a result, the EF-LMS algorithm will improve the convergence rate and the steady-state mean-squared error (MSE) as compared to the traditional least-mean-square algorithm. Moreover, in this work, we analyze the convergence behavior of the coefficient vector and the steady-state MSE performance of the EF-LMS algorithm. Through synthetic and real-world experiments, it has been seen that the EF-LMS algorithm can improve the convergence rate and the steady-state MSE whenever the hidden sparsity is revealed.

Speech Intelligibility for Cochlear Implant Users with the MMSE Noise-Reduction Time-Frequency Mask

Cochlear implants (CI) are the most successful electronic prostheses for human beings. They almost completely restore the communication capability in profound-to-severe hearing losses under quiet conditions. However, noisy scenarios still impose critical intelligibility limitations. Time-frequency masks have been widely used to improve signal to noise conditions (SNR) and, as a consequence, increase the speech intelligibility. The most studied methods in the literature are the Binary mask (BM) and the Wiener filter (WF), which were not originally designed for this application. This work analyzes the performance of the minimum mean square error (MMSE) magnitude estimator for increasing intelligibility in CI applications. Objective measures and psychoacoustic experiments with normal hearing volunteers and vocoded signals, as well as CI users indicate that the MMSE method outperfoms the intelligibility performance obtained by both BM and WF time-frequency masks, especially in low SNR conditions (SNR < 8 dB). This observation can be explained by the fact that CIs do not provide speech temporal-fine-structure information to the CI user, and that hard masks may suppress important speech information under low SNR. As a result, we provide strong indications that the MMSE magnitude-based time-frequency mask is more suitable to CI noise reduction applications, as compared to WF and BM, whenever the additional computation burden is tolerable by the CI processor.

Adaptive blind equalization for a MIMO chaotic communication system

There exist few blind solutions for chaotic MIMO channel equalization. In this work, a chaotic MIMO channel equalization framework is proposed. The objective function to be minimized in the proposed solution is obtained by adopting the objective function developed for chaotic SISO channel equalization. Furthermore, an optimum filter that minimizes the proposed cost function is designed to recover chaotic input signals assuming that the channel is known. The stationary point of the adaptive solution is equal to the optimal filter if the adaptive filter coefficients change sufficiently slowly. The adaptive solution is contrasted with the optimum filter in terms of mean-square error and bit error rate performances. In addition, the proposed solution reconstructs chaotic input signals at the same time. Consequently, it can be applied to multiple signal separation problems as well.

Optimal control of demand flexibility under real-time pricing for heating systems in buildings: A real-life demonstration

The identification, quantification, and control of demand flexibility is the major challenge for future grid operations and requires innovative methods and new control strategies. Optimal control strategies such as economic model predictive control have gained attention in building energy management systems. The present experimental case study demonstrates the application of an economic model predictive controller under real-time pricing, including day-ahead prices and imbalance prices. For real-time prices in balancing and spot markets, we introduce a method that presents a flexibility service to provide demand flexibility for a notification time of 1h<t<tdayend in advance. The flexibility service can be used as an ancillary service for innovative flexibility markets. The flexibility service includes a dynamic modification of the day-ahead prices to enable the adaption of energy consumption to errors in forecasting of renewable energy generation. The developed method was tested under real-life conditions, which also included the stochastic behaviour of occupants and the dynamic behaviour of the building and heating system. During the test periods, the controller managed the total operational costs of the heat pump’s electricity consumption and achieved a prediction performance of Root Mean Square Error between 0.17 and 0.22 kWh. To show the provision of demand flexibility, key performance indicators were quantified according to the categories 1) energy and power, 2) energy efficiency, and 3) energy costs. We introduce this categorization to present the benefits of using flexibility indicators along with conventional performance indicators in real-life applications.

Recursive Polynomial Minimum Mean-Square Error Estimation with Applications to Orbit Determination

This paper presents a systematic generalization of the linear update structure associated with the extended Kalman filter for high-order polynomial estimation of nonlinear dynamical systems. The minimum mean-square error criterion is used as the cost function to determine the optimal polynomial update during the estimation process. The high-order series representation is implemented effectively using differential algebra techniques. Numerical examples show that the proposed algorithm, named the high-order differential algebra Kalman filter, provides superior robustness and/or mean-square error performance as compared to linear estimators under the condition considered.

Refinement of TOA Localization with Sensor Position Uncertainty in Closed-Form.

The subject of localization has received great deal attention in the past decades. Although it is perhaps a well-studied problem, there is still room for improvement. Traditional localization methods usually assume the number of sensors is sufficient for providing desired performance. However, this assumption is not always satisfied in practice. This paper studies the time of arrival (TOA)-based source positioning in the presence of sensor position errors. An error refined solution is developed for reducing the mean-squared-error (MSE) and bias in small sensor network (the number of sensors is fewer) when the noise or error level is relatively large. The MSE performance is analyzed theoretically and validated by simulations. Analytical and numerical results show the proposed method attains the Cramér-Rao lower bound (CRLB). It outperforms the existing closed-form methods with slightly raising computation complexity, especially in the larger noise/error case.

Virtual Angular-Domain Channel Estimation for FDD Based Massive MIMO Systems with Partial Orthogonal Pilot Design

This article proposes a virtual angular-domain channel estimation scheme for massive multiple-input multiple-output systems operating in frequency division duplex (FDD) mode. Different from the conventional scheme where orthogonal pilots are transmitted on different antennas, we propose to transfer the channel estimation problem to the virtual angular domain and utilize the channel sparsity to reduce the training and feedback overhead. An orthogonal matching pursuit with Gram-Schmidt orthogonalization algorithm is proposed to construct the unitary transformation between the spatial domain and the virtual angular domain, which achieves higher sparsity than the existing approaches. Furthermore, we propose to estimate the downlink (DL) dominant angular set, which captures most of the channel power with only a few elements, by utilizing the directional reciprocity of FDD systems, where a calibration algorithm is introduced to handle the different wavelengths of uplink and DL transmissions. Based on the estimated dominant sets, we introduce a partial orthogonal criterion for virtual angular-domain pilot design and further propose two pilot assignment algorithms which minimize pilot overhead and pilot-reuse interference, respectively. Theoretical analyses on pilot overhead and the mean square error (MSE) performance are also presented. Simulation results demonstrate that our proposed virtual angular-domain channel estimation scheme provides excellent MSE performance with much reduced pilot overhead and, consequently, enjoys much larger per-user achievable rate in comparison to the conventional schemes.

Online Data Dimensionality Reduction and Reconstruction Using Graph Filtering

Graphs have become a popular toolbox for capturing similarity relationships among data vectors in a variety of networks. Diffusion processes are used to model the data using the graph topology. A novel approach is put forth that exploits data similarity and diffusion models, in a graph, to improve upon the reconstruction mean-square error (MSE) performance of principal component analysis. The tasks of data dimensionality reduction and reconstruction are formulated as graph matrix filtering operations, that are utilized in a dimensionality reduction framework optimal in the MSE sense. The novel formulation is seeking MSE-optimal filter matrices that minimize the reconstruction MSE of all the graph data. Block coordinate descent techniques are employed in the graph spectral domain MSE cost to recursively determine the MSE-optimal dimensionality reducing and reconstruction filters. Online implementations are put forth to work with continuous streams of graph data, while almost sure convergence to a stationary point of the ensemble reconstruction MSE cost is established. Analysis of the reconstruction MSE reveals that the lowest MSE achieved by the novel approach is not larger than the standard PCA MSE, while there is a bound for the filter orders after which no more MSE reduction occurs. Extensive numerical tests using synthetic data, as well as real image datasets demonstrate the advantage of the novel graph-based framework over standard PCA and related methods.

A Spatial–Temporal Subspace-Based Compressive Channel Estimation Technique in Unknown Interference MIMO Channels

Spatial–temporal (ST) subspace-based channel estimation techniques formulated with $\ell 2$ minimum mean square error (MMSE) criterion alleviate the multi-access interference (MAI) problem when the interested signals exhibit low-rank property. However, the conventional $\ell 2$ ST subspace-based methods suffer from mean squared error (MSE) deterioration in unknown interference channels, due to the difficulty to separate the interested signals from the channel covariance matrices (CCMs) contaminated with unknown interference. As a solution to the problem, we propose a new $\ell 1$ regularized ST channel estimation algorithm by applying the expectation-maximization (EM) algorithm to iteratively examine the signal subspace and the corresponding sparse-supports. The new algorithm updates the CCM independently of the slot-dependent $\ell 1$ regularization, which enables it to correctly perform the sparse-independent component analysis (ICA) with a reasonable complexity order. Simulation results shown in this paper verify that the proposed technique significantly improves MSE performance in unknown interference MIMO channels, and hence, solves the BER floor problems from which the conventional receivers suffer.

Nonlinear Adaptive Filtering With Kernel Set-Membership Approach

This paper develops nonlinear kernel adaptive filtering algorithms based on the set-membership filtering (SMF) framework. The set-membership-based filtering approach is distinct from the conventional adaptive filtering approaches in that it aims for the filtering error being bounded in magnitude, as opposed to seeking to minimize the time average or ensemble average of the squared errors. The proposed kernel SMF algorithms feature selective updates of their parameter estimates by making discerning use of the input data, and selective increase of the dimension in the kernel expansion. These result in less computational cost and faster tracking without compromising the mean-squared error performance. We show, through convergence analysis, that the sequences of parameter estimates of our proposed algorithms are convergent, and the filtering error is asymptotically upper bounded in magnitude. Simulations are performed which show clearly the advantages of the proposed algorithms in terms of lower computational complexity, reduced dictionary size, and steady-state mean-squared errors comparable to existing algorithms.

Burst-Mode Synchronization for SOQPSK

We consider a comprehensive synchronization strategy for burst-mode transmission of shaped-offset quadrature phase-shift keying (SOQPSK) signals over the additive white Gaussian noise channel. Due to the similarities between SOQPSK and continuous phase modulation (CPM), we make use of recent results for synchronization of burst-mode CPMs. We first derive a training sequence that is optimal in the sense that it jointly minimizes the Cramer–Rao bounds (CRBs) for frequency offset, phase offset, and timing offset estimation. Additionally, we develop a maximum likelihood data-aided algorithm for joint estimation of the synchronization parameters for SOQPSK signals. We show that the proposed algorithm for the optimal training sequence can be adapted to work with the suboptimal training sequence that is being considered for use in burst-mode integrated network enhanced telemetry. This demonstrates an immediate practical application for our approach. We present numerical results on the mean-squared error performance of the proposed algorithm for both training sequences and for different versions of SOQPSK. The numerical results show that our joint estimation algorithm yields performance that is very close to the CRBs for all three synchronization variables. Finally, we compare the overall performance of our proposed training sequence and synchronization algorithm for different sequence lengths by simulating a burst-mode SOQPSK receiver. This allows us to employ the right training sequence length based on the desired complexity and bit error rate performance.

Quality Evaluation of Speech Enhancement Algorithms for Normal and Hearing Loss Listeners

The subjective quality test of the enhanced speech from different enhancement algorithms for listeners with normal hearing (NH) capability as well as listeners with hearing impairment (HI) is reported. The subjective quality evaluation of speech enhancement methods in the literature survey is mostly done targeting NH listeners and fewer attempts are observed to subjectively evaluate for HI listeners. The algorithms evaluated are from four different classes: spectral subtraction class(SS), statistical model based class (minimum mean square error), subspace class(PKLT) and auditory class (ideal binary mask using STFT, ideal binary mask using gammatone filterbank and ideal binary mask using gammachirp filterbank). The algorithms are evaluated using four types of real world noises recorded in Indian scenarios namely cafeteria, traffic, station and train at -5, 0, 5 and 10 dB SNRs. The evaluation is being done as per ITU-T P.835 standard in terms of three parameters- speech signal alone, background noise and overall quality. The noisy speech database developed in Indian regional language, Marathi, at four SNRs -5, 0, 5 and 10 dB is used for evaluation. Significant improvement is observed in ideal binary mask algorithm in terms of overall quality and signal distortion ratings for NH and HI listeners. The performance of minimum mean square error is also observed comparable with the ideal binary mask algorithm in some cases.

Iterative Sparse Channel Estimation and Spatial Correlation Learning for Multichannel Acoustic OFDM Systems

This article addresses the problem of coherent detection of acoustic orthogonal frequency division multiplexing (OFDM) signals using a sparse channel estimation method based on a physical model of multipath propagation. Unlike the conventional sample-spaced and subsample-spaced methods, such as least squares and orthogonal matching pursuit (OMP), which target the taps of an equivalent discrete-delay channel response, the path identification (PI) method targets the physical propagation paths in a continuous-delay domain, and focuses on explicit estimation of delays and complex amplitudes of the channel paths in an iterative fashion. When multiple receive elements are available, two situations are possible: one in which the array elements see uncorrelated channel responses, and another in which the channel responses are correlated. In the first case, channel estimation must be accomplished element-by-element. This is done simply by applying the PI algorithm to each element individually. In the second case, correlation between the elements can be exploited. In doing so, our goal is to reduce the signal processing complexity without compromising the performance. Toward this goal, an adaptive precombining method is proposed. Without requiring any a priori knowledge about the spatial distribution of received signals, the method exploits spatial coherence between receive channels by linearly combining them into fewer output channels so as to reduce the number of subsequent channel estimators. The algorithm learns the spatial coherence pattern recursively over the carriers, thus effectively achieving broadband beamforming. The reduced-complexity precombining method relies on differential encoding that keeps the receiver complexity at a minimum and requires a very low pilot overhead. Using synthetic data as well as 210 experimental signals transmitted over a 3–7-km shallow-water channel in the 10.5–15.5-kHz acoustic band during a 3.5-h experiment, we study the system performance in terms of data detection mean-squared error (MSE), symbol error rate, and bit error rate (BER), and show that the PI algorithm achieves excellent MSE performance while its complexity is considerably lower than that of the OMP algorithm. We also demonstrate that the receiver equipped with the proposed reduced-complexity precombining scheme requires three times fewer channel estimators while achieving the same MSE and BER performance as the full-complexity receiver.

Application Analysis of LS Channel Estimation for Transform Domain Communication System

To reduce the influence of wireless transmission channel on transform domain communication system (TDCS) and improve the channel estimation accuracy, a system model of Least Square (LS) channel estimation on TDCS is established. In this paper, the influence of the unavailable frequency existing in basic function is analysed for the LS channel estimation, then the expression of mean square error (MSE) is derived. After that, the frequency pilot insertion pattern and the interpolation algorithm suitable for TDCS is proposed. The simulation results verify the paper’s theoretical analysis, and the effectiveness of the proposed algorithm is proved by the comparison of the MSE and BER performance.

Reduced complexity DOA and DOD estimation for a single moving target in bistatic MIMO radar

In this work, we propose a reduced dimension and low complexity algorithm to estimate the direction-of-arrival (DOA), direction-of-departure (DOD) and the Doppler shift of a moving target for a multiple-input-multiple-output (MIMO) radar. We derive two cost functions based on two different objective functions. We solve each of the derived cost function with a low complexity fast-Fourier-transform (FFT)-based solution in three dimensions. We further carry out a derivation to reduce the three-dimensional search to two-dimensional (2D) search and solve it with a 2D-FFT. Another reduced dimension algorithm is derived using the generalized eigenvalue method which finds the estimate of unknown parameters in one dimension with less memory constraints. This way, we propose three algorithms based on the first cost function and another three algorithms based on the second. Simulation results are used to validate the proposed algorithms. We compare the mean-square-error (MSE) performance and computational complexity of our proposed algorithms with existing ones as well. We show that our proposed algorithms have better MSE performance than existing ones and achieves the Cramér-Rao lower bound (CRLB) for all unknown target parameters. The proposed algorithms exhibit lower computational complexity than the existing ones and also provide an estimate for the Doppler shift.

A New Pentagon Search Algorithm for Fast Block-Matching Motion Estimation

The impact of search pattern is a crucial part in the block-based motion estimation for finding the motion vector. An issue of distortion performance and search speed heavily depends upon the size and shape of search strategy applied. Performing the deep analysis for motion vector distribution on standard test videos, it is desirable to have such type of algorithm that meets the requirement of searching motion vector in less time. Hence, a new kind of Pentagon algorithm is proposed in this paper for fast block-matching motion estimation (BMME). It is an easy and efficient technique for finding motion vector. Experimental results expose the proposed Pentagon algorithm sparsely surpass the noted Diamond search (DS) algorithm. The new Pentagon search algorithm is examined with the previously proposed Diamond search algorithm in terms of performance measure; the proposed algorithm attains better performance with the less complexity. The experimental examination also depicts that the pentagon algorithm is better than the previously proposed Diamond search (DS) in terms of mean-square error performance and required the number of search points. The overall speed improvement rate (SIR) is about 31% with respect to the DS.

Low Complexity Lattice Reduction Aided Detectors for High Load Massive MIMO Systems

In this paper, a very low complexity Lattice Reduction technique, called Dual Shortest Longest Vector algorithm (SLV), is adopted to improve the Bit Error Rate (BER) performance of the Minimum Mean Square Error (MMSE) detector in high-load Massive MIMO systems, whereby resulting in the so-called SLV-aided MMSE (MMSE–SLV) detector. An efficient combination scheme of Generalized Group Detection (GGD) algorithm and the MMSE–SLV, called MMSE–GGD–SLV, is further proposed to enhance BER performance of the system more significantly. In order to do so, we first convert the Group Detection approach to the generalized one (GGD) by creating an arbitrary number of sub-systems. Then, an additional operation, i.e., channel matrix sorting, is applied to the GGD to reduce the error propagation between sub-systems. To make the detection complexities of the MMSE–GGD–SLV detector more practical, the MMSE–SLV detection procedure is only applied to the first sub-system. Various BER performance simulations and complexity analysis show that both the MMSE–GGD–SLV and the MMSE–SLV detectors noticeably outperform their conventional MMSE counterpart, yet at the cost of higher detection complexities. However, their complexities are kept at acceptable levels, which are much lower than those of the conventional BLAST detector. Therefore, the proposed detectors are very good candidates for signal recovery in high load Massive MIMO systems.

Soft Decision Control Iterative Channel Estimation for the Internet of Things in 5G Networks

In the fifth generation mobile networks, generalized frequency division multiplexing (GFDM) is expected as the candidate waveform which can flexibly meet the requirements of diverse applications and scenarios for the Internet of Things (IoT) because of its advantages over orthogonal frequency division multiplexing (OFDM). In order to achieve the reliable data transmission in GFDM-based IoT systems, channel estimation (CE) is a prerequisite. However, the 2-D block modulation and the nonorthogonality between subcarriers for GFDM make it almost impossible that the conventional CE methods suitable for OFDM are directly applied to GFDM. To cope with this problem, a soft decision control strategy-based iterative CE (SDC-ICE) method is proposed in this paper. First, the received signal is equalized by the channel frequency response (CFR) from the pilot-based CE. After GFDM demodulation and Turbo decoding, the feedback log-likelihood ratio is utilized to rebuild symbols for data-aided CE by a redesigned Turbo receiver. Subsequently, the feedback information of both current and former iterations is used to improve the reliability of rebuilt symbols. The CFR obtained from SDC-ICE is used for equalization in the next iteration. The performance of SDC-ICE can be improved by increasing the iterations. Finally, the bit error rate (BER) and mean square error (MSE) performances of SDC-ICE and hard decision control strategy-based iterative CE (HDC-ICE) are simulated and evaluated. Simulation results demonstrate that the proposed method has better BER and MSE performance than HDC-ICE within fewer iterations.