Mean Square Error Analysis Research Articles

Steady-State Mean-Square Performance Analysis of a Relaxed Set-Membership NLMS Algorithm by the Energy Conservation Argument

This paper presents an analysis of the steady-state mean-square error (MSE) of the set-membership normalized least-mean square (SM-NLMS) algorithm with relaxation and regularization parameters. These parameters are introduced for the purpose of deriving in a unified way the steady-state MSE performances of the epsiv-normalized least mean square (epsiv-NLMS) algorithm and a special case of the adaptive parallel subgradient projection (PSP) algorithm. The approach of the paper is to employ the energy conservation relation as a starting point of our analysis. This relation enables us to avoid the transient analysis of the SM-NLMS algorithm, which is in general hard due to the nonlinearity of the SM-NLMS algorithm. As a result, a few nonlinear equations whose solutions are theoretical steady-state MSEs are derived, where two types of reasonable assumptions are introduced to overcome the nonlinearity of the SM-NLMS algorithm. Our results are generalizations of well-known results of the steady-state MSE of the epsiv-NLMS. Extensive simulations show the close agreement between our theories and experiments.

The Steady-State Mean-Square Error Analysis for Least Mean $p$-Order Algorithm

Based on series expansion, the steady-state mean-square error (MSE) analysis for real and complex least mean p-order (LMP) algorithm is developed, and some closed-form analytical expressions for the steady-state MSE and the corresponding restrictive conditions for step-size are given. Moreover, in Gaussian noise environments, its steady-state performance is also investigated. The analyses for some well-known algorithms and the computer simulation validate the accuracy of the results.

A simplified parametric channel estimation scheme for OFDM systems

This paper presents a novel parametric channel estimation scheme for orthogonal frequency-division multiplexing (OFDM) systems. In this scheme, two channel parameters, the number of channel paths and the multipath time delays, are estimated using the Hannan-Quinn (HQ) criterion and some simple operations; then the channel coefficients are estimated by utilizing these estimated channel parameters. Through complexity analysis and computer simulation, it is shown that our proposed scheme not only has better performance than the classical Yangpsilas scheme, but also has lower complexity. Additionally, two metrics, the mean square error (MSE) and the channel parameter error probability (CPEP), are analyzed. The MSE analysis gives the theoretical lower bound of the MSE that our scheme can attain, and the CPEP characterizes the rate with which our scheme estimates successfully the channel parameters.

Mean-Squared Error Analysis of Adaptive Subband-Based System Identification

Subband-based system identification is considered. The analysis in this paper gives geometric insights into the source of error in subband-based system identification. In particular, the mean-squared error arises as the residual after an orthogonal projection onto a subspace defined by the analysis filter impulse response. The minimum mean-squared identification error, which depends on the impulse response of the unknown system, is shown to be upper bounded by the error in a deterministic least-squares problem that involves the analysis filter response but is independent of the unknown system. The upper bounds may be used as criteria for analysis filter design that minimize the mean-squared error for the worst case unknown system, i.e., this is a minimax approach. An alternative subband processing structure is inspired by minimizing the projection residual. Examples show that significant reduction in minimum mean-squared error is possible.

Quantitative Performance Analysis of Scalogram as Instantaneous Frequency Estimator

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Instantaneous frequency (IF) estimation through the estimation of peak locations in the time-frequency plane is an important approach for signals contaminated with additive white Gaussian noise. In this paper, the forementioned analysis is carried out for continuous wavelet transform. The analysis of the scalogram as the instantaneous frequency estimator is performed for any FM signal regardless of the mother wavelet. Accurate expressions for the bias and the variance of the estimator are derived, and reveal that the bias and the variance are signal dependent. Results are statistically confirmed through the numerical analysis for several mother wavelets, and among considered wavelets, the Morlet wavelet produces the smallest estimation error. Furthermore, the performance of the IF estimator based on the scalogram and the spectrogram were compared through analysis of mean square error. These results showed that the scalogram with the Morlet wavelet exhibited good performance for the sample linear FM signal and the sample hyperbolic FM signal in comparison to the spectrogram. </para>

Reconstruction of Multidimensional Signals From Irregular Noisy Samples

We focus on a multidimensional field with uncorrelated spectrum and study the quality of the reconstructed signal when the field samples are irregularly spaced and affected by independent and identically distributed noise. More specifically, we apply linear reconstruction techniques and take the mean-square error (MSE) of the field estimate as a metric to evaluate the signal reconstruction quality. We find that the MSE analysis could be carried out by using the closed-form expression of the eigenvalue distribution of the matrix representing the sampling system. Unfortunately, such distribution is still unknown. Thus, we first derive a closed-form expression of the distribution moments, and we find that the eigenvalue distribution tends to the Marcenko-Pastur distribution as the field dimension goes to infinity. Finally, by using our approach, we derive a tight approximation to the MSE of the reconstructed field.

Analysis of Mean Square Errors of Spectral Reflectances Recovered by Multispectral Cameras

Analysis of Mean Square Errors of Spectral Reflectances Recovered by Multispectral Cameras

The Pure Numerical Graphical Study on The Errors of GM (1,1|α)

In the GM (1,1) study, the essential problem is ”how to establish the new known model to simulation the unknown given system, and to determine the best prediction results for the given system, in the other saying, the least error between real system to simulation model”. In this study, the graphical polygon area analyses are used to estimate the error of the number of equations in Least Square Method (LSQ Method) procedure and the value for the mean square error (MSE) analyses to determine a adjustable value of α in GM(1,1|α ). There are eight kinds of different data for original system’s output to analyze.

A new derivation of least-squares-fitting principle for OFDM channel estimation

Many channel estimation and data detection algorithms of the orthogonal frequency division multiplexing (OFDM) system have been proposed. Some of these algorithms are based on the principle of linear minimum mean-square error (LMMSE) estimation, which is theoretically optimal. There are also some algorithms developed based on the least-squares-fitting (LSF) principle, which finds a regression polynomial to fit a block of tentative channel estimates in the least-squares sense. The LSF principle is a non-statistical approach, while the LMMSE algorithm is statistical and it needs to known or estimate the channel statistics like correlation matrices and signal-to-noise ratio (SNR). This letter proposes a novel viewpoint of the LSF principle. We show that the non-statistical LSF principle can be derived alternatively from the statistical LMMSE principle by eigenvector approximation. This constructs a link between these two principles. The mean-square estimation error (MSEE) analysis shows that there are common terms in the MSEE expressions of these two principles. This further validates the constructed link. Based on the derived link and MSEE analysis, we also give some characteristics and discussions of the LSF principle.

An empirical investigation of the usefulness of ARFIMA models for predicting macroeconomic and financial time series

Abstract This paper addresses the notion that many fractional I ( d ) processes may fall into the “empty box” category, as discussed in Granger (Aspects of research strategies for time series analysis, Presentation to the Conference on New Developments in Time Series Economics, Yale University, 1999). We present ex ante forecasting evidence which suggests that ARFIMA models estimated using a variety of standard estimation procedures yield “approximations” to the true unknown underlying DGPs that sometimes provide significantly better out-of-sample predictions than AR, MA, ARMA, GARCH, and related models, based on analysis of point mean-square forecast errors (MSFEs), and based on the use of predictive accuracy tests. The strongest evidence in favor of ARFIMA models arises when various transformations of 5 major stock index returns are examined. Additional evidence based on analysis of the Stock and Watson (J. Bus. Econom. Stat. 20 (2002) 147–162) data set, the returns series data set examined by Ding et al. (J. Empirical Finance 1 (1993) 83–106), and based on a series of Monte Carlo experiments is also discussed.

Fixed-Point Error Analysis of CORDIC Processor Based on the Variance Propagation Formula

This paper presents a fixed-point mean-square error (MSE) analysis of coordinate rotation digital computer (CORDIC) processors based on the variance propagation method, whereas the conventional approaches provide only the error bound which results in large discrepancy between the analysis and actual implementation. The MSE analysis is aimed at obtaining a more accurate analysis of digital signal processing systems with CORDIC processor, especially when the design specification is given by the signal-to-noise ratio or MSE. For the MSE analysis, the error source and models are first defined and the output error is derived in terms of MSE in the rotation mode of the conventional CORDIC processor. It is shown that the proposed analysis can also be applied to the modified CORDIC algorithms. As an example of practical application, a fast Fourier transform processor using the CORDIC processor is presented in this paper, and its output error variance is analyzed with respect to the wordlength of CORDIC. The results show a close match between the analysis and simulation.

Blind OFDM Channel Estimation Through Simple Linear Precoding

A novel approach of blind channel estimation for orthogonal frequency-division multiplexing (OFDM) systems is proposed. A linear transformation is applied on each block before it enters the OFDM system. The transform imposes a correlation structure on the transmitted blocks, which is exploited at the receiver to recover the channel via simple cross-correlation operations. The proposed approach is computationally simple and converges fast, which makes it a good candidate for estimation of fast-varying channels. Its performance is tested analytically, through a mean-square error analysis, and also via simulations. Results show that it compares favorably to the training-based scheme used in the IEEE 802.11a wireless standard.

Desempenho do Algoritimo LMS na Identificação de Canais Variantes no Tempo e seu Emprego em Esquemas de Recepção MJSE-PSP

An analysis of the steady-state mean square error (MSE) performance of the Least Mean Square (LMS) algorithm in the identification of time-varying channels is presented. It is based on a set hypotheses usually adopted in the mobile communications context. The channel impulse response is modelled as time-varying transversal filter whose coefficients are wide-sense stationary stochastic processes with known power (Doppler) spectra. A generic expression of the steady state MSE as a function of the LMS step-size parameter is obtained. Two specific Doppler spectrum models are considered for which the optimization of the LMS step-size parameter in the sense of MSE minimization is also addressed. Close match between numerical and analytical results is shown in several examples. Besides, the impact of the LMS step-size parameter optimization on the performance of an adaptive MLSE (Maximum Likelihood Sequence Estimation) receiver is also addressed by Monte Carlo simulation.

MMSE detection in asynchronous CDMA systems: an equivalence result

The analysis of linear minimum mean-square error (MMSE) detection in a band-limited code-division multiple-access (CDMA) system that employs random spreading sequences is considered. The key features of the analysis are that the users are allowed to be completely asynchronous, and that the chip waveform is assumed to be the ideal Nyquist sinc function. It is shown that the asymptotic signal-to-interference ratio (SIR) at the detector output is the same as that in an equivalent chip-synchronous system. It is hence been established that synchronous analyses of linear MMSE detection can provide useful guidelines for the performance in asynchronous band-limited systems.

Analysis of mean-square error and transient speed of the LMS adaptive algorithm

For the least mean square (LMS) algorithm, we analyze the correlation matrix of the filter coefficient estimation error and the signal estimation error in the transient phase as well as in steady state. We establish the convergence of the second-order statistics as the number of iterations increases, and we derive the exact asymptotic expressions for the mean square errors. In particular, the result for the excess signal estimation error gives conditions under which the LMS algorithm outperforms the Wiener filter with the same number of taps. We also analyze a new measure of transient speed. We do not assume a linear regression model: the desired signal and the data process are allowed to be nonlinearly related. The data is assumed to be an instantaneous transformation of a stationary Markov process satisfying certain ergodic conditions.

Data Collection and Statistical Topics for the Preparation and Review of Manuscripts

The challenges encountered and discussions generated during the review process of the manuscripts submitted to the Variety Trials category of HortTechnology have revealed the need to review issues encountered during manuscript preparation and to provide flexible guidelines for authors and reviewers. Using a question/answer format, this manuscript discusses issues related to data collection and statistical methods available to compare varieties. Clear objectives and conclusions, adequate plot size, careful selection of entries, and sound statistical procedures are considered essential. Several additional factors (following standard production practices, using multiple seed sources, reporting analysis of variance table and mean square error, reporting multiyear/multilocation trials) are regarded as desirable, with different degrees of desirability, depending on the crop. These flexible guidelines should be viewed as recommendations for authors and reviewers rather than requirements. While defining the state-of-the-art in variety trialing is of interest to all those involved, it may be difficult to achieve when resources are limiting. It is ultimately the prerogative and responsibility of the author(s) to ensure that the work is scientifically sound.

Steady-state mean-square error analysis of the cross-correlation and constant modulus algorithm in a MIMO convolutive system

The cross-correlation and constant modulus algorithm (CC-CMA) has been proven to be an effective approach in the problem of joint blind equalisation and source separation in a multi-input and multi-output system. In the paper, the steady-state mean-square error performance of CC-CMA in a noise-free environment is studied, and a new expression is derived based on the energy preservation approach of Mai and Sayed (2000). Simulation studies are undertaken to support the analysis.

Closed-loop control of muscle length through motor unit recruitment in load-moving conditions

Neuroprostheses aimed at restoring lost movement in the limbs of spinal cord injured individuals are being developed in this laboratory. As part of this program, we have designed a digital proportional-integral-derivative controller integrated with a stimulation system which effects recruitment of motor units according to the size principle. This system is intended to control muscle length while shortening against fixed loads. Feline sciatic nerves were exposed and stimulated with ramp, triangular, sinusoidal, staircase and random signals as test inputs. Changes in muscle length and effective time delay under different conditions were measured and analyzed. Differences of tracking quality between open- and closed-loop conditions were examined through analysis of variance as well as the differences between small (250 g) and large (1 kg) loads. The results showed that parameters used to compare muscle length output to the input signals were dramatically improved in the closed-loop trials as compared to the open-loop condition. Mean squared correlation coefficients between input and output signals for ramp signals increased by 0.019, and for triangular signals by 0.12. Mean peak cross correlation between input and output signals for sinusoidal waveforms increased by 0.06, with decreases in time to peak cross correlation (effective time delay) from 195 to 38 ms. In slow random signals (power up to 0.5 Hz), peak cross correlation went from 0.74 to 0.89, and time-to-peak cross correlation decreased from 205 to 55 ms. In fast random signals (power up to 1 Hz), peak cross correlation went from 0.82 to 0.89, and time-to-peak cross correlation from 200 to 65 ms. For staircase signals, both rise times and mean steady-state errors decreased. It was found that, once the length range was set, the load weight had no effect on tracking performance. Analysis of mean square error demonstrated that for all signals tested, the feedback decreased the tracking error significantly, whereas, again, load had no effect. The results suggest that tracking is vastly improved by using a closed-loop system to control muscle length, and that load does not affect the quality of signal tracking as measured by standard control system analysis methods.

Performance analysis of adaptive MMSE reception for DS-CDMA in flat Nakagami fading channels

An efficient method is derived in this paper to obviate the integral involved in the performance analysis of the minimum mean square error (MMSE) detector for direct-sequence code-division multiple-access (DS/CDMA) systems in fading environments. The new method is shown to significantly reduce the necessary computational load for the performance analysis of the MMSE detector. We then examine the behavior of the MMSE detector in flat Nakagami fading channels.

Sinusoidal frequency estimation via sparse zero crossings

We consider estimation of the period of a sinusoid in additive Gaussian noise, based on observations of the zero-crossing (ZC) times. The problem is treated in a continuous-time framework. It is assumed that the signal-to-noise ratio is sufficient (approximately ⩾8 dB) such that the noise may be approximated as additive in the phase. An exact mean-square error analysis is provided for this approximation. We apply modified Euclidean algorithms (MEAs) and their least-squares refinements in this framework, to estimate the period of the sinusoid, with low complexity. Unlike linear regression methods based on phase samples, the proposed approach works with very sparse ZC measurements, and is resistant to outliers. The MEA-based approach is motivated by the fact that, in the noise-free case, the greatest common divisor (gcd) of a sparse set of the first differences of the zero crossing times is very highly likely to be the half-period of the sinusoid. The MEA acts to robustly estimate the gcd of the observed noisy data. The MEA period estimate may be refined via a least-squares approach, that asymptotically achieves the appropriate Cramer–Rao bound. Simulation results illustrate the algorithms with as few as 10 zero-crossing times. The algorithm behavior is also studied using Bernoulli and random burst models for the missing ZC times, and good performance is demonstrated with very sparse observations.