Mean Square Error Curve Research Articles

A Hybrid Digital-Analog Chaotic Code

Traditional analog chaotic codes suffer from the vulnerability of signs at the receiver, and this situation deteriorates at low signal-to-noise ratio (SNR). In order to solve this problem, this letter proposes a hybrid digital-analog chaotic code (HDACC), which uses chaotic states (analog value) and part of their signs (digital value) together as parity codewords. These signs are further encoded in a digital way. We derived the theoretical decoding mean square error (MSE) of HDACC over additive white Gaussian noise channel and gave the optimal digital/analog codewords allocation. Simulation results demonstrate that the derived theoretical MSE matches very well with simulated MSE curves. Evaluations show that HDACC outperforms existing related works in a wide SNR range.

Recursive inverse algorithm: Mean-square-error analysis

The recently proposed recursive inverse (RI) adaptive algorithm has shown improved performance compared to some well-known adaptive algorithms [1]. However, there has been no detailed study of its performance. In this paper, we provide an analytical treatment of the ensemble-average learning curve of the RI algorithm. A novel analytical result which describes the learning behavior of the RI algorithm is obtained. It is shown that within limits of approximation, the excess mean-square-error (MSE) of the algorithm approaches zero and the RI algorithm converges to a lower steady-state MSE than the LMS algorithm. The results show that the theoretical and experimental MSE curves of the RI algorithm are in agreement. Also, the MSE analysis of the RI algorithm in a nonstationary environment, where the optimum weight is assumed to be randomly changing about a fixed vector, is derived.

Tracking analysis of augmented complex least mean square algorithm

SummaryThe augmented complex least mean‐square (ACLMS) algorithm is a suitable algorithm for the processing of both second‐order circular (proper) and noncircular (improper) signals. In this paper, we provide tracking analysis of the ACLMS algorithm in the non‐stationary environments. Using the established energy conservation argument, we derive a variance relation that contains moments that represent the effects of non‐stationary environment. We evaluate these moments and derive closed‐form expressions for the excess mean‐square error (EMSE) and mean‐square error (MSE). The derived expressions, supported by simulations, reveal that unlike the stationary case, the steady‐state EMSE, and MSE curves are not monotonically increasing functions of the step‐size parameter. We also use this observation to optimize the step‐size learning parameter. Simulation results illustrate the theoretical findings and match well with theory. Copyright © 2015 John Wiley & Sons, Ltd.

Blind Channel Estimation Based on Multilevel Lloyd-Max Iteration for Nonconstant Modulus Constellations

In wireless communications, knowledge of channel coefficients is required for coherent demodulation. Lloyd-Max iteration is an innovative blind channel estimation method for narrowband fading channels. In this paper, it is proved that blind channel estimation based on single-level Lloyd-Max (SL-LM) iteration is not reliable for nonconstant modulus constellations (NMC). Then, we introduce multilevel Lloyd-Max (ML-LM) iteration to solve this problem. Firstly, by dividing NMC into subsets, Lloyd-Max iteration is used in multilevel. Then, the estimation information is transmitted from one level to another. By doing this, accurate blind channel estimation for NMC is achieved. Moreover, when the number of received symbols is small, we propose the lacking constellations equalization algorithm to reduce the influence of lacking constellations. Finally, phase ambiguity of ML-LM iteration is also investigated in the paper. ML-LM iteration can be more robust to the phase of fading coefficient by dividing NMC into subsets properly. As the signal-to-noise ratio (SNR) increases, numerical results show that the proposed method’s mean-square error curve converges remarkably to the least squares (LS) bound with a small number of iterations.

Study of the convergence behavior of the complex kernel least mean square algorithm.

The complex kernel least mean square (CKLMS) algorithm is recently derived and allows for online kernel adaptive learning for complex data. Kernel adaptive methods can be used in finding solutions for neural network and machine learning applications. The derivation of CKLMS involved the development of a modified Wirtinger calculus for Hilbert spaces to obtain the cost function gradient. We analyze the convergence of the CKLMS with different kernel forms for complex data. The expressions obtained enable us to generate theory-predicted mean-square error curves considering the circularity of the complex input signals and their effect on nonlinear learning. Simulations are used for verifying the analysis results.

Experimental and Analytical Studies on Improved Feedforward ML Estimation Based on LS-SVR

Maximum likelihood (ML) algorithm is the most common and effective parameter estimation method. However, when dealing with small sample and low signal-to-noise ratio (SNR), threshold effects are resulted and estimation performance degrades greatly. It is proved that support vector machine (SVM) is suitable for small sample. Consequently, we employ the linear relationship between least squares support vector regression (LS-SVR)’s inputs and outputs and regard LS-SVR process as a time-varying linear filter to increase input SNR of received signals and decrease the threshold value of mean square error (MSE) curve. Furthermore, it is verified that by taking single-tone sinusoidal frequency estimation, for example, and integrating data analysis and experimental validation, if LS-SVR’s parameters are set appropriately, not only can the LS-SVR process ensure the single-tone sinusoid and additive white Gaussian noise (AWGN) channel characteristics of original signals well, but it can also improves the frequency estimation performance. During experimental simulations, LS-SVR process is applied to two common and representative single-tone sinusoidal ML frequency estimation algorithms, the DFT-based frequency-domain periodogram (FDP) and phase-based Kay ones. And the threshold values of their MSE curves are decreased by 0.3 dB and 1.2 dB, respectively, which obviously exhibit the advantage of the proposed algorithm.

Steady-State Analysis of Diffusion LMS Adaptive Networks With Noisy Links

In this correspondence, we analyze the effects of noisy links on the steady-state performance of diffusion least-mean-square (LMS) adaptive networks. Using the established weighted spatial-temporal energy conservation argument, we derive a variance relation which contains moments that represent the effects of noisy links. We evaluate these moments and derive closed-form expressions for the mean-square deviation (MSD), excess mean-square error (EMSE) and mean-square error (MSE) to explain the steady-state performance at each individual node. The derived expressions, supported by simulations, reveal that unlike the ideal link case, the steady-state MSD, EMSE, and MSE curves are not monotonically increasing functions of the step-size parameter when links are noisy. Moreover, the diffusion LMS adaptive network does not diverge due to noisy links.

Steady-State Analysis of Incremental LMS Adaptive Networks With Noisy Links

Recently proposed adaptive networks assume perfect communication among the nodes. In this correspondence, we extend existing analysis to study the performance of incremental least mean square (LMS) adaptive networks in a more realistic case in which communication links between nodes are considered noisy. More precisely, using weighted spatial-temporal energy conservation relation, we arrive a variance relation which contains moments that represent the effects of noisy links. We evaluate these moments and derive closed-form expressions for the mean-square deviation (MSD), excess mean-square error (EMSE) and mean-square error (MSE) to explain the steady-state performance at each individual node. The derived expressions have good match with simulations. However, the main result is that unlike the ideal link case, the steady-state MSD, EMSE, and MSE curves are not monotonically increasing functions of the step-size parameter when links are noisy. We illustrate this behavior and also find the optimal step-size in a closed-form (for a special case) which minimizes the steady-state values of MSD, EMSE, and MSE in each node. Simulations are also provided to clarify the derived theoretical results.

A fast-converging space-time adaptive processing algorithm for non-Gaussian clutter suppression

The normalized fractionally-lower order moment (NFLOM) algorithm differs from the normalized least mean square (NLMS) algorithm in that it minimizes the lower order moment (p<2) of the error rather than the variance (p=2). This paper first evaluates the performances of the NFLOM for space-time adaptive processing in heavy-tailed compound K clutters in terms of the excess mean square error (MSE), misalignment, beampatterns, and output signal-to-interference-and-noise-ratio (SINR). The results show that the MSE curve of a small-order NFLOM exhibits faster convergence but higher steady-state error than a large-order NFLOM. Second, this paper proposes a new variable-order FLOM algorithm to dynamically change the order during adaptation, thus achieving both fast initial convergence and low steady-state error. The new algorithm is applied to STAP for Gaussian and non-Gaussian clutter suppression. The simulation results show that it achieves the best compromise between fast convergence and low steady-state error in both types of clutters.

Liver tissues classification by artificial neural networks

Abstract Applying artificial neural networks to the classification of ultrasonic images of liver tissue is a new attempt. Due to the poor quality of the ultrasonic images, classification of the liver tissue with ultrasonic imaging is always regarded as a difficult task. In order to utilize the neurons of the self-organizing mapping algorithm more efficiently and to improve accuracy rate, in this paper we propose the Dynamic Neurons-Adding (DNA) method, and point out two advantages, mean-square error curve and entropy curve with regard to deciding the stopping of the training procedure and to avoiding useless training. DNA is compared with the type one and type two of the learning vector quantization (LVQ) algorithms. Regarding the training time, the accuracy rate or the stability, DNA is always better than LVQs and, at least if not last, the Dynamic Neurons-Adding (DNA) method requires substantially fewer neurons.

Mean Square Error Versus Parameter Curves for Model Reference Adaptive Systems with Sinusoidal Input†

ABSTRACT The nature of performance index versus parameter curves determines the applicability of perturbation method. Mean square error (MSE) is taken as the PI in this paper. It is shown that with sinusoidal input there exists one maximum and one minimum for the MSE versus parameter curve irrespective of the order of the transfer function and irrespective of the model being an exact replica of the system or not. The location and value of the parameter corresponding to extremum performance index are found to be dependent on the adjustments made in the other adaptive loops (i.e. interaction between the adaptive loops) and the frequency of input. It is therefore concluded that the perturbation method in its simplest form cannot be applied without suitable modifications, and the necessary modifications are suggested. Three typical examples arc included to give a better understanding of the nature of adjustment to be done when two parameters are varying.