The random heat equation can be solved numerically by using **mean** **square** convergent Crank-Nicolson scheme. The random variable in the Crank-Nicolson scheme is must second order random variable and the random Crank-Nicolson scheme is unconditionally stable in the area of **mean** **square** **sense**. Many complicated equations in linear and nonlinear parabolic partial differential problems can be discussed using finite difference schemes in **mean** **square** **sense**.

Filtering in the most general terms is a process of noise removal from a measured process in order to reveal or enhance information about some quantity of interest. Any real data or signal measuring process includes some degree of noise from various possible sources. The desired signal may have added noise due to thermal or other physical effects related to the signal generation system, or it may noise may get added due to the measuring system or a digital data sampling process. Often the noise is a wide-**sense** stationary random process (has a constant finite **mean** and variance, and an autocorrelation function dependent only on the difference between the times of occurrence of the samples), which is known and therefore may be modelled by a common statistical model such as the Gaussian statistical model. It may also be random noise with unknown statistics. Otherwise, it may be a noise that is correlated in some way with the desired signal itself. Filtering, strictly means the extraction of information about some quantity of interest at the current time t by using data measured up to and including the time t. Smoothing, involves a delay of the output because it uses information extracted both after and before the current time t to extract the information. The benefit expected from introducing the delay is more to do with accuracy than filtering. Prediction, involves forecasting information some time into the future given the current and past data at time t and before. De convolution, involves the recovery of the filter characteristics given the filter’s input and output signals. Filters can be classified as either linear or nonlinear types. A linear filter is the one whose output is some linear function of the input. In the design of linear filters it is necessary to assume stationarity (statistical-time-invariance) and know the relevant signal and noise statistics a priori. The linear filter design attempts to minimise the effects of noise on the signal by meeting a suitable statistical criterion. The classical linear Wiener filter, for example, minimises the **Mean** **Square** Error (MSE) between the desired signal response and the actual filter response. The Wiener solution is said to be optimum in the **mean** **square** **sense**, and it can be said to be truly optimum for second-order stationary noise statistics (fully described by constant finite **mean** and variance). For non stationary signal and/or noise statistics, the linear Kalman filter can be used. Very well developed linear theory exists for both the Wiener and Kalman filters and the relationships between them.

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That is, assume that a linear combination of n uncorrelated families of random functions each of which is uniformly bounded in mean square and equicontinuous in mean square sense has at [r]

Compared to all LMS algorithms, RLS algorithm has a faster convergence speed and do not exhibit eigen value spread problem. It is a method of **least** squares for automatically adjusting the coefficient of a FIR filter without invoking assumptions on the statistics of input ECG signal. The estimation of RLS algorithm is done by minimizing the sum of squares of instantaneous error values [33]. The reason for faster convergence of this algorithm is, it denoises ECG signal by using inverse correlation matrix of the data assumed to be zero **mean** and also because it uses all the information contained in input data from, start of adaptation to the present. This improvement is achieved by increasing computational complexity of RLS filter. Inverse correlation matrix is computed directly in RLS algorithm [34]. Due to this feature RLS algorithm does not require any matrix inversion computations. The following equation provides the update coefficient for RLS algorithm [35].

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The LMS algorithm is the most efficient algorithm in terms of memory storage and calculations. In addition, the LMS algorithm has been the **least** suffering in terms of the numerical stability problem when compared with the problem inherent in RLS,and KF algorithms. This makes LMS algorithm is one of the first choices in several applications. The RLS algorithm is the good in terms of convergence properties. The RLS algorithm was proposed in order to provide superior performance compared to those of the LMS algorithm and its variants, with few parameters to be predefined, especially in highly correlated environments. In the RLS algorithm, an estimate of

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Adaptive filters are widely used in several digital signal processing applications. The most usually used adaptive filter is the tapped-delay line finite impulse response (FIR) filter whose weights are updated by the famous Widrow–Hoff **least** **mean** **square** (LMS) algorithm. Because it has not only simple in nature but also it has satisfactory convergence performance [1]. The direct form configuration on the forward path of the FIR filter results in a long critical path due to an inner-product computation to obtain a filter output. Therefore, it is necessary to minimize the critical path of the structure so that the critical path could not beat the sampling period, when the input signal sampling rate has a high. In current years, without multiplier DA-based system [2] has gained for thesignificant popularity for its high- throughput processing potential and reliability, which result in cost-effective and area–time efficient computing structures. Hardwarecapable DA-based design of adaptive filter has been suggested by Allred et al using two separate lookup tables (LUTs) for filtering and weight update. Author[3], have enhanced there system for filtering as well as weight updating by using only one lookup table. However, the system do not

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50% is the highest likely breakdown point of an estimator, which indicates that as many as half the observations could be discounted. A breakdown point higher than 0.5 is undesirable because it would **mean** that the estimate could be pertains to less than half of the data. Andersen (2012). 4. Application

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LMS incorporates an iterative procedure that makes successive corrections to the weight vector in the direction of the negative of the gradient vector which eventually leads to the **mean** **square** error which is of minimum value. LMS algorithm is relatively simpler compared to other algorithms; it does not require correlation function calculation nor does it require matrix inversions [3].

Overview: In this chapter, we investigate the maximum active noise control performance over a three-dimensional spatial space, by inves- tigating the capability of secondary sources in particular environment. We first formulate the spatial ANC problem in a 3-D room. Then we discuss a wave-domain **least** **square** method by matching the sec- ondary sound field to the primary sound field in wave domain. Fur- thermore, we extract the subspace from wave-domain coefficients of the secondary paths. We propose a subspace method by matching the secondary sound field to the projection of primary sound field in the subspace. Simulation results demonstrate the comparison between the wave-domain **least** **square** method and the subspace method, in terms of energy of the loudspeaker driving signals, noise reduction inside the region, and residual noise field outside the region. We also investigate the ANC performance under different loudspeaker configurations and noise source positions.

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where the total input power pertains to the sum of the **mean** squared value of the input data. When the adaptation parameter is little, the LMS technique consumes huge time to gain knowledge about its input with **least** **mean** **square** error and vice versa. Accordingly, a time changing step sized ordering of is desirable for optimal convergence [25].

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Abstract—In this paper, an eﬃcient method to obtain the elements current distribution for a non uniformly spaced array is presented. For a given far ﬁeld pattern, after sampling the array factor the proposed method uses the **least** **mean** **square** error technique to solve the system ofequations rather than solving the previously published Legendre function method. It’s shown that the average side lob level obtained by this proposed method is some 5 dB lower in comparison with the existing Legendre function method ofsolution. Ifthe Legendre function method published in the literature is to be used to solve for the current distribution, in the ﬁnal part ofthis paper, a criteria on how to choose suitable vectors that would result in a 3 dB lower side lobe level performance will be provided.

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By considering the purely sinusoidal voltage waveform, the system frequency is indicated by the time difference between two zero crossings. However, in general, the distorted forms of measured signals are available therefore numbers of techniques are there for the power system frequency estimation. In this paper **Least** **Mean** **Square** (LMS) filtering technique is used for the study of frequency estimation. In figure 1 the use of **least** **mean** **square** filter is used at signaling purposes.

In adaptive filtering [1], there is a class of algorithms specifically designed for sparse system identification, where the unknown system only has a few large coefficients while the remaining ones have a very small amplitude so that they can be ignored without significant effect on the overall performance of the system. A good example of them is the zero-attracting **least** **mean** **square** (ZA-LMS) algorithm proposed in [2]. This algorithm can achieve a higher convergence speed, and meanwhile, reduce the steady state excess **mean** **square** error (MSE). Compared to the classic LMS algorithm [3], the ZA- LMS algorithm introduces an l 1 norm in its cost function,

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Abstract: In this research the efficient and low computation complex signal acclimatizing techniques are projected for the improvement of Electroencephalogram (EEG) signal in remote health care applications. In clinical practices the EEG signal is extracted along with the artifacts and with some small constraints. Mainly in remote health care situations, we used low computational complexity filters which are striking. So, for the improvement of the EEG signal we introduced efficient and computation less Adaptive Noise Eliminators (ANE’s). These techniques simply utilize addition and shift operations, and also reach the required convergence speed among the other predictable techniques. The projected techniques are executed on real EEG signals which are stored and are compared with the effecting EEG arrangement. Our realizations visualize that the projected techniques offer the best concert over the previous techniques in terms of signal to noise ratio, mathematical complexity, convergence rate, Excess **Mean** **Square** error and Mis adjustment. This approach is accessible for the brain computer interface applications.

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Here, we consider the clinical data as the primary channel of an adaptive LMS filter. The OTC product groups are then used to estimate the daily clinical data in the follow- ing manner. Today's and several past days' OTC data are used together to find an estimate of today's clinical data, which is then compared to the actual value of today's clin- ical data to update the filter coefficients in such a way as to minimize the **mean** **square** error between the today's Map of the Urban National Capital Area (NCA) in the United

Two-stage training [17][22][36][264] is often used for constructing RBF neural networks. At the ﬁrst stage, the hidden layer is constructed by selecting the center and the width for each hidden neuron using various clustering algorithms. At the second stage, the weights between hidden neurons and output neurons are determined, for example by using the lin- ear **least** **square** (LLS) method [22]. For example, in [177][280], Kohonen’s learning vector quantization (LVQ) was used to determine the centers of hidden units. In [219][281], the k-means clustering algorithm with the se- lected data points as seeds was used to incrementally generate centers for RBF neural networks. Kubat [183] used C.4.5 to determine the centers of RBF neural networks. The width of a kernel function can be chosen as the standard deviation of the samples in a cluster. Murata et al. [221] started with a suﬃcient number of hidden units and then merged them to reduce the size of an RBF neural network. Chen et al. [48][49] proposed a constructive method in which new RBF kernel functions were added gradually using an orthogonal **least** **square** learning algorithm (OLS). The weight matrix is solved subsequently [48][49].

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Least Orthogonal Distance Estimator and Total Least Square Naccarato, Alessia and Zurlo, Davide and Pieraccini, Luciano Department of Economics - Roma Tre University.[r]

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It has been shown that the proposed VSSG algorithm is more efficient for 4G- LTE cell boundary users, where the transmission signal strength from eNodeB is less to provide the better QoS with less path loss and interference. Therefore, one can avoid the number iterations and time delay required to converge the **Mean** **square** error to zero during the UE’s handover between source eNodeB to target eNodeB, with respect to different beam steering algorithms , so that high data service users will maintain the required QoS for the data services without much packet loss. Using the proposed algorithm with more number of transmitting and receiving antenna (MIMO), the better Quality of service can be provided without any degradation for handover region users

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which is a rst order autoregressive (AR) process with a pole at . For the Gaussian case, w(n) is a white, zero-**mean**, Gaussian random sequence, having unit variance, and is set to 0.9. As a result, a highly colored Gaussian signal is generated. For the uniform case, w(n) is a uniformly distributed random sequence between -1.0 and 1.0 and is again set to 0.9. Measurement noise, v(n), with 2