Minimum Mean Square Error Method Research Articles

Optimization of Signal Detection Using Deep CNN in Ultra-Massive MIMO

This paper addresses the evolving landscape of communication technology, emphasizing the pivotal role of 5G and the emerging 6G networks in accommodating the increasing demand for high-speed and accurate data transmission. We delve into the advancements in 5G technology, particularly the implementation of millimeter wave (mmWave) frequencies ranging from 30 to 300 GHz. These advancements are instrumental in enhancing applications requiring massive data transmission and reception, facilitated by massive MIMO (multiple input multiple output) systems. Looking towards the future, this paper forecasts the necessity for faster data transmission technologies, shifting the focus toward the development of 6G networks. These future networks are projected to employ ultra-massive MIMO systems in the terahertz band, operating within 0.1–10 THz frequency ranges. A significant part of our research is dedicated to exploring advanced signal detection techniques, helping to mitigate the impact of interference and improve accuracy in data transmission and enabling more efficient communication, even in environments with high levels of noise, and including zero forcing (ZF) and minimum mean square error (MMSE) methods, which form the cornerstone of our proposed approach. Additionally, signal detection contributes to the development of new communication technologies such as 5G and 6G, which require a high data transmission efficiency and rapid response speeds. The core contribution of this study lies in the application of deep learning to signal detection in ultra-massive MIMO systems, a critical component of 6G technology. We compare this approach with existing ELMx-based machine learning methods, focusing on algorithmic efficiency and computational performance. Our comparative analysis included the regularized extreme learning machine (RELM) and the outlier robust extreme learning machine (ORELM), juxtaposed with ZF and MMSE methods. Simulation results indicated the superiority of our convolutional neural network for signal detection (CNN-SD) over the traditional ELMx-based, ZF, and MMSE methods, particularly in terms of channel capacity and bit error rate. Furthermore, we demonstrate the computational efficiency and reduced complexity of the CNN-SD method, underscoring its suitability for future expansive MIMO systems.

An Ordered QR Decomposition based Signal Detection Technique for Uplink Massive MIMO System

Signal detection turns out to be a critical challenge in massive MIMO (m-MIMO) system due to the deployment of large number of antennas at the base station. Although, the mini mum mean square error (MMSE) is one of the popular signal detection method, but, it requires matrix inversion with cubic complexity. In order to reduce computational complexity, sev eral suboptimal signal detection methods were proposed such as Gauss-Seidel, successive over relaxation, Jacobi, Richardson methods. Although, these methods provide low complexity but their performance are limited to MMSE method. In this paper, we have proposed two signal detection techniques namely QR decompositions (QRD) and ordered QRD (OQRD). Finally, the performances of proposed signal detection methods are com pared with various conventional methods in terms of symbol er ror rate (SER) and computational complexity. The simulation re sults validate that the proposed methods outperform the MMSE method with substantially lower computational complexity.

A novel method for estimating propagation pathloss in millimeter-wave communication systems

In millimeter wave communication systems, the construction of a path loss model that is close to empirical one plays an important role in planning coverage, system capacity, and link budgets. In this paper, we propose a novel approach applied unsupervised machine learning for propagation pathloss estimation in millimeter wave communication systems based on the iterative procedure of cooperative and iterative evaluation exchange (Co-IEE) algorithm. The propagation path loss models of the time city–an urban area in the center of Hanoi, Vietnam in both line-of-sight (LOS) and non line-of-sight (NLOS) are estimated and calculated using the proposed approach and then they are compared to the minimum mean square error (MMSE). The estimated results of the proposed model show the approximation with the optimum one returned by MMSE method. Moreover, the proposed model of estimating path loss can solve the problem of sensitivity with outliers existing in MMSE and give more choices for path loss exponents.

Enhancing channel estimation accuracy in polar-coded MIMO–OFDM systems via CNN with 5G channel models

In this paper, a novel channel estimation technique for multiple-input multiple-output orthogonal frequency-division multiplexing (MIMO–OFDM) systems in fifth generation (5G) networks is proposed, which leverages the power of convolutional neural networks (CNNs) along with the application of polar coding. By employing polar encoding at the transmitter and polar decoding at the receiver, the proposed system enhances error correction performance, reliability, and data transmission rate in 5G networks. A novel optimized CNN-based channel estimation technique, named CNN-CENet, has been integrated into the polar-coded MIMO–OFDM receiver. This integration aims to address the interference and noise challenges in 5G systems. The proposed channel estimation method utilizes multiple layers of CNN to accurately characterize the properties of the real channel by incorporating least squares (LS) estimations. Hence, the proposed CNN-CENet provides an accurate estimation of the channel coefficients, enabling efficient signal detection at the receiver. The MATLAB simulation has been conducted using deep learning and 5G toolboxes to obtain results for various channel parameters. These results are then compared with the LS and minimum mean square error (MMSE) methods for performance evaluation. The proposed model surpasses the performance of the LS and MMSE methods, achieving a significant 95.6% reduction in mean square error (MSE) compared to LS and a substantial 59.7% reduction compared to MMSE. Additionally, it delivers notable bit error rate (BER) reductions of 59.8% and 53.2% for LS and MMSE, respectively, in low mobility scenarios, and 57.4% and 48.5% reductions in high mobility scenarios. This improvement leads to enhanced reliability and overall system efficiency.

Suitable pilot search method for channel estimation in underwater acoustic OFDM systems

The underwater environment poses challenges to the Underwater Acoustic OFDM (UWA-OFDM) system, causing the tendency of lacking pilots to recover the channel impulse response (CIR). Our previous research (pilot enrichment estimator) supplemented potential pilots based on the distance to the nearest constellation point below a fixed threshold T. However, this does not guarantee that the extracted pilots have sufficiently good quality. Therefore, this article presents a suitable pilot search (SPS) method with the flexible threshold T and second minimum mean square error (MMSE) estimation to improve channel estimation effectiveness in UWA-OFDM systems. Our approach is compared to the MMSE and PE methods across various criteria, such as pilot spacing and different modulators. The experiments demonstrate that the SPS estimator performs better than the MMSE and PE techniques regarding bit error rate (BER).

Determination and use of external color filter for improved camera colorimetry

Most digital cameras have spectral characteristics that make it impossible for the camera to provide accurate colorimetric information of a recorded scene. We extend previous work on using multiple filtered exposures by the use of a newly available capability of creating manufacturable filters using online software. Multiple criteria are considered in the determination of an optimal color filter on a training dataset. These criteria include the Vora-Value, figure of merit (FOM), training average ΔE, and training maximum ΔE. Our method includes the use of a realistic imaging noise model. We use two optimization methods: minimum mean square error (MMSE), which requires knowledge of the color sensitivities of the camera’s internal RGB filters and knowledge of the covariance matrices of the data ensemble and the noise, and a pseudoinverse method that uses only the recorded data and knowledge of the values of the training data. Filters were selected using the training data and the performance is measured on independent testing data. The results from both optimization methods show that filters chosen using the training ΔE criteria consistently outperformed the theoretical FOMs. It is also shown that the MMSE method is very sensitive to small perturbations of the internal filter sensitivities. Real camera images were recorded using the optimally determined filters from both methods. Measured errors provide confirmation of the simulated results.

Multiuser Massive MIMO-OFDM for Visible Light Communication Systems

Visible light communication (VLC) has emerged as a viable supplement to existing radio frequency (RF) communication systems. The limited modulation bandwidth of sources is one of the significant challenges in the multiuser (MU) VLC systems, which prohibits transmission at high data rates for each user. Since an array of LEDs is used to illuminate a room, a MU VLC system utilizing massive multiple-input multiple-output (mMIMO) orthogonal frequency-division multiplexing (OFDM) is proposed for achieving high data rates in this paper. Since the distances of the multiple transmitter-receiver links are different, their temporal delays are also different, resulting in complex channel gain and phase differences when transformed to the frequency domain in the OFDM technique. Complex channel vectors associated with different users are mutually spatially orthogonal as the number of transmitters increases. Therefore, inter-user interference (IUI) can be eliminated with simple linear signal processing, and more users can simultaneously communicate in the same time-frequency resource. In this paper, three linear precoding methods, including the maximum ratio transmission (MRT), the minimum mean square error (MMSE), and the zero-forcing (ZF), are investigated for VLC systems based on proposed MU-mMIMO-OFDM in intensity-modulation and direct-detection link. We evaluate and compare the performance of the aforementioned precoding methods in terms of the achievable spectral efficiency and total downlink optical power for different scenarios. Moreover, we derive a closed-form expression for the lower bound on the average achievable sum-rate, which we confirm the accuracy of the derived expression with the numerical simulation. The results demonstrate that the performance of the proposed method in this paper for all precoder techniques, i.e., ZF, MMSE, and MRT, is better than the previous works since the complex channel in the frequency domain is used in this paper. Furthermore, the ZF and MMSE methods are better than MRT when the ratio of the number of LEDs to the number of users is large. At the same time, when this ratio is relatively small, the MRT precoding technique outperforms the ZF and MMSE techniques.

Low-complexity soft-output signal detector based on AI-SSOR preconditioned conjugate gradient method over massive MIMO correlated channel

Massive multiple-input multiple-output (MIMO) techniques with a large number of antenna elements at base station (BS) have been proved as an alternative to provide potential opportunity to increase the spectrum and energy efficiency. However, in the system, there generally exists a spatial correlation effect due to insufficient antenna elements spacing and/or the lack of rich scattering at BS. The minimum mean square error (MMSE) method performs signal detection at the expense of large-scale matrix inversion operation. Thus, the conjugate gradient (CG) method has received a lot of attentions to realize the MMSE detection efficiently. Unfortunately, this efficiency can be compromised due to the ill-conditioned equalization matrix of MMSE method over the correlated channel environments. Moreover, the hard output signal detection exhibits a sharply degradation in performance for higher-order quadrature amplitude modulation (QAM). Therefore, the modern communication systems use the soft-output information, i.e., log-likelihood ratio (LLR) along with the forward error-correcting code (FEC) to achieve satisfactory performance. The LLR computation along with a higher-order QAM remains challenging due to the exhaustive search of symbol in the modulation constellation. In this paper, a low-complexity soft-output signal detector based on approximate inverse symmetric successive over-relaxation preconditioned conjugate gradient (AI-SSOR-CG-SOD) method is proposed to realize MMSE method detection for uplink multiuser massive MIMO correlated channel. In the proposed method, a new preconditioner, an AI-SSOR, which is based on the Neumann series approximation of the inverse of the conventional SSOR preconditioner is firstly developed to handle ill-conditioned matrix, and then incorporated with CG method to improve the convergence rate and performance. According to the characteristic of the Gray-coding that adjacent symbols in the constellation set have only one different bit, the constellation set is divided multiple times based on the bits of the inphase and the quadrature components of the symbol, which reduces the complexity of the LLR computation of the transmitted bits by avoiding the exhaustive search process. Simulation results show that the AI-SSOR preconditioner is robust against spatial correlation effect, and the proposed detector converges at 3 iterations. Simulation results also show that the proposed detector achieves a better trade-off between the complexity and the performance compared to other existing detectors.

Throughput Maximization for IRS-Aided MIMO FD-WPCN With Non-Linear EH Model

This paper studies an intelligent reflecting surface (IRS)-aided multiple-input-multiple-output (MIMO) full-duplex (FD) wireless-powered communication network (WPCN), where a hybrid access point (HAP) operating in FD broadcasts energy signals to multiple devices for their energy harvesting (EH) in the downlink (DL) and meanwhile receives information signals from devices in the uplink (UL) with the help of an IRS. Taking into account the practical finite self-interference (SI) and the non-linear EH model, we formulate the weighted sum throughput maximization optimization problem by jointly optimizing DL/UL time allocation, precoding matrices at devices, transmit covariance matrices at the HAP, and phase shifts at the IRS. Since the resulting optimization problem is non-convex, there are no standard methods to solve it optimally in general. To tackle this challenge, we first propose an element-wise (EW) based algorithm, where each IRS phase shift is alternately optimized in an iterative manner. To reduce the computational complexity, a minimum mean-square error (MMSE) based algorithm is proposed, where we transform the original problem into an equivalent form based on the MMSE method, which facilities the design of an efficient iterative algorithm. In particular, the IRS phase shift optimization problem is recast as an second-order cone program (SOCP), where all the IRS phase shifts are simultaneously optimized. For comparison, we also study two suboptimal IRS beamforming configurations in simulations, namely partially dynamic IRS beamforming (PDBF) and static IRS beamforming (SBF), which strike a balance between the system performance and practical complexity.

Joint Transmit Precoding and Reflect Beamforming Design for IRS-Assisted MIMO Cognitive Radio Systems

In this paper, we consider an intelligent reflecting surface (IRS)-assisted downlink cognitive radio (CR) system, in which a secondary access point (SAP) communicates with multiple secondary users (SUs) without affecting multiple primary users (PUs) in the primary network and all nodes are equipped with multiple antennas. Our design objective is to maximize the achievable weighted sum rate (WSR) of SUs subject to the total transmit power constraint at the SAP and the interference constraints at PUs, by jointly optimizing the transmit precoding at the SAP and the reflecting coefficients at the IRS. To deal with the complex objective function, the problem is reformulated by employing the well-known weighted minimum mean-square error (WMMSE) method and an alternating optimization (AO)-based algorithm is proposed. Furthermore, a special scenario with only a single PU and multiple SUs is considered and AO algorithm is adopted again. It is worth mentioning that the proposed algorithm has a much lower computational complexity than the above algorithm without the performance loss. Finally, some numerical simulations have been provided to demonstrate that the proposed algorithm outperforms other benchmark schemes.

Asymptotic characterisation of regularised zero‐forcing receiver for imperfect and correlated massive multiple‐input multiple‐output systems

In this work, the authors present an asymptotic high-dimensional analysis of the regularised zero-forcing receiver in terms of its mean-squared error (MSE) and bit error rate (BER) when used for the recovery of binary phase-shift keying (BPSK) modulated signals in a massive multiple-input multiple-output communication system. The asymptotic analysis is carried out when the system dimensions grow simultaneously large at a fixed ratio. The authors assume that the channel matrix is spatially correlated and not perfectly known. The authors use the linear minimum mean-squared error (LMMSE) method to estimate the channel matrix. The asymptotic approximations of the MSE and BER enable us to solve various practical optimisation problems. Under MSE/BER minimisation, the authors derive (1) the optimal regularisation factor for RZF; (2) the optimal power allocation scheme. The numerical simulations match the derived asymptotic expressions, even when the problem dimensions are small.

Likelihood ascent search‐aided low complexity improved performance massive MIMO detection in perfect and imperfect channel state information

SummaryMassive multiple‐input multiple‐output (MIMO) systems improve spectral efficiency and link reliability. Linear minimum mean‐squared error (MMSE) detectors can achieve optimal performance in massive MIMO detection but require large dimension matrix inversion, which is computationally intensive. Therefore, low complexity iterative detection schemes are proposed in the literature as an alternative to the exact MMSE method. However, the performance of these schemes is greatly influenced by the choice of the initial solution. Therefore, to improve the detection performance in this paper, we proposed three hybrid detection schemes, which are Newton–Schultz–Richardson (NS‐RI), Newton–Schultz–Chebyshev (NS‐Cheby), and Newton–Schultz–Gauss–Seidel (NS‐GS). The proposed hybrid schemes show significant performance improvement and a higher convergence rate compared to their original counterpart. The performance of the proposed detectors is further improved by the likelihood ascent search (LAS) stage, which corrects the detected symbols obtained from iterative MMSE methods through a neighborhood search. However, the complexity of the LAS algorithm primarily depends on the initialization step. In this work, we introduce an efficient Gram matrix computation in the real domain. Additionally, we have applied a band approximation of the Gram matrix for the LAS initialization, which reduces the order of computational complexity of the Gram matrix from to O(ωNTNR) where ω < <2NT.

A comparative analysis of computational intelligence algorithms for estimation of LTE channels

Precise modelling and accurate estimation of long-term evolution (LTE) channels are essential for numerous applications like video streaming, efficient use of bandwidth and utilization of power. This deals with the fact that data traffic is increasing continuously with advances in Internet of things. Previous works were focused mainly on designing models to estimate channel using traditional minimum mean square error (MMSE) and least squares (LS) algorithms. The proposed model enhances LTE channel estimation. The designed model combines LS and MMSE methods using Taguchi genetic (GE) and Particle Swarm Intelligence (PSO) algorithms. We consider LTE operating in 5.8 GHz range. Pilot signals are sent randomly along with data to obtain information about the channel. They help to decode a signal in a receiver and estimate LS and MMSE combined with Taguchi GA and PSO, respectively. CI-based model performance was calculated according to the bit error rate (BER), signal-to-noise ratio and mean square error. The proposed model achieved the desired gain of 2.4 dB and 5.4 dB according to BER as compared to MMSE and LS algorithms, respectively.

Forecasting Methods to Reduce Inventory Level in Supply Chain

Based on the two-level supply chain composed of suppliers and retailers, we assume that market demand is subject to an ARIMA(1, 1, 1). The supplier uses the minimum mean square error method (MMSE), the simple moving average method (SMA) and the weighted moving average method (WMA) respectively to forecast the market demand. According to the statistical properties of stationary time series, we calculate the mean square error between supplier forecast demand and market demand. Through the simulation, we compare the forecasting effects of the three methods and analyse the influence of the lead-time L and the moving average parameter N on prediction. The results show that the forecasting effect of the MMSE method is the best, of the WMA method is the second, and of the SMA method is the last. The results also show that reducing the lead-time and increasing the moving average parameter improve the prediction accuracy and reduce the supplier inventory level.

NL-MMSE: A Hybrid Phase Optimization Method in Multimaster Interferogram Stack for DS-InSAR Applications

When the distributed scatterer interferometric synthetic aperture radar (DS-InSAR) technology is used for surface deformation monitoring, the accuracy strongly depends on the quality of phase optimization. Especially in the low coherence region, how to conveniently and effectively improve the quality of phase optimization has been a hot and difficult research topic in recent years. This paper proposes a hybrid phase optimization method for the DSInSAR technology, which chooses either non-local (NL) or minimum mean square error (MMSE) method for each pixel according to the distribution of statistically homogeneous pixel (SHP) in two windows of different sizes. This hybrid method (NL-MMSE) is not limited by the number of interferogram and is not influenced by multi-master images. The NL-MMSE method was applied to the deformation monitoring in the Jinsha River basin, Tibet, China, using 28 Sentinel-1A SAR images acquired between February and December 2020. Compared with the NL and MMSE methods, the NL-MMSE method provides better phase quality of the interferogram stack and is able to extract more temporal coherence points for subsequent deformation inversion. The deformation monitoring results showed there are three obvious large-scale landslide deformation and dozens of small-scale deformation in the study area. It is demonstrated that the NL-MMSE method based on DSInSAR technology can accurately monitor the detailed deformation characteristics of the low coherence surface, and can be applied and promoted as an effective means of identifying and monitoring geological hazards.

Efficient Channel Prediction Technique Using AMC and Deep Learning Algorithm for 5G (NR) mMTC Devices

Efficient utilisation of adaptive modulation and coding ensures the quality transmission of information bits through the significant reduction in bit error rate (BER). Channel prediction using parametric estimation is not efficient for massive machine-type communication (mMTC) devices under the 5G New Radio (NR). In this paper, we have proposed a channel prediction scheme based on a deep learning (DL) algorithm possessed by parametric analysis. In deep learning, the pipeline methodology is used along with the image processing technique to predict the channel condition for optimal selection of the adaptive modulation and coding (AMC) profile. The deep learning-based pipelining approach utilises image restoration (IR) and image super-resolution (SR). The super-resolution method is used to de-noise the low-pixel 2-D image that is obtained from the parametric value of the beacon to predict the channel condition. The estimation results are compared with the conventional minimum mean square error (MMSE) and an approximation to the linear MMSE (ALMMSE) method, which is obtained through channel state information (CSI). The comparison results show that the parametric-enabled deep learning approach is superior, especially in poorer channel conditions. The performance of BER through parametric estimation along with the DL approach is ~66% more efficient as compared to the conventional MMSE method for BPSK mapping.

A Two-Stage Deep Neuroevolutionary Technique for Self-Adaptive Speech Enhancement

This paper presents a novel self-adaptive approach for speech enhancement in the context of highly nonstationary noise. A two-stage deep neuroevolutionary technique for speech enhancement is proposed. The first stage is composed of a deep neural network (DNN) method for speech enhancement. Two DNN methods were tested at this stage, namely, both a deep complex convolution recurrent network (DCCRN) and a residual long short-term memory neural network (ResLSTM). The ResLSTM method was combined with a minimum mean-square error method to perform a preliminary enhancement. The ResLSTM network is used as an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori</i> signal-to-noise ratio (SNR) estimator. The second stage implements a self-adaptive multiband spectral subtraction enhancement method using tuning optimization based on a genetic algorithm. The proposed two-stage technique is evaluated using objective measures of speech quality and intelligibility. The experiments are carried out using the NOIZEUS noisy speech corpus using conditions of real-world stationary, colored, and nonstationary noise sources at multiple SNR levels. These experiments demonstrate the advantage of building a cooperative approach using evolutionary and deep learning-based techniques that are capable of achieving robust speech enhancement in adverse conditions. Indeed, the experimental tests show that the proposed two-stage technique outperformed a baseline implementation using a state-of-the-art deep learning approach by an average 13% and 6% improvement for six noise conditions at a −5 dB and a 0 dB input SNR, respectively.

Joint waveform design of cognitive radar based on LSTM network in the presence of signal-dependent interference

Waveform design is a key component of cognitive radar. However, waveforms designed using a single criterion have difficulty meeting the multi-working mode of radar. We present a joint waveform design method in the presence of signal-dependent interference based on a long short term memory network. The proposed method combines characteristics of mutual information (MI) and signal to interference-plus-noise-ratio (SINR) criteria and achieves a smooth trade-off between them. The minimum mean square error (MMSE) method is set as a comparison to prove the enhanced capabilities in detection and recognition. Here, we initiate a final recognition rate indicator of radar to measure the radar’s comprehensive performance. As a result, compared with the MI criterion, SINR criterion, and MMSE method, to use as a transmission signal, the proposed paradigm increased the final recognition rate by 4.08%, 6.89%, and 5.25%, respectively, verifying the effectiveness of the proposed method.

An Efficient Indoor Positioning Method with the External Distance Variation for Wireless Networks

This study strengthens the external distance variation for the indoor positioning performance. With the received signal strength (RSS) of the unknown node, a localization is performed to positioning its coordinates. The mean square error (MSE) of localization is deteriorated by the shadowing effect and the MSE depends on the location of reference nodes. Moreover, the minimum mean square error (MMSE) algorithm is also used with the RSS. The amount of variation in the distance between the reference point and the positioning node will also affect the accuracy. Therefore, this paper considers the distance between the reference point and the positioning node and also the distance variation between the reference points. MSE is used to estimate positioning performance and Monte Carlo is also used to simulate the average error of different shadowing and decay environments. When reference nodes have known distances, the distance is obtained separately and the estimated distances are identified by the MMSE method. In order to reduce the number of reference nodes and calculation cost, this paper uses adaptive reference node selection to improve the accuracy of positioning. Simulation results show that the external distance variation mechanism strengthens the indoor positioning performance. Moreover, this paper investigates the performance of several reference nodes (three, four, five, and six reference nodes) through 3D graphs to estimate the small range area. The differences are more clearly observed with fewer reference nodes and lower MSE. Finally, simulation results show that the MSE value of fixed three reference nodes is almost 100% better with external distance variation method compared to the random selected three reference nodes.

Channel Estimation Aware Performance Analysis for Massive MIMO With Rician Fading

In this paper, by considering the average mean squared error (AMSE) of channel estimation, we primarily obtain the closed-from expressions of the probability density function (PDF) and cumulative distribution function of AMSE for the least squares (LS)/minimum mean squared error (MMSE) estimation method as the line-of-sight (LOS) component is known, where the asymptotic analysis is executed in Rayleigh fading and strong LOS conditions. Secondly, the closed-form expressions for the expectation of AMSE ( Exp <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">amse</sub> ) and variance of AMSE ( Var <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">amse</sub> ) are acquired, where Var <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">amse</sub> is inversely proportional to the number of antennas ( M). As M becomes infinite, the PDF of AMSE at Exp <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">amse</sub> has an order of root M. When the pilot power decreases with M in a power law, the LS case keeps deteriorating while the MMSE case converges to a constant which basically depends on the Rician K-factor. Next, the spectral efficiency is investigated by considering AMSE. When Exp <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">amse</sub> accelerates, the spectral efficiency of the LS method keeps dropping and that of the MMSE method firstly is degraded and then is improved to a constant except Rayleigh fading. Finally, all results are validated via simulations.