Linear Minimum Mean Square Error Research Articles

Optimal pilot pattern for data‐aided channel estimation for MIMO‐OFDM wireless systems

AbstractThis article presents an optimal pilot pattern for the data‐aided channel estimation (DACE) scheme for both single‐input single‐output (SISO) and multiple‐input multiple‐output orthogonal frequency division multiplexing (MIMO‐OFDM) wireless systems. The research evaluates the performance of the DACE scheme using different comb‐type pilot patterns for both least square (LS) and linear minimum mean square error (LMMSE) channel estimators. In this regard, it is found that pilot spacing significantly influences system performance. Inserting pilot symbols in consecutive subcarriers cannot compensate for increased pilot spacing. Hence, the solution to this problem is to place pilot symbols at appropriate locations within the given spectrum. Moreover, data symbols which are reliably detected at the receiver are used as additional pilot signals to further enhance system performance. However, reliable data symbols need to be determined carefully because wrong detection results in severe performance degradation. In this respect, the proposed comb‐type pilot pattern using a single pilot subcarrier extracts the maximum number of reliable data symbols for the DACE scheme, improves channel estimation accuracy, and provides bandwidth optimization for MIMO‐OFDM systems. Furthermore, it outperforms all other pilot patterns in terms of system mean square error (MSE) and bit‐error‐rate (BER) performance.

Low complexity stationary iteration based approximate inversion for signal detection in OTFS system

Orthogonal time frequency space (OTFS) system offers a high data rate and effectively exploits full diversity compared to orthogonal frequency division multiplexing (OFDM), achieving a seamless trade-off between data rate and processing gain. However, detecting OTFS signals is challenging due to the complex conversion between delay-Doppler (DD) and time domains. In this article, we propose a stationary iteration-based approximate inversion (SIAI) technique for low-complexity detection in uplink OTFS systems.The proposed SIAI detection technique features square-order computational complexity and delivers performance close to that of a linear minimum mean square error (LMMSE) detector. Simulation results demonstrate that the SIAI technique outperforms several state-of-the-art detection methods in terms of both error performance and computational complexity. Additionally, the robustness of the SIAI technique is validated in scenarios with imperfect channel state information at the receiver.

Development of hybrid sewing training-based dingo optimization algorithm for massive MIMO spectral efficiency maximization system

ABSTRACT A Base Station (BS) with several antenna elements serves a number of users in the frequency of the same time- slot in a huge multiuser “Multiple-Input Multiple-Output (MIMO)”. It has emerged as a viable method for the future generation of wireless networks due to its capacity to enhance both spectrum and efficiency of energy, even with straightforward linear processing like “Maximum Ratio Combining (MRC), Linear Minimum Mean Square Error (LMMSE) and Zero Forcing (ZF)”. In a power allocation scheduling method, only users who do not interfere with one another can change their power at any one time. These studies all take into account orthogonal channels. However, the “Spectral Efficiency (SE)” is reduced by the channels’ non-orthogonality. This encourages us to research how to maximize spectral efficiency in “Massive MIMO (M-MIMO)” systems by allocating power for both uplink and downlink transmission. Here, the Hybrid of Sewing Training and Dingo Optimization (HSTDO) is developed for executing the best resource allocation in the M-MIMO system to improve SE performance which helps to enhance the SE of the M-MIMO system. The simulation is carried out to compare the developed model with the conventional works to reveal the better spectral efficiency of the suggested algorithm.

Adaptive filter design for cyclostationary processes associated with sine-wave extraction operation

Cyclostationary signals have been generally used in condition monitoring and fault diagnosis of bearings, for which the adaptive filter is important in obtaining the signal-of-interest cyclostationary process from a disturbed signal. However, due to the nonstationarity of cyclostationary signals, their adaptive filters are usually time-varying and difficult to be implemented. In this study, a new optimum filter is obtained by converting the time-varying optimization function to be time-invariant using a sine-wave extraction (SE) operator, for which the adaptive implementation is designed and analyzed. The time-varying linear minimum mean-square error is converted to be time invariant using the SE operator, and its solution is time invariant but without explicit expression. Subsequently, an adaptive implementation, called the SE least mean square (SE-LMS), is introduced, for which the mean and weighted mean-square-deviation (MSD) are explicitly expressed. The scope of step size is determined to obtain a mean-convergent and mean-square-stable SE-LMS. The steady-state SE-operated excess mean-square-error is analyzed, and its explicit expression is obtained. The SE-LMS algorithm uses only a single cyclic frequency, based on which, the combined SE-LMS algorithm is introduced for a cyclostationary signal with multiple cyclic frequencies. Finally, several simulations are designed to verify the superiority of the proposed estimators over the time-average operator-based estimator.

Hybrid Optimized LMMSE-Based Channel Estimation with Low Power Trellis Coded Modulation

Data transfer in a wireless channel environment is plagued by numerous security issues and power transmission loss. Channel state information is crucial to attaining the performance benefit of reconfigurable intelligent surfaces (RISs) in wireless systems. Because of the Doppler effect, mobility makes accurate channel estimate (CE) more difficult. The parameter estimation approach is employed as an alternative to direct estimation in the channel matrix to address the aforementioned problems. Also, to lower the Mean Square Error (MSE) between the estimated and original channel, training-based CE, such as Linear Minimum Mean Square Error (LMMSE), and Least Square Error (LSE), are widely used in some wireless standards. To optimize the channel, certain intelligent optimized strategies are presented. For the Optimal CE, a Trellis Coded Modulation (TCM) with hybrid optimization in LMMSE has been provided. Moth amalgamated Elephant Herding Optimization (MAEHO) algorithm is the proposed hybrid optimization. Finally, the performance of the accepted model is determined in comparison to other existing techniques using a variety of metrics.

Phase Noise Effects on OFDM Chirp Communication Systems: Characteristics and Compensation

Orthogonal frequency-division multiplexing (OFDM) chirp waveforms are an attractive candidate to be a dual-function signal scheme for the joint radar and communication systems. OFDM chirp signals can not only be employed to transmit communication data through classic phase modulation, but also can perform radar detection by applying linear frequency modulation for subcarriers. However, the performance of the OFDM chirp communication system under the phase noise environment still remains uninvestigated. This paper tries to discuss the influence of phase noise on OFDM chirp communication systems and proposes effective phase noise estimation and compensation methods. We find that the phase noise effect on OFDM chirp communication systems consists of a common phase error (CPE) and an inter-carrier interference (ICI). If not compensated, the performance of the dual-function systems can be seriously degraded. In particular, an exact expression for the signal-plus-interference to noise ratio (SINR) for the OFDM chirp communication system is derived and some critical parameters are analyzed to exhibit the phase noise effects on system performance. Moreover, two low-complexity estimation approaches, maximum likelihood (ML) and linear minimum mean square error (LMMSE), as well as two compensation approaches, the de-correlation and cancellation algorithms, are respectively utilized to eliminate the phase noise impairment. Finally, the phase noise effects and the effectiveness of the compensation approach are verified by extensive numerical results.

An inspired chaos-based estimation-theory optimization for low-density parity-check (LDPC) code decoding

Optimization techniques have emerged as powerful tools for solving complex engineering problems in real-time applications, surpassing traditional heuristic methods. This paper introduces a novel chaos optimization algorithm that integrates the bisection search method to enhance the convergence ability, solution accuracy, and search speed of conventional optimization problems. The proposed algorithm is applied to the design and analysis of a reduced complexity decoding scheme using Low-Density Parity-Check (LDPC) codes. By leveraging the Linear Minimum Mean Square Error (LMMSE) estimation criterion, the proposed method approximates the estimated parameters acquired during the message computation process. This combined approach effectively mitigates error propagation and premature convergence issues. The presented research offers a low-complexity decoding algorithm for LDPC codes, leveraging an optimized chaos bisection search method. The adoption of the LMMSE criterion overcomes data dependency and error propagation concerns during the estimation process. The modified chaos optimization technique, integrated with the bisection search, efficiently determines optimal values from perturbed observations, leading to improved convergence rates and data processing precision by reducing the observation dimension (search space). Simulation results validate the proposed approach, showcasing its ability to mitigate error propagation and premature convergence while maintaining coding gain accuracy. Experimental validation on an FPGA RF-SoC device (model XCZU27DR) demonstrates that the proposed method achieves a maximum throughput of 8.7 Gbps at a clock frequency of 386.72 MHz, with reduced hardware complexity. The experimental verification on an FPGA RF-SoC device highlights the algorithm's minimal hardware complexity, superior decoding throughput, and Bit Error Rate (BER) performance.

Gaussian Particle Filtering for Nonlinear Systems With Heavy-Tailed Noises: A Progressive Transform-Based Approach.

The Gaussian particle filter (GPF) is a type of particle filter that employs the Gaussian filter approximation as the proposal distribution. However, the linearization errors are introduced during the calculation of the proposal distribution. In this article, a progressive transform-based GPF (PT-GPF) is proposed to solve this problem. First, a progressive transformation is applied to the measurement model to circumvent the necessity of linearization in the calculation of the proposal distribution, thereby ensuring the generation of optimal Gaussian proposal distributions in sense of linear minimum mean-square error (LMMSE). Second, to mitigate the potential impact of outliers, a supplementary screening process is employed to enhance the Monte Carlo approximation of the posterior probability density function. Finally, simulations of a target tracking example demonstrate the effectiveness and superiority of the proposed method.

Channel reconstruction for mmWave Massive MIMO systems based on Channel Path Map

For the needs of high data rate and massive device access in current fifth-generation (5G) mobile communication networks and future sixth-generation (6G) mobile communication networks, Millimeter-Wave (mmWave) Massive Multiple-Input Multiple-Output (MIMO) is a crucial technical support. However, the conventional pilot based channel estimation methods are exceedingly time-consuming and substantially degrade system performance when the number of users or antennas is high. This paper proposes a Path Clustering based Channel Reconstruction (PCCR) scheme for channel reconstruction in order to solve this issue. By leveraging channel information (such as angle of arrival, angle of departure, and path gain) at a number of geographical locations that are recorded in the Channel Path Map (CPM), we aim to reconstruct the mmWave channel of any locations of interests. To be specific, the CPM is used to extract the neighborhood propagation path data, and an autonomous determination algorithm is used to find the number of path clusters at the location of interest. The recorded propagation path information is processed by the Equivalent Base Station (EBS) scheme, and the path clustering is accomplished by using the AGglomerative NESting (AGENS) algorithm. In each path cluster, Inverse Distance Weighted (IDW) algorithm is developed to forecast the path information at the user location. With all path information at user location are forecast, the channel can be reconstructed. The anticipated overhead is substantially lower than that of conventional methods because the computational cost of the procedure only depends on the total number of paths at the nearby locations. Ray-tracing based simulations are carried out to demonstrate the performance of the proposed algorithm. The results show that, when applied to a CPM with a spatial location precision of 5 m, the Normalized Mean Squared Error (NMSE) of the proposed algorithm is nearly identical to that of Linear Minimum Mean Square Error (LMMSE) estimation. The actual achievable date rate can match the needs of upcoming wireless communications and is superior over the conventional LMMSE.

Enhanced MIMO-DCT-OFDM system using cosine domain equaliser

ABSTRACT The Discrete Cosine Transform (DCT) can be used instead of the conventional Discrete Fourier Transform (DFT) for the Orthogonal Frequency Division Multiplexing (OFDM) construction, which offers many advantages. In this paper, the Multiple-Input-Multiple-Output (MIMO) DCT-OFDM is enhanced using a proposed Cosine Domain Equaliser (CDE) instead of a Frequency Domain Equaliser (FDE). The results are evaluated through the Rayleigh fading channel with Co-Carrier Frequency Offset (Co-CFO) of different MIMO configurations. The average bit error probability and the simulated time of the proposed scheme and the conventional one are compared, which indicates the importance of the proposed scheme. Also, a closed formula for the number of arithmetic operations of the proposed equaliser is developed. The proposed equaliser gives a simulation time reduction of about 81.21%, 83.74% compared to that of the conventional Linear Zero Forcing (LZF)-FDE and Linear Minimum Mean Square Error (LMMSE)-FDE, respectively, for the case of a 4 × 4 configuration.

Achievable Rate Analysis of the STAR-RIS-Aided NOMA Uplink in the Face of Imperfect CSI and Hardware Impairments

Reconfigurable intelligent surfaces (RIS) are capable of beneficially ameliorating the propagation environment by appropriately controlling the passive reflecting elements. To extend the coverage area, the concept of simultaneous transmitting and reflecting reconfigurable intelligent surfaces (STAR-RIS) has been proposed, yielding supporting 360° coverage user equipment (UE) located on both sides of the RIS. In this paper, we theoretically formulate the ergodic sum-rate of the STAR-RIS assisted non-orthogonal multiple access (NOMA) uplink in the face of channel estimation errors and hardware impairments (HWI). Specifically, the STAR-RIS phase shift is configured based on the statistical channel state information (CSI), followed by linear minimum mean square error (LMMSE) channel estimation of the equivalent channel spanning from the UEs to the access point (AP). Afterwards, successive interference cancellation (SIC) is employed at the AP using the estimated instantaneous CSI, and we derive the theoretical ergodic sum-rate upper bound for both perfect and imperfect SIC decoding algorithm. The theoretical analysis and the simulation results show that both the channel estimation and the ergodic sum-rate have performance floor at high transmit power region caused by transceiver hardware impairments.

Nonlinear Equalization for Performance Enhancement of FBMC Systems Using Walsh Transform

The Filter Bank Multi-Carrier (FBMC) is an interesting recent technology for next-generation wireless radio communication systems. Equalization is a fundamental technique for enhancing Bit Error Rate (BER) performance. The traditional Linear Zero Forcing (LZF) and Linear Minimum Mean Square Error (LMMSE) equalizers have large noise amplification and high complexity, respectively. As the estimated error of the Signal-to-Noise Ratio (SNR) grows, the BER performance of the LMMSE equalizer degrades. In this paper, we present a Joint Low Complexity Optimized ZF-Successive Interference Cancellation (JLCOZF-SIC) equalizer for FBMC communication systems that are based on the Walsh–Hadamard Transform (WHT) rather than the Fast Fourier Transform (FFT). As in the MMSE-Successive Interference Cancellation (MMSE-SIC) equalizer, the proposed equalizer avoids the estimating error induced by average power per symbol estimation. The simulation results show that the proposed algorithm improves the system performance more than the existing algorithms. Furthermore, the proposed equalization provides stability in the BER performance at high levels of the co-Carrier Frequency Offset (co-CFO). In terms of computational complexity, the proposed scheme has a lower computational complexity than traditional schemes.

Autoencoder assisted subcarrier optimization for nonlinear frequency division multiplexing.

Nonlinear frequency division multiplexing (NFDM) is a novel optical communication technique that can achieve nonlinear free transmission. However, current design of NFDM is analogous to orthogonal frequency division multiplexing (OFDM), where sinc function is utilized as subcarriers, which may not be optimal for nonlinear spectrums. In this paper, we propose an auto-encoder (AE) assisted subcarrier optimization scheme for dual-polarized (DP) NFDM systems. Numerical verifications show that our scheme can improve the Q-factor by 1.54 dB and 0.62 dB compared to sinc subcarrier and linear minimum mean square error (LMMSE) equalization, respectively, in a 960 km transmission scenario. We also analyze the characteristics of the optimized subcarriers and discuss how they enhance the performance. Furthermore, we demonstrate the robustness of the optimized subcarriers to different modulation formats, transmission distances and bandwidth. Our work provides a new idea in subcarrier design for NFDM.

Data-Aided CSI Estimation Using Affine-Precoded Superimposed Pilots in Orthogonal Time Frequency Space Modulated MIMO Systems

An orthogonal affine-precoded superimposed pilot (AP-SIP)-based architecture is developed for the cyclic prefix (CP)-aided single input single output (SISO) and multiple input multiple output (MIMO) orthogonal time frequency space (OTFS) systems relying on arbitrary transmitter-receiver (Tx-Rx) pulse shaping. The data and pilot symbol matrices are affine-precoded and superimposed in the delay Doppler (DD)-domain followed by the development of an end-to-end DD-domain relationship for the input-output symbols. At the receiver, the decoupled pilot and data symbol are extracted by employing orthogonal precoder matrices, which eliminates the mutual interference. Furthermore, a novel pilot-aided Bayesian learning (PA-BL) technique is conceived for the channel state information (CSI) estimation of SISO OTFS systems based on the expectation-maximization (EM) technique. Subsequently, a data-aided Bayesian learning (DA-BL)-based joint CSI estimation and data detection technique is proposed, which beneficially harnesses the estimated data symbols for improved CSI estimation. In this scenario our sophisticated data detection rule also integrates the CSI uncertainty of channel estimation into our the linear minimum mean square error (LMMSE) detectors. The AP-SIP framework is also extended to MIMO OTFS systems, wherein the DD-domain input matrix is affine-precoded for each transmit antenna (TA). Then an EM algorithm-based PA-BL scheme is derived for simultaneous row-group sparse CSI estimation for this system, followed also by our data-aided DA-BL scheme that performs joint CSI estimation and data detection. Moreover, the Bayesian Cramer-Rao bounds (BCRBs) are also derived for both SISO as well as MIMO OTFS systems. Finally, simulation results are presented for characterizing the performance of the proposed CSI estimation techniques in a range of typical settings along with their bit error rate (BER) performance in comparison to an ideal system having perfect CSI.

Deep Learning OFDM Receivers for Improved Power Efficiency and Coverage

In this article, we propose multiple machine learning (ML) based physical-layer receiver solutions for demodulating orthogonal frequency-division multiplexing (OFDM) signals that are subject to high level of nonlinear distortion. Specifically, three novel deep learning based convolutional neural network receivers are devised, containing layers in time- and/or frequency-domains, allowing to demodulate and decode the transmitted bits reliably despite the high error vector magnitude (EVM) in the transmit signal. Applicable training procedures are also described, such that the learned layers in the receiver processing properly generalize over different nonlinear distortion and multipath channel characteristics. Extensive set of numerical results is provided, in the context of 5G NR uplink (UL) incorporating also measured terminal power amplifier (PA) characteristics. The obtained results show that the proposed receiver systems are able to clearly outperform the classical linear minimum mean-squared error (LMMSE) receiver as well as the existing ML receiver approaches, especially when the EVM is high compared to modulation order. This is particularly so when the devised ML receiver is of hybrid nature with layers both in time and frequency. The proposed ML receivers can thus facilitate pushing the terminal PA systems deeper into saturation, and thereon improve the terminal power-efficiency, radiated power and network coverage. Through combining the obtained radio link performance results with link budget calculations, all carried out at the 28 GHz mmWave band, it is shown that the proposed ML receivers can enhance the network coverage in terms of maximum UL link distances by close to 100%, when compared to classical LMMSE receiver based networks.

MSTGAN: A Multi-Slot Conditional Generative Adversarial Network Based on Swin Transformer for Channel Estimation

Accurate channel estimation is the key to enhance the transmission performance of communication systems. In this letter, we propose a multi-slot conditional generative adversarial network (cGAN) based on Swin Transformer called MSTGAN for channel estimation in a single-input single-output (SISO) scenario. Specifically, the proposed MSTGAN could learn the temporal correlation feature from continuous channel slots by the 3D convolution and extract the deep feature well by the Swin Transformer to improve the accuracy of channel estimation. The model is trained with data augmentation to reduce the computational cost for offline training. The simulation results demonstrate that the proposed method outperforms the linear minimum mean square error (LMMSE) method and other deep learning methods. Furthermore, extension schemes of the MSTGAN to the multiple-input multiple-output (MIMO) case are also provided.

Super resolution direction finding technique of vortex electromagnetic wave radar in missing mode

Vortex electromagnetic waves have superior performance over electromagnetic waves. In order to improve the radar super-resolution lateralization technique in its missing modes, this study proposes to start from the perspective of mode missing. The missing modes are reconstructed using the Adaptive Step Size Gradient Descent (ASSGD) method by exploiting the features of the missing modes. The linear minimum mean square error (LMMSE) estimation method is also used to solve the problem of poor reconstruction accuracy due to the Missing Modes. The Missing Modes Iterative Adaptive Approach (MMIAA) algorithm and Missing Modes Sparse Learning via Iterative Minimization (MMSLIM) algorithm are then used. Minimization (MMSLIM) algorithm to recover missing modes. The results showed that the RMSEs of the recovery errors of MMSLIM, MMIAA and ASSGD were 0.16, 0.31 and 0.82 respectively at a modal missing ratio of 0.7, while ASSGD fails to recover the missing modal data at a modal missing ratio of 0.9. The overall data quality of the azimuthally estimated RMSE was average when the signal-to-noise ratio was at [–5, 10] dB. And the curve becomes flatter when it reaches 15 dB or more, indicating that MMSLIM, MMIAA has important theoretical and practical value.

Channel Estimation for RIS-Assisted MIMO Systems in Millimeter Wave Communications

The large number of estimated parameters in a reconfigurable intelligent surface (RIS) makes it difficult to achieve accurate channel estimation accuracy in 6G. Therefore, we suggest a novel two-phase channel estimation framework for uplink multiuser communication. In this context, we propose an orthogonal matching pursuit (OMP)-based linear minimum mean square error (LMMSE) channel estimation approach. The OMP algorithm is used in the proposed algorithm to update the support set and pick the columns of the sensing matrix that are most correlated with the residual signal, which successfully reduces pilot overhead by removing redundancy. Here, we use LMMSE’s advantages for handling noise to address the problem of inadequate channel estimation accuracy when the signal-to-noise ratio (SNR) is low. Simulation findings demonstrate that the proposed approach outperforms least-squares (LS), traditional OMP, and other OMP-based algorithms in terms of estimate accuracy.

Intelligently Wireless Batteryless RF-Powered Reconfigurable Surface: Theory, Implementation & Limitations

This work exploits ultra-low cost, commodity, radio frequency identification (RFID) tags as elements of a reconfigurable intelligent surface (RIS). Such batteryless tags are powered and controlled by a software-defined radio reader, with properly modified software, so that a source-destination link is assisted, operating at a different band. Signal model includes small-scale and large-scale fading, direct link, as well as specific parameters relevant to reflection (i.e., backscatter) radio, such as antenna structural mode and reflection efficiency, typically overlooked in the literature. An algorithm is offered that computes the optimal RIS configuration with complexity of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> log <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> ) in number of elements <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> , instead of intractable exponential complexity of exhaustive search, while accommodating any number <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> ≥ 2 of loads. With the proposed algorithm, it is shown that performance gains reach a plateau for constant element spacing and increasing number of elements, suggesting that the weak, passive nature of backscattered links limits the performance gains, even with perfect channel estimation. Channel estimation with linear minimum mean squared error (LMMSE) estimator is shown to be effective, provided that there are sufficient number of pilot symbols. A concrete way is offered to design and prototype a wireless, batteryless, RF-powered, reconfigurable surface and a proof-of-concept is experimentally demonstrated.

IRS-Assisted MISO System With Phase Noise: Channel Estimation and Power Scaling Laws

Recent studies have shown that intelligent reflecting surfaces (IRS) can significantly improve the spectral and energy efficiency of wireless communication links. However, most works assume perfect transceivers and IRS, which is impractical in real communication systems. In this work, we study the effect of the hardware impairments in IRS-assisted MISO systems with single user, where we consider both phase noise caused by the imperfect transceivers and IRS. To this end, we propose a linear minimum mean square error (LMMSE) channel estimation algorithm that takes the phase noise into account. Furthermore, we study the impact of phase noise on the downlink performance of an IRS-assisted system. Both analytical and numerical results are presented, where we prove that the transceiver phase noise can be compensated with the optimized IRS, while the IRS phase noise remains but is not exacerbated. As the number of reflective elements <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> approaches infinity, the IRS phase noise results in a constant loss in terms of the ergodic rate. Moreover, we find that the phase noise has no impact on the scaling laws. If the direct channel is blocked and as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> → ∞, the transmit power can be scaled down by 1/ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> and 1/ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> , respectively, for random and optimized IRS, without compromising the received signal to noise ratio (SNR).