Performance Analysis of Linear Precoding in Downlink Based on Polynomial Expansion on Massive MIMO Systems

The performance of linear precoding schemes in downlink Massive MIMO systems is dealt with in this paper. Linear precoding schemes are incorporated with zero-forcing (ZF) and maximum ratio transmission (MRT), truncated polynomial expansion (TPE), regularized zero force (RZF) in Downlink massive MIMO systems. Massive MIMO downlink output is evaluated with linear precoding included. This paper expresses the performance of achievable sum-rate linear precoding with variable signal-to-noise (SNR) ratio and achievable sum rate and several transmitter-receiver antennas, such as imperfect CSI, less complex processing, and inter-user interference. The transmitter has complete state information on the channel. The information narrates how a signal propagates to the receiver from the transmitter and reflects, for example, the cumulative effect of distance scattering, fading, and power decay. They show that the performance analysis of two linear precoding techniques, i.e., Maximum Ratio Transmission (MRT) and Zero Forcing (ZF) for downlink mMIMO output network over a perfect chain. The results show the improved ZF precoding achievable sum rate compared to the MRT precoding schemes and compared the average achievable rate RZF and TPE.


INTRODUCTION
Nowadays, the mobile communication network having several users is growing exponentially. Through mobile communications technology, users must have a more data rate low latency, and full mobility communications networks. Mobile communication technology needs to upgrade the infrastructure to satisfy the demands of the market. To reach a broader bandwidth, the mobile communications network is now moving into the 5th generation and working on the massive MIMO and millimeter wave spectrum [1].
However, the 5th generation organizes the recently developed subsystems on its network, one is an antenna subsystem and the other one is a beam-forming subsystem. A multi-input multi-output (MIMO) antenna subsystem is implemented by the latest mobile communications technology. When the massive MIMO there are many antennas. Therefore, one is a transmitter and the other one is a receiver. The massive MIMO has certain leads in terms of channel power, spectral efficiency, reducing interference [1]. Because of its ability to bit error rate, signal-to-noise ratio, feasible sum rate, and reduce interference, the mMIMO was a tropical research matter. The transmitter is fitted with large numbers of antennas assisting multiple users with single or multiplied receiver antennas. There will be a multipath between transmitter and

SYSTEM MODEL
Here we consider the downlink of an mMIMO scheme to be a single cell, in which BS with M antennas transfers data in a single antenna to K user terminals (UTs). The channel has a Rayleigh fading channel with zero mean and variance of λk (Fig. 1). This work employs a single-cell model. Fig 1 represents an mMIMO model of the single-cell downlink. For all users, the transmitter has perfect CSI. The Massive MIMO platform uses configuration Rayleigh channel [18]. The channel which has the transmitter and user coupling is shown in Fig.1 Furthermore, the precoding locality is also viewed in Fig.2. As illustrated in Fig.1, mMIMO device consisting of the BS equipped with M antennas and K(« N) UTs, each UT provided with one antenna, We assume in this paper that the BS supporting UTs over the Rayleigh fading channel will obtain perfect channel state information on a certain frequency or subcarrier The signal vector is transmitted to the K users during the downlink transmission, where M > K can be expressed as Where W ∈ C M × K is the linear precoding matrix S ∈ C M × 1 is the precoding source data, and ρ is the moderate transmission power of the BS. Here, both M and K are huge, and their ratio is supposed to be persistent [ 24]. The precoding matrix W is a justification of the channel matrix defines by H ∈ C M × K The power of the original signal being transmitted is normalized, i.e., s 2 = 1 Let 'r' is M ×1 Precoded vector with a complex information symbol transmitted from the base station antenna. The signal that the y ∈ C K × 1 user antenna receives is then given as [13].

ACHIEVABLE SUM RATE
This The achievable sum rate of precoders ZF and MRT is argued in [12], Assuming full downlink power is set and divided uniformly among all users. The Shannon theorem acquires the achievable rate over the additive white Gaussian noise (AWGN) as a function of the signal-to-noise ratio (SNR) [12].
The SNR is the signal-to-noise ratio. The channel state information (CSI) is a very crucial matter in multiuser communication systems. Normally each user emits data streams of multiple transmitters periodically and systematically to the CSI [11]. All the transmitters receive the channel evaluation response from the receiver to the reverse path, so the transmitter acquires CSI. The transmitter, therefore, communicates with only the complete CSI with all receivers [9]. As can be seen in equation (2), additive noise and interference between the users themselves is the signal emitted to each unit. Therefore, in a single cell downlink mMIMO network the obtainable information rate per user is defined with perfect channel state information.
Where SINR of the k th user is SINRk. The achievable sum rate of ZF and MF precoders with optimal CSI for huge values of M and K [5] The achievable sum rate of K users as formulated as: 3.1 ACHIEVABLE SUM RATE WITH ZF The Formula (12) has been applied in zero-forcing; the following are described [7] [8].
Substituting (6) into (13) gives The zero-forcing using vector normalization/matrix normalization methods are given in the following equation

Achievable sum Rate with MRT
The MRT of the achievable sum-rate is also deductible from (12) as 2 log (1 ) MRT is using methods of vector normalization/matrix normalization is given by below As the number of transmitting antennas hikes with Z > > K, Equations (14) and (17) indicate that the same downlink transmits power available and a rigid number of mobile users. ZF reaches data rates higher than MRT. where X ∈ C M × 1 is transmitted signal and hk Represent a specific variable to the BS channels and the k th UT. The spatially linear additive Gaussian noise k th UT is expressed by the nk ∼ C Ɲ(0,σ 2 ) for k=1,..., K, where σ 2 is the noise variance The BS utilizes Gaussian code and precoding, In light of this suspicion, the transmitted signal in (19) can be communicated as

Regularized Zero Forcing Precoding
The matrix representation is defined by charter U=[u1……uK] ∈ C M×K be matrix of the precoding and v =[v1……….vK] ∼ C Ɲ(0K×1,IK) is the vector that carries all symbols of UT data.
So, the signal received (19) can be indicated as The signal-to-interference noise ratio (SINR) at the k th UT [17] By taking that each perfect instantaneous CSI has UT, the achievable rates in the UTs are Regularized zero-forcing (RZF) precoder was known as a linear precoder for mMIMO wireless communication systems due to its ability to trade compensation for MRT and ZF precoders [13], [14], [15], [16].
assume the total power constraints 1 ( ) H tr UU P K (24) we specify the scaling factor 1/K counteracts the channel variance scaling, and tr(.) is the trace function. We density the total power P is set, though we allow antennas number M and UTs K to grow large.
Like that to [22], we specify the ZRF precoding as Where the variable for power normalization γ is set so that UZRF achieves the power limit in (21). Regularization of the scalar coefficient. The ζ can be chosen the various ways, based on P, σ 2 , τ, and device proportion.
The user efficiency characteristic in SINRk in (22). Whereas the SINR is a randomized quantity. The SINR be conditional on the instant random user channel values in H and the instant estimate of Ĥ the large (M, K) regime [23][24][25][26] can be used to estimate deterministic quantities. Such tests change based on channel statistics and are frequently mentioned as finding equivalents, as they are within the asymptotic limit almost definitely.
This process of stiffening property is due in part to the law of large numbers. The speaker Hachem has been proposed for first deterministic equivalents. In [ 23], Who also demonstrated their capture capability essential measures of system performance. Once applied to finite M and K the deterministic equivalents are pointed to as huge scale extrapolations.
A) Imperfect Channel State Information at BSs: Uplink pilot transmissions are used to receive instant CSI based on the TDD protocol at each BS. Yet each of the UT in such a cell detects a reciprocally orthogonal pilot pattern, Due to the incomplete channel coherence of the fading channels, The equal set of orthogonal chains is reused in each cell. Thus, pilot interference originating from neighboring cells weakens the channel estimate [27]. When testing the User Terminal k channel in cell j the subsequent BS takes its pilot signal obtained and compares it to the pilot sequence of this UT. , , ,

B) Issues of RZF in complexity:
Where the precoding of the SINR achieved by RZF converges in the major reign. However, random quantities of precoding matrices that need to be retrieved at the same rate as channel command are modified, so with the typical consistency of fewer milliseconds, they are essential to reverse hundreds of times per second to calculate the large-dimensional matrix. The number of arithmetic activities needed for matrix expression grows cubically in the matrix range, rendering this matrix operation inflexible in huge-scale devices, reducing the complexity of implementation and retaining the majority of RZF performance; Precoding of low-complexity TPEs for single-cell systems was proposed in [28] and [29]. The latest precoding strategy has two advantages over RZF precoding 1) at the beginning of each coherence interval the pre-coding matrix is not pre-computed, so there are no mathematical loops, and the mathematical processes are distributed over time uniformly.
2) The precoding method is classified into easy matrix-vector integer arithmetic which can be extremely parallel and implemented.

TPE Precoding
The idea of truncated polynomial expansion (TPE) is, we furnish the multi-cell scenario with a new type of low complication linear precoding strategy. We recalled that the definition of TPE comes from those in the   (27) where αl is the coefficient of the habitual polynomial. The simplified precoding is evaluated by taking only a truncated number of the matrix capabilities.
For Zj = 0M×M and truncated order of TPE precoding Jj.  (28) is Jj -1. For an ability that would choose to maximize proper device performance metric [28]. An starting option is

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This equation is determined by calculating a Taylor expansion of the matrix inverse. The coefficients in (29) provide performance near to that of ZRF precoding when Jj →∞ [28]. However, we can acquire far and away superior execution than the imperfect RZF, using just little TPE orders.

SIMULATION RESULTS
Six cases represent the performance analysis for linear precoding techniques i.e., ZF and MRT, based on the method of vector normalization, the method of matrix normalization, and the contrast of the two methods of normalization. The findings are presented in displayed achievable sum-rate (bit/s/Hz) vs the number of base station antennas and the achievable sum-rate vs the number of users.    This segment gives a numerical approval of the proposed TPE precoding in fig. 8 a down-to-earth arrangement situation. We consider a four-part site L =4 made out of cells and BSs. see Fig.1. Like the channel model introduced in [32], we expect that the UTs in every cell is separated into G= 2 gatherings in fig. 9. UTs of a gathering share roughly a similar area and factual properties.  Let's see Fig. 8. It sees a J= 4 TPE order and three separate CSI quality levels at the BS: From Fig. 8, We see it when a poor channel estimate is available, RZF and TPE accomplish nearly the same typical UT performance Additionally, at low SNR values, TPE and RZF perform almost similarly for any τ. The understandable observation is that the difference in the rate increases at high SNRs and when τ is small Fig. 9 The relationship between the average attainable UT level and the TPE order J is more clearly seen. To be in a regime where TPE performs poorly (see Fig.8) and the precoding complexity becomes a problem, we consider the case τ= 0.3, M= 512, and K= 256. From the figure, we can see that choosing a greater value for J gives a TPE performance closer to RZF's. The proposed TPE precoding never surpasses the RZF efficiency, which is remarkable because TPE has J degrees of freedom that can be optimized, while RZF has only one design parameter.

CONCLUSION
This paper offers an examination and investigation of direct precoding in mMIMO in a single-cell downlink. The parameters contemplated are the achievable sum-rate with a distinction in the number of dynamic users and the signal to noise ratio. Simulation results show a superior rate of error created by the MRT precoding scheme. The ZF precoding strategy, in the meantime, gives a superior achievable sum rate. A massive MIMO organize offers the chance to increment the achievable sum rate. The achievable sum rate changes for ZF and MRT when numbering the base station antenna 512. The achievable sum rate upgrades for ZF and MRT (18.8207 dBm and 16.6465dBm respectively at 0dBm and 10.2418dBm and 10.4415 dBm at 15dBm); Therefore vector/matrix normalization for ZF gives better performance at high downlink transmission power, while above normalizations MRT provides better performance at low downlink transmission power. This paper sums up the recently proposed TPE precoder to MIMO systems with multicell huge scale. This type of precoder series from the exceptional mind complex RZF precoding system through a truncated polynomial creation approximates the regularized channel reversal. The model contains basic multi-cell highlights, for example, client explicit channel measurements, different TPE arranges in various cells, and force requirements explicit to the cells. We acquired SINR expressions asymptotic.