INTRODUCTION As the need for mobile broadband services expands

INTRODUCTION
As the need for mobile broadband services expands, many operators are challenged with the necessity to get plentiful extra capacity from their spectrum, because of this many wireless technologies have been introduced in the last few years but still today users are facing issues such as poor signal strength and low call quality when used indoors leading to call drops 1.
To solve the problem users are facing in poor signal strength and low signal quality when used in certain environment like indoors, scientists are coming up with many techniques to combat such a kind of problems, one being the introduction of Massive multiple-input multiple-output or usually known as massive MIMO (M-MIMO) antennas which help in ensuring that the capacity and signal quality is of high standards by incorporating many base station antennas serving the particular cell in which the user is allocated.
The massive MIMO perception is grounded on deploying base stations (BSs) with many antenna (let say hundreds or thousands of antennas) elements with different conventional cellular technology, that are operated in a coherent mode which can deliver an amazing array gains, and a spatial resolution which permits multi-user MIMO communication to tens or hundreds of user equipments (UEs) per cell, but maintaining robustness to inter-user interference 2. The main role of these additional antennas is for directing the signal energy into the zones where it has minimum signal strength 3.
Massive MIMO has come at this moment because, conventional MIMO technology has confirmed incompetent to provide the spectral efficiencies that 5G applications are calling for and the assurance in the outstanding value of the technology have been spread rapidly from the moment when impressive real-life prototypes exposed the record spectral efficiencies, and the robust operation with low-complexity RF and baseband circuits has been substantiated.
By equipping a considerable number of antennas on the base station (BS) side, massive MIMO system has proved to achieve extraordinary spectral efficiency, but still, its performance is restricted by pilot contamination due to inevitable reuse of pilot sequences from user terminals in other cells 4.
With the aim of reaping all the benefits fetched by the massive antenna arrays and making sure about uplink system performance, the power control among the users is essential, and this can be done by varying or fluctuating the power of different users to increase the sum achievable rates, deliver services with certain fairness, or balance between these objectives 5.
Massive MIMO can similarly, decrease intra-cell interference existing between users who are served in the same time-frequency resource, owing to its focus of transmitted power to anticipated users 6. With the entrance of this antenna technology in the wireless network field, some of the old problems are confirmed as irrelevant problems, but it exposes completely new problems that instantly need attention 3, and among the new problems includes making low cost and low precision types of equipment work together proficiently, effective utilization of additional degrees of freedom (DoF) that come up with an extra service antennas, the necessity for effective schemes for channel state information (CSI), resources allocation and for reducing internal power consumption for attaining total energy efficiency reductions.
Arguments constructed on random matrix theory proves that the impacts of uncorrelated noise and small-scale fading are eradicated, the number of users per cell are free from the size of the cell, and the required transmitted energy per bit disappears as the number of antennas in a massive MIMO cell raises to infinity; likewise, simple linear signal processing methods, such as matched filter (MF) or Maximum Ratio Transmission (MRT), Zero-forcing (ZF) precoding/detection, can be applied in massive MIMO systems to accomplish these advantages 7.
According to Ngo 8, each single-antenna user in a massive MIMO technology can scale down its transmit power proportional to the number of antennas (M) at the BS with perfect channel state information (CSI) or to the square root ( ) of the number of BS antennas for imperfect CSI, to acquire similar performance as that of single-input-single-output (SISO) technology.
In practice, the application of massive MIMO is significant, if both hardware cost and power consumption are kept very low and in such a kind of case, hardware impairments or deficiencies become a substantial degrading factor leading to imperfect CSI which may affect the quality of spectral efficiency 9.
For the downlink massive MIMO system, channel estimation can be achieved by either using frequency division duplex (FDD) or time division duplex (TDD) modes, in an FDD mode, the BS transmits a pilot sequence to all UEs, then, each UE estimates its own channel and after estimating the channel, the UE gives the feedback to the BS through feedback channel, but in the TDD mode, each UE transmits its own pilot symbols to the BS, the BS then estimates the uplink channel and allocate the conjugate of the estimated channel as its downlink version 10.
Massive MIMO usually operates in the time-division-duplex (TDD) mode, in which channel reciprocity is exploited to acquire downlink channel estimations from the uplink channel estimations.
The performance of massive MIMO mostly depends on the quality of CSI at the transmitter, so it is of supreme importance to examine realistic situations with imperfect CSI, attained during the uplink training phase in time-division duplex (TDD) mode.
While most of the works done in this area delivers good results about the performances of the linear precoding schemes, but they did not provide and compare the performances of ZF and MRT in terms of ergodic achievable sum rate and downlink transmit power under the same conditions and in a single-cell downlink scenario for imperfect channel state information (CSI), so this must be done in order to have clear results for the case of imperfect CSI.
The main aim of this paper is to investigate and analyze the performance of single-cell massive MIMO on the downlink ergodic achievable sum rate under imperfect CSI with linear precoding schemes namely Zero-forcing (ZF) and maximum ratio transmission (MRT) with minimum mean square error used to give the estimation error.
In this paper the authors restrict themselves in using maximum ratio transmission (MRT) and zero-forcing (ZF) because of their low-complexity RF and baseband circuits and the TDD model is employed because of the nature of channel reciprocity it offers compared to FDD. So the major contributions this paper offers is the analysis of the impact of imperfect CSI on the ergodic achievable sum rates for MRT and ZF precoding techniques, deriving the ergodic achievable sum rate and hence evaluating its performance by variation of users (K), BS antennas (M) and the estimation error.
BACKGROUND AND LITERATURE SURVEY
Lim et al 11, derived simple closed-form formulas for the sum rate of zero-forcing (ZF) and maximum ratio transmission (MRT) with vector/matrix normalization for downlink transmission and ZF and maximum ratio combining (MRC) for uplink transmission and they concluded that, for the downlink transmission, vector normalization was better for ZF where matrix normalization was better for MRT.
Jin et al 12, derived users’ achievable sum ergodic rates who are served by a relay station with a massive number of antennas, which uses (MRC)/MRT and a fixed amplification factor for reception/ transmission, they then analyzed the rate gain when the transmit power of the senders and the relay is sufficiently large and they found the ergodic achievable sum rates to increase with the number of antennas at the relay, i.e., , but it decreases with the number of user pairs, they then further discovered that the achievable ergodic sum-rate can be maintained while the users’ transmit power is scaled down by a factor of .
Qiu et al 13, addressed the problem of downlink pre-coding scheme by exploitation of statistical and imperfect instantaneous CSI by analyzing the effect of the mutual interference between the user with statistical CSI and the one with imperfect instantaneous CSI on their achievable rate and they proposed an extended ZF and an extended MRT precoding methods which were observed to reduce the total transmit power of BS and to maximize the received signal power of users.
Mathematical derivation, with perfect CSI and imperfect CSI, the lower capacity bounds for (MRC), Zero Forcing (ZF) and Minimum Mean Square Error (MMSE) receivers were presented by Zourob et al 14, and they found through analysis and simulations that the users’ transmission power can be minimized by the number of BS antennas ( ) for the perfect CSI, and can also be minimized by the square root of the number of BS antennas ( ) for imperfect CSI, by also providing desired Quality-of-Service (QoS).
Cheng et al 15, proposed non-orthogonal-multiple-access (NOMA) scheme that uses shared pilots and they compared their proposed scheme (NOMA) with the traditional orthogonal access scheme having orthogonal pilots and they found that having downlink CSI at the users, the proposed NOMA scheme outperforms orthogonal schemes but with more crowds of users present in the cell, it is desirable to use multi-user beamforming as an alternative to NOMA.
Cheng et al 16, proposed non-orthogonal-multiple-access (NOMA) scheme that uses shared pilots and they compared their proposed scheme (NOMA) with the traditional orthogonal access scheme having orthogonal pilots and they concluded that the proposed NOMA scheme offers strong performance gains in training based multiuser MIMO systems and this gain increases as the number of antennas at the BS increases thus showing that there is advantage of using the developed NOMA scheme in massive MIMO systems.
Tebe Parfait and Yujun Kuang 17, analyzed and compared the performances of ZF and MRT in a downlink massive MIMO system with perfect CSI and their results found that ZF performs better than MRT under the same conditions for the achievable sum rates but for the downlink transmit power there are contrasting results as it seems that with the number of antennas let say less than 100, the MRT outperforms the ZF with the same ergodic achievable sum rate but as the number of antennas increases lets say above 200 the two precoding schemes were found to have almost the same performance.
Quintero et al 18, suggested that in order to satisfy a high amount of data demand for a large number of antennas in the BS in contrast to the number of active users, it is desirable to use the ZF precoder, which needs less computation and offers an optimal performance in these conditions but for circumstances with lower data traffic, it is reasonable to use the MRT precoder because of its capability to satisfy such demand using lower transmission power and low computational complexity.
Israr et al19, analyzed and then made comparisons on the achievable sum rate and the downlink transmit power of ZF and MRT precoding techniques and they finally found that if the ratio of BS antennas ( ) to the number of users (K) is large then ZF performance was better than MRT and if the ratio is quite small then MRT performance was better than ZF under the same conditions.
Eakkamol Pakdeejit 20, compared the performance between MRT and ZF, in which he found out that MRT provides superior performance at low signal to noise ratio (SNR) compared to ZF and ZF performs better than MRT at high SNR while the Minimum Mean Square Error (MMSE) offers the best performance through the entire SNR.
Li et al 21, derived a controllable, but accurate closed-form expressions for the rate bounds with (MRT) and (ZF) beamforming in distributed massive MIMO systems in which their results showed that ZF attains superior performance gain and faster convergence speed than MRT but when the coherence interval size is large, then the downlink beamforming training scheme becomes more desirable for the distributed massive MIMO systems.
System analysis and derivation of the asymptotic SINR equation when a matched filter (MF) having a pilot contaminated estimation is employed at the receiver was performed and analyzed by Krishnan et al 22, where they showed that when the number of users (K) is similar to the number of BS antennas then the performance of the MF with corrupt estimate is limited by the interference.
According to Tebe et al 23, reducing the size of the cell have significant mitigation effects on the pilot contamination which hence enhances the system performance.
Ergodic achievable rate when MRC/MRT is exploited at the relay with respect to imperfect channel state information (CSI) was analyzed by Wang et al 24, they then derived an analytical expression of the ergodic rate, based on the asymptotic analysis and power scaling law which are conducted when the number of antennas grows large and from their results it was observed that the ergodic rate logarithmically increases when the number of relay antennas also increases.
Gao et al 25, evaluated the ergodic sum-rate performance for two linear pre-coding schemes, (ZF) and (MMSE) as a function of the number of BS antennas and they found that even at 20 BS antennas these pre-coding schemes reach 98% of the optimum dirty-paper coding (DPC) ergodic sum-rate for the measured channels.
Low complexity hybrid analog/digital precoding algorithm for downlink multiuser mmWave systems was proposed by Alkhateeb et al 26, the results through simulations showed that the proposed algorithm delivers higher achievable sum rates in comparison with analog-only beamforming, and its performance approaches that of the unconstrained digital precoding solutions.
CONCEPTS AND THEORIES
In this paper the downlink transmission which is the transmission from the base station antenna to the user equipment which may be mobile station or any kind of computer devices receiving signal from the BS antennas, also we considered the linear precoding techniques (Linear precoding is a modest processing by which the transmitted signal vector is obtained by multiplying the information data vector with a linear precoding matrix) in the form of Maximum transmit ratio or sometimes known as Matched filter (MF) at the BS and the zero forcing (ZF) precoding techniques because of their low complexity in deployment.
Precoding algorithms like MRT/MF and ZF are used to reduce the number of errors that may occur at the receiver output, they are basically useful in case of multilayer transmission where numerous transmitters, with multiple antennas, are used to transmit data to the multiple receivers but the need for single stream beam-forming is not that much because there is the single antenna that is used to transmit signal and has proper gain and phase.
There are many assumptions made in this paper one of it being that the user equipment (UEs) are positioned randomly in the cell region of the BS, and the components of the channel matrix are assumed to be independent and identically distributed (i.i.d) complex Gaussian variables with zero mean ( ) and unit variance ( ).
A. DOWNLINK CHANNEL ESTIMATION
Channel estimation is the technique for measuring the characteristics of a wireless channel and the easy way to do this is by relating known transmitted (pilots) signals and the received signals 27.
Downlink channel state estimation usually depends on the type of system duplexing mode which can either be time division duplex (TDD) or frequency division duplex (FDD). But for the case of this paper TDD mode is assumed because of its reciprocity in which the uplink and downlink channels use similar frequency spectrum for the uplink transmission and downlink transmissions but with different time slots.
In line for channel reciprocity in TDD mode, the channel estimated at the BS in the uplink transmission can be used to pre-code the transmitted symbols in the downlink transmission (Usually, the BS transmits multiple data streams to each user concurrently and selectively with CSI) 28.

Figure 2: Downlink operation of a Massive MIMO transmission 29.
If is defined as the channel matrix, and as the channel vector between BS and user , then the received vector at the users is given by:

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where is the normalized downlink transmission power, is a user data vector where is a data symbol of user and is the noise vector and is independent and identically distributed (i.i.d) complex Gaussian random variables with zero mean ( ) and unit variance ( ), that is, .
After using precoding/linear beamforming scheme the signal received by the user is:

Where and .
Signal-Interference-Noise Ratio (SINR)
This is the ratio of signal to the sum of interference and noise of the channel, in which for the quality signal this ratio must be large enough so as to cancel out any noises and interference associated with the channel. In this description, the SINR for MRT and ZF precoding under imperfect channel state information are derived. From 30,

Where , and are the power of the desired signal, interference and the noise respectively. For a signal received by the user , the signal to interference plus noise ratio (SINR) of that user k can be written as 31:

There are two simple (with low complexity) linear precoding techniques usually used in downlink transmission which are ZF and MRT/MF respectively.
ZF Precoding
This is a linear precoding scheme which cancel out inter- user- interference at each user, and it is assumed to implement a pseudo-inverse of the channel matrix.
ZF precoding at BS is given as:

Where is a scaling factor to satisfy the transmit power constraint and it is given as

Where and is the downlink transmit power while is the pre-coding matrix, and is the column vector of the user and represents the complex conjugate transpose of the estimated channel matrix. In order to satisfy the power control, the precoding matrix must be normalized.
But for the imperfect CSI, the channel matrix is given by 32:

Where is the reliability of the estimation and represents the error matrix and this is used to calculate the accuracy of channel estimation. For , means that the channel information is perfect (that is no channel estimation errors) but for means that the channel is completely imperfect ( that is the channel totally contains estimation error matrix)

The approximated kth user under Imperfect CSI for large values of , and with lower bound vector is given as 33, 34:

Where then
Maximum ratio transmission (MRT) precoding
MRT is one technique of linear precoding which maximizes the signal gain at the intended user 20, 31.The MRT pre-coding at the BS is given as:

Where is a scaling factor to gratify the transmit power and is given as

Where and is the downlink transmit power.
The approximated kth user under Imperfect CSI for large values of , and with lower bound vector is given as 33, 34:

For , then
B. PERFORMANCE EVALUATION
This section describes the performances evaluation of downlink massive MIMO which includes SINR, achievable rate and total downlink transmit power. By means of this performance evaluation which is under the imperfect channel state information (CSI), the comparison of different performance parameters like SINR, achievable sum rate and downlink transmit power are investigated in the downlink transmission with linear precoding schemes. The investigation between the SINR, achievable sum rate and downlink transmit power with different number of base station antennas M, number of users K and estimation errors have been explained in this part.
Ergodic Achievable sum rate
This is average rate in which the transmitter can transmit signals over the channel and it can be used to describe the effectiveness and efficiency of a massive MIMO system, it is usually derived from the Shannon theorem, by taking the average of the capacity obtained from Shannon theorem from 30:

From the above equation, the rate of user k can be expressed as

Then for K number of users, the ergodic sum-rate is given as

From the formula above, the ergodic sum rate with MRT and ZF can be approximated as
Where
For ZF pre-coding scheme at high SNR
The ergodic sum rate is given by:

Where and substituting in (10) above:

For MRT pre-coding scheme at high SNR
The ergodic sum rate is given by:

Where and substituting in (12) above:

RESULTS AND DISCUSSION
The analysis of massive MIMO downlink system with ZF and MRT pre-coding schemes over Rayleigh fading channel was made by considering the variations of achievable sum rate with Signal to Noise Ratio (SNR), the number of antennas in the BS, number of UEs and the estimation errors.
Figure 1 explains the relationship between ergodic sum-rate against the Signal to Noise Ratio (SNR) under imperfect channel state information where the number of users was fixed to 20, and the number of base station antennas M was assumed to be M= 200 with the channel estimation error also assumed to be 0.3, 0.5 and 0.7. The results show that as SNR increases, the ergodic sum-rate for both imperfect MRT and ZF increases but ZF linear precoder achieves higher ergodic sum-rate than MRT at different values of channel estimation errors. Also according to the figure 1, the smaller the estimation errors shows the better the ergodic rate which in other words it can be explained that the lower the estimation error approximates to that of the perfect CSI while the larger the estimation errors the lower the ergodic sum-rate.

Figure 1: Showing the graph of ergodic sum rate with the signal to Noise Ratio (SNR) at M=200
Increasing the value of M=200 to M=500 have the practical implications on the ergodic sum rate because with the same number of users and the same estimation errors the ergodic sum-rate seemed to increase suddenly for example at SNR=0dB and M=200; the ergodic sum rate at e=0.7 was found to be approximately 25bits/s/Hz for MRT and 35bits/s/Hz for ZF while at e=0.5 the ergodic sum-rate was 35bits/s/Hz for MRT and 50bits/s/Hz for ZF and at 0.3 the ergodic sum- rate was 50bits/s/Hz for MRT and 65bits/s/Hz for ZF as shown in figure 1 while for M=500 and SNR=0dB as in figure 2:

Figure 2: Showing the graph of ergodic sum rate with the signal to Noise Ratio (SNR) at M=500
In figure 2, there is an increase in ergodic sum-rate when the number of base station antennas M have been increased to 500, for example the ergodic sum-rate was found to be approximately 48bits/s/Hz for MRT linear precoding and 60bits/s/Hz for ZF at estimation error e=0.7 while the ergodic sum-rate was found to be 58bits/s/Hz for MRT and 75bits/s/Hz at estimation error e=0.5 and at e=0.3, the ergodic sum-rate was found to be 73bits/s/Hz for MRT and 90bits/s/Hz for ZF. Therefore there is an increase in ergodic sum-rate when the number of base station antennas M is increased while keeping all other simulation parameters constant, and for the case of linear precoding, the ZF linear precoding scheme outperforms MRT linear precoding as shown in the two graphs above.

Figure 3: Showing the graph of ergodic sum rate against number of users K at M=200
In figure above, the relationship between ergodic sum-rate and the number of users is clearly shown in which the lower channel estimation error gives the appreciable ergodic sum-rate compared to the higher estimation error which in turn means that the lower value of estimation error i.e. let say it tends to zero then the imperfect channel becomes approximately perfect channel under the same conditions. As it can be seen in fig. 3, the ZF linear precoder outperforms MRT precoder under the same conditions. So in general increasing the number of users for MRT tends to increase the ergodic sum-rate but at a certain point let say optimal number of users, increasing the number of users causes the decrease in the ergodic sum-rate while for the case of ZF precoder, increasing the number of users causes an appreciable increase in the ergodic sum-rate.

Figure 4: Showing the graph of ergodic sum-rate against the number of users K at M=500
Increasing the number of base station antenna to let say M=500 as shown in fig.4 tend also to increase the ergodic sum-rate by an appreciable amount under the same conditions of estimation error and downlink transmit power (in the case of figure 3 and 4 the downlink transmit power was assumed to be 0dB for simulation purpose). The lower the estimation error, as usual, tend to give the better the ergodic sum-rate as compared to the higher estimation error.

Figure 5: Showing the graph of ergodic sum-rate against the number of base station antenna M at Pd=0dB.
As shown in fig.5, the ergodic sum-rate tends to increase as the number of base station antennas M increases but at the same time the increase differs according to the estimation errors as shown i.e. as the estimation error becomes very low the ergodic sum-rate tends to be higher and in proximity it approaches that of the perfect channel state information. But increasing the downlink transmit power to 15dB causes an increase in the ergodic sum-rate with respect to the number of base station antennas as shown in fig. 6

Figure 6: Showing the graph of ergodic sum-rate against the number of base station antenna M at Pd=15dB.
The relationship between ergodic sum-rate and the channel estimation error have been also analysed as shown in the figure below in which the presence of estimation errors shows that the ergodic sum-rate fluctuates according to change in the number of user equipment, also as the channel estimation error increases, the ergodic sum-rate tends to decrease and the vise verse is true. Also according to fig. 7 in which the number of users was 10 (i.e. K=10, M=300), the ergodic sum-rate was seen to decrease as the error increases but when the number of users was increased to 50 (i.e. K=50, M=300), the ergodic sum-rate was large compared to when K=10 but as the rule of thumb, it tend to decrease as the error increases. In general, the ZF linear precoding scheme still outperforms the MRT linear precoding scheme at higher SNR compared to lower SNR.

Figure 7: Showing the graph of ergodic sum-rate against estimation error at K=10.

Figure 8: Showing the graph of ergodic sum-rate against estimation error at K=50.
Also seen in fig.7 and fig.8, the ergodic sum-rate tends to be zero (R=0) for both ZF and MRT linear precoders when the estimation is 1 because at this value it is practically impossible for any transmission to take place because the channel is totally poor thus it does not allow transmission of any kind of signals from the base station antenna to the user equipment antenna.
CONCLUSION
The analysis of ergodic sum-rate with different parameters in the time division duplex (TDD) in the downlink transmission using linear precoding schemes under imperfect channel state information has been made in this paper. The presence of estimation errors was found to be an obstacle in achieving higher ergodic sum-rate as shown in simulation results above, for example in fig. 7 and fig. 8 the ergodic sum rate seemed to be decreasing as the the estimation error increases up to the point the ergodic rate became zero when the estimation error was 1 which hence concludes that in th presence of estimation errors whatever parameter let say number of base station antennas M, number of user equipment K or downlink transmit power increases in the presence of error still the ergodic sum-rate will not be as high as that of the perfect channel state information.
On the other hand, the ZF linear precoding scheme was much better compared to MRT linear precoding scheme in whatever conditions for example when ergodic sum-rate was found in relation to SNR, ZF outperformed MRT also when ergodic sum-rate was found in relation to number of users K still ZF outperformed MRT but in both cases the SNR was assumed to be high.
RECOMMENDATION AND FUTURE WORK
In the future further research must be conducted in the analysis of downlink transmission power of massive MIMO in the TDD mode under imperfect channel state information, in which the variation of downlink transmit power with parameters like number of base station antennas, number of user equipments and the channel estimation errors must be analysed using linear precoding scheme (ZF, MRT, and MMSE) so as to have a clear picture of which linear precoding scheme works well with such kind of analysis.
Also, the analysis of ergodic sum rate in FDD mode must be conducted in future under imperfect channel state information and the comparison must be made between the FDD and TDD transmission mode to find which mode gives better results using linear precoding schemes.