1 Introduction

The data traffic has grown exponentially during the last years, because of the dramatic growth of many wireless data consuming devices such as smartphones, laptops, tablets, etc. It is expected that global mobile data traffic by 2018 will increase 15.9 Exabyte per month, which is about 6 times increase over 2014 [1]. In addition, by 2018 the number of mobile devices and connections are expected to grow to 10.2 billion [1]. The increasing demand of these wireless data consuming devices and wireless mobile connection also enhancing the demand of more throughputs and the concerns about energy consumption. Thus, future wireless comunication has to satisfy the three main demands, high throughput; serving many users simultaneously; and less energy consumption [2].

In order to meet this demand new technologies are required which will be able to provide optimal quality of service, spectral efficient, link reliable and energy efficient system in future wireless communication. The demand for 5G is already increasing because it is expected that 5G will be able to resolve the huge capacity and connectivity challenges brought by the increasing mobile traffic and data usage [3]. To increase the spectral efficiency, a well-known way is using multiple antennas at the transceivers. MIMO (multiple input multiple output) technique attracted the researchers over the past decade.

The efforts in order to utilize multiplexing gain have been shifted from MIMO to multiuser MIMO (MU-MIMO), where a multiple-antenna base station (BS) serves several users simultaneously. In MU-MIMO system, even with users having single antenna, spatial multiplexing gain can be obtained [4]. MU MIMO not only reaps all the benefits of MIMO systems but it also overcomes its limitations. The more degrees of freedom can be offered if the BS is equipped with more antennas, hence it results a huge sum throughput [5]. This high signal dimensions make the signal processing techniques prohibitively complex. So the main question is whether with low-complexity signal processing, the huge multiplexing gain can be obtained. Marzetta [5] showed that the simple linear processing becomes nearly optimal if number of BS antennas is large as compared with the number of users.

Massive MIMO, a system in which transmitter uses large number of antennas about hundred or more and serving many users simultaneously in the same time frequency resource [5]. Massive MIMO has been considered as a promising technology for next generation wireless systems and to address a significant challenges in 5G [6, 7]. The design and analysis of massive multi user MIMO systems is a fairly novel research area that is attracting substantial interest [8,9,10,11,12,13]. Before putting Massive MIMO into practice, several issues still need to be solved despite the much research work on it [19,20,21,22,23,24,25,26]. If base station communicates in same frequency or time resources with the users, the higher data rates can be obtained [14]. Inter user interference is a problem in data transmission in downlink channel of massive MIMO and require more transmit power, thus, interference cancellation at BS is required. Precoding methods are important to pre-subtract interference from users in transmitter side before sending it through the channel [15]. Precoding is one of the hot and active research topics of massive MIMO [16, 17]. A practical approach that has received considerable attention due to its simplicity is represented by linear precoding [18]. Proper signal processing can be performed at the transmitter to separate the multiple users in space if CSI is available at the transmitter (CSIT) [19]. Exact channel state information, or perfect CSIT, is ideal, but in real scenario, acquiring perfect CSIT is difficult in a severe fading channel [15]. The authors in [8] presented various challenges for massive MIMO in next generation wireless systems, especially to scale up the number of transmit antennas. Among them precoding is one of the hot and active research topic for massive MIMO. Especially for imperfect channel state information where estimation errors play vital role [16, 17]. The performance of linear and nonlinear precoding methods in faded environment investigated in [15] showed that how different precoding techniques improve the bit error rate using perfect CSI. It is found that implementation of nonlinear precoding methods require a tremendous computational complexity at both BS and user equipment (UEs). The performance of massive MIMO system in single cell downlink with different precoding schemes under perfect CSI is analyzed in [20]. The authors used the same value of signal to interference to noise ratio for each precoders which should be avoided for good analysis because each precoding has different signal-to-interference-plus-noise (SINR).

The matrix and vector normalization for ZF and MRT precoding is compared in [21]. The ergodic performance of these precodings with perfect CSI in a cell boundary users scenario is investigated. In [22], the authors investigated the performance of downlink massive MIMO with ZF precoding in term of outage probability and bit error ratio using perfect CSI. System performance using perfect CSI with ZF and MRT is analyzed in [23]. The power efficiency on the uplink of massive MIMO system under imperfect and perfect CSI using minimum mean square error (MMSE), ZF and maximal ratio combining (MRC) receivers is analyzed in [24]. In [25], the authors analyzed and compared the performance of eigen beaforming (BF) and RZF in term of achievable data rate in presence of channel estimation errors in a multi cell downlink scenario. The uplink achievable rate of massive MIMO incorporated the ZF and MRC receivers is evaluated in Ricean fading channels with both imperfect and perfect CSI in [26].

However, most of the above work considered perfect channel state information which is hard to obtain in practical scenarios. The difference of our work with respect to [20, 21, 23, 27] is that the effect of estimation error is considered which was neglected in these works. Due to estimation error consideration, the analysis will be different and more practical. While in [24, 26], channel estimation was considered for the perfromance of uplink channel and in [25] BF and RZF precoding was compared in the presence of imperfect CSI. Whereas, we consider downlink channel with ZF and MRT precoding.

In this paper, the performance of Massive MIMO downlink including linear precoding is evaluated. Spectral efficiency in terms of achievable rate and energy efficiency in terms of transmit power of zero force (ZF) and maximum ratio transmission (MRT) linear precoding is investigated under imperfect channel state information (CSI). Since ZF and MRT precoding can balance the system performance and complexity. Different channel estimation values are considered in order to compare the performance of these precoding techniques in the given system. The achievable rate and the downlink transmit power of ZF and MRT precoding techniques are derived, analyzed and compared under the same conditions and assumptions. Several scenarios are considered to study these performance parameters.

The paper makes the following specific contributions:

  • We consider a massive MIMO system where the number of BS antennas M and number of users K grow large and examine the achievable rates of ZF and MRT precoding techniques for large M and large K.

  • We also derive and evaluate downlink transmit power of these precoding techniques and perform the comparison by investigating the effect of large M and K on it.

  • We explore the impact of imperfect CSI on achievable rates and downlink transmit power of ZF and MRT precoding techniques.

The remainder of this paper is organized as follows. Section 2 describes the system & channel model. Section 3 derives achievable rate and downlink transmit power of ZF and MRT. In Sect. 4, we provide a set of simulation results, while Sect. 5 concludes the main results of this paper.

Notation Throughout the paper, matrices are denoted by uppercase boldface letters while vectors are expressed in lowercase boldface letters. \({\mathbf {X}}^H\), \({\mathbf {X}}^T\) and \({\mathbf {X}}^{-1}\) are used to denote conjugate transpose, transpose, and inverse of \({\mathbf {X}}\), respectively. Moreover, \([{\mathbf {X}}_{ij}]\) gives the (i, j)th entry of \({\mathbf {X}}\) and \({\mathbf {Z}} \in {\mathbb {C}}{\mathbb {N}} (a, b)\) denotes that \({\mathbf {Z}}\) is a complex Gaussian matrix with mean a and variance b.

2 System and Channel Model

The downlink of a multi user massive MIMO system is considered where the BS (base station) and each MS (mobile station) are equipped with M and N antennas, respectively, and BS serves K single antenna users (\(N = 1\)). The multiuser massive MIMO system is shown in Fig. 1

Fig. 1
figure 1

Multiuser massive MIMO systems, here, M-antenna BS serves K single-antenna users in the same time-frequency resource

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The downlink transmission occurs in two phases: training phase and downlink data transmission phase [28]. The base station estimates the CSI from K users based on the received pilot sequences in the uplink during the training phase. The base station uses this CSI and linear precoding schemes to process the transmit data [27].

2.1 Channel Estimation

The channel estimation techniques depend upon the system duplexing mode either time division duplex (TDD) or frequency division duplex (FDD) [29]. In this work we assume TDD mode, that the channel is estimated at the BS via uplink pilots, assuming channel reciprocity. Uplink and downlink channels are reciprocal in TDD because it uses the same frequency spectrum for the uplink and downlink transmissions but with different time slots. M is large in massive MIMO therefore, TDD operation is preferable in Massive MIMO [29]. The downlink and uplink channels can not be perfectly reciprocal in practice because of hardware chains mismatch. But with proper calibration [30,31,32], this non-reciprocity can be removed. In this work, it is assumed that the hardware chain calibration is perfect. It is assumed that the channel characteristics do not change and are constant for T symbol durations. There are two phases during each coherence interval [29]. In the first phase, during part t of the coherence interval the training or pilot sequences are transmitted to estimate the channel of each user. During the second phase, the data of all K users is simultaneously transmitted to the BS. The channel estimates obtained during first phase are then used by the BS to detect the transmitted signals. Due to channel reciprocity in TDD system, the channel estimated at the base station in the uplink can be used to precode the transmit symbols in the downlink. Estimated channel may contain estimation errors and therefor it is not perfect which is given as

$$\begin{aligned} {\mathbf {H}} = \bar{{\mathbf {H}}} + {\mathbf {E}} \end{aligned}$$
(1)

where \({\mathbf {H}}\in {\mathbb {C}}^{K\times M}\) is a Rayleigh fading MIMO channel with independent and identically distributed zero mean and unit variance complex Gaussian entries. \(\bar{{\mathbf {H}}}\) is the imperfect CSI or estimated channel matrix obtained through minimum mean square error (MMSE) channel estimation [33] and \({\mathbf {E}} = {\mathbf {H}} - \bar{{\mathbf {H}}}\) is the error matrix. \(\bar{{\mathbf {H}}}\) and \({\mathbf {E}}\) are uncorrelated. Each element of error matrix \({\mathbf {E}}\) is i.i.d zero mean complex Gaussian random variables with variance \(\sigma _e^2 = MMSE = \epsilon \{[{\mathbf {H}}]_{ij}- [\bar{{\mathbf {H}}}_{ij}]^2\}\)

Each element of \(\bar{{\mathbf {H}}}\) is independent and identically distributed as \({\mathbb {C}}{\mathbb {N}} (0, 1- \sigma _e^2 )\) while the error is complex white Gaussian with variance \(\sigma _e^2\).

CSI is imperfect if it contains the estimation errors. Channel estimation error can be modeled by specifying the channel estimation error matrix elements using i.i.d zero mean complex Gaussian variables [33]. Hence the imperfect CSI or estimated CSI (\(\bar{{\mathbf {H}}}\)) using minimum mean square error (MMSE) channel estimation is [33]

$$\begin{aligned} \bar{{\mathbf {H}}} = \sqrt{1- \sigma _e^2} {\mathbf {H}} + \sigma _e{\mathbf {E}} \end{aligned}$$
(2)

The parameter \(\sigma _e\) is used to compute the accuracy of channel estimation. When \(\sigma _e = 0\), it means that there is no channel estimation error and channel information is perfect while \(\sigma _e = 1\) indicates it is a complete failure of channel estimation and estimation totally contains estimation error matrix and it is complete imperfect CSI.

2.2 Downlink Data Transmission

The base station uses the channel estimates obtained from channel estimation phase to process the signals before transmitting them to K users and the base station uses linear precoding techniques. Using linear precoding the received signal \({\mathbf {y}}\in {\mathbb {C}}^{K\times 1}\) will be

$$\begin{aligned} {\mathbf {y}} = \bar{{\mathbf {H}}}{\mathbf {W}}{\mathbf {x}}+{\mathbf {n}} \end{aligned}$$
(3)

where \({\mathbf {x}}\in {\mathbb {C}}^{M\times 1}\) is the transmit signal from the base station, \({\mathbf {W}}\in {\mathbb {C}}^{M\times K}\) is a linear precoding matrix and \({\mathbf {n}}\in {\mathbb {C}}^{K\times 1}\) is the AWGN (additive white Gaussian noise) with zero-mean and variance.

The received signal by kth user using linear precoding technique is given by

$$\begin{aligned} y_k = \bar{{\mathbf {h}}_k}\mathbf {w}_k x_k + \sum ^k_{i=1,i\ne k} \bar{{\mathbf {h}}_k} {\mathbf {w}}_i x_i + n_k \end{aligned}$$
(4)

\(\bar{{\mathbf {h}}_k}\mathbf {w}_k x_k\) is the desired signal, \(\sum ^k_{i=1,i\ne k} \bar{{\mathbf {h}}_k} {\mathbf {w}}_i x_i\) is the interference from the other users and \({\mathbf {n}}\) is the noise.

The received signal to interference plus noise ratio of the of the kth user can be expressed as [20]

$$\begin{aligned} SINR_k= \frac{|\bar{{\mathbf {h}}_k}\mathbf {w}_k|^2}{\sum ^k_{i=1,i\ne k}|\bar{{\mathbf {h}}_k}\mathbf {w}_i|^2 + 1} \end{aligned}$$
(5)

2.3 Linear Precoding

In this work, we consider ZF and MRT linear precoding techniques.

2.3.1 ZF Precoding

ZF precoding is a linear precoding technique which cancel out the inter user interference at each user. This precoding is assumed to implement a pseudo-inverse of the channel matrix. ZF precoding at BS is given as [20]

$$\begin{aligned} {\mathbf {W}} = \beta \left\{ \bar{{\mathbf {H}}}^H(\bar{{\mathbf {H}}}\bar{{\mathbf {H}}}^H)^{-1}\right\} \end{aligned}$$
(6)

where \(\beta\) is a scaling factor to satisfy the transmit power constraint and it is given as

$$\begin{aligned} \beta = \sqrt{\frac{P_d}{tr(\mathbf {AA}^H)}} \end{aligned}$$
(7)

where \({\mathbf {A}} = \bar{{\mathbf {H}}}^H(\bar{{\mathbf {H}}}\bar{{\mathbf {H}}}^H)^{-1}\) and \(P_d\) is the downlink transmit power.

The Kth user \(SINR^{zf_{csi}}_k\) under Imperfect CSI for large values of M and K is given as (see Appendix 1)

$$\begin{aligned} SINR^{zf_{csi}}_k = \left\{ \frac{(\alpha - 1) (1 - \sigma _e^2)P_d}{(P_d\sigma _e^2)+1}\right\} \end{aligned}$$
(8)

where \(\alpha = \frac{M}{K}\)

2.3.2 MRT Precoding

MRT Maximizes signal gain and SNR at the intended user. MRT precodning at BS is given as [20]

$$\begin{aligned} {\mathbf {W}}= \beta \left\{ {\mathbf {H}}^H\right\} \end{aligned}$$
(9)

where \(\beta\) is same as (8), but \({\mathbf {A}} = {\mathbf {H}}^H\)

The Kth user \(SINR^{MRT_{csi}}_k\) under Imperfect CSI for large values of M and K is given as (see Appendix 2)

$$\begin{aligned} SINR^{MRT_{csi}}_k = \frac{(1-\sigma ^2)P_d\alpha }{(P_d + \sigma ^2) + 1} \end{aligned}$$
(10)

3 Performnce Analysis

The system performance under imperfect CSI is analysed using achievable rate and downlink transmit power.

3.1 Achievable Rate

Achievable rate is the one method to quantify the system performance. The achievable rate is followed by Shannon theorem. This theory tells the maximum rate which the transmitter can transmit over the channel [34]

$$\begin{aligned} C=log_2\left( 1+SNR\right) \quad \text {(bits/s/Hz)} \end{aligned}$$
(11)

In MU-Massive MIMO downlink system, the transmitter transmits multiple data streams to each user simultaneously and selectively with CSI [35], so transmitter must know this CSI. With MU-MIMO interference among the users and additive noise remains a common factor. Therefore, kth user will experience the inference of the different users and the additive noise [15]. If the power of the desired signal, interference and the noise is defined as \(S_n, I_n\) and \(N_n\) respectively, then the SINR at the kth user becomes [34]

$$\begin{aligned} SINR_k = \left( \frac{S_n}{I_n + N_n}\right) \end{aligned}$$
(12)

The achievable rate for K number of users is given as [20]

$$\begin{aligned} C_k=Klog_2 \left\{ 1+SINR_k \right\} \end{aligned}$$
(13)

From (13) the achievable rate with ZF precoding is calculated as

$$\begin{aligned} C^{zf}_k=Klog_2\left\{ 1+SINR^{zf_{csi}}_k \right\} \end{aligned}$$
(14)

Substituting (8) into (14), we get;

$$\begin{aligned} C^{zf}_k=Klog_2\left( 1+\frac{(\alpha - 1) (1 - \sigma _e^2)P_d}{(P_d\sigma _e^2)+1}\right) \end{aligned}$$
(15)

The achievable rate with MRT precoding from (13) can be calculated as

$$\begin{aligned} C^{MRT}_k=Klog_2\left\{ 1+SINR^{MRT_{csi}}_k\right\} \end{aligned}$$
(16)

Substituting (10) into (16), we get

$$\begin{aligned} C^{MRT}_k=Klog_2\left( 1+\frac{(1-\sigma ^2)P_d\alpha }{(P_d + \sigma ^2) + 1}\right) \end{aligned}$$
(17)

3.2 Downlink Transmit Power

The dowlink transmit power makes the system energy efficient or inefficient. Since the energy efficiency of the system depends on transmit power, with increasing dowlink transmit power, higher capacity can be achieved. but it dereases the energy efficiency of the system [27]. The system becomes more energy efficient when less transmit power is required to achieve the targeted information rate. Here we calculated the dowlink transmit power of ZF and MRT to obtain the same achievable rate for the system under consideration.

Downlink Transmit Power with ZF from (15) can be calculated as

$$\begin{aligned} ln2(\frac{C_k}{K}) = ln\left( 1+\frac{(\alpha - 1) (1 - \sigma _e^2)P_d}{(P_d\sigma _e^2)+1}\right) \end{aligned}$$
(18)

By taking exponential on both sides, we get

$$\begin{aligned} e^{ln2(\frac{C_k}{K})} = 1+\left( \frac{(\alpha - 1) (1 - \sigma _e^2)P_d}{(P_d\sigma _e^2)+1}\right) \end{aligned}$$
(19)

But \(e^{ln2} = 2\)

$$\begin{aligned} \therefore 2^{(\frac{C_k}{K})} = 1+\left( \frac{(\alpha - 1) (1 - \sigma _e^2)P_d}{(P_d\sigma _e^2)+1}\right) \end{aligned}$$
(20)

Hence, the total downlink transmit power with ZF under imperfect CSI is given as

$$\begin{aligned} P_d^{ZF} = \frac{\left( 2^{\left( \frac{C_k}{K}\right) } - 1\right) }{(\alpha - 1) (1 - \sigma _e^2) - \sigma _e^2\left( 2^{\left( \frac{C_k}{K}\right) } - 1\right) } \end{aligned}$$
(21)

Similary downlink transmit power with MRT using (17) can be written as

$$\begin{aligned} ln2\left( \frac{C_k}{K}\right) = ln\left( 1+\frac{(1-\sigma ^2)P_d\alpha }{(P_d + \sigma ^2) + 1}\right) \end{aligned}$$
(22)

Taking exponetial on both sides

$$\begin{aligned} e^{ln2\left( \frac{C^{MRT}_k}{K}\right) } = 1+\frac{(1-\sigma ^2)P_d\alpha }{(P_d + \sigma ^2) + 1} \end{aligned}$$
(23)

Finally, the total downlink transmit power with MRT under imperfect CSI is given as

$$\begin{aligned} P_d^{MRT} = \frac{\sigma _e^2\left( 2^{\left( \frac{C_k}{K}\right) } - 1\right) + \left( 2^{\left( \frac{C_k}{K}\right) } - 1\right) }{(1 - \sigma _e^2)\alpha - \left( 2^{\left( \frac{C_k}{K}\right) } - 1\right) } \end{aligned}$$
(24)

4 Simulation Results

The performance of massive MU-MIMO downlink system with ZF and MRT precoding over Rayleigh fading channel is analyzed in terms of achievable rate and downlink transmit power. Using the theoretical results of Sect. 3, Matlab simulation is performed to provide the numerical results. Three different scenarios are considered to analyze achievable rate and downlink transmit power.

4.1 Achievable Rate

To analyze the achievable rate it is assumed that downlink power of 0 dB is equally divided among all users.

4.1.1 First Scenario

During the first scenario, we change the number of antennas M from 20 to 200 while keep the number of users fixed, \(K=20\) with channel estimation error of 0.3.

Figure 2 shows the achievable rate verus the number of BS antennas M for both ZF and MRT under perfect and imperfect CSI. From Fig. 2, it is clear that the as M increases the achievable rate also increases for both precoding techniques. Achievable rate for ZF increases rapidly with M while for MRT it increases slowly. The performance of both precoding techniques degrades in presence of estimation error. This is because when the number of BS antennas M increases without limit, uncorrelated noise, fast fading and intra-cell interference tend to vanish. Therefore, improvement in achievable rate is greater with large number of M. Moreover, this also increases the value of signal to noise ratio (SNR) which makes the performance of ZF better than MRT. Since ZF performs better at high SNR [26].

Fig. 2
figure 2

Achievable rate of ZF and MRT for different number of antennas keeping the number of users constant

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4.1.2 Second Scenario

During the second scenario, the M is kept fixed while we increase K. We set the \(M=200\) and increase K from 20 to 200 with channel estimation error of 0.3.

Figure 3 shows the achievable rate verus the number of users K for both ZF and MRT under perfect and imperfect CSI. From Fig. 3, it is observed that as K increases, the achievable rate for ZF starts to decrease and it is concave function of K, while it keeps on increasing for MRT. The findings are similar for both Perfect and imperfect CSI. As long as \(M\gg K\), the performance of ZF is much better than MRT. For larger value of K, performance of MRT is better than ZF. This is due to the fact that, ZF works well at high SNR while MRT performs well at low SNR [26]. When K is large it increases the intra-cell interference which causes the SNR to decrease and hence the performance of ZF degrades.

Fig. 3
figure 3

Achievable rate of ZF and MRT for different number of users while keeping the number of antennas constant

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4.1.3 Third Scenario

In this scenario effect of channel imperfectness is analyzed on achievable rate. For achievable rate the channel estimation error are considered from 0 to 1. Achievable rate under different estimation errors is investigated when (1) ratio of M and K is not large, (2) ratio of M and K is large enough.

Figure 4 shows the achievable rate for ZF and MRT with different channel estimation errors. It is observed from Fig. 4 that the achievable rate for both precoding techniques decreases with estimation errors but the decrease in achievable rates of ZF is more as compared to MRT with increase in estimation error. Further, the MRT achievable rate is greater than ZF when M and K are closer to each other, while ZF achievable rate is much higher when M is large enough than K. This is because \(P_d\) is kept constant while estimation error noise increases that causes the achievable rates of both precodings to decrease. Moreover, when ratio of M and K is large, SNR value is high that causes performance of ZF superior over MRT. This is further validated by reducing M to smaller value, that caused the SNR value to go low and performance of MRT is better.

Fig. 4
figure 4

Achievable rate of ZF & MRT verus channel estimation error

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4.2 Downlink Transmit Power

To analyze the downlink transmit power it is assumed that the total achievable data rate of 15 bits/s/Hz is equally shared among the users.

4.2.1 First Scenario

To observe the downlink transmit power we change M from 20 to 200 and choose \(K=10\) with channel estimation error of 0.3 during the first scenario.

Figure 5 shows the downlink transmit power (dB) versus the number of BS antennas M for both ZF and MRT under perfect and imperfect CSI with channel estimation error of 0.3. It is obseved from Fig. 5 that the downlink transmit power decreases as M increases for both precoding techniques. But ZF is more power efficient in this case as compare to MRT. This is because when M is small, \(P_d\) has not been cut down so much. However, as M gets larger, \(P_d\) is cut down more.

Fig. 5
figure 5

Downlink transmit power of ZF and MRT for different number of antennas keeping the number of users constant

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4.2.2 Second Scenario

During the second scenario, downlink transmit power of ZF and MRT precoding is analyzed for \(M=200\) and K from 20 to 200 with channel estimation error of 0.3.

Figure 6 shows the downlink transmit power (dB) versus the number of user K for both ZF and MRT under perfect and imperfect CSI with channel estimation error of 0.3. It is observed from Fig. 6 that as the K increases the downlink power of ZF keeps on increasing rapidly while the downlink power of MRT decreases slowly. With the increase of K the downlink transmit power of ZF is quite large as compare to MRT. This is due to the same fact that, ZF works well at high SNR while MRT performs well at low SNR [26]. When K is large, value of SNR is low and performance of MRT is better.

Fig. 6
figure 6

Downlink transmit power of ZF and MRT for different number of users keeping the number of antennas constant

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4.2.3 Third Scenario

For downlink transmit power, we change the channel estimation errors from 0 to 1. Downlink transmit power under different estimation errors is investigated when (1) ratio of M and K is not large, (2) ratio of M and K is large enough.

Figure 7 shows the downlink transmit power for ZF and MRT with different estimation errors. From Fig. 7 it is clear that downlink transmit power of both precoding techniques increases with estimation error. It is also observed that when M is not quits large, MRT needs less power to transmit the same number of bits as compare to ZF regardless of the estimation error. When we increase the number of antennas from 100 to 500 for same estimation error model, ZF becomes more power efficient for higher estimation error. This is because ZF performs better than MRT at high SNR. When M is large, value of SNR is high. When M is small, SNR is low, as a result MRT performs better.

Fig. 7
figure 7

Downlink transmit power of ZF and MRT for different estimation errors, (1) when ratio of \(M \& K\) is not large, (2) when ratio of \(M \& K\) is large

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5 Conclusion

This paper provides performance of MU-Massive MIMO system with ZF and MRT precoding under imperfect CSI. The key performance parameters are achievable rate and downlinks transmit power which is theoretically derived for imperfect CSI. The simulation results show that performance of ZF is better when M is much larger than K, in this case ZF will achieve higher data rate and will be more power efficient as compare to MRT. But when the ratio of M and K is not large, performance of MRT is superior over ZF. The effect of increasing K is more on the performance of ZF while MRT performance is robust in this case. The effect of channel estimation errors is more on ZF as compare to MRT. Hence, we can say that ZF performs better when BS antennas are large as compare to users, while MRT performs better even when BS antennas are small.