RZF=RMRT, K= 20, M=100

Figure above shows the relationship between the downlink transmit power and the reliability of channel estimation in which the downlink transmit power was found to increase when the reliability of the channel estimates increases from 0 to 1 for both linear precoding schemes (MRT and ZF) under the same conditions i.e. Rzf=Rmrt and K=20 while M=100. Furthermore it was found that for the low values of achievable sum rate (R=-5), the downlink transmit power is better for both pre-coders compared to the high value of achievable sum rate but ZF linear pre-coder provides better result as compared to MRT under the same conditions.

According to figure 2 above, increasing the number of serving BS antenna causes to have a very good results of downlink transmit power as compared to few antennas for example when the ratio of M/K becomes very large, the downlink transmit power decreases as compared to low ratio of M/K that is to say as the number of BS antennas increases the downlink transmit power decreases gradually as shown in figure above but the relationship with channel reliability is not affected as when the channel reliability increases also the power increases. And once again for higher number of BS antennas (M=500), the ZF linear precoder outperforms the MRT linear precoder but for the low number of BS antennas (M=100), the MRT linear precoder outperforms ZF precoder as table 1 shows the extract of figure 2.

Table 1: Showing the values of downlink transmit power with reliability of channel estimation and the number of BS antennas at constant achievable sum rate (R= 5 bits/Hz/s) and constant number of users (K=50)

Number of BS antennas M=100 M=500

Reliability of Channel Estimation e=0.4 e=0.8 e=0.4 e=0.8

Downlink Transmit power for ZF (dB) -11 -5.5 -22 -16

Downlink Transmit power for MRT (dB) -13 -7.5 -20 -14

R=5, K=50 and M varies

R=10, K=10

Figure 3 above shows the relationship between the downlink transmit power and the number of BS antennas (M), in which the transmit power is decreasing as the number of BS grows large which is required for energy efficiency and better performance but again as it cab been seen in the figure with different values of reliability of channel estimation, the downlink transmit power favors the channel with small value of channel reliability compared to the one with high value as it can be seen in the figure, the downlink transmit power for reliability e=0.3 is better than that with reliability e=0.7 under the same condition of achievable sum rate (5 bits/Hz/s) and number of users (K=20). However, in both cases ZF linear pre-coder gave better performance compared to MRT linear pre-coder.

R= 5, k=20

Table 2: Showing the values of downlink transmit power with respect to number of users and with varying reliability of channel estimation at constant achievable rate(R=5 bits/Hz/s) and constant number of BS antennas (M=100)

Number of users (K) K=40 K= 90

Reliability of Channel Estimation e=0.3 e=0.7 e=0.3 e=0.7

Downlink Transmit power for ZF (dB) -13 -9.5 -5.5 2

Downlink Transmit power for MRT (dB) -14.5 -8.5 -15 -8.9

As it can be observed in the table above and the figure 4 above, as the number of users (K) increases it has got detrimental effect on the downlink transmit power depending on the reliability of the channel estimation and the type of linear pre-coding scheme, for the case of MRT, increasing the number of users tends to cause the downlink transmit power to decrease which is better for energy efficiency compared to ZF linear pre-coding which tends to increase the downlink transmit power as the number of users increases and this takes place in both cases when the reliability of channel estimation changes while keeping the number of BS antennas and achievable sum rate constant. So in general for few number of BS antennas, the MRT outperforms ZF linear pre-coding and in this case the concept that MRT performs better when the ratio of M/K is low is easily proved.

R=5, M=100

Table 2: Showing the values of downlink transmit power with respect to number of users and with varying reliability of channel estimation at constant achievable rate(R=5 bits/Hz/s) and constant number of BS antennas (M=500)

Number of users (K) K=40 K= 90

Reliability of Channel Estimation e=0.3 e=0.7 e=0.3 e=0.7

Downlink Transmit power for ZF (dB) -22.2 -17.8 -21.7 -17.3

Downlink Transmit power for MRT (dB) -21.5 -15.7 -21.8 -15.9

Increasing the number of serving BS antennas (M=500) had an effect on the downlink transmit power as it can be seen in the table above and in figure 5 above in which the downlink transmit power was observed to be decreasing as the number of users increases for both MRT and ZF linear precoding schemes under the constant number of BS antenna and constant achievable sum rate. When the reliability of channel estimation was 0.7, the ZF precoding scheme performed better even though the number of users was kept increasing compared to MRT but the same case also happens when the reliability of the channel estimation was 0.3 which provided better results for both linear precoding techniques (ZF and MRT) compared to when e=0.7 but also the ZF precoding scheme outperformed MRT precoding scheme under the same conditions (R=5bits/Hz/s and M=500).

R=5, M=500