## 3 Performance Metric for Measuring EE Gain  
In this section, we present a performance metric for measuring EE gain. It is well-known that most signal transmission systems use SE or TP as a performance metric. However, EE could be more important than TP depending on the applications.  
The justification of using the EE metric for LSASs is comes from [PERSON]'s channel capacity equation that is represented in AWGN channel as:  
\[C=\log_{2}(1+\Gamma), \tag{18}\]  
where \(\Gamma\) is the signal-to-noise ratio (SNR) that shows how much signal power is received against noise power at RX. Equation (18) is plotted in Figure 1.  
As observed, there are roughly two regions in the figure of channel capacity versus \(\Gamma\). In the region where \(\Gamma\) is not sufficiently high, the channel capacity increases significantly as \(\Gamma\) increases. We usually call this region as power limited region. The channel capacity logarithmically increases as \(\Gamma\) increases. Due to this reason, once \(\Gamma\) is high enough, even if \(\Gamma\) continue to increase, it does little help to increase \(C\). We call this region as bandwidth limited region. The bandwidth limited region is the region where \(\Gamma\) is high enough that increasing \(\Gamma\) is not particularly helpful in increasing \(C\), while increasing \(B\) is very helpful. If we can increase the signal-to-noise ratio using a large amount of TX antennas, \(C\) goes to the bandwidth limited region, and thus we can reduce \(\Gamma\) with little channel capacity loss. \(\Gamma\) is tightly connected to the PA which is one of the most power hungry devices. This is the logic of using the EE metric.  
The simplified channel capacity with beamforming effect can be represented as:  
\[C=\log_{2}(1+\chi\cdot\Gamma), \tag{19}\]  
where \(\chi\) is the beamforming factor from a large amount of TX antennas, which usually depends on the number of TX antennas, and is typically much larger than 1. With high \(\chi\), we can get room to reduce the TX power consumption, which consequently increases EE.  
With this in mind, based on (17) and (19), the TP can be represented as follows:  
\[TP\approx\alpha\cdot K\cdot B\cdot\log_{2}(1+\frac{N_{t}}{K}\cdot\frac{\rho}{N _{0}B}), \tag{20}\]  
where \(\alpha\) is the scaling factor to reflect guard interval, and reference signal overhead, etc. Although (20) has been used as a performance metric for a long time, the current form is not sufficient for some applications. The reason is that Equation (20) does not represent power consumption related performance. For example, based on (20), increasing TX power is always helpful, but that is not true on real systems. Equation (20) has had very high impact so far because the wireless TP has been always less than demand. Of course, there is still high TP demand, but today's TP demand is a little bit different from the past. Moreover, environmental pollution from signal transmission systems is growing, and there is a consensus that reducing the carbon footprint has become much more important than previously.  
For these reasons, the EE performance metric can be defined as the ratio of TP to power consumption:  
\[EE\approx\frac{\alpha\cdot K\cdot B\cdot\log_{2}\left(1+\frac{N_{t}}{K}\cdot \frac{\rho}{N_{0}B}\right)}{\frac{1}{\eta}P_{tx}+P_{RF}+P_{bb}}. \tag{21}\]  
Continuous increase of power consumption can increase EE only up to a certain point which is in the power limited region. Figure 2 shows the performance metric comparison and change of TP and EE.  
Figure 1: Shannon’s channel capacity versus \(\Gamma\).