## 2 Analysis
### Basic Theory and EE vs. SE Relationship  
Specifically, \(SE\) is a performance measure of wireless communication systems, which presents transmission rate \(R\) [b/s] achievable for given bandwidth \(B\) [Hz], i.e., \(SE\) is expressed as the ratio \(R/B\). On the other hand, the radio channel \(EE\) [b/J] is described as the dimensionless ratio \(E_{b}/N_{0}\) between the energy per bit and the noise spectral density, which is in fact the count of the transmitted information bits per energy unit. Hence, as the two very basic resources of a radio channel--bandwidth \(B\) and mean power \(P_{S}\)--determine its Shannon capacity \(C\), we can express the latter by \(SE\) and \(EE\) as follows:  
\[C=B\cdot log_{2}\bigg{(}1+\frac{P_{S}}{P_{N}}\bigg{)}=B\cdot log_{2}\bigg{(}1+ \frac{E_{b}}{N_{0}}\frac{R}{B}\bigg{)}=B\cdot log_{2}(1+SE\cdot EE) \tag{1}\]  
Specifically, in case of a Additive-White-Gaussian-Noise (AWGN) channel, it is well known that for given \(P_{S}\) and \(B\), \(EE\) can be expressed as the ratio of \(C\) and \(B\), which applied to (1) implies that:  
\[SE=\log_{2}(1+SE\cdot EE) \tag{2}\]  
Hence, the explicit relationship between \(EE\) and \(SE\) is:  
\[EE=\frac{2^{SE}-1}{SE} \tag{3}\]  
Applying (3) to the simplest wireless network consisting of a single BS and a single UE, we can analyze relations in various power and energy regions of the channel, which are determined by a \(SE\) and \(EE\) relationship, with the goal of enabling a significantly larger throughput and data rate. In this regard, let us consider two exemplar cases, which are characteristic for the linear and the non-linear region of the \(EE\) vs. \(SE\) relationship. In the first case, let us assume the spectral efficiency as low as \(SE\) = 0.1 b/s/Hz. In order to increase the throughput eight times, the received signal power needs to be increased about ten times. This example reflects the almost linear relationship between the increment of the information data rate and the required power at the receiver frontend, which, however, is not applicable to demanding services requiring higher data rates, Figure 2.  
Nevertheless, for higher \(SE\) in the order of 2 b/s/Hz, to increase the data rate only four times while keeping the same bandwidth, the received signal power needs to be increased about 85 times, whereas to increase the data rate ten times, the necessary received signal power must ramp up about 300,000 times, Figure 2.  
From these examples, it is obvious that the increasing data rate requires a significant increase of the received signal power. This implies reducing distances between BS and UE down to tens of meters, whereas the linear-region models tolerate considerably longer distances, but with significantly lower \(SE\), as a consequence of reducing \(EE\), due to channel propagation effects. Figure 2: Spectral efficiency (SE) and energy efficiency (EE) (Eb/N0) relationship. However, if the above identified problem is considered at a network level with multiple UEs using the selected multiple access scheme, very complex processes can arise, where _EE_ plays a dominant role:  
* In linear-region models, such as is the case with 2G mobile networks, _EE_ significantly decreases towards the cell edge, which also implies the reduction of _SE_ and therefore also of the throughput. A particular negative aspect of such models is the almost uniform distribution of the signal energy across the cell, as equal coverage of areas with few active UEs is definitely not a rational use of energy resources. Hence, e.g., reducing the cell radius from 1000 m to 250 m results in _EE_ increasing from 0.11 Mb/s/J to 1.92 Mb/s/J (for a 3G system), which is equivalent to increasing _EE_ 17.5 times [25]. * Moreover, the models with huge cell dimensions require large BS power, which is not welcome in zones around the BS, due to high electromagnetic radiation density and its potentially dangerous impact in highly populated areas.