## 4 Multi-Objective Optimization  
The demand of transmission is maximizing the Bit Rate (\(BR\)) (bit/s), which is equal to the number of bits per frame multiplied by the number of OFDM symbols per second. Two other conflicting objectives must be minimized at once. The first is the total power of the signal \(P_{tot}\), which is the sum of the powers of all the sub-carriers. The second conflicting objective is the error bit rate (\(BER\)), which depends on the noise spectrum: a probability density function can be associated to each point of the constellation centered in these points and the \(BER\), for a given point, will be equal to the probability that the distance between a received point and its reference position be greater than the semi-distance between two adjacent points. In this paper, Gaussian distribution is assumed, so that there is a direct relationship between \(BER\) and the least distance among the points of the constellation. Different assumptions on the statistical distribution of points around the reference position do not change the application of the proposed method.  
Note that, by setting the maximum transmission power of each constellation (\(P_{MAX}\)), its area is univocally assigned (area of the outer circle in Figure 4). Then, by setting the minimum distance between the points of the constellation (i.e., the \(BER\)), also its number of points is determined, hence, also the \(BR\).  
Figure 4 refers to the 16-APSK scheme for a generic sub-carrier [25]. In the same figure, the shadow circles centered in the points of the constellation have a radius equal to 3 times the standard deviation \(\sigma\) of the Gaussian distribution. The distance between two adjacent points is greater than \(6\sigma\).  
The requirements on the \(BER\) are particularly stringent, therefore, the APSK constellations [25] are adopted, which are increasingly considered in 5G mobile communications; for a given power, APSK gives the maximum distance between adjacent points. Without loss of generality, different constellation schemes can be assumed because the inter-dependence among power, \(BR\) and \(BER\) holds valid.  
The water-filling (WF) algorithm [26; 27] is used to allocate the bits to the sub-carriers. Note that, WF can be used either to design the constellations or to dynamically allocate the bits to the sub-carriers. In this work, it has been used for the former purpose.  
The multi-objective optimization of the bit loading consists in finding the solutions that reconcile the conflicting demands. In particular, a solution is said nondominated, or Pareto optimal, if none of  
Figure 4: The 16-APSK constellation.  
the objective functions can be improved in value without worsening some of the others. The set of such solutions is called Pareto Front [14] and, in the problem at hand, it represents a discrete subspace of the surface of the feasible solutions. In fact, as \(BR\) is a discrete variable, except for a discrete set of points in the (\(P_{tot}\), \(BER\)) plane, the increase of one or both of them does not correspond to an increase of \(BR\).  
The multi-objective optimization aims to find this set of Pareto solutions and then to present them to the designer who will choose among them [14; 15]. In the problem at hand, the surface \(BR=f(P_{tot},BER)\) has been densely sampled, and the set of non-dominated points has been selected, which is assumed as the Pareto front.