 that turns out to be a hyperellipsoid whose main kth semi-axis is Ok long (Fig. 2.4). Now, let us come back to the communication problem. Information is encoded in different values of the Xk coefficients (Fig. 2.4) and is retrieved observing the field distribution on S2, i.e., the values of the yk coefficients. However, following the Kolmogorov approach, due to the presence of noise and measurement uncertainties, the received field is Physical Deep Physical Layer Physical layer layer Fig. 2.4 Scheme of a functional communication system; A is the radiation operator, X is an infinite-dimensional ball having unit radius, Y is an almost finite-dimensional hyperellipsoid having effective NDFé dimensions and whose semi-axes have length equal to Ok; the receiver divides Y in subsets given by e-balls centered on an E distinguishable set 2 Information Flows at the DPL Level (2.5) Intuitively, since 11xkll 1, if Ok < E, we have < E for any value of Xk, making it impossible to transmit information using the kth coefficient of the expansion. We define "Number of Degrees of Freedom of the field at the E level" (NDF) the number of singular values higher than E [34]. It is worth noting that the definition of the Number of Degrees of Freedom of the field at E level of uncertainty adopted in our discussion is different compared to the NDF defined by Bucci and Franceschetti in [20]. The problem discussed in [20] regards the approximation of the electromagnetic field, and its solution involves a covering problem [35] (i.e., the minimum number of balls having radius E required to cover the Y set), instead of a packing problem. The dimension of the effective spaces in the two cases differs by a singular value. Furthermore, Bucci's definition of NDF does not explicitly consider the noise level since Bucci's theory was developed mainly in the case of free-space propagation in radiative region, where the singular values gently decrease until reaching a value, e.g., after which there is an extremely rapid decrease toward zero. In this c