press the vectorization. For a matrix Z E COXP, vec = = (4.19) UP,Q is a unitary matrix and IIQ,P is a permutation matrix known as the commutative matrix of size QP x QP defined by IIQ,p[p + qPq + pQ] = 1. Note that = IIP,Q Using the TD relation, XFM M = [FDFM]O[[FF] Let Ag = /Kdiag{vec{GFu}}, and because x = Ad, then (4.20) The matrices Vt and U t are unitary matrices, and therefore, the condition of A is influenced by the diagonal elements of Ag. 4 Similarly, the FD matrix , where x = Ad can be derived 4 Different permutations of the elements of A(g) lead to different Vt and U Thus, different formulas can be found in the literature. 4 Generalized Frequency Division Multiplexing: Unified Multicarrier Framework from XFH OFF = (4.21) where Mdiag(vec(G)F)|.Any matrix that fulfills one of those structures is called a GFDM matrix. As a result, the condition of the modulation matrix is determined from the TD or FD Zak transform of g[n] given by Zm,k = (GFM)[k, m] and = respectively. For instance, a real symmetric pulse produces a singular matrix when both K and M are even numbers [21]. Orthogonal Modulation In order to achieve an orthogonal modulation matrix, it is necessary that = const. Thus we get different orthogonal waveforms by designing the phases in the time domain = in the frequency domain. Two corner cases arise; the first is whendmk=k,which results in a frame of M consecutive OFDM symbols. In other words, D[ m] can be seen as the frequency domain data symbol mapped to the m-th OFDM symbol of length K. In this case, there is clear separation of the signals corresponding to each subsymbol. In the other case, when we get the DFT-S-OFDM, and for the special case of K = 1, we get the single carrier. Therefore, D[k, : ] can be seen as the time-domain symbols that use the k-th subchannel of bandwidth M. Thereby, the subcarriers can be sep- arated. In all other cases, the subsymbols and subcarriers overlap, i.e. the data are spread over the time and frequency domains (Figure 4.8). 4.2.2 Architecture and Extended Fl