the waveform features, such as low out-of-band (OOB). More details on using GFDM for OFDM enhancement are introduced in Section 4.3. 4.2.1.3 2D-Block Representation To clarify the time-frequency structure, the entries of the modulation block x can be rear- ranged in a matrix using the inverse of vectorization X = unveckxm{x} , CKXM where X[k, m] = x[k + mK]. The FD block is given by X = m-th col- umn of X represents the samples transmitted within the subsymbols duration, and the k-th column of X corresponds to a subcarrier width of M samples. Using the notation X[q,p] = x[q + pK], from (4.13) we get (4.14) where G E CKXM, G[q,p] = g[q + pK]. Hence, the 2D-block results from the circular filter- ing of the rows of FHD and the rows of G. This is essentially the polyphase representation of the filter bank represented in Figure 4.6, where the q-th row of G is the q-th polyphase of g[n]. By computing the M-DFT of the rows and using the element-wise product notation o, we get (4.15) Relation to Zak Transform The matrix ZMK=FMX = is known as the Zak transform of X, which is invertible. The k-th row of X, xT=X[k,:] results from the downsampling of x by factor K starting from the k-th sample. The matrix XT stacks the polyphase signals with respect to the sampling fac- tor K, as illustrated in Figure 4.7. This transform couples the subsymbols and provides a MK-DFT {.} e-12rm Figure 4.7 Zak transform. 3 Here, we use Matlab notations for indexing. 4.2 GFDM As a Flexible Framework time-frequency relation. To determine the relation between the TD and FD block, we use the definition of N-DFT, and X[p, q] = x[p + qM], K-1M-1 = (XFk,pp = As a result, the time-frequency relation is given by (4.16) Thus ZRA = 1FHX is the dual FD Zak transform. Following similar derivation for the FD block X, from (4.15), (4.16), and using ZRM = FMGT, (4.17) where G E (4.18) 4.2.1.4 GFDM Matrix Structure The structure of the GFDM modulation matrix can be derived by the vectorization of (4.15) and (4.17). First, we define the notations required to ex