exibility The TD and FD blocks in (4.15) and (4.17) can be rewritten in the form (4.22) Accordingly, the two cases can be presented in the form (4.23) where Ux E CNxxNx is a normalized DFT or IDFT matrix of size Nx; for X= 1, 2, 3, N3 = N2, Wtx E CN2XN1 corresponds to the Zak transform, D0 E N represents the data matrix, x0 E CN1N2X1 is the GFDM block, and X E CN1xN2 is the GFDM block matrix. The concrete values corresponding to each domain are summarized in Table 4.2. Figure 4.8 Orthogonal GFDM. (a) Separate subsymbols and overlapped subcarriers. (b) Separate subcarriers and overlapped subsymbols. (c) Overlapped subcarriers and subsymbols. 4.2 GFDM As a Flexible Framework Table 4.2 Flexible GFDM parameters. Parameter [U1,U,U] Transformation Time domain Spreading Windowing {.} T {{.} }} K-DFT Frequency domain Figure 4.9 GFDM modulator in four steps. 4.2.2.1 Alternative Interpretation of GFDM Based on the model (4.23), GFDM modulation can be achieved in four steps, as illustrated in Figure 4.9 for both TD and FD. 1. Data spreading: Spreading is achieved by applying DFT/IDFT on the columns and rows of D0 E CN1XN2. Therefore, we get the spread data matrix Ds = U2DTU1 E CN2XN1 2. Windowing: The spread data matrix DS is element-wise multiplied with a transmitter windowing matrix is generated based on the prototype pulse shape. 3. Transformation: The columns of the matrix WixODS are transformed by means of IDFT or DFT, where the samples are expressed by matrix 4. Sample allocation: The final vector is achieved by allocating the generated samples to the corresponding indexes, specifically, x0 = vec{X}. In the case of FD modulation, an addi- tional N-IDFT is required to convert the signal to the time domain, i.e. The demodulator performs the inverse operations, as illustrated in Figure 4.10. Let y be the equalized block, and W x be the receiver window. First, the vector is reallocated to the matrix Y = unvecn2xN1 {y}, then DO is computed by performing the inverses of transform, windowing, and spreading, as (4.24) Transfo