architecture is enhanced with additional flexibility that extends GFDM beyond the scope of conventional waveforms. 4.2 GFDM As a Flexible Framework Figure 4.6 Circular convolution filter bank representation. do [m] * og[n] dk [m] dk_1[m] * g[n] 4.2.1 GFDM Representations Recall that the pulse shapes in GFDM are generated using a prototype pulse g[n], whose N-DFT is denoted as g(s) such that (4.12) The pulse shapes have a similar formulation in TD and FD, with the exchange of param- eters. In the following, we use the data matrix D E CKXM, where D[k, m] = dk.m. The TD-GFDM block can be expressed as K-1M-1 (4.13) 4.2.1.1 Filter Bank Representation The GFDM block can be seen as a result of circular convolution with a filter g[n] followed by frequency upconversion, as seen in Figure 4.6: where * denotes the circular convolution. Circular filtering was introduced in the origi- nal GFDM publication [15] as an approach for the digital implementation of the classical filter bank. Circular convolution is equivalent to linear filtering with tail biting. It is used to reduce the overhead due to the filter tail and to allow inserting CP. This representation corresponds to the conventional GFDM, where the focus is on a specific type of prototype pulses that spans two subcarriers spacing [21]. 4.2.1.2 Vector Representation The TD block (4.13) can be presented in a vector formx = Ad, where d = vec{D} is the vec- torization of D, with d[k + mK] = D[k, m]. The modulation matrix A is defined by A[n, m + kM] = gk,m[n]. The FD block is accordingly x = FAd. Thus, A = FNA represents the FD modulation matrix. Note that x = Ad and can be seen as the core block corre- sponding to the precoded data vector x and the modulation matrix 1FH Thereby, any linear modulation can be technically considered as precoded OFDM. Nevertheless, this precoding is required to achieve certain waveform properties, and the design and implementation of 4 Generalized Frequency Division Multiplexing: Unified Multicarrier Framework the precoding matrix A provides 