block spacing and the number of discrete channel taps, a received block might be 4 Generalized Frequency Division Multiplexing: Unified Multicarrier Framework W=BHHH Block y(rx) Windowing/ Remove Equalization/ Symbol }(on) demultiplexing filtering overhead demodulation demapping Figure 4.5 Stages of multicarrier waveforms receiver. interfered with the previous and the next block. Let Hb be the discrete channel matrix representing the linear channel. The received block of N samples can be expressed in the vector where ZIX is the inter-block-interference (IBI) and LIX the additive noise. 2. Further processing: Receiver filtering and/or windowing can be used at the receiver, and the CP/CS are removed. This linear processing, which is expressed in a matrix G(rx) E CNXN1, produces the core block, yi = which can be expressed as (4.10) where HGtECNNis = the effective channel matrix, Zb the IBI, and Vb the additive noise. 3. Equalization and demodulation: Several approaches can be used. In order to exploit the design architecture, channel equalization can be performed first using the equal- izer matrix HH eq' and the equalized block is fed to the demodulator with matrix BH. To simplify the channel equalization, it is convenient to design the waveforms to force the channel H to become circular, and Zb = 0. This is feasible when the channel variation is negligible within the block and the CP is selected to be longer than the filter length plus the channel length. With that, single-tap FD channel equalization can be performed on the FD block similar to OFDM. Using the DFT matrix FN, Fn[m, n] = e-j2 we get yb = = (4.11) 4. Data demapping: After the demodulation, the estimated symbols are extracted from the demodulated vector such that don = db[Non]. = 4.2 GFDM As a Flexible Framework In this section, we focus on the GFDM-based core modulator and demodulator. First, we provide several different GFDM representations. Then, we develop a uni- fied time-frequency framework followed by a hardware implementation architecture. This 