rmation Time domain Despreading Windowing unveckxm{.}7 TD-equalizer FD-equalizer unvecmxx(+}7 K-IDFT Frequency domain Figure 4.10 GFDM demodulator in four steps. 4 Generalized Frequency Division Multiplexing: Unified Multicarrier Framework The receiver window is determined based on the transmitter window in the correct domain. For instance, at the transmitter, TD processing is performed, whereas FD processing is more appropriate at the receiver when FD channel equalization is used. Accordingly, the FD window needs to be determined from the equivalent FD transmitter window. Note that the linear demodulation is equivalent to the operation, do = Boy where B is the demod- ulation matrix, which also has the GFDM matrix structure, as discussed in Section 4.2.1.4. 4.2.2.2 Extended Flexibility In addition to the original flexibility in selecting the prototype pulse (the transmitter win- dow), the number of subcarriers and subsymbols, the flexibility can be extended without additional computational cost as follows: 1. Flexible spreading: It is achieved by switching one or both the spreading matrices. This is equivalent to replacing the spreading matrix by identity matrix. It can also be interpreted as GFDM precoding [22]. On the other hand, the DFT spreading can be replaced by Walsh-Hadamard transform, which requires a smaller complexity (only addition oper- ation). 2. Bypassing windowing and/or transform: This flexibility can be exploited to reduce the latency when generating specific waveforms, such as OFDM and DFT-spread-OFDM. 3. Flexible allocation: Instead of transmitting X0 = vec{XT}, other types of mapping, such as x0 = vec {X}, which essentially produces orthogonal time frequency space modula- tion (OTFS) signal [23, 24] can be used. Refer to Chapter 6 of this book for an in-depth discussion on OTFS. On the other hand, flexible allocation allows direct integration of FD-GFDM within an OFDM system. For that, the additional IDFT block used in the FD modulation needs to be flexible. 4.2.2.3 Flexible Hardware Architec