Optimal Radix-2 FFT Compatible Filters for GFDM

For a linear waveform, a finite condition number of the corresponding modulation matrix is necessary for the waveform to convey the message without ambiguity. Based on the Zak transform, this letter presents an analytical approach to compute the condition number of the modulation matrix for the multi-carrier waveform generalized frequency division multiplexing (GFDM). We further propose a filter design that yields non-singular modulation matrices for an even number of subcarriers and subsymbols, which is not achievable for any previous work. Such a new design has a significant impact on implementation complexity, as the radix-2 FFT operations for conventional multicarrier waveforms can readily be employed for GFDM. Also, we analytically derive the optimal filter that minimizes the condition number. We numerically evaluate the signal-to-interference ratio (SIR) and noise-enhancement factor (NEF) for matched filter (MF) and zero-forcing (ZF) GFDM receivers for the proposed design, respectively.