On the Configuration-Dependent Singularity of GFDM Pulse-Shaping Filter Banks

It is known that, under some configurations, the generalized frequency-division multiplexing (GFDM) pulse-shaper matrix is singular. Moreover, a GFDM pulse-shaper matrix may have unequal magnitude gains in different vector-space dimensions. This letter presents a Fourier-transform-based decomposition of the GFDM pulse-shaper matrix from which the above GFDM properties, as well as some other ones, can be readily appreciated. An upper bound on the nullity of the pulse-shaper matrix is obtained. We also show that, for some commonly conceivable designs, the nullity of the pulse-shaper matrix, if singular, is one. In the latter case, we also specify the corresponding null space. From this, GFDM signals can be designed, so that they do not fall in the null space.