orously proved in [19] that Wexler-Raz duality condition cannot be fulfilled if aB > 1. Based on that, the choice of P and Q is influenced by QP N, which is necessary to achieve Wexler-Raz duality condition. AfK <F,= = = QK < N, which is necessary to confine the signal within a bandwidth < Fs. Consequently, the design needs to fulfill the conditions Q M, P<K. 4.1.2 GFDM As a Flexible Waveform The case P = K and Q = M, i.e. PQ = N, in (4.8) corresponds to a critically sampled system, where Afat===1,whichisacually = GFDM [14]: (4.8) GFDM structure enables efficient implementation, as discussed in Section 4.2.2 and the related parameters (Table 4.1), which can be customized to produce different waveforms. For example, OFDM corresponds to the case where M = 1 and g[n] is a rectangular pulse. SC is achieved with K = 1 and g[n] = S[n]. FMT can be generated with proper configuration of Mon (see Figure 4.1). 2 In simple words, letx = A[k + mK, n] = gka,mg[n], d[k + mK] = dk.m' and B, where d = BHx. Therefore BH A = I. Wexler-Raz duality condition states that there exist h such that B[k + = 4.1 Overview of Multicarrier Waveforms Table 4.1 GFDM-based waveform parameters. Number of subcarriers Number of subsymbols Block length Active subcarriers Active subsymbols Active symbols Non=(k+mk,dkm#0) Prototype pulse Pulse shapes GFDM block kekonmeMon 4.1.2.1 GFDM with Multiple Prototype Pulses The case PQ < N corresponds to an over-sampled system where AfAT <1 does not fit in the GFDM representation. However, by managing the sampling frequency, the block length, and properly defining the active sets, it can be represented as a superposition of two or more GFDM blocks. For example, the block length and the sampling frequency can be adjusted to fulfill the conditions Q = TAf = M and P = ATF = = = where L is a positive integer. The set of active subcarriers Kon is configured with respect to the available bandwidth B Fs. The modulation pulses (4.8) can be redefined in the form 8.[]n where = - 1. Each pulse shape is used to generate a