Table 2 Number of multiplications of DFT models. Modulation and demodulation complexity are the same with this criterion

OFDM

\( \mathbf {x} = \frac {1}{N}\mathbf {W}^{\mathrm {H}}_{N} \mathbf {d} \), \(\, \hat {\mathbf {\!d}} = \mathbf {W}_{N} \mathbf {y} \)

\( C_{\text {OFDM}} = C_{\text {DFT,N}}=N\log _{2}(N)\)

K=2048, M=1

22,528

GFDM

\( \mathbf {x} = \frac {1}{N}\mathbf {W}_{N}^{\mathrm {H}} \sum \limits _{k=0}^{K-1} \mathbf {P}^{(k)} \mathbf {G} \mathbf {R}^{(K,M)} \mathbf {W}_{M} \mathbf {d}^{\text {r}}_{k} \)

C _GFDM,ZF=C _DFT,N+L(N+C _DFT,M)

K=128, M=15

274,202

Time conv.

\( \,\hat {\mathbf {d}}^{\text {r}}_{k} = \frac {1}{M}\mathbf {W}_{\!\!M\,\,}^{\mathrm {H}} \left (\mathbf {R}^{(K,M)}\right)^{\mathrm {T}} \mathbf {\Gamma } \left (\mathbf {P}^{(k)} \right)^{\mathrm {T}} \mathbf {W}_{N} \mathbf {y}, \forall k\)

L=K

GFDM

\(\mathbf {x} = \sum \limits _{m=0}^{M-1} \mathbf {P}^{(m)} \text {diag}(\boldsymbol {g}) \mathbf {R}^{(M,K)} \mathbf {W}_{K}^{H} \mathbf {d}^{\text {c}}_{m} \)

C _GFDM=M(N+C _DFT,K)

K=128, M=15

42,240

Freq. conv.

\( \,\hat {\mathbf {d}}^{\text {c}}_{m} = {\mathbf {W}_{K}}\left (\mathbf {R}^{(M,K)}\right)^{\mathrm {T}} \text {diag}(\boldsymbol {\gamma }) \left (\mathbf {P}^{(m)} \right)^{\mathrm {T}} \mathbf {y}, \forall m\)