In order to meet the requirements of high system throughput, massive connectivity, and asynchronous transmission for 5G and beyond, non-orthogonal transmission techniques, including modulation techniques and multiple access techniques, have attracted a great deal of attention in both academia and industry. Traditional non-orthogonal multiple access (NOMA) schemes are investigated mainly based on the orthogonal waveform, i.e., orthogonal frequency-division multiplexing (OFDM). However, OFDM is not the most suitable waveform for the foreseeable mMTC scenarios because it is vulnerable to carrier frequency offset (CFO). To employ the advantages of both NOMA and non-orthogonal waveforms, in this paper, we focus on NOMA transmissions with the non-orthogonal waveform modulation. Specifically, an uplink transmission scheme for pattern-division multiple access (PDMA) based on discrete Fourier transform spread generalized multi-carrier (DFT-S-GMC) modulation, DFT-S-GMC-PDMA for short, is studied. First, implementation schemes of the proposed DFT-S-GMC-PDMA in the time domain and frequency domain are presented, respectively. Second, the equivalent channel response matrix and noise formulas of both implementation schemes are derived. Furthermore, simulation results are given to show that the DFT-S-GMC-PDMA can achieve a comparable performance to PDMA based on discrete Fourier transform spread OFDM (DFT-S-OFDM-PDMA), while the improvement of complexity is less than 3%. system performance under different equalizers and different PDMA patterns are also evaluated where neglectable performance loss is observed. Thanks to the robustness against CFO, the multiple-access interference performance of the DFT-S-GMC-PDMA scheme is about 0.5 dB superior to that of the corresponding the DFT-S-OFDM-PDMA scheme. It indicates by simulation that the proposed DFT-S-GMC-PDMA is robust to CFO at the cost of neglectable performance loss, and thus it is a more attractive candidate for the uplink transmission of mMTC.