An arbitrary waveform generator (AWG) architecture suited for 5G transmission is presented. The digital-to-analog (DA) conversion principle is discussed, underlining its theoretical features toward radio frequency (RF) applications. The signal generation is based on a piecewise linear approximation, resulting from the use of a differential digital coding associated with a custom digital-to-analog converter (DAC), named here the Riemann Pump. The intrinsic performances of this architecture in terms of quantization noise make the Riemann Pump an efficient DAC for multi-carrier applications. Simulations have been carried out on the considered architecture, with a configuration that covers more than 3 GHz bandwidth with a slight oversampling ratio (OSR) and several input bits. Carrier aggregation capabilities are shown with a 5G handset transmission scheme of 10 synchronized 64-QAM modulated signals between 1.8 GHz and 3.6 GHz. This disruptive system exhibits promising performances as for the realization of a 5G handset transmitter with moderate hardware complexity and low power consumption.