Audio bandwidth extension (BWE) is a subtask of audio enhancement, whose goal is to extend the audio spectrum to higher frequencies, based on the low-band content of the input signal. Applications include old music restoration or blind audio coding, and many audio signal processing methods, focusing on relatively low sampling rate, could benefit from such an algorithm afterwards. In this work, we propose to explore a differentiable digital signal processing (DDSP) system based on an additive synthesizer plus noise generator, in order to generate high-frequency content. We first consider monophonic signals, and show that our system outperform a conventional signal processing method (SBR) and a state-of-the-art deep-learning system based on Resnet, while being much faster than the later. Then, we address BWE for polyphonic signals by proposing two extensions of the monophonic DDSP system. The first one relies on a cyclic processing of the polyphonic low-band input signal, where each iteration consists in generating an harmonic+noise signal with the monophonic DDSP system, based on the polyphonic low-band signal, and removing the generated harmonic low-band content from the polyphonic signal, in order to reiterate the same scheme. The second idea proposes to directly adapt the architecture of the neural network to a polyphonic additive synthesizer.