A system, the synthesis convolver, is described that can process the output of an analysis filter bank and: • Reconstruct the analysis filter bank input signal, • Implement continuous convolution on the reconstructed signal, and • Resample the reconstructed signal to different sample rates. The synthesis convolver combines the capabilities of a synthesis filter bank and a continuous convolver. The synthesis convolver is based on earlier work but improves upon it, adding convolution and resampling. As well as reconstructing filterbank data, convolution allows dechirping of pulsar signals and resampling allows synthesized data to conform to the VLBI VDIF standard. The spectral overlap-add approach described, compared to earlier work, reduces errors and is more robust to channel gain errors. The system uses windows with smoothed or apodized edges, with the classical Tukey window being used previously. Here the Tukey window is generalized leading to a class of apodized windows. This class of windows is explored and one is found that is close to optimal in all conditions and can reduce errors by up to 40[Formula: see text]dB compared to an equivalent Tukey window. Achievable aliasing errors are lower than those of a standard polyphase synthesis filter bank. The synthesis convolver provides a high quality and versatile replacement for polyphase synthesis filter banks.