A waveform synthesizer comprising for synthesizing RF lightwave waveforms in the optical domain. These waveforms are constructed by generating their constituent Fourier frequency components or tones and then adjusting the amplitudes of those frequency components or tones. The apparatus includes: a RF-lightwave frequency-comb generator; and a multi-tone, frequency selective amplitude modulator coupled to the RF-lightwave frequency-comb generator for generating a continuous-wave comb comprising a set of RF tones amplitude modulated onto a lightwave carrier.