We present an innovative photonic strategy to generate arbitrary microwave and millimeter-wave signals with maximal time-bandwidth product capability and broadly tunable center frequency. The proposed approach incorporates high-resolution pulse shaping, optical interferometry, and the concept of frequency-to-time mapping in order to enable independent control over the temporal amplitude, temporal phase, and center frequency of the generated waveforms. Numerical simulation and experimental results validate that the time-bandwidth product of these pulses is equal to the upper bound set by the number of independent pulse shaper control elements, extending to more than twice that of conventional frequency-to-time mapping techniques. We thus demonstrate a record photonic arbitrary waveform generation time-bandwidth product of ∼589. Also, a length 15 Costas sequence realization is implemented to further portray the potentials of this technique. Detailed analysis of the repeatability and stability of these waveforms as well as higher order dispersion compensation is provided.