In this work, multiple radar waveforms are simultaneously transmitted, emitted from different ldquovirtualrdquo antennas. The goal is to process the returns in such a way that the overall ambiguity function is a sum of ambiguity functions better approximating the desired thumbtack shape. A 4times4 example involves two spatially separated antennas with each able to transmit and receive simultaneously on two different polarizations. The 4times4 unitary design dictates the scheduling of the waveforms over the four virtual antennas over four PRIs (pulse repetition intervals), and how the matched filtering of the returns over four PRIs is combined in to achieve both perfect separation (of the superimposed returns) and perfect reconstruction. Perfect reconstruction means the sum of the time-autocorrelations associated with each of the four waveforms is a delta function. Conditions for both perfect separation and perfect reconstruction are developed, and a variety of waveform sets satisfying both are presented.