Time-frequency waveform synthesis using a least-squares approach

The optimal solution for least-squares ambiguity function synthesis in the continuous-time-frequency domain is found as a solution to a homogeneous Fredholm integral equation of the second kind. The solutions for time- or band-limited waveforms are examined. It is shown that the least-squares cost function is not convex. An exact expression for the continuous ambiguity function of a time-limited waveform in terms of the discrete ambiguity function of the same waveform is given, and the resulting aliasing problem is investigated. The least-squares synthesis of ambiguity functions for time-limited waveforms in the discrete-time-frequency domain is solved. A practical suboptimal design algorithm is given. A design obtained by employing the proposed algorithm is presented. A relation is given between discrete Wigner distributions obtained using samples taken at the Nyquist rate and at twice the Nyquist rate, and it is shown that the corresponding discrete Wigner distribution synthesis can be performed using essentially the same algorithm proposed for ambiguity function synthesis