Spectrally-Constrained Sequences: Bounds and Constructions

We investigate spectrally-constrained sequences (SCSs), which are applicable to the communication and radar systems operating over non-contiguous carriers or frequency slots. Typical examples of such systems are overlay cognitive radio or cognitive radar networks. First, we derive the periodic- and aperiodic-correlation lower bounds for single-channel SCSs and multi-channel SCSs by convex optimization in the frequency domain. Each of these bounds reduces to a Welch bound when the number of forbidden carriers is set to zero. We then propose systematic constructions of optimal unimodular single-channel SCSs with the aid of cyclic difference sets and the theory of maximal-length shift register sequences.