Optimal spectrally-constrained sequences

A sequence is said to be spectrally-constrained if it has to satisfy a spectral map consisting of several non-contiguous nulled frequency-slots. Such sequences play a key role in emerging spectrally-constrained systems such as cognitive radio and cognitive radar. In this paper, we study two types of spectrally-constrained sequences (SCSs), one with low periodic auto-correlation function (PACF) sidelobe, the other with zero auto-correlation zone (ZACZ). By deriving a correlation lower bound, we show that Type-I SCSs are optimal with minimum total PACF sidelobe energy provided that uniform power allocation is applied to all active (non-nulled) frequency-slots. We also propose optimal Type-II SCSs, each having maximum ZACZ width, for certain spectral map patterns.