Good Code Sets from Complementary Pairs via Discrete Frequency Chips

Abstract

It is shown that replacing the sinusoidal chip in Golay complementary code pairs by special classes of waveforms that satisfy two conditions, symmetry/anti-symmetry and quazi-orthogonality in the convolution sense, renders the complementary codes immune to frequency selective fading and also allows for concatenating them in time using one frequency band/channel. This results in a zero-sidelobe region around the mainlobe and an adjacent region of small cross-correlation sidelobes. The symmetry/anti-symmetry property results in the zero-sidelobe region on either side of the mainlobe, while quasi-orthogonality of the two chips keeps the adjacent region of cross-correlations small. Such codes are constructed using discrete frequency-coding waveforms (DFCW) based on linear frequency modulation (LFM) and piecewise LFM (PLFM) waveforms as chips for the complementary code pair, as they satisfy both the symmetry/anti-symmetry and quasi-orthogonality conditions. It is also shown that changing the slopes/chirp rates of the DFCW waveforms (based on LFM and PLFM waveforms) used as chips with the same complementary code pair results in good code sets with a zero-sidelobe region. It is also shown that a second good code set with a zero-sidelobe region could be constructed from the mates of the complementary code pair, while using the same DFCW waveforms as their chips. The cross-correlation between the two sets is shown to contain a zero-sidelobe region and an adjacent region of small cross-correlation sidelobes. Thus, the two sets are quasi-orthogonal and could be combined to form a good code set with twice the number of codes without affecting their cross-correlation properties. Or a better good code set with the same number codes could be constructed by choosing the best candidates form the two sets. Such code sets find utility in multiple input-multiple output (MIMO) radar applications.