  In this paper, we develop new fast and efficient algorithms for designing
single/multiple unimodular waveforms/codes with good auto- and
cross-correlation or weighted correlation properties, which are highly desired
in radar and communication systems. The waveform design is based on the
minimization of the integrated sidelobe level (ISL) and weighted ISL (WISL) of
waveforms. As the corresponding optimization problems can quickly grow to large
scale with increasing the code length and number of waveforms, the main issue
turns to be the development of fast large-scale optimization techniques. The
difficulty is also that the corresponding optimization problems are non-convex,
but the required accuracy is high. Therefore, we formulate the ISL and WISL
minimization problems as non-convex quartic optimization problems in frequency
domain, and then simplify them into quadratic problems by utilizing the
majorization-minimization technique, which is one of the basic techniques for
addressing large-scale and/or non-convex optimization problems. While designing
our fast algorithms, we find out and use inherent algebraic structures in the
objective functions to rewrite them into quartic forms, and in the case of WISL
minimization, to derive additionally an alternative quartic form which allows
to apply the quartic-quadratic transformation. Our algorithms are applicable to
large-scale unimodular waveform design problems as they are proved to have
lower or comparable computational burden (analyzed theoretically) and faster
convergence speed (confirmed by comprehensive simulations) than the
state-of-the-art algorithms. In addition, the waveforms designed by our
algorithms demonstrate better correlation properties compared to their
counterparts.
