A Limited Memory BFGS Based Unimodular Sequence Design Algorithm for Spectrum-Aware Sensing Systems

Unimodular sequences with good correlation and spectral properties are desirable in numerous applications such as active remote sensing and communication systems. Therefore, designing sequences with stopband and correlation sidelobe constraints has gained a lot of attention in the last few decades. In this paper, we propose a fast and efficient iterative algorithm to design unimodular and sparse frequency waveforms with low aperiodic/periodic autocorrelation sidelobes and desired stopband properties. In our approach, the bi-objective optimization problem which minimizes both the integrated sidelobe level (ISL) of the autocorrelation function and the power density in the spectral stopbands is first turned into an unconstrained single objective optimization problem and then is treated as a nonlinear large-scale problem. For the solution of the problem, we develop an algorithm based on Limited Memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) Quasi-Newton optimization method. Unlike most gradient based algorithms which employ line searches to deduce the step length, owing to L-BFGS method, unit step length is taken as a general rule to avoid the cost of computation at every iteration with very few exceptions. The calculation of gradient is based on Fast Fourier Transform and Hadamard product operations and thus the algorithm is fast and computationally efficient. Moreover, the algorithm is space efficient and its low-memory feature makes it possible to generate long sequences. Several numerical examples are presented to validate the efficacy of the proposed method and to show its superiority over other state-of-art algorithms.