Multiobjective Sequence Design via Gradient Descent Methods

Gradient descent provides a simple and flexible approach to sequence design. Simplicity is inherent in the gradient approach itself—The negative of the gradient of a given cost function gives an indication of the change to make to a sequence to find the local minimum of the cost function. Flexibility is provided by the lack of constraints on the cost function, allowing different aspects of a sequence to be optimized, e.g., autocorrelation or cross-correlation sidelobe energy, or spectral characteristics. Additional flexibility is provided by the linearity of gradients. The gradient of a linear combination of cost functions is simply an equivalent linear combination of their gradients. This allows multiple objectives to be optimized simultaneously. Gradient computational complexity has been one impediment to the use of gradient descent in sequence design in the past. In this paper, it is shown that gradients for a large variety of cost functions can be computed very efficiently. For a sequence of length <inline-formula><tex-math notation="LaTeX">$N$</tex-math></inline-formula>, the computational complexity of computing the full gradient (the gradient with respect to all <inline-formula><tex-math notation="LaTeX">$N$</tex-math> </inline-formula> elements) is <inline-formula><tex-math notation="LaTeX">$O(N^3)$</tex-math></inline-formula> in a “naive” implementation. In this paper, equations are derived that allow the gradients to be computed with <inline-formula><tex-math notation="LaTeX">$O(N \log N)$</tex-math></inline-formula> operations, a substantial improvement over previous methods.