Extending the Capacity of 1 / f Noise Generation

From the emissions of massive quasars scattered across the universe, to the fluctuations in the stock market and the melodies of music, several real world signals have a power spectral density (PSD) that follows an inverse relationship with their frequency. Specifically, this type of random process is referred to as a \(1/f\) signal, and has been of much interest in research, as sequences that have this property better mimic natural signals. In the context of constraint programming, a recent work has defined a constraint that enforces sequences to exhibit a \(1/f\) PSD, as well as a corresponding constraint propagator. In this paper we show that the set of valid solutions associated with this propagator misses an exponential number of \(1/f\) solutions and accepts solutions that do not have a \(1/f\) PSD. Additionally, we address these two issues by proposing two non-exclusive algorithms for this constraint. The first one can find a larger set of valid solutions, while the second prevents most non-\(1/f\) solutions. We demonstrate in our experimental section that using the hybrid of these two methods results in a more robust propagator for this constraint.