This paper proposes an EEG processor for sleep spindle detection algorithms. It non-linearly separates the raw EEG signal into non-oscillatory transient and sustained rhythmic oscillation components using long and short windows for the short-time Fourier transform. The processor utilizes the fact that sleep spindles can be sparsely represented via the inverse of a short-time Fourier transform. Five sleep spindle detectors were tested on the EEG database with and without the proposed EEG processor. We achieved an improvement of 13.3% in the by-sample F1 score, and 13.9% in the by-sample Matthews Correlation Coefficient score of these algorithms when the processed EEG was used for spindle detection. The processor was able to improve the scores by reducing the number of false positive spindles and increasing the number of true positive spindles detected.