The ability to understand speech varies significantly among cochlear implant (CI) listeners as it depends on a variety of individual and environmental factors. Especially in adverse listening situations, good cognitive and linguistic (“top-down”) skills and the use of signal pre-processing algorithms are considered beneficial. However, not much is known about the interactions between these top-down and perceptive bottom-up processes in CI listeners. To shed light on the relation between the spectral representation of speech sounds and the ability of CI listeners to decode these signals, two methods for the simplification of speech signals through spectral sparsification were developed and evaluated in listening tests with postlingually deaf adult CI listeners. Speech signals were separated into transient and harmonic parts. After sparsification of the harmonic spectrum by principal component analysis (PCA) or by an individualized spectral peak picking approach, the transient and the sparsened harmonic parts were remixed. Furthermore, cognitive parameters of the subjects were assessed via a neurocognitive test battery (ALAcog) and their correlation with recognition scores evaluated. The PCA-based sparsification method showed a significant benefit in speech recognition relative to the unprocessed signal. Furthermore, subjects with better performance in working memory and mental flexibility showed larger improvements.