The spectral kurtosis: a useful tool for characterising non-stationary signals

The spectral kurtosis (SK) is a statistical tool which can indicate the presence of series of transients and their locations in the frequency domain. As such, it helpfully supplements the classical power spectral density, which as is well known, completely eradicates non-stationary information. In spite of being particularly suited to many detection problems, the SK had rarely been used before now, probably because it lacked a formal definition and a well-understood estimation procedure. The aim of this paper is to partly fill these gaps. We propose a formalisation of the SK by means of the Wold–Cramér decomposition of “conditionally non-stationary” processes. This definition then engenders many useful properties enjoyed by the SK. In particular, we establish to which extent the SK is capable of detecting transients in the presence of strong additive noise by finding a closed-form relationship in terms of the noise-to-signal ratio. We finally propose a short-time Fourier-transform-based estimator of the SK which helps to link theoretical concepts with practical applications. This paper is also a prelude to a second paper where the SK is shown to find successful applications in vibration-based condition monitoring.