Compact S‐transform for analysing local spectrum

Abstract

The Fourier transform of a N point time series is a N point complex series, while the S-transform (ST) of the same time series is a N × N 2D time–frequency complex matrix. The computation and storage of N 2 − N additional points are a major drag on the usage of ST. In this study the compact S-transform (cST) is presented, with efficiencies brought about through computation of only selected voices (frequencies). The cST spectrum has uncomputed voice gaps that increase in width towards the higher frequencies. Plot of the cST magnitude spectrum is virtually indistinguishable from the ST magnitude plot. Local spectrum at any spot on the cST can be quickly examined in detail through interpolation. The cST requires the computation of approximately 3 N voices compared to ⌊ N / 2 ⌋ + 1 for the ST. The proportion of computed voices decrease for larger N. For N = 1024, ∼20% of the voices in the time-frequency spectrum is computed; for N = 2048 only 14% of the voices is computed. For applications, such as audio and speech signal processing where segments of one million samples are not uncommon, <1% of the voices are computed, thereby reducing the computation time by ∼99%.