Exponential spectro-temporal modulation generation

Abstract

Traditionally, real-time generation of spectro-temporally modulated noise has been performed on a linear amplitude scale, partially due to computational constraints. Experiments often require modulation that is sinusoidal on a logarithmic amplitude scale as a result of the many perceptual and physiological measures which scale linearly with exponential changes in the signal magnitude. A method is presented for computing exponential spectro-temporal modulation, showing that it can be expressed analytically as a sum over linearly offset sidebands with component amplitudes equal to the values of the modified Bessel function of the first kind. This approach greatly improves the efficiency and precision of stimulus generation over current methods, facilitating real-time generation for a broad range of carrier and envelope signals.

FIG. 1.

(Color online) Spectrographic analysis of sideband-based STM. Multiple comparisons of the proposed sideband-based generation method with a classic numerical solution for a midpoint-to-peak modulation depth of 20 dB (40 dB peak-to-valley). (a) Spectrogram of the STM created with the proposed Bessel function sideband approach using a sideband extent of five (ten sidebands), (b) spectrogram of the STM created by exhaustively evaluating the explicit form for exponential modulation, and (c) ratio of the Bessel function sideband-generated stimulus to the explicit form of the stimulus. Note that the colormap limits are $\pm 6\times {10}^{-4}$ dB for the spectral power ratio.

FIG. 2.

(Color online) The ratio between the energy of each term in the partial sum over a limited number of sidebands and the energy of the complete infinite sum, expressed in decibels and shown for several midpoint-to-peak modulation depths. Because sidebands are distributed symmetrically about the carrier tone, they are counted in terms of “sideband extent,” which is half of the total number of sidebands. The visualized “power ratio” value can be interpreted as the energetic contribution of the omitted terms relative to the entire sum. The values converge to zero quickly enough such that few total datapoints are visible when visualized with a linear ordinate axis.

FIG. 3.

(Color online) Metrics of spectral envelope fluctuations in SM compared among three generation methods: black open circles for the explicit evaluation, red open squares for the existing method, and green filled triangles for the proposed sideband method. The calculations were performed on 100 exemplars across each of 20 modulation depths for each stimulus generation method. (a) The fourth moment of the spectrum (ordinate) as a function of the SM depth (abscissa). Error bars indicate the standard deviation across the 100 exemplars. (b) The crest factor of the spectral envelope (ordinate) as a function of the SM depth (abscissa). Error bars indicate the standard deviation across 100 samples.