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Sines + Noise Modeling

As mentioned in the introduction to this chapter, it takes many sinusoidal components to synthesize noise well (as many as 25 per critical band of hearing under certain conditions [85]). When spectral peaks are that dense, they are no longer perceived individually, and it suffices to match only their statistics to a perceptually equivalent degree.

Sines+Noise (S+N) synthesis [249] generalizes the sinusoidal signal models to include a filtered noise component, as depicted in Fig.10.7. In that figure, white noise is denoted by $ u(t)$ , and the slowly changing linear filter applied to the noise at time $ t$ is denoted $ h_t(\cdot)$ .

The time-varying spectrum of the signal is said to be made up of a deterministic component (the sinusoids) and a stochastic component (time-varying filtered noise) [246,249]:

$\displaystyle s(t) \eqsp \sum_{i=1}^{N} A_i(t) \cos[ \theta_i(t)] + e(t),$ (11.21)

where $ A_i(t)$ and $ \theta_i(t)$ are the instantaneous amplitude and phase of the $ i$ th sinusoidal component, and $ e(t)$ is the residual, or noise signal, assumed to be well modeled by filtered white noise:

$\displaystyle e(t) \eqsp (h_t \ast u)(t) \isdefs \int_0^t h_{t}(t-\tau)u(\tau)d\tau,$ (11.22)

where $ u(t)$ is the white noise, and $ h_t(\tau)$ is the impulse response of a time varying linear filter at time $ t$ . Specifically, $ h_t(\tau)$ is the response at time $ t$ to an impulse at time $ t-\tau$ .

Filtering white-noise to produce a desired timbre is an example of subtractive synthesis [186]. Thus, additive synthesis is nicely supplemented by subtractive synthesis as well.

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University