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 [74]). 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 Synthesis generalizes the sinusoidal signal models to include a filtered noise component, as depicted in Fig.7.6.

The time-varying spectrum of the signal is assumed to be made up of a deterministic component (the sinusoids) and a stochastic component (time-varying filtered noise):

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

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:

$$e(t) = (h_t \ast u)(t) \isdef \int_0^t h(t-\tau,\tau)u(\tau)d\tau,$$

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

Note that filtered white noise is what is generally known in computer music as subtractive synthesis [168]. Thus, the best additive synthesis involves some subtractive synthesis as well.