Time Series

In a finite difference setting, continuously variable functions of $t$, such as $u(t)$, are approximated by time series, often indexed by integer $n$. For instance, the time series $u^{n}$ represents an approximation to $u(t_{n})$, where $t_{n} = nk$, for a time step $k$. In audio applications, the sampling frequency $F_{s}$ is defined as

$$F_{s} = 1/k$$ (2.1)

Note here that the same dependent variable name ($u$) has been used here to denote both the continuously variable function $u(t)$ and the approximating time series $u^{n}$; this is simply an attempt at avoiding the proliferation of notation, and leads to little confusion, as such forms rarely appear together in the same expression.

Before introducing difference operators and examining discretization issues, it is worth making two comments which relate specifically to audio. First, consider a function $u(t)$ which appears as the solution to an ODE, such as that defined by the simple harmonic oscillator. If some difference approximation to the ODE is derived, which generates a solution time series $u^{n}$, it is important to be aware that $u^{n}$ is not simply a sampled version of the true solution. Though obvious, it is especially important for those with an electrical or audio engineering background (i.e., those accustomed to dealing with sampled data systems) to be conscious of this so as to avoid arriving at false conclusions based on familiar results such as, e.g., the Shannon sampling theorem. In fact, one can indeed incorporate such results into the simulation setting, but in a manner which may be counterintuitive (see §#a#>nd Problems §). In sum, it is best to remember that in the standard physical modeling sound synthesis framework, there occurs no sampling of recorded audio material. Second, in audio applications, as opposed to standard simulation in other domains, the sample rate $F_{s}$ and thus the time step $k$ are generally set before run time, and are not varied; in audio, in fact, one nearly always takes $F_{s} = 44 100$. This, in contrast to the first comment above, is intuitive for audio engineers, but not for those involved with numerical simulation in other areas, who often are interested in developing numerical schemes which allow a larger time step with little degradation in accuracy. Though the benefits of such schemes may be interpreted in terms of numerical dispersion, in an audio synthesis application, there is no point in developing a scheme which runs with increased efficiency at a larger time step (i.e., at a lower sampling rate), as such a scheme will be incapable of producing potentially audible frequencies in the upper range of human hearing.