1.2: Sampled functions

In mathematics we become familiar with the notion of a function. Consider the simplest case of a function \(y(t)\) which takes a real scalar \(t\) and returns a real scalar value \(y(t)\). This is frequently written formally as \(y:\mathbb{R}\rightarrow \mathbb{R}\). This means, take any arbitrary real number \(t\), plug it into \(y\), and out will come a new real number \(y(t)\). The important idea to glean from this is that the function is like a machine into which one can input any \(t\) and expect to get an output. That is, the function is defined on every real number \(t\).

Unfortunately, when working on real data using a computer this nice abstraction doesn’t always hold. Rather, it is very common for us to have only certain, discrete points \(t_n\) where the function is defined. Digital audio is a simple example: the sounds we hear with our ear are related to the sound pressure waves moving our eardrums. The sound pressure has a value for every point in time. But when processing audio using a computer the sound pressure is sampled at regular, discrete times and turned into numbers representing the sound pressure at each sample time. This process is often referred to as "discretization", meaning that the continuously-defined \(y(t)\) is replaced by a set of discrete samples \(y_n\). This is depicted in 1.2, which shows the relationship of a continuous-time function and its discrete-time (discretized) replacement.

Figure 1.2: A sampled function. The continuous-time version of the function is \(y(t)\). Sampling replaces \(y(t)\) with samples taken at discrete times, \(t_n\). The samples \(y_n\) may be handled by the computer as a vector or list of values. In the case shown here the function is sampled at evenly-spaced intervals separated by \(h\). Evenly-spaced samples are the most frequent case – for example, digital audio commonly samples an input sound signal with a period \(h\) of about 22.7usec (44.1kHz sampling rate).

It turns out that the ODE solvers we will study all work with sampled functions. That is, the solvers compute the solution \(y(t)\) at a sequence of discrete time values \(t_n\) similar to the situation shown in 1.2. The output of the solver will be a vector of discrete values \(y_n\) representing samples of the actual, underlying continuous function \(y(t)\).

Here is an implementation hint when you write programs processing sampled functions: The sampled function itself is generally represented by a vector, \(y_n\). This is the object your algorithm will use during its work. On top of \(y_n\) I recommend also carrying around a vector representing the sample times, \(t_n\), in your program. Having your sample times readily accessible can help decrease confusion when you want to plot \(y_n\) vs. time, for example. For moderately sized vectors, the extra memory required to hold \(t_n\) is a small price to pay to keep confusion at a minimum while writing your code.