We have heretofore discussed digital audio signals as if they were capable of describing any function of time, in the sense that knowing the values the function takes on the integers should somehow determine the values it takes between them. This isn't really true. For instance, suppose some function (defined for real numbers) happens to attain the value 1 at all integers: for . We might guess that for all real . But perhaps happens to be one for integers and zero everywhere else--that's a perfectly good function too, and nothing about the function's values at the integers distinguishes it from the simpler . But intuition tells us that the constant function is in the spirit of digital audio signals, whereas the one that hides a secret between the samples isn't. A function that is ``possible to sample" should be one for which we can use some reasonable interpolation scheme to deduce its values for non-integers from its values for integers.
It is customary at this point in discussions of computer music to invoke the famous Nyquist theorem. This states (roughly speaking) that if a function is a finite or even infinite combination of REAL SINUSOIDS, none of whose angular frequencies exceeds , then, theoretically at least, it is fully determined by the function's values on the integers. One possible way of reconstructing the function would be as a limit of higher- and higher-order polynomial interpolation.
The angular frequency , called the Nyquist frequency, corresponds
to cycles per second if is the sample rate. The corresponding period
is two samples. The Nyquist frequency is the best we can do in the sense that
any real sinusoid of higher frequency is equal, at the integers, to one whose
frequency is lower than the Nyquist, and it is this lower frequency that will
get reconstructed by the ideal interpolation process. For instance, a REAL
SINUSOID with angular frequency between and , say ,
can be written as
We conclude that when, for instance, we're computing an EXPLICIT SUM OF SINUSOIDS,
either as a wavetable or as a real-time signal, we had better drop any sinusoid
in the sum whose frequency exceeds . But the picture in general is not
this simple, since most techniques other than additive synthesis don't lead to
neat, band-limited signals (ones whose components stop at some limited
frequency.) For example, a sawtooth wave of frequency , of the form
put out by Pd's
object but considered as a continuous
function , expands to:
Many synthesis techniques, even if not strictly band-limited, give partials which may be made to drop off more rapidly than as in the sawtooth example, and are thus more forgiving to work with digitally. In any case, it is always a good idea to keep the possibility of foldover in mind, and to train your ears to recognize it.
The first line of defense against foldover is simply to use high sample rates; it is a good practice to systematically use the highest sample rate that your computer can easily handle. The highest practical rate will vary according to whether you are working in real time or not, CPU time and memory constraints, and/or input and output hardware, and sometimes even software-imposed limitations.
A very non-technical treatment of sampling theory is given in [Bal03]. More detail can be found in [Mat69, pp. 1-30].