of Microprocessors

516 MUSICAL ApPLICATIONS OF MICROPROCESSORS
waveform samples is of interest and has, in fact, been mentioned in passing
several times so far. In this section, such interpolation will be looked at more
closely.
In general, the interpolation problem can be stated as follows: Given
one or more tabulated points (X and Y coordinates) describing some sort of
curve and an arbitrary value of X, determine as "accurately" as possible the
corresponding Y. Accurately was quoted because unless an equation is known
that exactly describes the curve in the region from which the tabulated points
were taken, there is nothing to compare the interpolated result to. In cases in
which such an equation is not known, interpolation only gives a guess of
what the untabulated value should be. One can, however, evaluate the guess
by how close it comes to the "most likely" untabulated value. Clearly, a
procedure that painstakingly evaluates all of the points in the table (or
record) should be able to make a good guess. For our purposes, an interpolation
algorithm can be characterized by how many data points and how much
calculation is used to make the guess and how good it is likely to be relative
to an exhaustive procedure.
The general approach to performing interpolation is to first find an
equation that "fits" the data, that is, a plot of the equation would pass
through the data points being considered. After this is done, the X for which
a value is needed is plugged into the equation and out comes its corresponding
Y. The accuracy of this Y depends entirely on how well the "interpolation
function" models the physical process that created the data points in the
first place. This is important because there are many different equations that
will pass through the tabulated data points. Note that in the general case the
tabulated points need not be equally spaced. In synthesis applications, however,
they usually are, which simplifies the calculations.
Generally speaking, the interpolation will be most accurate when the
unknown point is in the middle of the cluster of known points. If it is
completely outside the tabulated points, the process is known as extrapolation,
which is sheer speculation as to what the point might be if the curve
continued to behave as it did in the tabulated interval.
One class of equations that can be used in interpolation is the standard
algebraic polynomials. In general an N -lth degree polynomial can be found
that will exactly pass through N data points. Once found, the polynomial can
be easily evaluated at the unknown point. The higher the degree of the
polynomial, that is the more data points that are considered, the better the
interpolation results.
The simplest example, which is what most people call interpolation, is
linear interpolation. Here, a first-degree polynomial, which is a straight line,
is fit to two data points; one on each side of the unknown point. The
procedure for doing this was described in detail in the previous chapter. One
could also fit a quadratic curve through three points and so forth. When a