of Microprocessors

436 MUSICAL ApPLICATIONS OF MICROPROCESSORS
As we shall see, a block of samples representing a segment of sound can
be transformed into the frequency domain via Fourier transform and appear
as a bunch of harmonic amplitudes and phases. This data can then be transformed
back into a block of samples unscathed and indistinguishable (within
round-off error) from the original block via an inverse Fourier transform!
Thus, Fourier transformation is a reversible operation. However, while in the
frequency domain, we can manipulate the spectrum directly by altering individual
frequency component amplitudes as desired; no filrers to design and
tune are necessary. After the manipulation has been accomplished, the inverse
transform returns the data to the time domain. This is one way to
implement a filter with a completely arbitrary amplitude response.
One can also synthesize a spectrum directly and convert it to the time
domain for output. This can be valuable, since most sounds are more easily
described in terms of their spectrum rather than in terms of their waveforms.
Although a method was just discussed for Fourier series synthesis, the Fourier
transform can require far less computation if the required spectral detail is
great.
In source-signal analysis using digital techniques, the first step is
nearly always Fourier transformation of the input into spectral form. Since
the ear hears in the frequency domain, it is logical that audible features of the
source signal will be much more apparent than when in the time domain. In
most cases, transformation greatly reduces the quantity of data to be processed
as well.
Characteristics of the
Discrete Fourier Transform
Fourier transforms applied to sampled data are usually called discrete
Fourier transforms, since the time domain data are at discrete instants of time
and the frequency data are likewise at discrete frequencies. One very important
property of the discrete transform is that the waveform data are specificsized
chunks called records, each consisting of a specified number of samples.
The discrete transform assumes that the samples in the record represent
exactly one cycle of a periodic waveform. This assumption must be made
regardless of whether or not it is actually true, as in Fig. 13-9. The transform
then gives all of the harmonic amplitudes and phases of the assumed
periodic waveform.
This record-oriented property has several important ramifications when
constantly changing arbitrary sounds are to be synthesized or analyzed. For
most applications, the record size is fixed and often is a power of two. Thus,
even if a periodic waveform were being analyzed, it is unlikely that a tecord
would exactly span a single cycle. In order to reduce the resulting error, the
record size is chosen to be great enough to span se!Jeral cycles of the lowest
frequency expected. Then the partial cycle at the beginning and end of the