Introduction to Acoustics

Some might notice that linear interpolation is not
quite adequate, so they might assume that the correct solution
must lie in more elaborate curve-fitting techniques
using quadratics, cubics, or higher-order splines, and indeed
these types of interpolation can be adequate for
some applications. To arrive at the correct answer (provably
correct from theory) the interpolation task should
be viewed and accomplished as a filtering problem, with
the filter designed to meet some appropriate error criterion.
Linear time-invariant filtering is accomplished by
convolution with a filter function. If the resampling filter
is defined appropriately, we can exactly reconstruct
the original analog waveform from the samples.
The correct (ideal in a provable digital signal
processing sense) way to perform interpolation is convolution
with the sinc function, defined as:
sinc(t/T ) = sin(πt/T )/(πt/T ) ,
17.2 Pulse Code Modulation Synthesis
The majority of digital sound and music synthesis today
is accomplished via the playback of stored pulse
code modulation (PCM) waveforms. Single-shot playback
of entire segments of stored sounds is common for
sound effects, narrations, prompts, segments of music,
etc. Most high-quality modern electronic music synthesizers,
speech synthesis systems, and PC software
systems for sound synthesis use pre-stored PCM as the
basic data. This data is sometimes manipulated to yield
the final output sound(s).
There are a number of different ways to look at
sound for computer music, with PCM being only one.
We can look at the physics that produce the sound and
try to model those. We could also look at the spectrum
of the sound and other characteristics having to do with
the perception of those sounds. Indeed, much of the
legacy of computer music has revolved around parametric
(using mathematical algorithms, controlled by a few
well-chosen/-designed control parameters) analysis and
synthesis algorithms. We will discuss most of the commonly
used algorithms later, but first we should look at
PCM in more depth.
For speech, the most common synthesis technique is
concatenative synthesis [17.3]. Concatenative phoneme
synthesis relies on the concatenation of roughly 40
pre-stored phonemes (for English). Examples of vowel
phonemes are /i/ as in beet, /I/ as in bit, /a/ as in father,
etc. Examples of nasals are /m/ as in mom, /n/
Computer Music 17.2 Pulse Code Modulation Synthesis 717
where
T = 1/SRATE .
The sinc function is the ideal low-pass filter with
a cutoff of SRATE/2, where SRATE is the sampling
rate. Figure 17.4 shows reconstruction of a continuous
waveform by convolution of a digital signal with the sinc
function. Each sample is multiplied by a corresponding
continuous sinc function, and those are added up to
arrive at the continuous reconstructed signal [17.2].
Resampling: This is usually accomplished at the
same time as interpolation, because it is not necessary to
reconstruct the entire continuous waveform in order to
acquire new discrete samples. The resampling ratio can
be time varying, making the problem a little more difficult.
However, viewing the problem as a filter-design and
implementation issue allows for guaranteed tradeoffs of
quality and computational complexity.
as in none, /ng/ as in sing, etc. Examples of fricative
consonant phonemes are /s/ as in sit, /sh/ as in
ship, /f/ as in fifty, etc. Examples of voiced fricative
consonants are /z/, /v/ (visualize), etc. Examples of
plosive consonants are /t/ as in tat, /p/ as in pop, /k/
as in kick, etc. Examples of voiced plosives include /d/,
/b/, /g/ (dude, bob, gag) etc. Vowels and nasals are
essentially periodic pitched sounds, so the minimal required
stored waveform is only one single period of
each. Consonants require more storage because of their
noisy (non-pitched, aperiodic) nature.
The quality of concatenative phoneme synthesis is
generally considered quite low, due to the simplistic
assumption that all of the pitched sounds (vowels, etc.)
are purely periodic. Also, simply gluing /s/ /I/ and
/ng/ together does not make for a high-quality realistic
synthesis of the word “sing”. In actual speech, phonemes
gradually blend into each other as the jaw, tongue, and
other articulators move with time.
Accurately capturing the transitions between
phonemes with PCM requires recording transitions from
phoneme to phoneme, called diphones. A concatenative
diphone synthesizer blends together stored diphones.
Examples of diphones include see, she, thee, and most of
the roughly 40x40 possible combinations of phonemes.
Much more storage is necessary for a diphone synthesizer,
but the resulting increase in quality is significant.
PCM speech synthesis can be improved further by stor-
Part E 17.2