of Microprocessors

DIGITAL TONE GENERATION TECHNIQUES
423
16-BIT ANGLE WORD (X I
Is, !!!, I ! ! , I , ! ,
~'---------v----\
QUADRANT
SELECT
TABLE
LOOKUP
ARGUMENT
(XI)
FIRST DERIVATIVE
TABULATED FUNCTION VALUE
L......J---'---'---'--'---'-...L-L-L...J-'----'---'--'---L....J = F(XI)~
ADD
I
F(X)
Fig. 13-2. Linear interpolation in a sine table
16-BIT
RESULT
improved noise level are worth the extra computation time, particularly if
hardware multiply is not available.
Alias Distortion
So far everything sounds great; all of the normal analog synthesizer
waveforms are available in digital form with adequate frequency resolution
and are just waiting to be processed further. There exists, however, one
problem that can only be minimized and that is alias distortion from the
upper harmonics of most of these waveforms. When the algorithms just
described are used to generate samples on a waveform, the samples generated
are exactly the same as would have come from an ADC sampling the equivalent
analog waveform with no low-pass jilter! Thus, one would expect the
higher harmonics to fold back into the audio range and create distortion.
Before deciding what to do about the problem, its severity should be
determined. As a "best-case" example, let us assume a sample rate of 50 ksls,
the use of a sharp I5-kHz low-pass filter in the output DAC, and a I-kHz
sawtooth wave. The idea is to add up the power in all of the unaliased
harmonics in the I5-kHz range and compare this with the sum of the aliased
harmonics that are in the 15-kHz range.
The power spectrum of a sawtooth wave is well known and is
Pn = Pdn 2 , where Pn is the power in the nth harmonic relative to the