of Microprocessors

424 MUSICAL ApPLICATIONS OF MICROPROCESSORS
power 10 the fundamental. Thus, the "signal" is 1 + 1/4 + 1/9 . ..
+ 1/225 for the first 15 harmonics that lie within the range of the low-pass
filter. Harmonics 16 through 34 do not contribute to the signal or the
distortion, since they or their aliases are above 15 kHz in this example.
Harmonics 35 to 65, 85 to 115, etc., contribute to the distortion. If the
calculations are carried out, it is found that the signal power is 1.58 units,
while the distortion due to the first-two groups offoldovers is 0.01717 unit
giving a SiN ratio of about 20 dB. Actually, if the example frequencies were
exact, the distortion frequencies would exactly overlap the signal frequencies
and would not be heard. However, if the signal frequency is not a submultiple
of the sample rate, then the distortion would be apparent.
This does not seem to be very good and in fact does not sound all that
good either. What's worse, a 2-kHz signal can be expected to be almost 6 dB
worse, although lower frequencies can be expected to be better by about 6
dBloctave of reduction. The square wave is about 2 dB better, but a
rectangular waveform approaches 0 dB SiN as the width approaches zero.
The triangle wave SiN ratio is an acceptable 54 dB, and the sine generates no
alias distortion at all. In conclusion, the results are usable if high-amplitude,
high-frequency sawtooth and rectangular waveforms are avoided.
Besides restrictions in use, there are few options available for lowering
the distortion figure. One thing that can be done is to generate the
troublesome waves at a higher sample rate, pass rhem rhrough a digital lowpass
filter operating at the higher rate, and then re-sample the filter output at
the lower system sample rate. For example, the sawtooth might be generated
at 400 ksls, which is eight times the system rate and then fed to a simple
digital low-pass filter that cuts off at 15 kHz. Only every eighth sample
emerging from the filter would actually be used. With this setup, the SiN
ratio for the I-kHz sawtooth would be improved to 38 dB. While the
computation time necessary to do this in software is much greater than that
required for some of the more sophisticated tone generation techniques, it
can be a viable hardware technique whereby simplicity of the algorithm often
outweighs computation time considerations because the digital hardware is
so fast.
Table Lookup Method
If direct computer synthesis is to live up to its promise of nearly infinite
flexibility and very high sound quality, then better tone-generation techniques
than the simulation of analog synthesizer oscillators will have to be
used. One of these involves scanning of precomputed waveform tables. An
important advantage is that the sample values stored in the table can in many
instances be selected so that alias distortion is not a problem. Another is that
microprocessors are far more efficient in looking up waveform samples than
in computing them from scratch.