of Microprocessors

222 MUSICAL ApPLICATIONS OF MICROPROCESSORS
than eight different voltage levels as its output. With an ideal DAC, these
eight levels would be equally spaced across the range of output voltages. If
the output is to span the range of - 10 V to + 10 V, for example, these eight
levels might be assigned as:
Binary Analog Binary Analog
000 -10.00 100 + 1.42
001 - 7.14 101 + 4.29
010 - 4.29 110 + 7.14
011 - 1.42 111 +10.00
Actually, since twos-complement binary inputs are usually desirable, the
eight levels should probably be assigned instead as:
000
001
010
011
+ 0.00
+ 2.50
+ 5.00
+ 7.50
100
101
110
111
-10.00
- 7.50
- 5.00
- 2.50
Unfortunately, neither assignment is ideal. The first has no code for
a zero output and puts out rather odd voltages anyway. The second has a zero
point and nice round levels but falls short of the full ± 1O-V range desired.
Actually, practical DACs have considerably more resolution than this example
so that last missing level on the positive side is generally of no consequence.
Using the twos-complement assignment, the resolution of this 3-bit
DAC would be a very coarse 2.5 V. The maximum error in converting an
arbitrary number (with rounding) to a voltage would be only half of this or
1.25 V.
Moving up to an 8-bit DAC improves things considerably. The
resolution now would be 20/(2""-1) or 0.078 V. The largest negative output
would still be - 10 V, but the positive limit would be one step short of
+ 10.0 or +9.922 V. Even with 8 bits, the step size of0.078 V controlling a
voltage-controlled oscillator with a sensitivity of one octave per volt would
yield a pitch step size of about one semitone. A 12-bit DAC would have a
step size of 0.00488 V, which would give a nearly inaudible 1/17 semitone
increment. Even higher resolutions are available, but the expense would
limit extensive use.
Usually it is convenient to consider the binary input to a DAC as being
a signed binary fraction between -1.000 ... and +0.9999 .... The
output voltage of the DAC then is the binary fraction (rounded or truncated
to the DAC's resolution) times 10 V. This representation has the advantage
that calculations leading up to the value to be converted are not affected by
the actual resolution of the DAC used. For example, if a 16-bit computer is
being used, it would be convenient to perform all calculations using 16-bit
fractional arithmetic. (Fractional arithmetic is really the same as integer