of Microprocessors

112 MUSICAL ApPLICATIONS OF MICROPROCESSORS
termed quantization error because a sample pulse, which can be any amplitude
whatsoever, has been quantized to the nearest available numerical representation.
Background noise due to quantization error is termed quantization
nOIse.
An actual ADC or DAC usually has a definite maximum signal
amplitude that it can handle. A typical value is from -10 to + 10 V. Inputs
to an ADC beyond this range are effectively clipped, and numerical inputs to
a DAC beyond this range are converted into a value that fits. The 20-V range
is then broken up into a large number ofquantization levels, which is usually a
power of two, since DACs and ADCs are typically binary devices. The
number of quantization levels generally ranges from 256 to 65,536, which
corresponds to 2 8 and 2 16 or 8 and 16 bits, respectively.
Let us try to estimate the amount of background noise present in a
12-bit (4,096 level) DAC output so that the number of levels necessary for
high-fidelity reproduction can be determined. Twenty volts divided by 4,096
equals 4.88 mV per level. Thus, the 500th level would be 2.44141 V and
would have to be used for all desired voltages between 2.43895 and 2.44384
V. It is reasonable to assume that if the desired voltage level falls in this
range, it is equally likely to be anywhere in the range and the same should be
true for any other quantization level. Thus, the average error (difference
between desired level and the nearest quantization level) is one-quarter of the
difference between quantization levels. This works out to an average error
amplitude of 1.22 mY. The maximum signal amplitude without distortion
is 10 V. Therefore, the maximum signal-to-noise ratio is 10/
0.00122 = 8,192. Expressed in decibels, this is about 78 dB. Actually, due
to an oversimplified analysis, the average noise amplitude is really one-third
of the quantization interval; thus, a more accurate figure is 76 dB.
If the number of quantization levels were doubled (one bit added to
make 13), the denominator of the previous equation would be halved, resulting
in an increase in the signal-to-noise ratio of 6 dB. Eliminating a bit
would subtract 6 dB from the signal-to-noise ratio. In real DACs and ADCs,
the quantization levels are not perfectly spaced or perfectly equal in size.
Such imperfections add about 4 dB to the noise level. Thus, the signal-tonoise
ratio that can be expected is approximately 6 dB times the number of
bits in the DAC or ADC.
Compared with analog audio equipment, the 72 dB available from a
12-bit DAC is quite good, in fact better than any program source available to
consumers and better even than much professional recording equipment. An
increase to 16 bits, which is achieved by the newer direct synthesis
installations, and one has dynamic range far exceeding any available analog
recording device!
As a result, direct computer synthesis utilizing a 50-ks/s to 60-ks/s
sample rate and 16-bit DACs is capable of unsurpassed audio quality. At the
other end of the spectrum, 15 ks/s to 20 ks/s with 10 to 12 bits gives AM