Amplitude Quantizationhttp://cnx.org/contenthttp://cnx.org/content/m0051/latest/m0051Amplitude Quantization2.232000/07/272010/11/18 10:02:30.741 US/CentralCarolBettoneyCarol Bettoneycarolrb@alumni.rice.eduDonJohnsonDon Johnsondhj@rice.edudhjdhj carolrbdhjanalog-to-digital (A/D) conversionquantization intervalquantizedsignal-to-noiseScience and TechnologyAnalog-to-digital conversion.en
The Sampling Theorem says that if we sample a bandlimited
signal
st
fast enough, it can be recovered without error from its
samples
snTs,
n…-101….
Sampling is only the first phase of acquiring data into a
computer: Computational processing further requires that the
samples be quantized: analog values are converted
into
digital
form. In short, we will have performed analog-to-digital
(A/D) conversion.
A phenomenon reminiscent of the errors incurred in
representing numbers on a computer prevents signal amplitudes
from being converted with no error into a binary number
representation. In analog-to-digital conversion, the signal is
assumed to lie within a predefined range. Assuming we can
scale the signal without affecting the information it
expresses, we'll define this range to be
11.
Furthermore, the A/D converter assigns amplitude values in
this range to a set of integers. A
B-bit
converter produces one of the integers
01…2B1
for each sampled input.
shows how a three-bit A/D converter assigns input values to
the integers.
We define a quantization interval to be the range
of values assigned to the same integer. Thus, for our example
three-bit A/D converter, the quantization interval
Δ
is
0.25;
in general, it is
22B.
Recalling the plot of average daily highs in
this
frequency domain problem,
why is this plot so jagged? Interpret this effect in
terms of analog-to-digital conversion.
The plotted temperatures were quantized to the nearest
degree. Thus, the high temperature's amplitude was
quantized as a form of A/D conversion.
Because values lying anywhere within a quantization interval
are assigned the same value for computer processing,
the original amplitude value cannot be recovered
without error. Typically, the D/A converter, the
device that converts integers to amplitudes, assigns an
amplitude equal to the value lying halfway in the quantization
interval. The integer 6 would be assigned to the amplitude
0.625 in this scheme. The error introduced by converting a
signal from analog to digital form by sampling and amplitude
quantization then back again would be half the quantization
interval for each amplitude value. Thus, the so-called
A/D error equals half the width of a
quantization interval:
12B.
As we have fixed the input-amplitude range, the more bits
available in the A/D converter, the smaller the quantization
error.
To analyze the amplitude quantization error more deeply, we
need to compute the signal-to-noise ratio, which
equals the ratio of the signal power and the quantization
error power. Assuming the signal is a sinusoid, the signal
power is the square of the rms amplitude:
powers12212.
The
illustration
details a single quantization interval.
Its width is Δ and the
quantization error is denoted by
ε. To find the power in
the quantization error, we note that no matter into which
quantization interval the signal's value falls, the error will
have the same characteristics. To calculate the rms value, we
must square the error and average it over the interval.
rmsε1ΔεΔ2Δ2ε2Δ21212
Since the quantization interval width for a
B-bit converter equals
22B2B1,
we find that the signal-to-noise ratio for the
analog-to-digital conversion process equals
SNR1222B1123222B6B10101.5 dB
Thus, every bit increase in the A/D converter yields a 6 dB
increase in the signal-to-noise ratio.
The constant term
10101.5
equals 1.76.
This derivation assumed the signal's amplitude lay in the
range
-11.
What would the amplitude quantization signal-to-noise
ratio be if it lay in the range
AA?
The signal-to-noise ratio does not depend on the signal
amplitude. With an A/D range of
AA,
the quantization interval
Δ2A2B
and the signal's rms value (again assuming it is a
sinusoid) is
A2.
How many bits would be required in the A/D converter to
ensure that the maximum amplitude quantization error was
less than 60 db smaller than the signal's peak value?
Solving
2B.001
results in
B10
bits.
Music on a CD is stored to 16-bit accuracy. To what
signal-to-noise ratio does this correspond?
A 16-bit A/D converter yields a SNR of
61610101.597.8 dB.
Once we have acquired signals with an A/D converter, we can
process them using digital hardware or software. It can be
shown that if the computer processing is linear, the result of
sampling, computer processing, and unsampling is equivalent to
some analog linear system. Why go to all the bother if the
same function can be accomplished using analog techniques?
Knowing when digital processing excels and when it does not is
an important issue.