Summary: Analogtodigital conversion.
The Sampling Theorem says that if we sample a bandlimited
signal

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 analogtodigital 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
Recalling the plot of average daily highs in this frequency domain problem, why is this plot so jagged? Interpret this effect in terms of analogtodigital 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 socalled
A/D error equals half the width of a
quantization interval:
To analyze the amplitude quantization error more deeply, we
need to compute the signaltonoise 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:
This derivation assumed the signal's amplitude lay in the
range
The signaltonoise ratio does not depend on the signal
amplitude. With an A/D range of
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
Music on a CD is stored to 16bit accuracy. To what signaltonoise ratio does this correspond?
A 16bit A/D converter yields a SNR of
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.
"Electrical Engineering Digital Processing Systems in Braille."