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
We define a quantization interval to be the range of values assigned to the same
integer. Thus, for our example threebit A/D converter, the quantization
interval
Exercise 1
Recalling the plot of average daily highs in Problem 4 why is this plot so jagged? Interpret this effect in terms of analogtodigital conversion.
Solution
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
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:
Its width is
Exercise 2
This derivation assumed the signal's amplitude lay in the
range
Solution
The signaltonoise ratio does not depend on the signal
amplitude. With an A/D range of
Exercise 3
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?
Solution
Solving
Exercise 4
Music on a CD is stored to 16bit accuracy. What signaltonoise ratio does this correspond to?
Solution
A 16bit A/D converter yields a SNR of 97.8 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.