In this subsection, we present three important processes associated with the ADC. We first start
with the subject of sampling. Imagine yourself as a photographer in an Olympic diving stadium.
Your job is taking a sequence of pictures of divers jumping off from a diving board 10 meters above
the surface of the diving pool. Your goal is to put the sequence of pictures together to reconstruct
the motion of each diver. The sequence of pictures makes up samples of divers’ motions. If a diver
tries a complex motion and you want to faithfully reconstruct his motion, you must take enough
pictures from the start to the end of the dive. If a diver makes a simple routine dive, you only need
to take a few pictures over the period of the dive.Two very different cases of motions generated by a
diver is shown in Figure 3.3. The same time sequence is used to capture samples for both motions.
As can be seen from figure, frame (a) motion cannot be regenerated from the samples, whereas the
motion shown in frame (b) can clearly be reconstructed from the same number of samples used to
capture both motions.
Sampling is the process of taking ‘‘snapshots’’ of a signal over time. Naturally, when we
sample a signal, we want to sample it in an optimal fashion such that we can capture the essence
of the signal while minimizing the use of resources. In essence, we want to minimize the number
of samples while faithfully reconstructing the original signal from the samples. As can be deduced
fromour discussion above, the rate of change in a signal determines the number of samples required
to faithfully reconstruct the signal, provided that all adjacent samples are captured with the same
sample timing intervals.
Harry Nyquist from Bell Laboratory studied the sampling process and derived a criterion
that determines the minimum sampling rate for any continuous analog signals. His, now famous,
minimum sampling rate is known as the Nyquist sampling rate, which states that one must sample
a signal at least twice as fast as the highest frequency content of the signal of interest. For example,
if we are dealing with the human voice signal that contains frequency components that span from
about 20 Hz to 4 kHz, the Nyquist sample theorem tells us that we must sample the signal at
least at 8 kHz, 8000 ‘‘snapshots’’ every second. Engineers who work for telephone companies
must deal with such issues. For further study on the Nyquist sampling rate, refer to Barrett and Pack [2] listed in the References section. Sampling is important because when we want to
represent an analog signal in a digital system, such as a computer, we must use the appropriate
sampling rate to capture the analog signal for a faithful representation in digital systems.
Now that we understand the sampling process, let us move on to the second process of the
ADC, quantization. Each digital system has a number of bits, which it uses as the basic units to
represent data. A bit is the most basic unit where single binary information, 1 or 0, is represented.
A nibble is made up of 4 bits put together. A byte is 8 bits.
In the previous section, we tacitly avoided the discussion of the form of captured signal
samples. When a signal is sampled, digital systems need some means to represent the captured samples.The quantization of a sampled signal is how the signal is represented as one of quantization
level. Suppose you have a single bit to represent an incoming signal. You only have two different
numbers, 0 and 1. You may say that you can distinguish only low from high. Suppose you have
2 bits. You can represent four different levels, 00, 01, 10, and 11. What if you have 3 bits? You
now can represent eight different levels: 000, 001, 010, 011, 100, 101, 110, and 111. Think of it as
follows.When you had 2 bits, you were able to represent four different levels. If we add one more
bit, that bit can be 1 or 0, making the total possibilities 8. Similar discussion can lead us to conclude
that given n bits, we have 2n
different numbers or levels one can represent.
Figure 3.4 shows how n bits are used to quantize a range of values. In many digital systems,
the incoming signals are voltage signals. The voltage signals are first obtained from physical signals
with the help of transducers, such as microphones, angle sensors, and infrared sensors. The voltage
signals are then conditioned to map their range with the input range of a digital system, typically
0to5V.InFigure 3.4, n bits allow you to divide the input signal range of a digital system into 2n
different quantization levels.
Quantization
As can be seen from the figure, higher quantization levels means
better mapping of an incoming signal to its true value. If we only had a single bit, we can only
represent levels 0 and 1. Any analog signal value in between the range had to be mapped either as
level 0 or level 1, not many choices. Now imagine what happens as we increase the number of bits
available for the quantization levels. What happens when the available number of bits is 8? How
many different quantization levels are available now? Yes, 256. How about 10, 12, or 14? Notice
also that as the number of bits used for the quantization levels increases for a given input range the
‘distance’ between two adjacent levels decreases with a factor of a polynomial.
Finally, the encoding process involves converting a quantized signal into a digital binary
number. Suppose again we are using 8 bits to quantize a sampled analog signal. The quantization
levels are determined by the 8 bits, and each sampled signal is quantized as one of 256 quantization
levels. Consider the two sampled signals shown in Figure 3.5. The first sample is mapped to
quantization level 2 and the second one is mapped to quantization level 198.
Quantization with fewer bits
Note the amount of quantization error introduced for both samples. Now consider Figure 3.5. The same signal is
sampled at the same time but quantized using a less number of bits. Note that the quantization
error is inversely proportional to the number of bits used to quantize the signal.
Once a sampled signal is quantized, the encoding process involves representing the quanti-
zation level with the available bits. Thus, for the first sample, the encoded sampled value is 0000
0001, whereas the encoded sampled value for the second sample is 1100 0110. As a result of the
encoding process, sampled analog signals are now represented as a set of binary numbers. Thus,
the encoding is the last necessary step to represent a sampled analog signal into its corresponding
digital form, shown in Figure.
