Digital Signal Processing Reference
In-Depth Information
on our hands. Consider our temperature example once more: we can use
a mercury thermometer and decide to write down just the number of de-
grees; maybe we can be more precise and note the half-degrees as well; with
a magnifying glass we could try to record the tenths of a degree - but we
would most likely have to stop there. With a more sophisticated thermo-
couple we could reach a precision of one hundredth of a degree and possibly
more but, still, we would have to settle on a maximum number of decimal
places. Now, if we know that our measures have a fixed number of digits,
the set of all possible measures is actually countable and we have effectively
mapped the codomain of our temperature function onto the set of integer
numbers. This process is called
quantization
and it is the method, together
with sampling, to obtain a fully digital signal.
In a way, quantization deals with the problem of the continuum in a
much “rougher” way than in the case of time: we simply accept a loss of
precision with respect to the ideal model. There is a very good reason for
that and it goes under the name of
noise
. The mechanical recording devices
we just saw, such as the thermograph or the phonograph, give the illusion
of analytical precision but are in practice subject to severe mechanical lim-
itations. Any analog recording device suffers from the same fate and even
if electronic circuits can achieve an excellent performance, in the limit the
unavoidable thermal agitation in the components constitutes a noise floor
which limits the “equivalent number of digits”. Noise is a fact of nature that
cannot be eliminated, hence our acceptance of a finite (i.e. countable) pre-
cision.
l
g
r
,
y
i
d
.
,
©
,
L
s
Figure 1.8
Analog and digital computers.
Noise is not just a problem in measurement but also in processing.
Figure 1.8 shows the two archetypal types of analog and digital computing
devices; while technological progress may have significantly improved the
speed of each, the underlying principles remain the same for both. An ana-
log signal processing system, much like the slide rule, uses the displacement
of physical quantities (gears or electric charge) to perform its task; each el-
ement in the system, however, acts as a source of noise so that complex or,
