Digital Signal Processing Reference
In-Depth Information
1.1.2
The Hellenic Shift to Analog Processing
“Digital” representations of the world such as those depicted by the Palermo
Stone are adequate for an environment in which quantitative problems are
simple: counting cattle, counting bushels of wheat, counting days and so
on. As soon as the interaction with the world becomes more complex, so
necessarily do the models used to interpret the world itself. Geometry, for
instance, is born of the necessity of measuring and subdividing land prop-
erty. In the act of splitting a certain quantity into parts we can already see
the initial difficulties with an integer-based world view ;
(2)
yet, until the Hel-
lenic period, western civilization considered natural numbers and their ra-
tios all that was needed to describe nature in an operational fashion. In the
6th century BC, however, a devastated Pythagoras realized that the the side
and the diagonal of a square are incommensurable, i.e. that
2is
not
asim-
ple fraction. The discovery of what we now call irrational numbers “sealed
the deal” on an abstract model of the world that had already appeared in
early geometric treatises and which today is called
the continuum
. Heavily
steeped in its geometric roots (i.e. in the infinity of points in a segment), the
continuummodel postulates that time and space are an uninterrupted flow
which can be divided arbitrarily many times into arbitrarily (and infinitely)
small pieces. In signal processing parlance, this is known as the “analog”
world model and, in this model, integer numbers are considered primitive
entities, as rough and awkward as a set of sledgehammers in a watchmaker's
shop.
In the continuum, the infinitely big and the infinitely small dance to-
gether in complex patterns which often defy our intuition and which re-
quired almost two thousand years to be properly mastered. This is of course
not the place to delve deeper into this extremely fascinating epistemologi-
cal domain; suffice it to say that the apparent incompatibility between the
digital and the analog world views appeared right from the start (i.e. from
the 5th century BC) in Zeno's works; we will appreciate later the immense
import that this has on signal processing in the context of the sampling the-
orem.
Zeno's paradoxes are well known and they underscore this unbridgeable
gap between our intuitive, integer-based grasp of the world and a model of
(2)
The layman's aversion to “complicated” fractions is at the basis of many counting sys-
tems other than the decimal (which is just an accident tied to the number of human fin-
gers). Base-12 for instance, which is still so persistent both in measuring units (hours in
a day, inches in a foot) and in common language (“a dozen”) originates from the simple
fact that 12 happens to be divisible by 2, 3 and 4, which are the most common number
of parts an item is usually split into. Other bases, such as base-60 and base-360, have
emerged from a similar abundance of simple factors.
