Digital Signal Processing Reference
In-Depth Information
THEOREM 2.1 (Sampling Theorem)
If
x
is a signal that obeys requirement 1 above, it may be perfectly
determined by its samples
x
(
t
)
(
nT
S
)
,
n
integer, if
T
S
obeys requirement 2.
If these requirements are not complied with, the reconstruction process
will be adversely affected by a phenomenon referred to as aliasing [219].
2.2.2 The Filtering Problem
There are many practical instances in which it is relevant to process informa-
tion, i.e., to treat signals in a controlled way. A straightforward approach to
fulfill this task is to design a filter, i.e., a system whose input-output relation
is tailored to comply with preestablished requirements. The project of a filter
usually encompasses three major stages:
•
Choice of the filtering structure, i.e., of the general mathematical
form of the input-output relation.
•
Establishment of a filtering criterion, i.e., of an expression that
encompasses the general objectives of the signal processing task at
hand.
•
Optimization of the cost function defined in the previous step with
respect to the free parameters of the structure defined in the first
step.
It is very useful to divide the universe of discrete-time filtering structures
into two classes: linear and nonlinear. There are two basic types of linear dig-
ital filters: finite impulse response filters (FIR) and infinite impulse response
filters (IIR). The main difference is that FIR filters are, by nature, feedforward
devices, whereas IIR filters are essentially related to the idea of feedback.
On the other hand, nonlinearity is essentially a negative concept. There-
fore, there are countless possible classes of nonlinear structures, which
means that the task of treating the filtering problem in general terms is far
from trivial.
Certain classes of nonlinear structures (like those of neural networks and
polynomial filters, which will be discussed in Chapter 7) share a very rele-
vant feature: they are derived within a mathematical framework related to
the idea of universal approximation. Consequently, they have the ability
of producing virtually any kind of nonpathological input-output mapping,
which is a remarkable feature in a universe as wide as that of nonlinear
filters.
A filtering criterion is a mathematical expression of the aims subjacent
to a certain task. The most direct expression of a filtering criterion is its
