While any mapping from signals
to real numbers
can be called a filter
we normally work with filters which have more structure than that.
Some of the main structural features are illustrated in the following
The filter analyzed in Chapter 1
was specified by
Such a specification is known as a difference equation
simple filter is a special case of an important class of filters called
linear time-invariant (LTI) filters
. LTI filters
in audio engineering because they are the only
preserve signal frequencies.
The above example remains a real LTI filter if we scale the input
samples by any real coefficients
If we use complex coefficients, the filter remains LTI, but it becomes
a complex filter
The filter also remains LTI if we use more input samples in a
The use of ``future'' samples, such as
difference equation, makes this a non-causal
filter example. Causal filters
present and/or past input samples
, and so on.
Another class of causal LTI filters involves using past output
in addition to present and/or past input samples. The
past-output terms are called feedback
and digital filters employing feedback are called
recursive digital filters
An example multi-input, multi-output
digital filter is
where we have introduced vectors and matrices
inside square brackets.
This is the 2D generalization of the SISO filter
The simplest nonlinear
digital filter is
, it squares each sample of the input signal to produce the output
signal. This example is also a memoryless
because the output at time
is not dependent on past
inputs or outputs. The nonlinear filter
is not memoryless.
Another nonlinear filter example is the
which assigns the middle value of
input samples centered about time
to the output at time
It is useful for ``outlier'' elimination. For example, it will reject
spikes, and preserve steps.
An example of a linear time-varying
It is time-varying because the coefficient of
time. It is linear because no coefficients depend on
These examples provide a kind of ``bottom up'' look at some of the
major types of digital filters. We will now take a ``top down''
approach and characterize all
linear, time-invariant filters
mathematically. This characterization will enable us to specify
analysis tools that work for any
Next Section: Linear FiltersPrevious Section: Definition of a Filter