Examples of Digital Filters

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 examples.

The filter analyzed in Chapter 1 was specified by

Such a specification is known as a

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 shift-invariant way:

The use of ``future'' samples, such as in this difference equation, makes this a

Another class of causal LTI filters involves using *past output
samples* 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* (MIMO)
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

is not memoryless.

Another nonlinear filter example is the
*median smoother* of order
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
isolated noise spikes, and preserve steps.

An example of a linear *time-varying* filter is

It is time-varying because the coefficient of changes over time. It is linear because no coefficients depend on or .

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
frequency-domain analysis tools that work for *any* LTI digital
filter.

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