## 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

*difference equation*. This simple filter is a special case of an important class of filters called

*linear time-invariant (LTI) filters*. LTI filters are important in audio engineering because they are the

*only*filters that preserve signal frequencies. The above example remains a real LTI filter if we scale the input samples by any real

*coefficients*:

*complex filter*:

*non-causal*filter example. Causal filters may compute using only

*present and/or past input samples*, , , and so on. 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*:

*multi-input, multi-output*(MIMO) digital filter is

*nonlinear*digital filter is

*i.e.*, it squares each sample of the input signal to produce the output signal. This example is also a

*memoryless nonlinearity*because the output at time is not dependent on past inputs or outputs. The nonlinear filter

*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

*all*linear, time-invariant filters mathematically. This characterization will enable us to specify frequency-domain analysis tools that work for

*any*LTI digital filter.

**Next Section:**

Linear Filters

**Previous Section:**

Definition of a Filter