Holger Boche and Gunther Reißig.

The structure of BIBO stable LTI-filters - A norm type approach.

1st Int. Symp. Communication Systems and Digital Signal Proc., Sheffield, UK, April 6-8, 1998, vol. 2, pp. 511-514.

**Abstract:**

In this paper the behaviour of discrete-time linear filters is investigated.
The main idea of the theory of discrete-time single-input single-output
linear filters S is that every such filter has an input-output map that can
be represented by an expression that relates the input and the impulse
response to the output of the system. Filters of this type are called
filters of the sum type. If the system is time invariant, then the
expression can be reduced to a filter of the convolution type. They are
especially important for signal processing. The expression for the
convolution filter contains a term that is the impulse response of the
filter, and thus a term for the Kronecker-delta-function. It is almost
always emphasized that these representations hold for all linear discrete-time filters. It was recently discovered, however, that not all
discrete-time single-input single-output linear filters are of the sum type or the
convolution type. The first examples were constructed by Boyd (1985) and
Sandberg (1996). Since not all discrete-time single-input single-output LTI
filters (linear and time invariant filters) are of the convolution type it
would be interesting to characterize the filters of the convolution type
(Boche, 1997, Boche and Reissig, 1997). In this paper such a
characterization is given with the help of the BIBO (bounded input bounded
output) norm of the filter. The consequences of the result for the filter
design are also discussed in the paper.

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