Digital Signal Processing Reference
In-Depth Information
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Chapter 7
Filter Design
In discrete-time signal processing, filter design is the art of turning a set of
requirements into a well-defined numerical algorithm. The requirements,
or
specifications
, are usually formulated in terms of the filter's frequency
response; the design problem is solved by finding the appropriate coeffi-
cients for a suitable difference equation which implements the filter and
by specifying the filter's architecture. Since realizable filters are described
by rational transfer functions, filter design can usually be cast in terms of
a polynomial optimization procedure for a given error measure. Additional
design choices include the
computational cost
of the designed filters, i.e. the
number of mathematical operations and storage necessary to compute each
output sample. Finally, the structure of the difference equation defines an
explicit operational procedure for computing the filter's output values; by
arranging the terms of the equation in different ways, we can arrive at dif-
ferent algorithmic structures for the implementation of digital filters.
7.1 Design Fundamentals
As we have seen, a realizable filter is described by a rational transfer func-
tion; designing a filter corresponds to determining the coefficients of the
polynomials in transfer function with respect to the desired filter character-
istics. For an FIR filter of length
M
,thereare
M
coefficients that need to be
determined, and they correspond directly to the filter's impulse response.
Similarly, for an IIR filter with a numerator of degree
M
−
1andadenomina-
tor of degree
N
−
1, there are
M
+
N
−
1 coefficients to determine (since we
always assume
a
0
=
1). The main questions are the following:
