Filter Design - Signal Processing for Communications - page 165

• How do we specify the characteristics of the desired filter? This ques-

tion effectively selects the domain in which we will measure the dif-
ference (i.e. the error ) between the desired filter and the achieved im-
plementation. This can be the time domain (where we would be com-
paring impulse responses) or the frequency domain (where we would
be comparing frequency responses). Usually the domain of choice is
the frequency domain.

l g r , y i d . , © , L s

• What are the criteria tomeasure the quality of the obtained filter? This

question defines theway inwhich the above-mentionned error is mea-
sured; again, different criteria are possible (such as minimum square
error or minimax) and they do depend on the intended application.

• How do we choose the filter's coefficients in order to obtain the desired

filtering characteristics? This question defines an optimization prob-
lem in a parameter space of dimension M

1 with the optimality
criterion chosen above; it is usually answered by the existence of a
numerical recipe which performs the task.

+ N −

• What is the best algorithmic structure (software or hardware) to imple-

ment a given digital filter? This last question concerns the algorithmic
design of the filter itself; the design is subject to various application-
dependent constraints which include computational speed, storage
requirement and arithmetic precision. Some of these design choices
will be addressed at the end of the Chapter.

As is apparent, real-world filters are designed with a variety of practi-
cal requirements in mind, most of which are conflicting. One such require-
ment, for instance, is to obtain a low “computational price” for the filtering
operation; this cost is obviously proportional to the number of coefficients
in the filter, but it also depends heavily on the underlying hardware architec-
ture. The tradeoffs between disparate requirements such as cost, precision
or numerical stability are very subtle and not altogether obvious; the art of
the digital filter designer, although probably less dazzling than the art of the
analog filter designer, is to determine the best design strategy for a given
practical problem.

7.1.1 FIR versus IIR

Filter design has a long and noble history in the analog domain: a linear
electronic network can be described in terms of a differential equation
linking, for instance, the voltage as a function of time at the input of the
network to the voltage at the output. The arrangement of the capacitors,