IEEE Circuits and Systems Magazine - Q1 2018 - 14
Appendix 2
[Regarding approximations and, e.g., inequality (12)]
FIR filter having sufficiently small errors in its passband
We will frequently assess as "quite unlikely" that an FIR
and stopband [e.g., to have practical implementation errors
filter of a certain degree (order) can be realized for a par-
in the passband and stopband that are sufficiently small,
ticular example, or that a region of, e.g., the(x, y)-plane is
compared with a desired min (d p, d s)], we know that W
"practically unreachable" for a filter in a particular example.
must be sufficiently large. By starting with the "approxi-
Such references are intimately related to the notion of our
mate" inequality (7), which expresses such a relationship
digital filter being a "practical" FIR filter. In this regard in-
and then, via (8) through (11), this requirement is embed-
equality (7) shows a relationship between the wordlength
ded into our fundamental (but necessarily "approximate")
(W ) of the data words that are processed by a filter and a
relation (12). Thus, FA + FF (expressing the total hardware
desired passband and stopband quality of the filter. (Please
required) must be large enough to accommodate a desired
notice that it is pointed out in (7) that the relationship is
(high enough) filter order, along with a desired high filter
"approximate.") Fortunately, we are dealing with situations
quality, expressed by min (d p, d s) . That is, there is a non-
in which exact precision is not likely to be essential. We
trivial (and likely not immediately obvious) interconnection
are ultimately dealing with the approximate "cost" of imple-
between the three entities:
menting, in practice, a physical circuit. Other issues will of-
* filter quality:
ten come into the picture as well, and a "ball-park" estimate
* size of filter (i.e., filter order, or degree): order min
of a circuit's intrinsic implementation cost is all that one is
* amount of physical hardware required:
likely to require. This freedom allows us to arrive at "rather
(A "threesome" not often found explicitly in FIR-filter
simple" relationships between entities, relationships that
work.) Our Fig. 3, for example, shows how these three fea-
will surely suffice when, say, one must estimate the cost of
tures interrelate, following reasonably simple, and useful,
a proposed project. For example, in order to build a desired
mathematical relationships, e.g. (12).
and hence, the number of multiplier adders MA = 0.
Therefore:
Total HW complexity =
FA + FF $ -^1 + a h ordermin log 2 ;
min ^d p, d s h
E.
7
(11)
From (11) we obtain:
ordermin #
FA + FF
min ^d p, d s h
E
-^1 + a h log 2 ;
7
(12a)
ordermin #
HW budget
.
min ^d p, d s h
E
-^1 + a h log 2 ;
7
(12b)
The outcome in (11) and (12) predicts that, given a fixed
hardware budget, in terms of the total number of full adders and flip-flops, a practical realization of an FIR filter
with a Remez order larger than the upper bound defined
in (12) is highly improbable (see Appendix 2)-where
the parameter a is defined in (2). The practical hardware complexity bound in (12) is visually illustrated in
Fig. 3 both in linear and log scales. The z-axis represents
14
IEEE cIrcuIts ANd systEMs MAgAzINE
FA + FF
the highest Remez order of the practically realizable FIR
filter, as a function of the hardware budget (x-axis) and
filter spec (y-axis) . These results are further illustrated
using contour plots in Fig. 4. The predictive capabilities
of the proposed upper bound, as defined in (11) and
(12), will be discussed in the following sections. While
the choice of parameter a = 1 is appropriate for the majority of FIR filter design cases, we offer an extension of
(2) for a more conservative definition of parameter a:
a . C#'
or, alternatively:
min (d p, d s)
1 - 1/M
1
for Mth band FIR filters
otherwise
(13)
where scaling factor C is a function of ~ p (0 < ~ p < 1)
to take into account the fact that the average coefficient
complexity is more sensitive to the filter's passband zeros than to its stopband zeros [46], [47]. The extension
in (13) provides a slightly more conservative prediction
for extremely narrowband filters (i.e., ~ p % 1) .
To further extend the result in (11) so that the minimum required total HW complexity is defined directly as
a function of the filter specifications, we can use Kaiser's
approximation (14) for the FIR filter's order [35], [41].
Alternatively, filter-order prediction theories, including
Bellanger's formula [35], the Mintzer-Liu estimation [35],
fIrst quArtEr 2018
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