IEEE Circuits and Systems Magazine - Q1 2018 - 17

```When a stopband attenuation of 80-dB is required, achieving a Remez
order of higher than 155 is impractical if the total hardware
complexity budget is less than 5000.

[42] or the Hermann-Rabiner-Chan approximation [35],
[43], [44] can be used. Notice that this further extension and the corresponding outcome in (15) [and also in
(17), (18)] are limited (in terms of accuracy) due to the
inherent inaccuracies in such approximations.
According to Kaiser [35], [41]:
ordermin ,

-10 log 10 ^d p d s h - 13
.
14.6 ~ - ~
s
p
2r

(14)

Applying (14) in (11) we obtain:
Total HW complexity = FA + FF
min ^d p, d s h
-10 log 10 ^d p d s h - 13
E
log 2 ;
14.6 ~ - ~
7
s
p
2r
1 .3 ^ 1 + a h
min ^d p, d s h
E.
log 2 ^20d p d s h log 2 ;
=2 FA + FF \$
7
~s - ~ p
(15)

\$ -^1 + a h

Alternatively, according to Bellanger [35]:
ordermin ,

-2 log 10 ^10d p d s h
- 1.
3 ~s - ~ p
2r

(16)

Applying (16) in (11) we obtain:
Total HW complexity = FA + FF
min ^d p, d s h
-2 log 10 ^10d p d s h
E
- 1 log 2 ;
7
3 ~s - ~ p
H
2r
min ^d p, d s h
E.
(17)
=2 FA + FF \$ log 2 ^10d p d s h log 2 ;
7

\$ - ^1 + ah

>

It is also possible to use the Hermann-Rabiner-Chan FIR
filter order approximation [35], [43], [44]:
ordermin ,

D ∞ ^d p, d s h
~s - ~ p
- F ^d p, d s h
2r
~s - ~ p
2r

(18)

where the functions D 3 ^d p, d s h and F ^d p, d s h are outlined in [35]. We can then apply (18) in (11) in the same
manner as in Kaiser's case in (15) to arrive at an approximation of the lower bound for the required total
hardware complexity (FA + FF ) as a function of the target filter specifications.
Fig. 4 displays the 3D and 2D contour maps showing
the minimum level of the hardware complexity, given a
fIrst quArtEr 2018

target d p and d s, that is required to realize a specific
Remez order (color of lines) for the FIR filter.
Fig. 5 illustrates the highest attainable Remez order
(z-axis) of a practically realizable FIR filter, given a target hardware budget (HW Resource) in terms of the total number of full adders and flip-flops (x-axis) for the
five filter design cases with stopband attenuations of
20, 30, 40, 60 and 80 dB where, for simplicity, we have
assumed that min ^d p, d s h = d s . The insights from this
plot have broad implications. For example, this plot
predicts that when a stopband attenuation of 80-dB is
required, achieving a Remez order of higher than 155
is impractical if the total hardware budget is less than
5000 (FA + FF).
As another example, for the case of a target 60-dB attenuation, it is predicted that achieving a Remez order of
higher than 100 requires a total hardware budget of more
than 2500 (FA + FF). The gray area in Fig. 5 highlights
the region that is predicted to be impractical when
min ^d p, d s h # 0.1 (which covers the majority of practical
filter design cases). Note that a potential further extension of this work may attempt to explore the proximity
of the hardware complexity to this bound as a function
of the maximum sampling rate that the specific filter can
achieve in a practical implementation.
III. How Close are the Hardware Complexities
of the Reported Filter Designs, Compared
with the Limit?
To further verify the validity of the presented bound and
also to examine how close the hardware complexities of
the announced filter designs are, compared with such
limit, a few examples are presented in this section using highly-cited filter designs that have been employed
to benchmark a variety of FIR filter-design approaches.
Example 1: Order-59 narrowband FIR filter [45]:
We first consider the highly cited order-59 example, a
narrowband FIR filter [45] where:
■ Passband edge ~ p = 0.042r rad.; Stopband edge
~ s = 0.14r rad.;
■ Ripple d p = 0.012 (! 0.1035 dB); Attenuation d s =
0.001 (60 dB);
This filter has been used to benchmark many filter-design methods [45]−[53] since its introduction in [45]. An
optimally-factored IFIR implementation (Fig. 6) of this
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17

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